Geometric and Harmonic Analysis on Homogeneous Spaces and Applications: TJC 2019, Djerba, Tunisia, December 15–19 (Springer Proceedings in Mathematics & Statistics, 366) 3030783456, 9783030783457

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Springer Proceedings in Mathematics & Statistics

Ali Baklouti Hideyuki Ishi   Editors

Geometric and Harmonic Analysis on Homogeneous Spaces and Applications TJC 2019, Djerba, Tunisia, December 15–19

Springer Proceedings in Mathematics & Statistics Volume 366

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

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Ali Baklouti Hideyuki Ishi •

Editors

Geometric and Harmonic Analysis on Homogeneous Spaces and Applications TJC 2019, Djerba, Tunisia, December 15–19 In Honor of Professor Takaaki Nomura

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Editors Ali Baklouti Department of Mathematics Faculty of Sciences of Sfax Sfax, Tunisia

Hideyuki Ishi Department of Mathematics Osaka City University Osaka, Japan

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-030-78345-7 ISBN 978-3-030-78346-4 (eBook) https://doi.org/10.1007/978-3-030-78346-4 Mathematics Subject Classification: 22D05, 54B20, 22E40, 22E25, 22E27, 22E45, 33C67, 43A90, 53D05, 53D12, 53C12, 15A04, 17A99, 17D99, 22E46, 32M05, 22E60, 14M17, 81S10 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume collects a series of papers on harmonic analysis and representation theory. All the authors participated in the sixth Tunisian-Japanese conference “Geometric and Harmonic Analysis on homogeneous spaces and Applications” held at Djerba Island in Tunisia from December 16, 2019 to December 19, 2019. Each paper submitted to the proceedings has been peer-reviewed through the standard process as an academic work, and the papers in this volume are all agreed for publication with the referees after necessary revisions. The conference was organized also to have the opportunity to celebrate Takaaki Nomura for his retirement from Kyushu University. He was a prominent mathematician working on analysis with group representation and had organized this series of Tunisian-Japanese conferences with great effort and enthusiasm. Nomura was very pleased with our invitation, but he could not participate in the conference because of his sudden onset. Except for his absence, the conference was successful. We had 27 talks of 40 minutes by primary speakers and 10 talks of 20 min by mainly young researchers in parallel sessions. There were over 100 participants in the conference with stimulating discussion on their mathematical ideas. Nomura sent a message expressing his gratitude to the conference, while some speakers presented their pleasant memory with Nomura at the beginning of their talks. Despite his brave struggle against the disease, Nomura passed away on January 27, 2020. In our deepest sadness to lose him, we decided to dedicate these proceedings to the memory of Takaaki Nomura. The volume contains 13 original papers, all of which give important contributions to the area of non-commutative harmonic analysis and related topics. The subjects of the papers are as follows: coherent state representations, Jacobi hypergroups, invariant differential operators, double coset decompositions, closed subgroup lattices, orbital measures, Hom-Sabinin algebras, pre-Lie-Rinehar algebras, semi-invariant vectors, rapid decay properties, compression semigroups, Dunkl operators, and q-hypergeometric equations. The fact that the range of topics is so wide implies that our long-standing interactions between Tunisia and Japan are really fruitful. We shall continue to

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encourage active collaborations in various subjects as above among mathematicians from Tunisia, Japan, and other countries, especially of young generations. We wish Takaaki Nomura should share this ambitious dream. Sfax, Tunisia Osaka, Japan

Ali Baklouti Hideyuki Ishi

Contents

Takaaki Nomura: A Prominent Figure of the Tunisia-Japan Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ali Baklouti and Hideyuki Ishi

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Coherent State Representations of the Holomorphic Automorphism Group of the Tube Domain over the Dual of the Vinberg Cone . . . . . . Koichi Arashi

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Variety of Hom-Sabinin Algebras and Related Algebra Subclasses . . . . Daniel de la Concepción and Abdenacer Makhlouf

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Eigenvalues of Positive Pseudo-Hermitian Matrices . . . . . . . . . . . . . . . . Jacques Faraut

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On the Subgroup Lattices of Lie Groups with Finitely Many Connected Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hatem Hamrouni and Zouhour Jlali

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Semi-invariant Vectors Associated with Holomorphically Induced Representations of Exponential Lie Groups . . . . . . . . . . . . . . . . . . . . . . Junko Inoue

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Singular Integral Operators of Convolution Type on Jacobi Hypergroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Takeshi Kawazoe The Compression Semigroup of the Dual Vinberg Cone . . . . . . . . . . . . 123 Hideyuki Ishi and Khalid Koufany The Universal Pre-Lie–Rinehart Algebras of Aromatic Trees . . . . . . . . 137 Gunnar Fløystad, Dominique Manchon, and Hans Z. Munthe-Kaas Variants of Confluent q-Hypergeometric Equations . . . . . . . . . . . . . . . . 161 Ryuya Matsunawa, Tomoki Sato, and Kouichi Takemura

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Rapid Decay Property for Algebraic p-Adic Groups . . . . . . . . . . . . . . . 181 Sami Mustapha Rings of Invariant Differential Operators on Homogeneous Cones and Capelli-Type Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Hideto Nakashima An Extension of Pizzetti’s Formula Associated with the Dunkl Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Nobukazu Shimeno and Naoya Tani On Double Coset Decompositions of Real Reductive Groups for Reductive Absolutely Spherical Subgroups . . . . . . . . . . . . . . . . . . . . 229 Yuichiro Tanaka

Contributors

Koichi Arashi Graduate School of Mathematics, Nagoya University, Nagoya, Japan Ali Baklouti Faculté des Sciences de Sfax, Départment de Mathématiques, Sfax, Tunisie Daniel de la Concepción Departamento de Matemáticas y Computación, Universidad de la Rioja, Logrono, Spain Jacques Faraut Institut de Mathématiques de Jussieu, Sorbonne Université, Paris, France Gunnar Fløystad Matematisk Institutt, Universitetet i Bergen, Bergen, Norway Hatem Hamrouni Faculty of Sciences at Sfax. Department of Mathematics., Sfax University, Sfax, Tunisia Junko Inoue Center for Data Science Education, Organization for Educational Support and International Affairs, Tottori University, Tottori, Japan Hideyuki Ishi Osaka City University, Sumiyoshi-ku, Osaka, Japan Zouhour Jlali Faculty of Sciences at Sfax. Department of Mathematics., Sfax University, Sfax, Tunisia Takeshi Kawazoe Department of Mathematics, Keio University at SFC, Tokyo, Japan Khalid Koufany CNRS, IECL, University of Lorraine, Nancy, France Abdenacer Makhlouf IRIMAS-Département de Mathématiques, Université de Haute-Alsace, Mulhouse, France Dominique Manchon Laboratoire de Mathématiques Blaise Pascal, CNRS and Université Clermont-Auvergne, Aubière, France

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Contributors

Ryuya Matsunawa Department of Mathematics, Faculty of Science and Engineering, Chuo University, Tokyo, Japan Hans Z. Munthe-Kaas Matematisk Institutt, Universitetet i Bergen, Bergen, Norway Sami Mustapha Institut Mathématique de Jussieu, Sorbonne Université, Paris, France Hideto Nakashima The Institute of Statistical Mathematics, Midori-cho 10-3, Tokyo, Japan Tomoki Sato Department of Mathematics, Faculty of Science and Engineering, Chuo University, Tokyo, Japan Nobukazu Shimeno School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo, Japan Kouichi Takemura Department of Mathematics, Ochanomizu University, Tokyo, Japan Yuichiro Tanaka Graduate School of Mathematical Sciences, The University of Tokiyo, Meguro-ku, Tokyo, Japan Naoya Tani Graduate School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo, Japan

Takaaki Nomura: A Prominent Figure of the Tunisia-Japan Cooperation Ali Baklouti and Hideyuki Ishi

Abstract We remember Takaaki Nomura’s outstanding contribution to international collaborations of mathematicians as well as his significant works on representation theory of Lie groups and non-commutative harmonic analysis on homogeneous spaces. Keywords Paley-Wiener type Theorem · Invariant differential operator · Symmetric space · Jordan algebra · Semi-simple Lie group · Solvable Lie group · Berezin transform · Bounded homogeneous domain · Homogeneous Siegel domain · Homogeneous cone

A. Baklouti Faculté des Sciences de Sfax, Départment de Mathématiques, Route de Soukra, 3038 Sfax, Tunisie e-mail: [email protected] H. Ishi (B) Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_1

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Takaaki Nomura was born in 1953 at Osaka in Japan. He was enrolled in Kyoto University in 1972. He started the study of representation theory of Lie groups and non-commutative harmonic analysis at graduate school under the supervision of Professors Nobuhiko Tatsuuma and Takeshi Hirai. He obtained a doctor degree of science at Kyoto University in 1982. The title of his doctoral thesis was “The Paley-Wiener type theorem for the oscillator group” [1]. In 1980, he was employed as a research assistant at Kyoto University. After working for twenty-five years at Kyoto, he moved to Kyushu University for a full professor in 2005, and retired in 2019. Then he became an emeritus professor of Kyushu University and a member of Osaka City University Advanced Mathematical Institute to continue his research. In 1983, he was employed by French government scholarship (Bourses du gouvernement français) to study in Paris. One of his motivations to go to France was

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works by Michel Duflo, who visited Japan in 1982. Since then, Nomura visited France from time to time and made contributions to development of interactions between French and Japanese mathematicians. Moreover, he organized international collaboration projects on harmonic analysis with several countries: Germany (2003, 2011), Russia (2007–2008), Netherland (2010, 2013), and Tunisia (2009, 2011, 2013, 2015, 2017). It is then clear that the cooperation with the Tunisian side was so multiple.

1 The Tunisian-Japanese Cooperation Thanks to recommendation from Hidenori Fujiwara, the cooperation between Sfax University and Kyushu University under the guidance of Takaaki Nomura and Ali Baklouti started in 2008. Accordingly, a reciprocity agreement was signed between the Faculty of Sciences of Sfax and the Faculty of Mathematics of Kyushu, which was followed by the organization of the first conference, which took place in November 2009 in Kerkenah Islands. The success of the first conference was a major incentive for both Nomura and Baklouti to organize the second in Sousse in December 2011 and the third in Hammamet in December 2013 in honor of Hidenori Fujiwara on the occasion of his retirement. The fourth seminar took place in Monastir in December 2015 in honor of Jean Ludwig, one of the important mathematicians in our fields, who made substantial mathematical achievements with both sides of Tunisia and Japan. In December 2017, Baklouti and Nomura hold the fifth seminar in Mahdia in memory of Majdi Ben Halima, an eminent researcher of the laboratory LAMHA, who passed away in February 2016. All these conferences aimed at exploring topics such as commutative harmonic analysis (Fourier analysis on Euclidean spaces), analysis on homogeneous spaces, uncertainty principles, geometric analysis, and some of their interactions with other representation theory and operator algebras. Takaaki Nomura had been an efficient engine throughout the organizations of all these seminars and worked hard toward their success. Being so, we were so proud to organize the sixth seminar in his honor and this was the least that can be offered to thank him for this great endeavors to make thriving the reciprocity between Tunisia and Japan. All the details about the events can be found through the link: https://sites.google.com/view/ tunisian-japanese-conference/home.

2 About the Scientific Activities of Takaaki Nomura Concerning the scientific activities, Nomura worked on both semi-simple and solvable Lie groups and homogeneous spaces. He published his first academic paper [2] on finite covering groups of SU (1, 1) when he was a graduate student. There are several important works [3–9] by him on representation theory of semi-simple Lie groups and harmonic analysis including his description of algebraically independent generators of invariant differential operators on a symmetric cone [10]. On

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the other hand, since Hidenori Fujiwara invited Nomura in doctor course to study representations of solvable Lie groups, he continued to have interest in solvable objects. Actually, after returning from Paris, he published a series of papers [5, 11–13] on representations of solvable Lie groups and Lie algebras. In this respect, harmonic analysis on a bounded homogeneous domain (BHD) and a homogeneous Siegel domain is a fascinating subject for Nomura’s mathematics. In general, the full automorphism group of a BHD is complicated, whereas its Iwasawa subgroup (a maximal connected split solvable Lie subgroup) has a rich and feasible structure. A homogeneous Siegel domain is an unbonded realization of BHD, and the Iwasawa subgroup acts simply transitively on the Siegel domain as affine transforms. Therefore, a general BHD is usually studied in a Siegel domain realization as a solvable homogeneous space. On the other hand, a bounded symmetric domain (BSD) has a structure of non-compact Hermitian symmetric space, so it is a semi-simple homogeneous space. As a splendid application of this bilaterality of BSD, Nomura gave a description of holomorphic discrete series representations of Hermitian Lie groups making use of harmonic analysis of solvable Lie groups [4, 5]. Other than the Siegel domain realization, a BSD has a canonical Harish-Chandra realization, on which the isotropy subgroup at the origin acts as linear transforms. Although the first example of non-symmetric BHD was found by Piatetskii-Shapiro [14] about a quarter-century after E. Cartan’s classification of BSD [15] historically, it turns out that the class of BSD is a very special subclass of BHD. Nomura established several characterization theorems of BSD among BHD (equivalently, symmetry characterization for homogeneous Siegel domains) [16–21]. In particular, he proved two analytic characterization theorems: a homogeneous Siegel domain is symmetric if and only if (1) its Berezin transform (cf. [22, 23]) commutes with the Laplace-Beltrami operator with respect to the Bergman metric [16], or, (2) the Poisson kernel defined from Cauchy-Szegö kernel is vanished by the Laplace-Beltrami operator [20]. Inspired by Richard Penney’s work [24], Nomura introduced a family of Cayley transforms mapping a homogeneous Siegel domain onto bounded domains [25, 26]. A remarkable fact is that the conditions (1) and (2) are, respectively, equivalent to certain geometric conditions for images of appropriate Cayley transforms [17, 20]. Indeed, the Cayley transform image can be regarded as a generalization of Harish-Chandra realization, which had been lacking for the study of a general BHD. We refer to his excellent survey article [27] for the details about these topics. After he moved to Kyushu University, his main research object was a homogeneous cone (HC) [28–35], which is one of the defining data of homogeneous Siegel domain. Here, the contrast between semi-simple and solvable objects appears again. A symmetric cone, i.e., a self-dual homogeneous cone, is described efficiently by using Euclidean Jordan algebra, which arises from a semi-simple automorphism group action. On the other hand, a general homogeneous cone is described by compact normal left symmetric algebra (CLAN), which arises from the action of the Iwasawa subgroup of the automorphism group of the cone. Besides the symmetric characterization theorems for HC [28, 29], Nomura found interesting examples of HC by using both Euclidean Jordan algebra and CLAN [30, 31]. Recently HCs are also studied by applied mathematicians working on convex programming and math-

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ematical statistics. Nomura’s work [32] about matrix realization of HC will provide a concise and convenient tool to such researchers not necessarily familiar with Lie theory. As is already mentioned above, Jordan algebra structure arises from actions of semi-simple Lie groups on symmetric spaces, while Jordan theory can be regarded as a tool for geometry and analysis complementary to Lie theory. Some of Nomura’s works (e.g. [6, 10]) were based on his expertise in Jordan algebra. In particular, he studied the infinite-dimensional manifold of idempotents of finite rank in JordanHilbert algebras [36, 37]. Nomura’s last academic work was a beautiful paper [38] giving a short proof of Hobson’s formula. He found the result in preparation for his book on spherical harmonics [39]. We remember that Nomura really enjoyed writing the book, which is quite inspiring to both students and researchers.

References 1. T. Nomura, The Paley-Wiener type theorem for the oscillator group, doctoral thesis, Kyoto University, 1982. J. Math. Kyoto Univ. 22, 71–96 (1982/83) 2. T. Nomura, The Paley-Wiener type theorems for finite covering groups of SU (1, 1). Proc. Jpn. Acad. Ser. A 53, 195–197 (1977) 3. T. Nomura, The Paley-Wiener type theorem for finite covering groups of SU (1, 1). J. Math. Kyoto Univ. 18, 273–304 (1978) 4. T. Nomura, Fourier transform of a space of holomorphic discrete series. Proc. Jpn. Acad. Ser. A 61, 133–136 (1985) 5. T. Nomura, A description of a space of holomorphic discrete series by means of the Fourier transform on the Šilov boundary. J. Funct. Anal. 82, 1–37 (1989) 6. T. Nomura, Algebraically independent generators of invariant differential operators on a bounded symmetric domain. J. Math. Kyoto Univ. 31, 265–279 (1991) 7. E. Fujita, T. Nomura, Spectral decompositions of Berezin transformations on Cn related to the natural U (n) -action. J. Math. Kyoto Univ. 36, 877–888 (1996) 8. E. Fujita, T. Nomura, Berezin transforms on the 2 × 2 matrix space related to the U (2) × U (2)action. Integral Equ. Oper. Theory 32, 152–179 (1998) 9. T. Nomura, Harmonic analysis on the universal covering group of certain semidirect product Lie group. Jpn. J. Math. 6, 129–145 (1980) 10. T. Nomura, Algebraically independent generators of invariant differential operators on a symmetric cone. J. Reine Angew. Math. 400, 122–133 (1989) 11. T. Nomura, Representations of a solvable Lie group on ∂¯b cohomology spaces. Proc. Jpn. Acad. Ser. A 62, 331–334 (1986) 12. T. Nomura, Existence of a noninductive linear form on certain solvable Lie algebras. J. Math. Soc. Jpn. 39, 667–675 (1987) 13. T. Nomura, Harmonic analysis on a nilpotent Lie group and representations of a solvable Lie group on ∂¯b cohomology spaces. Jpn. J. Math. 13, 277–332 (1987) 14. I.I. Piatetskii-Shapiro, On a problem proposed by E. Cartan, Dokl. Akad. Nauk SSSR 124, 272–273 (1959) 15. E. Cartan, Sur les domaines bornés homogènes de l’espace de n variables complexes. Abh. Math. Sem. Univ. Hamburg 11, 116–162 (1935) 16. T. Nomura, Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains. Differ. Geom. Appl. 15, 91–106 (2001)

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17. T. Nomura, A characterization of symmetric Siegel domains through a Cayley transform. Transform. Groups 6, 227–260 (2001) 18. T. Nomura, A symmetry characterization for homogeneous Siegel domains related to Berezin transforms, Geometry and analysis on finite- and infinite-dimensional Lie groups (B¸edlewo, 2000), Banach Center Publ., vol. 55, Polish Acad. Sci. Inst. Math. (2002), pp. 323–334 19. T. Nomura, Cayley transforms and symmetry conditions for homogeneous Siegel domains, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie (Kyoto/Nara, 1999). Adv. Stud. Pure Math. 37, Math. Soc. Japan (2002), pp. 253–265 20. T. Nomura, Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains. J. Funct. Anal. 198, 229–267 (2003) 21. C. Kai, T. Nomura, A characterization of symmetric tube domains by convexity of Cayley transform images. Differ. Geom. Appl. 23, 38–54 (2005) 22. T. Nomura, Berezin transforms and group representations. J. Lie Theory 8, 433–440 (1998) 23. T. Nomura, Invariant Berezin transforms. Harmonic analysis and integral geometry (Safi, 1998), pp.19–40, Chapman and Hall/CRC Res. Notes Math., vol. 422 (Chapman and Hall/CRC, Boca Raton, 2001) 24. R. Penney, The Harish-Chandra realization for non-symmetric domains in Cn , Topics in geometry, Progr. Nonlinear Differential Equations Appl. vol. 20 (Birkhäuser, 1996), pp. 295–313 25. T. Nomura, On Penney’s Cayley transform of a homogeneous Siegel domain. J. Lie Theory 11, 185–206 (2001) 26. T. Nomura, Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms. Differ. Geom. Appl. 18, 55–78 (2003) 27. T. Nomura, Focusing on symmetry characterization theorems for homogeneous Siegel domains. Sugaku Expo. 23, 47–67 (2010) 28. C. Kai, T. Nomura, A characterization of symmetric cones through pseudoinverse maps. J. Math. Soc. Jpn. 57, 195–215 (2005) 29. T. Yamasaki, T. Nomura, A characterization of symmetric cones by the degrees of basic relative invariants. Kyushu J. Math. 70, 237–257 (2016) 30. H. Nakashima, T. Nomura, Clans defined by representations of Euclidean Jordan algebras and the associated basic relative invariants. Kyushu J. Math. 67, 163–202 (2013) 31. H. Nakashima, T. Nomura, Basic relative invariants on the dual clans obtained by representations of Euclidean Jordan algebras. Rev. Roumaine Math. Pures Appl. 59, 443–451 (2014) 32. T. Yamasaki, T. Nomura, Realization of homogeneous cones through oriented graphs. Kyushu J. Math. 69, 11–48 (2015) 33. H. Ishi, T. Nomura, Tube domain and an orbit of a complex triangular group. Math. Z. 259, 697–711 (2008) 34. H. Ishi, T. Nomura, Irreducible homogeneous non-symmetric cones linearly isomorphic to the dual cones, Contemporary geometry and topology and related topics (Cluj University Press, 2008), pp. 167–171 35. H. Ishi, T. Nomura, An irreducible homogeneous non-selfdual cone of arbitrary rank linearly isomorphic to the dual cone, Infinite dimensional harmonic analysis IV (World Scientific Publishing, 2009), pp. 129–134 36. T. Nomura, Manifold of primitive idempotents in a Jordan-Hilbert algebra. J. Math. Soc. Jpn. 45, 37–58 (1993) 37. T. Nomura, Grassmann manifold of a JH-algebra. Ann. Global Anal. Geom. 12, 237–260 (1994) 38. T. Nomura, A proof of Hobson’s formula with the Euler operator. Kyushu J. Math. 72, 423–427 (2018) 39. T. Nomura Spherical Harmonics and Group Representations (in Japanese) (Nippon Hyoron sha co., Ltd., 2018)

Coherent State Representations of the Holomorphic Automorphism Group of the Tube Domain over the Dual of the Vinberg Cone Koichi Arashi

Abstract We classify all irreducible coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone. Keywords Coherent state representation · Homogeneous bounded domain · Momentum mapping · Reproducing kernel · Multiplier representation

1 Introduction Let G 0 be a connected Lie group, and let (π, H) be a unitary representation of G 0 . We regard the projective space P(H) as a (possibly infinite-dimensional) Kähler manifold. We call a G 0 -orbit of P(H) a coherent state orbit (CS orbit for short) if it is a complex submanifold of P(H), and we call π a coherent state representation (CS representation for short) if there exists a CS orbit in P(H) that does not reduce to a point (see [13, Definition 4.2]). In this case, we say that π is generic if π is irreducible and ker π is discrete. By Lisiecki [11], the generic CS representations coincide with the irreducible highest weight representations with discrete kernels for a semisimple Lie group. Thus, CS representations can be considered as generalizations of the highest weight representations of semisimple Lie groups to a wider class of groups. Also, the generic CS representations of connected unimodular Lie groups were studied and classified by Lisiecki [12]. After this remarkable advance, CS representations were also studied in the setting of Lie groups which have compactly embedded Cartan subalgebras by Neeb [14]. The purpose of the present article is to give classifications of irreducible CS representations and generic CS representations for a Lie group which has not been considered. Let 5 be the dual cone of the Vinberg cone, and let D5 be the tube Dedicated to the memory of Professor Takaaki Nomura. K. Arashi (B) Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_2

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domain over 5 . Let G be the identity component of the holomorphic automorphism group of D5 . In Sect. 2, we review the theory of CS representations studied in [11–13]. In Sect. 3, we review the explicit description of G studied in [5, 8]. In Sect. 4, we show that every generic CS representation of G is unitarily equivalent with a unitarization of a holomorphic multiplier representation of G over D5 or the complex conjugate representation of it. In Sect. 5, we review the classification of the unitarizations of holomorphic multiplier representations of G over D5 studied in [1]. In Sect. 6, we classify all generic CS representations of G. In Sect. 7, we classify all irreducible non-generic CS representations of G. In Sect. 8, we consider an intertwining operator between the external tensor product of a one-dimensional unitary representation of R>0 and an irreducible highest weight representation of S L(2, R) × S L(2, R) and the unitarization of a holomorphic multiplier representation of G over D5 . The author would like to thank Professor H. Ishi for a lot of helpful advice on this paper.

2 General Theory of CS Representations Throughout this paper, for a Lie group, we denote its Lie algebra by the corresponding Fraktur small letter. Let G 0 be a connected Lie group. For a G 0 -equivariant holomorphic line bundle L 0 over a complex manifold M0 , let us denote the natural representation of G 0 on the space  hol (M0 , L 0 ) of holomorphic sections of L 0 by τ L 0 . We introduce a notion of unitarizability for τ L 0 . Definition 2.1 We say that the representation τ L 0 of G 0 is unitarizable if there exists a nonzero Hilbert space H ⊂  hol (M0 , L 0 ) satisfying the following conditions: (i) the inclusion map ι : H →  hol (M0 , L 0 ) is continuous with respect to the open compact topology of  hol (M0 , L 0 ), (ii) τ L 0 (g)H ⊂ H (g ∈ G 0 ) and τ L 0 (g)sH = sH (g ∈ G 0 , s ∈ H). In this case, we call the subrepresentation (τ L 0 , H) a unitarization of the representation (τ L 0 ,  hol (M0 , L 0 )) of G 0 . A Hilbert space H satisfying the condition (i) is a reproducing kernel Hilbert space. We note that when G 0 acts on M0 transitively, a Hilbert space giving a unitarization of τ L 0 is unique if it exists, and any unitarization is irreducible (see [7, 9, 10]). Thus, we write π L 0 instead of (τ L 0 , H). Let (π, H) be a CS representation of G 0 , and let L be the natural holomorphic line bundle over P(H), such that the fiber over [v] = Cv ∈ P(H) is given by the dual space [v]∗ . Then we can identify the dual space H∗ with  hol (P(H), L). By the following proposition, we can see that if π is irreducible, then π is equivalent with π L 0 for a G 0 -equivariant holomorphic line bundle L 0 over a CS orbit in P(H), where π denotes the complex conjugate representation.

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Proposition 2.2 ([12, Proposition 2]) Suppose that π is irreducible, and let M ⊂ P(H) be a CS orbit. Then the map H∗ →  hol (M, L) given by the composition of the map H∗ →  hol (P(H), L) and the restriction map  hol (P(H), L) →  hol (M, L) is injective. Let M be a CS orbit, let α0 : G 0 × M → M be the action of G 0 on M, and let Z g0 be the center of g0 . When π is generic, it holds that Lie(ker α0 ) = Z g0 ,

(2.1)

where ker α0 = {g ∈ G 0 ; α0 (g, x) = x for all x ∈ M}. Next let us see the relationship between CS orbits and coadjoint orbits. Let μπ : P(H∞ ) → g0 ∗ be a moment map defined by x, μπ ([v]) = −i

(dπ(x)v, v)H (v ∈ H∞ \{0}, x ∈ g0 ). (v, v)H

Then the image of M under μπ coincides with a coadjoint orbit. We note that M has the natural structure of a Kähler manifold which is induced by the Fubini–Study metric on P(H). As a consequence of this property, we have the following theorem. Theorem 2.3 ([16, Theorem 2.17]) The isotropy subgroup of G 0 at any point of μπ (M) is connected. In particular, the coadjoint orbit μπ (M) is simply connected, and μπ defines a diffeomorphism of M onto the coadjoint orbit.

3 The Holomorphic Automorphism Group of the Tube Domain over the Dual of the Vinberg Cone Let

⎧⎡ 1 ⎫ ⎤ ⎨ x 0 x4 ⎬ V = ⎣ 0 x 2 x 5 ⎦ ∈ M3 (R); x 1 , . . . , x 5 ∈ R , ⎩ 4 5 3 ⎭ x x x

and let 5 = V ∩ P(3, R), where P(3, R) denotes the homogeneous convex cone consists of all 3-by-3 real positive-definite symmetric matrices. We consider the following Siegel domain D5 in VC : ⎧ ⎨

⎫ ⎤ z1 0 z4 ⎬ D5 = z = ⎣ 0 z 2 z 5 ⎦ ∈ VC ; Im z ∈ 5 . ⎩ ⎭ z4 z5 z3 ⎡

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K. Arashi

Let Aut hol (D5 ) be the holomorphic automorphism group of D5 . We note that D5 is holomorphically equivalent to a complex bounded domain, and Aut hol (D5 ) has the unique structure of a Lie group compatible with the compact open topology. Let G be the identity component of Aut hol (D5 ). Theorem 3.1 G: ⎧⎡ a1 0 ⎪ ⎪ ⎪ ⎢ 0 a2 ⎪ ⎪ ⎪ ⎨⎢ ⎢ λ1 λ2 ⎢ ⎢ ⎪ ⎪⎢ c1 0 ⎪ ⎪ ⎣ 0 c2 ⎪ ⎪ ⎩ 0 0

([5], [8, Theorem 2.2]) The following linear group is isomorphic to 0 0 a3 0 0 0

b1 0 μ1 d1 0 0

0 b2 μ2 0 d2 0

⎫ ⎤ μ1 ⎪   ⎪ , b , c , d , λ , λ , μ , μ , κ ∈ R, a ⎪ i i i i i i i i ⎪ μ2 ⎥ ⎪ ⎪ ⎥ ∈ R , a d − b c = 1, a ⎬ 3 >0 i i i i ⎥   κ ⎥   b a . ∈ M (R); i i 6 −λ1 ⎥ [λi μi ] = a3 λi μi ⎪ ⎥ ⎪ c d ⎪ i i ⎪ −λ2 ⎦ ⎪ ⎪ (i = 1, 2) ⎭ −1 a3

In more detail, the linear group acts on D5 by linear fractional transformations, and the natural map from the linear group to G gives rise to an isomorphism between the Lie groups. Let us denote by G  the linear group given in Theorem 3.1. Let E 1 , E 2 , E 3 , E 3,1 , E 3,2 , A1 , A2 , A3 , A3,1 , A3,2 , W1 , and W2 be the elements of M6 (R) satisfying e1 E 1 + e2 E 2 + e3 E 3 + e3,1 E 3,1 + e3,2 E 3,2 + a1 A1 + a2 A2 + a3 A3 + a3,1 A3,1 + a3,2 A3,2 + k1 W1 + k2 W2 ⎤ ⎡ a1 0 0 e1 − k1 0 e3,1 2 a 2 ⎢ 0 0 0 e2 − k2 e3,2 ⎥ 2 ⎥ ⎢ ⎢ a3,1 a3,2 a3 e3,1 e3,2 e3 ⎥ 2 ⎥ ⎢ =⎢ a1 0 −a3,1 ⎥ ⎥ ⎢ k1 0 0 − 2 ⎣ 0 k2 0 0 − a22 −a3,2 ⎦ 0 0 0 0 0 − a23 for e1 , e2 , e3 , e3,1 , e3,2 , a1 , a2 , a3 , a3,1 , a3,2 , k1 , k2 ∈ R. Then {E 1 , E 2 , E 3 , E 3,1 , E 3,2 , A1 , A2 , A3 , A3,1 , A3,2 , W1 , W2 } form a basis of g , and we use the same symbols E 1 , E 2 , E 3 , E 3,1 , E 3,2 , A1 , A2 , A3 , A3,1 , A3,2 , W1 , and W2 for the corresponding elements of g. Let G i I3 be the isotropy subgroup of G at i I3 ∈ D5 . Theorem 3.2 ([5]) We have G i I3 = expW1 , W2 . We have the following bracket relations:

Coherent State Representations of the Holomorphic Automorphism …

[E 3,1 , A1 ] = − 21 E 3,1 , [E 1 , A1 ] = −E 1 , [E 3,1 , A3 ] = − 21 E 3,1 , [E 1 , A3,1 ] = −E 3,1 , [E 3,1 , A3,1 ] = −2E 3 , [E 1 , W1 ] = 2 A1 , [E 3,1 , W1 ] = A3,1 , [E 2 , A2 ] = −E 2 , [E 3,2 , A2 ] = − 21 E 3,2 , [E 2 , A3,2 ] = −E 3,2 , [E 3,2 , A3 ] = − 21 E 3,2 , [E 2 , W2 ] = 2 A2 , [E 3,2 , A3,2 ] = −2E 3 , [E 3 , A3 ] = −E 3 , [E 3,2 , W2 ] = A3,2 ,

11

[A1 , A3,1 ] = − 21 A3,1 , [A1 , W1 ] = −(W1 + 2E 1 ), [A2 , A3,2 ] = − 21 A3,2 , [A2 , W2 ] = −(W2 + 2E 2 ), [A3 , A3,1 ] = 21 A3,1 , [A3 , A3,2 ] = 21 A3,2 , [A3,1 , W1 ] = −E 3,1 , [A3,2 , W2 ] = −E 3,2 .

4 CS Orbits of Generic CS Representations Let M be a CS orbit of a generic CS representation π of G, and let K be the isotropy subgroup of G at some point m 0 of M. For a connected Riemannian manifold, every isotropy subgroup of the isometry group is compact. Thus exp adg k ⊂ Int g is a compact subgroup, where for a Lie algebra g0 , we denote by Int g0 the subgroup exp ad g0 ⊂ G L(g0 ). It is known [6] that G has trivial center and that G i I3 = expW1 , W2 is a maximal compact subgroup of G. Thus Int g is isomorphic to G. Moreover, any two maximal compact subgroups of G are conjugate (see [15, Chap. 4, Theorem 3.5]), so that we may and do assume that k ⊂ W1 , W2 . We then have k = 0 or W1 , W2 because M is an even-dimensional differentiable manifold. We shall show that k must equal W1 , W2 . Arguing contradiction, assume that k = 0. Then M is diffeomorphic to G. We have the following theorem. Theorem 4.1 ([15, Chap. 4, Proposition 4.4 and Theorem 4.7]) (a) Let G 0 be a connected linear Lie group. If G 0 equals K 0 D0 for some compact subgroup K 0 of G 0 and some connected real split solvable Lie subgroup D0 of G 0 , then K 0 is a maximal compact subgroup of G 0 . (b) Let G 0 be a real algebraic linear group. Then the identity component of G 0 can be topologically decomposed into the direct product of the groups K 0 and D0 , where K 0 is a maximal compact subgroup of G 0 and D0 a maximal real split solvable Lie subgroup of G 0 . Thus, it follows from Theorem 4.1(b) that G is homeomorphic to D5 × G i I3 . Hence, π1 (G, e) = π1 (G i I3 , e) = Z2 , which contradicts that M is simply connected. Therefore, we conclude that k = W1 , W2 . Now we have a G-equivariant diffeomorphism D5 → M. Let us consider the Kähler structure ( j, g) on D5 which is the pullback, by the diffeomorphism, of the Kähler structure on M. Also, we can regard D5 as a Kähler manifold by means of the Bergman

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K. Arashi

metric on D5 . Then it follows from [4, Theorem 6.1] that there exists a biholomorphism D5 → M since G acts on (D5 , j, g) by holomorphic isometries. Thus, the action of G on M induces an action of G on D5 by holomorphic automorphisms, and the action is given by G × D5 (g, z) → ϕ(g)z ∈ D5 for some automorphism ϕ of G. Let ψ be the automorphism of g satisfying ψ 2 = idg , ψ(E 1 ) = E 2 , ψ(E 3 ) = E 3 , ψ(E 3,1 ) = E 3,2 , ψ(A1 ) = A2 , ψ(A3 ) = A3 , ψ(A3,1 ) = A3,2 , and ψ(W1 ) = W2 , and let σ be the automorphism of g satisfying σ(E 1 ) = −E 1 , σ(E 2 ) = −E 2 , σ(E 3 ) = −E 3 , σ(E 3,1 ) = −E 3,1 , σ(E 3,2 ) = −E 3,2 , σ(A1 ) = A1 , σ(A2 ) = A2 , σ(A3 ) = A3 , σ(A3,1 ) = A3,1 , σ(A3,2 ) = A3,2 , σ(W1 ) = −W1 , and σ(W2 ) = −W2 . For an automorphism ϕ of g, let ϕ0 = idg , and let ϕ1 = ϕ. 

Proposition 4.2 Any automorphism ϕ of g can be written as ϕ = ψ ε ◦ σ ε ◦ Ad(g) for some g ∈ G and some ε, ε ∈ {0, 1}. We postpone the proof to Sect. 7. The automorphisms ψ and σ lift to the automorphisms of G. To simplify the notation, we use the same symbols ψ and σ for the lifts. Then we see that ψ induces a biholomorphism D5 (z 1 , z 2 , z 3 , z 4 , z 5 ) → (z 2 , z 1 , z 3 , z 5 , z 4 ) ∈ D5 and σ a biholomorphism D5 (z 1 , z 2 , z 3 , z 4 , z 5 ) → (−z 1 , −z 2 , −z 3 , −z 4 , −z 5 ) ∈ D5 , where D5 denotes the conjugate manifold. Thus, by Proposition 2.2, π or π is unitarily equivalent with π L 0 for some G-equivariant holomorphic line bundle L 0 over D5 . Here for a subgroup G 0 ⊂ G, by a G 0 -equivariant bundle over D5 , we mean a G 0 -equivariant bundle over D5 , such that the action of G 0 on the base space D5 is given by G 0 × D5 (g, z) → gz ∈ D5 . We note that D5 is a Stein manifold (see [3]), and hence every holomorphic line bundle over D5 is trivial by the Oka–Grauert principle. Hence, the representation τ L 0 can be realized on the space O(D5 ) of holomorphic functions on D5 . We call such a representation of G on O(D5 ) a holomorphic multiplier representation of G over D5 . We get the following theorem. Theorem 4.3 Let π be a generic CS representation of G. Then π or π is unitarily equivalent with a unitarization of a holomorphic multiplier representation of G over D5 .

5 Holomorphic Multiplier Representations over D5 Let g− be the complex subalgebra of gC given by     d d  0,1 ty e i I ∈ T D g− = x + i y ∈ gC ;  et x i I3 + i 3 5 , i I3 dt t=0 dt t=0

Coherent State Representations of the Holomorphic Automorphism …

13

where Ti0,1 I3 D5 denotes the antiholomorphic tangent vector space at i I3 . By Tirao and Wolf [17], the isomorphism classes of G-equivariant holomorphic line bundles over D5 stand in one-one correspondence with the one-dimensional complex representations of g− whose restrictions to gi I3 lift to representations of G i I3 . For a basis {xλ } of g, we shall denote the dual basis by {xλ∗ }. Let M be the set consists of all linear forms ξ on g given by ξ = ξ(ξ3 , η3 , n, n  ) = ξ3 E 3∗ + η3 A∗3 +

n n (2W1∗ − E 1∗ ) + (2W2∗ − E 2∗ ), 2 2

with ξ3 , η3 ∈ R and n, n  ∈ Z. If ξ is extended to a complex linear form on gC , then iξ|g− (ξ ∈ M) defines a one-dimensional complex representation of g− whose restriction to gi I3 lifts to a representation of G i I3 . For ξ ∈ M, let L 0 be a G-equivariant holomorphic line bundle over D5 whose isomorphism class corresponds to iξ|g− , and put τξ = τ L 0 . Also, we put πξ = π L 0 when τ L 0 is unitarizable. Let G (n, n  ) = {ξ(ξ3 , η3 , n, n  ); ξ3 < 0, η3 ∈ R} (n, n  ∈ Z>0 ), G (η3 , n, n  ) = {ξ(0, η3 , n, n  )} (η3 ∈ R, n, n  ∈ Z≥0 ).

(5.1)

Then we have the following theorem. Theorem 5.1 ([1], [7, Theorem 13(i) and (iii)]) (a) For ξ ∈ M, the representation τξ is unitarizable if and only if ξ belongs to any of the sets in (5.1). (b) For ξ, ξ  ∈ M with τξ , τξ  unitarizable, the representations πξ and πξ  are unitarily equivalent if and only if ξ and ξ  belong to the same set in (5.1). (c) Every holomorphic multiplier representation of G over D5 is unitarily equivalent with πξ for some ξ ∈ M. From now on, for ξ ∈ M such that τξ is unitarizable, we think of πξ as any of the holomorphic multiplier representations of G over D5 . We shall mention the converse of Theorem 4.3. Let Hξ be the representation space of πξ , let Kξ : D5 × ξ D5 → C be the reproducing kernel of Hξ , and let Ki I3 ∈ Hξ be the function given ξ by Ki I3 (z) = Kξ (z, i I3 ) (z ∈ D5 ). If the representation dπξ of g is extended to a complex representation, then we have ξ

ξ

dπξ (x)Ki I3 = iξ(x)Ki I3 (x ∈ g− ),

(5.2)

which implies that πξ is an irreducible CS representation of G if dim Hξ > 1 (see [13, Proposition 4.1]).

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6 Generic CS Representations For n, n  ∈ Z>0 , let ξn,n  be any of the elements of G (n, n  ). Proposition 6.1 For any n, n  , l, l  ∈ Z>0 , the representations πξn,n and πξl,l  are not unitarily equivalent. Proof Let b = E 1 , E 2 , E 3 , E 3,1 , E 3,2 , A1 , A2 , A3 , A3,1 , A3,2 , and let B = exp b ⊂ G. It is enough to show that πξn,n and πξl,l  are not equivalent as unitary representations of B. Note that B is an exponential solvable Lie group, so that the equivalence classes of irreducible unitary representations of B are in one-one correspondence with the coadjoint orbits of B in b∗ (see [2]). By [7, Theorem 13(ii)], the equivalence classes of πξn,n | B and πξl,l  | B correspond to the coadjoint orbit through −(E 1∗ + E 2∗ + E 3∗ )|b ∈ b∗ (see Remark 6.2 below for more detail). Then we see that the equivalence class of πξl,l  corresponds to the coadjoint orbit through (E 1∗ + E 2∗ + E 3∗ )|b ∈ b∗ . Let η be a linear form on b, and suppose that E 3 , η > 0. We have Ad(et A3 ) E 3 = t e E 3 (t ∈ R), and E 3 commutes with E 1 , E 2 , E 3 , E 3,1 , E 3,2 , A1 , A2 , A3,1 , and A3,2 . Thus, E 3 , Ad∗ (b) η = Ad(b−1 ) E 3 , η > 0 for b ∈ B. This implies that the coadjoint orbit through −(E 1∗ + E 2∗ + E 3∗ )|b and the one through (E 1∗ + E 2∗ + E 3∗ )|b are different. The proof is complete.  Remark 6.2 For ξ = ξ(ξ3 , η3 , n, n  ) ∈ G (n, n  ), we shall show that the equivalence class of πξ | B corresponds to the coadjoint orbit through −(E 1∗ + E 2∗ + E 3∗ )|b .  For s = (s1 , s2 , s3 ) ∈ C3 , let αs = 3k=1 sk A∗k |b ∈ (b∗ )C , and let χs be the character of B given by χs (exp x) = exp αs (x) (x ∈ b). Let us consider the action of B on the holomorphic line bundle D5 × C given by B × D5 × C (b, z, ζ) → (bz, χ−s/2 (b)ζ) ∈ D5 × C, and we denote the B-equivariant holomorphic line bundle by L s . Now the isomorphism classes of B-equivariant holomorphic line bundles over D5 stand in oneone correspondence with the one-dimensional complex representations of bC ∩  g− , and L s corresponds to − 2i ( 3k=1 Re sk E k∗ + Im sk A∗k )|bC ∩g− , where we extend 3 ∗ ∗  k=1 Re sk E k + Im sk Ak to a complex linear form on gC . Hence, for s = (n, n , . For α ∈ −2(ξ3 + iη3 )), the representation π L s of B is unitarily equivalent with πξ | B g∗ , let bα = {x ∈ b; [y, x] = α(y)x for all y ∈ A1 , A2 , A3 }. Put qk = 3≥l>k≥1 dim b(Al∗ −A∗k )/2 (k = 1, 2, 3). Then we have q1 = dimA3,1 = 1, q2 = dimA3,2 = 1, q3 = 0, and hence (6.1) Re sk > qk /2 (k = 1, 2, 3). According to [7, Theorem 13(ii)], we can obtain the desired result from (6.1). Let us consider the set of equivalence classes of irreducible unitary representations of G. For a unitary representation π0 of a Lie group G 0 , we denote the equivalence class of π0 by [π0 ].

Coherent State Representations of the Holomorphic Automorphism …

15

Theorem 6.3 The set of equivalence classes of generic CS representations of G is given by {[πξn,n ]; n, n  ∈ Z>0 }  {[πξn,n ]; n, n  ∈ Z>0 }. Proof By Theorems 4.3 and 5.1, it is enough to show that (a) For any n, n  ∈ Z>0 , and ξ ∈ G (n, n  ), the representation πξ is generic, (b) For any η ∈ R, n, n  ∈ Z≥0 , and ξ ∈ G (η3 , n, n  ), the representation πξ is not generic. ξ

For ξ ∈ M with τξ unitarizable, we have μπξ ([Ki I3 ]) = ξ, and hence we can idenξ tify the coadjoint orbit through ξ ∈ g∗ with the CS orbit through [Ki I3 ] ∈ P(Hξ ). We denote by α the action of G on the coadjoint orbit through ξ. Let G ξ be the isotropy subgroup of G at ξ. We note that gξ = {x ∈ g; ξ([x, y]) = 0 for all y ∈ g}. The matrix of the skew-symmetric bilinear form ξ([x, y]) with respect to the basis {E 1 , E 2 , E 3 , E 3,1 , E 3,2 , A1 , A2 , A3 , A3,1 , A3,2 , W1 , W2 } is given by ⎡

0 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ n ⎢− ⎢ 2 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 0 0 0 0 0  − n2 0 0 0 0 0

0 0 0 0 0 0 0 ξ3 0 0 0 0

0 0 0 0 0 0 0 0 2ξ3 0 0 0

0 0 0 0 0 0 0 0 0 2ξ3 0 0

n 2

0 0 0 0 0 0 0 0 0 0 0

0 

n 2

0 0 0 0 0 0 0 0 0 0

⎤ 0 0 0 00 0 0 0 0 0⎥ ⎥ 0 0 0⎥ −ξ3 0 ⎥ 0 −2ξ3 0 0 0 ⎥ ⎥ 0 0 −2ξ3 0 0 ⎥ ⎥ 0 0 0 0 0⎥ ⎥. 0 0 0 0 0⎥ ⎥ 0 0 0 0 0⎥ ⎥ 0 0 0 0 0⎥ ⎥ 0 0 0 0 0⎥ ⎥ 0 0 0 0 0⎦ 0 0 0 00

(a) Since ξ3 < 0 and n, n  ∈ Z>0 , it follows that gξ = W1 , W2 . Now we have Lie(ker πξ ) ⊂ Lie(ker α) = 0, and hence πξ is generic. (b) We have E 3 ∈ gξ . Thus, dim ker α ≥ 1. We see from (2.1) that πξ is not generic. 

7 Irreducible Non-generic CS Representations Let h5 = E 3 , E 3,1 , E 3,2 , A3,1 , A3,2 , h3 = E 3 , E 3,1 , A3,1 , h3 = E 3 , E 3,2 , A3,2 , a1 = A3 , s3 = E 1 , A1 , W1 , s3 = E 2 , A2 , W2 . Then we have the following lemma. Lemma 7.1 (a) Every nontrivial ideal in g contains E 3 . (b) Let h be an ideal in g such that E 3  h. Then h contains h3 or h3 . (c) Let h be an ideal in g such that h3  h. Then h contains h5 or h3 ⊕ s3 .

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(d) Let h be an ideal in g such that h3 ⊕ s3  h. Then h contains h5 ⊕ s3 . Proof Let x = e1 E 1 + e2 E 2 + e3 E 3 + e3,1 E 3,1 + e3,2 E 3,2 + a1 A1 + a2 A2 + a3 A3 + a3,1 A3,1 + a3,2 A3,2 + k1 W1 + k2 W2 with e1 , e2 , e3 , e3,1 , e3,2 , a1 , a2 , a3 , a3,1 , a3,2 , k1 , k2 ∈ R. / h. Since [E 3 , x] = (a) Suppose that x is contained in an ideal h of g such that E 3 ∈ −a3 E 3 , we have a3 = 0. Then we have [E 3,1 , x] = − a21 E 3,1 − 2a3,1 E 3 + k1 A3,1 , [E 3,1 , [E 3,1 , x]] = −2k1 E 3 , [A3,1 , [E 3,1 , x]] = −a1 E 3 , a2 [E 3,2 , x] = − 2 E 3,2 − 2a3,2 E 3 + k2 A3,2 , [E 3,2 , [E 3,2 , x]] = −2k2 E 3 ,

[A3,2 , [E 3,2 , x]] = −a2 E 3 ,

so that a1 = a2 = a3,1 = a3,2 = k1 = k2 = 0. Next [A3,1 , x] = e1 E 3,1 + 2e3,1 E 3 , [A3,2 , x] = e2 E 3,2 + 2e3,2 E 3 ,

[A3,1 , [A3,1 , x]] = 2e1 E 3 , [A3,2 , [A3,2 , x]] = 2e2 E 3 ,

which imply that e1 = e2 = e3,1 = e3,2 = 0. Thus, x = e3 E 3 = 0 and h = 0. Therefore, every nontrivial ideal of g contains E 3 . (b) Let h = E 3 . It is enough to show that h˜ = h/h contains s E 3,1 + t A3,1 +  h with s 2 + t 2 = 0 or s E 3,2 + t A3,2 + h with s 2 + t 2 = 0. Arguing contradiction, assume that h˜ does not contain either of them. Let x ∈ h. We have [E 3,1 , x] = − 21 (a1 + a3 )E 3,1 + k1 A3,1 , [E 3,2 , x] = − 21 (a2 + a3 )E 3,2 + k2 A3,2 , [A3 , x] = 21 (e3,1 E 3,1 + e3,2 E 3,2 + a3,1 A3,1 + a3,2 A3,2 ), [W1 , [A3 , x]] = 21 (a3,1 E 3,1 − e3,1 A3,1 ) (mod h ),

(7.1)

so that e3,1 = e3,2 = a1 + a3 = a2 + a3 = a3,1 = a3,2 = k1 = k2 = 0. We also have [A3,1 , x] = e1 E 3,1 +

a1 −a3 2

A3,1 , [A3,2 , x] = e2 E 3,2 +

a2 −a3 2

A3,2 (mod h ),

so that e1 = e2 = a1 = a2 = a3 = 0. Hence, h˜ = 0, which contradicts the assumption. (c) Let h = h3 . It is enough to show that h˜ = h/h contains s E 3,1 + t A3,1 + h with s 2 + t 2 = 0 or s E 2 + t A2 + h with s 2 + t 2 = 0. Arguing contradiction, assume that h˜ does not contain either of them. Let x ∈ h. We have [E 1 , x] = −a1 E 1 − a3,1 E 3,1 + 2k1 A1 , [A3,1 , [E 1 , x]] = −a1 E 3,1 + k1 A3,1 , 1 [A3 , x] = (e3,1 E 3,1 + a3,1 A3,1 ), [E 2 , x] = −a2 E 2 + 2k2 A2 (mod h ), (7.2) 2

Coherent State Representations of the Holomorphic Automorphism …

17

so that e3,1 = a1 = a2 = a3,1 = k1 = k2 = 0. Next [A2 , x] = e2 E 2 , [A3,1 , x] = e1 E 3,1 −

a3 2

A3,1 (mod h ),

which implies that e1 = e2 = a3 = 0. Hence, x ∈ h , which contradicts the assumption. (d) Let h = h3 ⊕ s3 . It is enough to show that h˜ = h/h contains s E 3,1 + t A3,1 + h with s 2 + t 2 = 0. Arguing contradiction, assume that h˜ does not contain such an element. Let x ∈ h. From (7.1) and (7.2), we see that e3,1 = a1 = a3 = a3,1 = k1 = 0. Since [W1 , x] = −2e1 A1 , [A3,1 , [W1 , x]] = −e1 A3,1 (mod h ), we obtain e1 = 0. Hence, x ∈ h , which contradicts the assumption.



If we take into account Lemma 7.1 and that s3 ⊕ s3 is semisimple, it is not hard to determine all ideals of g. Figure 1 gives the Hasse diagram of the set of all nontrivial ideals of g, ordered by inclusion. Proof of Proposition 4.2 We have ϕ(h3 ) = h3 or ϕ(h3 ) = h3 , and it is enough to  show that if ϕ(h3 ) = h3 , then ϕ can be written as ϕ = σ ε ◦ Ad(g) for some g ∈ G and some ε ∈ {0, 1}. Let ϕ(h3 ) = h3 . Let us consider the adjoint action of G on g. The subgroups exp a1 , exp s3 ⊂ G act on the ideal h3 of g by dilations h3 e3 E 3 + e3,1 E 3,1 + a3,1 A3,1 → r 2 e3 E 3 + r e3,1 E 3,1 + ra3,1 A3,1 ∈ h3 (r > 0) and symplectic maps h3 e3 E 3 + e3,1 E 3,1 + a3,1 A3,1 → e3 E 3 + (e3,1 α + a3,1 β)E 3,1 + (e3,1 γ+a3,1 δ)A3,1 ∈ h3 (αδ − βγ = 1),

respectively. It is well known that the automorphism group of h3 is generated by inner automorphisms, symplectic maps, dilations, and inversion h3 e3 E 3 + e3,1 E 3,1 + a3,1 A3,1 → −e3 E 3 + a3,1 E 3,1 + e3,1 A3,1 ∈ h3 . Thus, we have ϕ ◦ Ad(g) |h3 = idh3 or ϕ ◦ Ad(g) |h3 = σ|h3 for some g ∈ exp h3 ⊕ a1 ⊕ s3 ⊂ G. Now it is enough to show that if ϕ|h3 = idh3 , then ϕ ◦ Ad(g) = idg for some g ∈ G. Let ϕ|h3 = idh3 . We have ϕ ◦ Ad(g) |h5 = idh5 for some g ∈ G. Hence, we may and do assume that ϕ|h5 = idh5 . Let us consider the subrepresentation (ad, h5 ) of the adjoint representation ad of g. Then the kernel of the subrepresentation equals E 3 (see Remark 7.2 below), and hence it follows that ϕ(A3 ) = A3 + e3 E 3 with e3 ∈ R. Moreover, we see that ϕ ◦ Ad(g) |h5 ⊕a1 = idh5 ⊕a1 for some g ∈ expE 3 ⊂ G. Let us consider the subrepresentation (ad, h5 ⊕ a1 ) of the adjoint representation ad of g. Then the kernel of the subrepresentation equals {0} (see Remark 7.2 below), and

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Fig. 1 The Hasse diagram of the set of all nontrivial ideals of g

hence it follows that an automorphism of g which is the identity on h5 ⊕ a1 is the identity on g. The proof is complete.  Remark 7.2 For x = e1 E 1 + e2 E 2 + e3 E 3 + e3,1 E 3,1 + e3,2 E 3,2 + a1 A1 + a2 A2 + a3 A3 + a3,1 A3,1 + a3,2 A3,2 + k1 W1 + k2 W2 with e1 , e2 , e3 , e3,1 , e3,2 , a1 , a2 , a3 , a3,1 , a3,2 , k1 , k2 ∈ R, let us see the matrix of ad(x) : h5 ⊕ a1 → h5 ⊕ a1 with respect to the basis {E 3 , E 3,1 , E 3,2 , A3 , A3,1 , A3,2 }. The matrix is given by ⎡

a3 2a3,1 2a3,2 ⎢ 0 a3 /2 + a1 /2 0 ⎢ ⎢0 /2 + a2 /2 0 a 3 ⎢ ⎢0 0 0 ⎢ ⎣0 0 −k1 0 0 −k2

⎤ −e3 −2e3,1 −2e3,2 ⎥ −e3,1 /2 k1 − e1 0 ⎥ −e3,2 /2 0 k2 − e2 ⎥ ⎥. ⎥ 0 0 0 ⎥ ⎦ −a3,1 /2 a3 /2 − a1 /2 0 −a3,2 /2 0 a3 /2 − a2 /2

By the definition of CS representation, if all generic CS representations of all the quotient groups of G by connected closed normal subgroups are given, then we can obtain all irreducible CS representations of G by composing the quotient maps. Let G˜ be the quotient group by a connected closed normal subgroup of G, and let G˜ = G, {e}. According to Fig. 1, it is enough to consider the following cases: (i) g˜  R, (ii) g˜  sl(2, R), (iii) g˜  R ⊕ sl(2, R), (iv) g˜  sl(2, R) ⊕ sl(2, R), (v) g˜  R ⊕ sl(2, R) ⊕ sl(2, R), (vi) g˜ = g/h3 ⊕ s3 , (vii) g˜ = g/h3 , (viii) g˜ = g/E 3 .

However, G˜ does not admit generic CS representations in the cases (vi)–(viii). We shall prove this. Suppose that M is a CS orbit of a generic CS representation of ˜ Let K be the isotropy subgroup of G˜ at some point m 0 of M. We shall seek a G. maximal compact subgroup of Int g˜ .

Coherent State Representations of the Holomorphic Automorphism …

19

Proposition 7.3 (a) We have the following isomorphisms: Int g˜  G/ exp h3 ⊕ s3 in the case (vi), Int g˜  G/ exp h3 in the case (vii), and Int g˜  G/ expE 3 in the case (viii). (b) The maximal compact subgroups of G/ exp h3 ⊕ s3 , G/ exp h3 , and G/ expE 3 are conjugate to the images of expW1 , W2 ⊂ G under the quotient maps. (c) We have π1 (Int g˜ , e) = Z in the case (vi) and π1 (Int g˜ , e) = Z2 in the cases (vii) and (viii). Proof We shall prove (a) and (b) for the case (viii) and (c) for the case (vi). For the other cases, this can be proved in the same way. (a) It is enough to show that G/ expE 3 has a trivial center. Let ⎡

0 a1 ⎢ 0 a2 ⎢ ⎢ (c1 μ + a1 λ )a3 (c2 μ + a2 λ )a3 1 1 2 2 g=⎢ ⎢ 0 c1 ⎢ ⎣ 0 c2 0 0

⎡

Then

g −1

d1 ⎢ 0 ⎢ ⎢ −λ 1 =⎢ ⎢ −c1 ⎢ ⎣ 0 0

0 0 d2 0 −λ2 1/a3 0 0 −c2 0 0 0

0 b1 0 0 0 b2 a3 (d1 μ1 + b1 λ1 )a3 (d2 μ2 + b2 λ2 )a3 0 d1 0 0 0 d2 0 0 0

−b1 0 −μ1 a1 0 0

0 −b2 −μ2 0 a2 0

⎤ μ1  μ2 ⎥ ⎥ κ ⎥ ⎥ ∈ G. −λ1 ⎥ ⎥ −λ2 ⎦ 1/a3

⎤ −(d1 μ1 + b1 λ1 )a3 −(d2 μ2 + b2 λ2 )a3 ⎥ ⎥ ⎥ −κ ⎥.   (c1 μ1 + a1 λ1 )a3 ⎥ ⎥ (c2 μ2 + a2 λ2 )a3 ⎦ a3

We have Ad(g −1 ) E 1 = (d12 − c12 )E 1 + λ1 E 3 − d1 λ1 E 3,1 + 2c1 d1 A1 − c1 λ1 A3,1 − c12 W1 , 2

Ad(g −1 ) E 2 = (d22 − c22 )E 2 + λ2 E 3 − d2 λ2 E 3,2 + 2c2 d2 A2 − c2 λ2 A3,2 − c22 W2 , 2

Ad(g −1 ) A3,1 = (2μ1 E 3 + b1 E 3,1 + a1 A3,1 )/a3 , Ad(g −1 ) A3,2 = (2μ2 E 3 + b2 E 3,2 + a2 A3,2 )/a3 , Ad(g −1 ) W1 = (−d12 + c12 − b12 + a12 )E 1 + (−μ1 − λ1 )E 3 + (d1 λ1 − b1 μ1 )E 3,1 2

2

+ (−2c1 d1 − 2a1 b1 )A1 + (c1 λ1 − a1 μ1 )A3,1 + (c12 + a12 )W1 , Ad(g −1 ) W2 = (−d22 + c22 − b22 + a22 )E 2 + (−μ2 − λ2 )E 3 + (d2 λ2 − b2 μ2 )E 3,2 + (−2c2 d2 − 2a2 b2 )A2 + (c2 λ2 − a2 μ2 )A3,2 + (c22 + a22 )W2 . 2

2

Suppose that Ad(g −1 ) induces the identity map of g/E 3 onto itself. Then we have

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K. Arashi

d12 − c12 = 1, d1 λ1 = c12 = 0, d22 − c22 = 1, d2 λ2 = c22 = 0, b1 = 0, a1 /a3 = 1, b2 = 0, a2 /a3 = 1, c1 λ1 c2 λ2

− a1 μ1 − a2 μ2

= 0, c12 + a12 = 1, = 0, c22 + a22 = 1.

Hence, it follows that g −1 ∈ expE 3 ⊂ G, which implies that G/ expE 3 has trivial center. (b) By (a), we see that G/ expE 3 is linearlizable. By Theorem 4.1, we conclude that the image of the subgroup expW1 , W2 of G under the quotient map is a maximal compact subgroup of G/ expE 3 . Note that the image of a compact subgroup or a real split solvable Lie subgroup under a homomorphism of a Lie groups is also a compact subgroup or a real split solvable Lie subgroup, respectively. (c) The group exp h3 ⊕ s3 ⊂ G is a topological product of H3 (R) and S L(2, R).  By (a), it follows that π1 (Int g˜ , e) = Z2 /Z = Z. In the case (vi), we have π1 (Int g˜ , e) = Z. We may assume that k ⊂ E 2 , E 3 , E 3,2 , A2 , A3,2 , W1 , W2 /E 2 , E 3 , E 3,2 , A2 , A3,2 , W2 , and we then have dim k = 0. Since M is diffeomorphic to a coadjoint orbit, the group Int g˜ acts transitively on M, and the isotropy subgroup (Int g˜ )m 0 at m 0 equals {e}. Thus, π1 ((Int g˜ )m 0 , e) = {e}. This contradicts that M is simply connected. Similarly, we have π1 (Int g˜ , e) = Z2 , π1 ((Int g˜ )m 0 , e) = Z in the case (vii), and we have π1 (Int g˜ , e) = Z2 , π1 ((Int g˜ )m 0 , e) = Z in the case (viii). These results contradict that M is simply connected. We obtain the following theorem. Theorem 7.4 Every irreducible non-generic CS representation of G is given by the composition of the external tensor product of a one-dimensional unitary representation of R>0 and a nontrivial irreducible highest weight representation of S L(2, R) × S L(2, R) with the map G  G  → R>0 × S L(2, R) × S L(2, R) given by ⎡

a1 ⎢ 0 ⎢ ⎢ λ1 G ⎢ ⎢ c1 ⎢ ⎣ 0 0

0 a2 λ2 0 c2 0

0 0 a3 0 0 0

b1 0 μ1 d1 0 0

0 b2 μ2 0 d2 0

⎤ μ1  μ2 ⎥ ⎥      κ ⎥ ⎥ → a 3 , a 1 b1 , a 2 b2 ∈ R>0 × S L(2, R) × S L(2, R).  ⎥ −λ1 ⎥ c 1 d1 c 2 d2 −λ ⎦ 2

a3−1

8 Intertwining Operators 

For n, n  ∈ Z, let (πn,n  , Hn,n ) be any irreducible unitary representation of S L(2, R)  × S L(2, R) such that there exists v ∈ (Hn,n )∞ \{0} satisfying

Coherent State Representations of the Holomorphic Automorphism …

 dπn,n  (x, y) v = 0 for (x, y) ∈ C

21

   −i 1 −i 1 ×C 1 i 1 i

and  πn,n 

    cos θ − sin θ cos τ − sin τ , v = ei(nθ+n τ ) v (θ, τ ∈ R). sin θ cos θ sin τ cos τ

Then the set of equivalence classes of irreducible highest weight representations of S L(2, R) × S L(2, R) is given by {[πn,n  ]; n, n  ∈ Z}. Let G˜ = R>0 × S L(2, R) ×  S L(2, R). For η3 ∈ R, let (πη3 ,n,n  , Hn,n ) be the external tensor product of the onedimensional representation of R>0 given by R>0 γ → γ 2iη3 ∈ C× and the representation πn,n  of S L(2, R) × S L(2, R). Composing with the map G → G˜ given in Theorem 7.4, we regard πη3 ,n,n  as a representation of G. Let n, n  ∈ Z≥0 . By (5.2), we can take πξ(0,η3 ,n,n  ) | SL(2,R)×SL(2,R) to be the irreducible unitary representation πn,n  , and hence πξ(0,η3 ,n,n  ) is unitarily equivalent with πη3 ,n,n  as representations of G. Therefore, we get the following theorem. Theorem 8.1 The set of unitary equivalence classes of irreducible non-generic CS representations of G is given by {[πη3 ,n,n  ]; (η3 , n, n  ) ∈ R × Z × Z\R × {0} × {0}}. For (η3 , n, n  ) ∈ R × Z≥0 × Z≥0 , we have πη3 ,n,n   πξ(0,η3 ,n,n  ) and π−η3 ,−n,−n   πξ(0,η3 ,n,n  ) . We fix a triple (η3 , n, n  ) with η3 ∈ R and n, n  ∈ Z≥0 . We shall give an explicit description of an intertwining operator between the unitary representations πη3 ,n,n  and πξ(0,η3 ,n,n  ) of G. Using the realization of G as a linear group in Sect. 3, we shall define a holomorphic multiplier representation of G. Let m : G × D5 → C× be the holomorphic multiplier given by 

m(g, z) = (c1 z 1 + d1 )n (c2 z 2 + d2 )n a3 2iη3 (g ∈ G, z ∈ D5 ), and let τm be the holomorphic multiplier representation given by τm (g) f (z) = m(g −1 , z)−1 f (g −1 z) (g ∈ G, z ∈ D5 , f ∈ O(D5 )). Then πξ(0,η3 ,n,n  ) can be considered as a unitarization of τm . Next we see a natural holomorphic multiplier representation of G in which πη3 ,n,n  is realized. Let D1 be the unit disk in C, and let m˜ : G˜ × D1 × D1 → C× be the holomorphic multiplier given by 

m((γ, ˜ g1 , g2 ), (w 1 , w 2 )) = (c1 w 1 + d1 )n (c2 w 2 + d2 )n γ 2iη3 ˜ (w 1 , w 2 ) ∈ D1 × D1 ), ((γ, g1 , g2 ) ∈ G,

22

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 ai bi ∈ S L(2, R) for i = 1, 2. We denote by D1 wi → gi wi ∈ D1 ci di the action of S L(2, R) by linear fractional transformations for i = 1, 2. Then we can define the following holomorphic multiplier representation τm˜ of G˜ on the space O(D1 × D1 ) of holomorphic functions on D1 × D1 : 

where gi =

τm˜ (g) f (w 1 , w 2 ) = m(g −1 , (w 1 , w 2 ))−1 f (g1−1 w 1 , g2−1 w 2 ) ˜ (w 1 , w 2 ) ∈ D1 × D1 , f ∈ O(D1 × D1 )). (g = (γ, g1 , g2 ) ∈ G, We regard τm˜ as a representation of G which exp h5 ⊂ G acts by the trivial representation. Then we have the following theorem. Theorem 8.2 The map F : O(D1 × D1 ) f → F f ∈ O(D5 ) defined by F f (z) = f (z 1 , z 2 ) (z ∈ D5 ) intertwines τm˜ with τm , and hence F gives rise to an intertwining operator between the unitarizations.

References 1. K. Arashi, Holomorphic multiplier representations for bounded homogeneous domains. J. Lie Theory 30, 1091–1116 (2020) 2. P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard, M. Vergne, Représentations Des Groupes de Lie Résolubles. Monographies de La Société Mathématique de France, vol. 4 (Dunod, Paris, 1972) 3. S.-S. Chen, Bounded holomorphic functions in Siegel domains. Proc. Am. Math. Soc. 40, 539–542 (1973) 4. I.G. Dotti, Rigidity of invariant complex structures. Trans. Am. Math. Soc. 338, 159–172 (1993) 5. L. Geatti, Holomorphic automorphisms of some tube domains over nonselfadjoint cones. Rend. Circ. Mat. Palermo (2), 36, 281–331 (1987) 6. S.G. Gindikin, I.I. Pjatecki˘ı-Šapiro, È.B. Vinberg, Homogeneous Kähler manifolds, in Geometry of Homogeneous Bounded Domains, ed. by E. Vesentini. C.I.M.E. Summer Schools, vol. 45 (Springer, Berlin, 1968), pp. 1–87 7. H. Ishi, Unitary holomorphic multiplier representations over a homogeneous bounded domain. Adv. Pure Appl. Math. 2, 405–419 (2011) 8. H. Ishi, K. Koufany, The compression semigroup of the dual Vinberg cone (2020), 12 p. 9. S. Kobayashi, Irreducibility of certain unitary representations. J. Math. Soc. Jpn. 20, 638–642 (1968) 10. R.A. Kunze, On the irreducibility of certain multiplier representations. Bull. Am. Math. Soc. 68, 93–94 (1962) 11. W. Lisiecki, Kaehler coherent state orbits for representations of semisimple Lie groups. Ann. Inst. H. Poincaré Phys. Théor. 53, 245–258 (1990) 12. W. Lisiecki, A classification of coherent state representations of unimodular Lie groups. Bull. Am. Math. Soc. (N.S.), 25, 37–43 (1991) 13. W. Lisiecki, Coherent state representations. a survey. Rep. Math. Phys. 35, 327–358 (1995) 14. K.-H. Neeb, Holomorphy and Convexity in Lie Theory. De Gruyter Expositions in Mathematics, vol. 28 (Walter de Gruyter, Berlin, 2000) 15. A.L. Onishchik, È.B. Vinberg, Lie Groups and Lie Algebras, III: Structure of Lie Groups and Lie Algebras. Encyclopaedia of Mathematical Sciences, vol. 41 (Springer, Berlin, 1994)

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16. J. Rosenberg, M. Vergne, Harmonically induced representations of solvable Lie groups. J. Funct. Anal. 62, 8–37 (1985) 17. J.A. Tirao, J.A. Wolf, Homogeneous holomorphic vector bundles. Indiana Univ. Math. J. 20, 15–31 (1970/71)

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses Daniel de la Concepción and Abdenacer Makhlouf

Abstract The purpose of this paper is to study Sabinin algebras of Hom-type. It is shown that Lie, Malcev, Bol and other algebras of Hom-type are naturally Sabinin algebras of Hom-type. To this end, we provide a general key construction that establish a relationship between identities of some class of Hom-algebras and ordinary algebras. Moreover, we discuss a new concept of Hom-bialgebra, in relation with universal enveloping Hom-algebras. A study based on primitive elements is provided. Keywords Hom-Sabinin algebra · Hom-Q-algebra · Hom-Lie-algebra · Sabinin algebra · Hom-bialgebra Introduction Hom-type algebras appeared first in physics literature, in quantum deformations of algebras of vector fields, mainly Witt and Virasoro algebras. It turns out that if one replaces usual derivation by σ-derivations, the Jacobi condition is no longer satisfied. The identity satisfied is a twisted version of Jacobi condition. This was the motivation to introduce the so-called Hom-Lie algebras in [5] and their corresponding associative algebras called Hom-associative algebras in [10]. Since then, many algebraic structures were extended to Hom-setting. Hom-type analogues of alternative algebras, Jordan algebras or Malcev algebras were defined and discussed in [9, 21]. As well as n-ary algebras for which Hom-type versions were introduced in [2, 22]. On the other hand there has been work done in relation to the generalization of Lie algebras in such a way that these generalizations would classify differential manifolds with locally defined loop structures; as Lie algebras classify local Lie groups. This resulted in Malcev algebras in relation to Moufang loops; for instance, and eventually to Sabinin algebra and the following results from [14]: • Analytic local loops (Q i , ·, \, /, e) are locally isomorphic if an only if their corresponding Sabinin qi = Te (Q i ) algebras are isomorphic. D. de la Concepción Departamento de Matemáticas y Computación, Universidad de la Rioja, Logrono, Spain e-mail: [email protected] A. Makhlouf (B) IRIMAS-Département de Mathématiques, Université de Haute-Alsace, Mulhouse, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_3

25

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D. de la Concepción and A. Makhlouf

• Any finite-dimensional Sabinin algebra over R whose structure constants satisfy certain conditions (can be found in [14]), is the Sabinin algebra of some analytic local loop. • Let (Q, ·, \, /, e) be an analytic local loop and q its Sabinin algebra. If R is a local subloop of Q then Te (R) = r is a Sabinin sublagebra of q. Conversely, for any Sabinin subalgebra r of q, there is a unique local subloop R in Q such that Te (R) = r. The subloop R is normal if and only if r is an ideal of q. The structure of Sabinin algebras is extensive; as a Sabinin algebra may have an infinite number of operations. A Sabinin algebra is a vector space V and two sets of operations < −; −, − >: V ⊗n ⊗ V ⊗ V → V for n ≥ 0 and n,m : V ⊗n ⊗ V ⊗m → V for n > 0 and m > 1; that satisfy the following identities: < x; a, b >= − < x; b, a >,  < xaby; c, e > − < xbay; c, e > + < x(1) < x(2) ; a, b >; c, e >= 0, σa,b,c (< xc; a, b > + < x(1) ; < x(2) ; a, b >, c >) = 0, (x, y) = (τ · x, σ · y) where τ ∈ Sn and σ ∈ Sm ,  where Sn is the permutation group of n elements, x(1) ⊗ x(2) is the natural coproduct of T (V ), the tensor algebra, that makes it a bialgebra and where V is given by primitive elements. The main purpose of this paper is to unify the constructions of certain classes of Hom-algebras and study Hom-type generalization of Sabinin algebras and related structures. A Hom-type algebra is an algebra (multiplication) together with an endomorphism called twisting map. The main feature of Hom-type generalizations of algebra varieties is that the endomorphism appears in the identities that define the variety; in such a way that, when the twisting map is the identity map; one recovers the original variety of algebras. Moreover, we discuss a new concept of Hom-bialgebra, in relation with universal enveloping Hom-algebras and provide a study based on primitive elements. Throughout this paper; unless specifically stated, every field K is of characteristic zero and an algebra is understood as a vector space A with a countable set of operations μi : A⊗ni → A. We mean by a Hom-algebra an algebra together with a homomorphism. 1. 2. 3. 4.

1 Main Construction and Hom-Sabinin Algebras In this section, we state a theorem providing a suitable way to establish a Hom-type generalization of a given ordinary algebraic structure. This will be used to study HomSabinin algebras and their subclasses. Let’s define some concepts that are relevant to understand the extent of this result. Firstly, let’s recall an idea given by Yau in [20] explaining how to define Homalgebra varieties from algebra varieties using operads and PROPs. The main idea is

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

27

to consider a defining identity of the variety as a function obtained using the identity function, the algebra operations; and tensor product, sum and composition of functions; then replacing a subset of instances of the identity function by endomorphisms αi : A → A to get a new identity of Hom-type. Then the Hom-algebras of the new variety are defined as those algebras with endomorphisms (A, {μi }i∈I , {αi }i∈J ) that satisfy the new identities. Specifically, let’s recall first what a variety of algebras is Definition 1.1 ([4]) A variety of algebras is the class of all algebraic structures of a given signature satisfying a set of identities; or equivalently a class of algebraic structures of a given signature closed by homomorphic images, subalgebras and direct products. The result that states the equivalence of those two definitions is known as Birkhoff’s Theorem. A signature is a map from a fixed set S to N; σ : S → N. An algebraic structure of a signature σ : S → N is a vector space A with a set of maps identified with S such that every map s ∈ S is given as s : A⊗σ(s) → A. In other words, the signature of an algebraic structure is a family of operations considering for each one its arity; meaning that two algebraic structures have the same signature if they have families of operations { f i }i∈I and {gi }i∈J of the same cardinal I = J and the operation f i has the same arity as gi ; i.e. σ( f i ) = σ(gi ). Example 1.2 There are many examples of varieties of algebras on vector spaces: • • • • •

Lie algebras are given by one binary operation and two identities. Associative algebras are given by one binary operation and one identity. Jordan algebras are given by one binary operation and two identities. 3-Lie algebras are given by one ternary operation and three identities. Bol algebras are given by one binary and one ternary operation and five identities.

Varieties of algebras are interesting because of the following classical result: Proposition 1.3 Every non-empty variety of algebras has a free algebra generated by any vector space. To make everything easier, it is natural to think of free algebra elements as trees, where every node has one output and n inputs. A node would then represent an n-ary operation of the algebraic structure. The identities can be thought as linear combinations of those trees Definition 1.4 Let (A, {μi }) be an algebraic structure and α : A → A an endomor phism of such structure. Consider an identity as a linear combination of trees Ti with its internal nodes labelled by the set of operations in such a way that if the node N j is labelled with the operation s, then N j has σ(s) inputs and one output. The leaves represent elements in the algebra. The Hom-type identities would then be obtained by the following procedure: For every internal node N j on every tree Ti , apply ασ(s)−1 to every leaf which is not comparable to it; considering the partial order defining the tree. Given a list of identities defining a variety of algebras, its Hom-algebra variety is defined by the Hom-type identities of those.

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The previous procedure can be applied to different sets of identities. Example 1.5 In case of associative algebras, there is only one identity (x y)z − x(yz) = 0 that is given by the following linear combination:

Every internal node in this trees is binary, hence σ(N j ) − 1 = 1. Nevertheless, only one internal node in each tree has a leaf not comparable to it. The final trees are then defined as

The identity obtained is: (x y)α(z) − α(x)(yz) = 0, which is the identity of Hom-associative algebras. In the sequel, we denote the Hom-type associator by (x, y, z)α = (x y)α(z) − α(x)(yz).

(1)

Example 1.6 In case of Lie algebras there are two identities [x, y] + [y, x] = 0 and [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 that is given by the following linear combination:

Every internal node in this trees is binary, hence σ(N j ) − 1 = 1. In the first equation, there is no change, since the only internal node is comparable to both leaves in both trees. Nevertheless, one internal node in each tree of the second equation has a leaf not comparable to it. The final trees are then defined as

The identities obtained are:

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

29

[x, y] + [y, x] = 0 [[x, y], α(z)] + [[y, z], α(x)] + [[z, x], α(y)] = 0, which define exactly Hom-Lie algebras. In the sequel, we denote the Hom-type Jacobiator by Jα (x, y, z) = [[x, y], α(z)] + [[y, z], α(x)] + [[z, x], α(y)].

(2)

We will only consider cases of homogeneous identities in our examples; i.e., p p identities f (x1 , . . . , xn ) such that f (λ1 x1 , . . . , λn xn ) = λ1 1 . . . λn n f (x1 , . . . , xn ). It may seem restrictive, but a wide range of examples of algebras fall into this definition, in particular, any Sabinin algebra. The reason is to get free algebras which are N+ graded. Comparing our procedure with the one used in [20]; in our case, we reach an algorithm which computes the Hom-type identities by changing every instance of the identity map by αn ; where n depends on the arity of the operations of the algebra. Definition 1.7 Let (A, {μi }) be an algebra. Consider τ to be a tree representing some monomial in A on elements x1 , . . . , xn , then we denote the monomial obtained by the previous procedure as (x1 ⊗ · · · ⊗ xn )ατ . Lemma 1.8 Consider Q to be a variety of algebras. Let (B, {μ j }, α B ) be a multiplicative Hom-Q-algebra which is N+ -graded. We define the product μ j,α : Ba1 ⊗ · · · ⊗ Ban j → B ai as μ j,α (x1 , . . . , xn j ) =

  ( i=n ai )−n j +1 ( i=1 ai )−n j +1 μ j (α B (x1 ), . . . , α B j (xn j )),

for ai > 0. Under these assumptions, (B, {μ j,α }) is a Q-algebra. If (B, {μ j,α }) is a Q-algebra for some (B, {μ j }, α B ), N+ -graded multiplicative Hom-algebra; then (B, {μ j }, α B ) is in fact a Hom-Q-algebra. Proof The proof follows from Definition 1.4 and the definition of μα from the multilinear operator μ.  Our definition of Hom-type structures differs from some already defined Homalgebras varieties; making α less generic in our definitions. This approach is suitable to discuss enveloping structures and the multilinear operations. Here is an example of mismatching definitions, first we consider Lie triple systems and then 3-Lie algebras Hom-types Example 1.9 A Hom-Lts with this definition is given by a triple (L , [−, −, −], α), where 1.

σx,y,z [x, y, z] = 0,

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2. [α2 (u), α2 (v), [x, y, z]] = [[u, v, x], α2 (y), α2 (z)] + [α2 (x), [u, v, y], α2 (z)] + [α2 (x), α2 (y), [u, v, z]]. This definition differs from that in [22], as we are considering only the case where α1 = α2 = α2 in that paper. Nevertheless, if from a Hom-Lie algebra (L , [ , ], α) is defined a new triple product [x, y, z] = [[x, y], α(z)]; then (L , [ , , ], α) is a Hom-Lts in the sense defined above or (L , [ , ], (α2 , α2 )) is a Hom-Lts as defined in [22]; meaning that our definition isn’t as restrictive as it may seem. Example 1.10 A 3-Hom-Lie algebra with this definition is given by a triple (L , [−, −, −], α), where [x, y, z] = −[y, x, z],

1.

[x, y, z] = [y, z, x],

2. [α (u), α (v), [x, y, z]] = 2

3.

2

[[u, v, x], α2 (y), α2 (z)] + [α2 (x), [u, v, y], α2 (z)] + [α2 (x), α2 (y), [u, v, z]].

Again this definition differs from that in [2]; as we are considering only the case where α1 = α2 = α2 in that paper. In the previous examples, one can define the algebra variety using the following α-twisted fundamental identity: [α(u), α(v), [x, y, z]] = [[u, v, x], α(y), α(z)] + [α(x), [u, v, y], α(z)]+ [α(x), α(y), [u, v, z]], but the results obtained in the following sections cannot be applied to this more general case; as there might not be an endomorphism β such that α = β 2 . The definition of Hom-Sabinin algebras results in Definition 1.11 A Sabinin algebra of Hom-type, or Hom-Sabinin algebra (S, {< −; −, − >}n , {n,m }n,m , α) is defined as • a vector space S, • a family of maps < −; −, − >: S ⊗n ⊗ S ⊗ S → S for n ≥ 0 and n,m : S ⊗n ⊗ S ⊗m → S for n > 0 and m > 1, • an endomorphism α of the algebra (S, {< −; −, − >: S ⊗n ⊗ S ⊗ S → S}n , {n,m : S ⊗n ⊗ S ⊗m → S}n,m ), such that the following identities hold: < x; a, b > + < x; b, a >= 0,

(3)

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

< x[a, b]y; c, e > +

31



< αk (x(1) ) < x(2) ; a, b > αk (y); αk (c), αk (e) >= 0, (4)  < αk (x(1) ); < x(2) ; a, b >, αk (c) >) = 0, (5) σa,b,c (< xc; a, b > + (x, y) = (τ · x, σ · y),

(6)

where τ ∈ Sn , σ ∈ Sm and k = |x(2) | + 1. The following result provides the main theorem of this paper; not because of its elaborate proof, which is rather easy, but because it has very deep consequences. Theorem 1.12 (Main Theorem) Consider some variety of algebras, Q, where every free Hom-Q-algebra is N+ -graded. Then any Hom-Q-algebra satisfies the Hom-type identities satisfied by the Q-algebras. Proof By our first assumption, we only need to prove the result for free algebras. Consider any free Hom-Q-algebra B(V, α). This Hom-algebra is the quotient of the free Hom-algebra F(V, α) by the ideal generated by the identities that define Hom-Q-algebras. By our second assumption, it is graded by N+ . It is then possible to define {μ j,α } as in the previous results. By Lemma 1.8, (B(V ), μ j,α ) is a Q-algebra; and hence satisfies all identities of Q-algebras. The result follows from unfolding the definition of μ j,α in any identity  of Q-algebras satisfied by (B(V ), μ j,α ). Note that, as stated before, if the defining relations of the variety Q are given by homogeneous multilinear identities, the free Q-algebras are N+ -graded and so are the free Hom-Q-algebras. In conclusion, this result can be directly applied to those algebra classes. One may provide a generalization of the Yau’s twisting principle, given in [17, 20]. Proposition 1.13 Let (S, α) be a Hom-Sabinin algebra and β : S → S be an endomorphism of the Hom-Sabinin algebra. Then (S, {β n+1 ◦ < −; −, − >}n , {β n+m−1 ◦ n,m }n,m , β ◦ α) is a Hom-Sabinin algebra. In particular, one may construct a Hom-Sabinin algebra starting from a Sabinin algebra and a Sabinin algebra endomorphism. Corollary 1.14 Let (S, {< −; −, − >: S ⊗n ⊗ S ⊗ S → S}n ) be a Sabinin algebra and α : S → S be a Sabinin algebra endomorphism. Then (S, {αn+1 ◦ < −; −, − >: S ⊗n ⊗ S ⊗ S → S}n , {αn+m−1 ◦ n,m }n,m , α) is a Hom-Sabinin algebra.

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2 Consequences of the Main Theorem 2.1 Hom-Sabinin Algebra Subclasses Lie algebras, Malcev algebras and Bol algebras are particular examples of Sabinin algebras. The natural question then is whether the Hom-type algebras of Lie, Malcev and Bol are Hom-Sabinin and the answer is affirmative. Example 2.1 Let (L , [ , ], α) be a multiplicative Hom-Lie algebra. Define < 1; a, b >=< a, b >= −[a, b], < x; a, b >= 0, (x, y) = 0, for a, b ∈ L and x, y ∈ L ⊗n , where n > 0. Then L becomes a Hom-Sabinin algebra. Example 2.2 A Hom-Malcev algebra, see [21], is a Hom-algebra (M, [ , ], α) where [ , ] : M × M → M is a skew-symmetric bilinear map and α : M → M an algebra map with the respect to the bracket satisfying for all x, y, z ∈ M Jα (α(x), α(y), [x, z]) = [Jα (x, y, z), α2 (x)],

(7)

where Jα is the Hom-type Jacobiator. Let (M, [ , ], α) be a Hom-Malcev algebra. Define < 1; a, b >=< a, b >= −[a, b], 1 < c; a, b >= − Jα (a, b, c), 3  < xc; a, b >= < α|x(2) |+1 (x(1) ); α|x(2) |+1 (c), < x(2) ; a, b , (x, y) = 0, for a, b, c ∈ M and x, y ∈ M ⊗n , where n > 0. Then M becomes a Hom-Sabinin algebra. Example 2.3 A Hom-Bol algebra (see [3]) is a quadruple (M, [ , ], {, , , }, α) where [ , ] : M × M → M is a skew-symmetric bilinear map, { , , } : M × M × M → M is a trilinear map and α : M → M an algebra map with the respect to the brackets satisfying for all x, y, z, u, v, w ∈ M

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

33

{x, y, z} = −{y, x, z},

(8) {x, y, z} + {z, x, y} + {y, z, x} = 0, (9) {α(x), α(y), [u, v]} = [{x, y, u}, α2 (v)] + [α2 (u), {x, y, v}] + {α(u), α(v), [x, y]} −[[α(u), α(v)], [α(x), α(y)]], (10) 2 2 2 2 2 2 {α (x), α (y), {u, v, w}} = {{x, y, u}, α (v), α (w)} + {α (u), {x, y, v}, α (w)} +{α2 (u), α2 (v), {x, y, w}}. (11) Let (B, [ , ], [ , , ], α) be a left Hom-Bol algebra. Define < 1; a, b >=< a, b >= −[a, b], < c; a, b >= {a, b, c} − [[a, b], α(c)],  < α|x(2) |+1 (x(1) ); α|x(2) |+1 (c), < x(2) ; a, b ,

< cx; a, b >= − (x, y) = 0,

for a, b, c ∈ B and x, y ∈ B ⊗n , where n > 0. Then B becomes a Hom-Sabinin algebra.

Example 2.4 A Hom-Lie Yamaguti algebra (see [3]) is a quadruple (A, [ , ], {, , , }, α) where [ , ] : A × A → A is a skew-symmetric bilinear map, { , , } : A × A × A → A is a trilinear map and α : A → A an algebra map with the respect to the brackets satisfying for all x, y, z, u, v ∈ A {x, y, z} = −{y, x, z}, [[x, y], α(z)] + [[z, x], α(y)] + [[y, z], α(x)] +{x, y, z} + {z, x, y} + {y, z, x} = 0,

(12)

{[x, y], α(z), α(u)} + {[z, x], α(y), α(u)} + {[y, z], α(x), α(u)} = 0, {α(x), α(y), [u, v]} = [{x, y, u}, α2 (v)] + [α2 (u), {x, y, v}],

(14) (15)

(13)

{α2 (u), α2 (v), {x, y, z}} = {{u, v, x}, α2 (y), α2 (z)} + {α2 (x), {u, v, y}, α2 (z)} +{α2 (x), α2 (y), {u, v, z}}. (16) Let (A, [ , ], { , , }, α) be a Hom-Lie-Yamaguti algebra. Define < 1; a, b >=< a, b >= −[a, b], < c; a, b >= {a, b, c}, < xc; a, b >=< α(x); α(c), [a, b] >  + < α|x(2) |+1 (x(1) ); α|x(2) |+1 (c), < x(2) ; a, b , (x, y) = 0, for a, b, c ∈ A and x, y ∈ A⊗n , where n > 0. Then A becomes a Hom-Sabinin algebra.

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2.2 Hom-Akivis Algebras and Hom-BTQQ Algebras There are examples of more algebras where their Hom-type structure has been defined and that are related to Sabinin algebras, one such examples is Akivis algebras; since every Sabinin algebra is an Akivis algebra. There are also examples of algebras related to Sabinin algebras whose Hom-type structure have not yet been defined; for instance, BTQQ algebras. Their name comes from the fact that they have a Binary, a Ternary and two Quaternary products. This algebras appeared when considering algebras that are Sabinin up to some degree d; i.e., they satisfy the Sabinin defining equations that involve up to d-linear operations and forget about the rest. This type of algebras were worked on up to degree 4 in [8], using computational methods. In this setting, Sabinin algebras of degree 2 are skew-symmetric algebras, Sabinin algebras of degree 3 are Akivis algebras and Sabinin algebras of degree 4 are BTQQ algebras. Definition 2.5 An Akivis algebra is a vector space V endowed with two linear maps [ , ] : V ⊗2 → V and ( , , ) : V ⊗3 → V , such that the following identities hold: [a, b] + [b, a] = 0,

(17)

[[a, b], c] + [[b, c], a] + [[c, a], b] = (a, b, c) − (a, c, b) − (b, a, c) + (b, c, a) +(c, a, b) − (c, b, a). (18) Definition 2.6 A BTQQ algebra is a vector space V endowed with four linear maps [ , ] : V ⊗2 → V , ( , , ) : V ⊗3 → V , { , , , } : V ⊗4 → V and [| , , , |] : V ⊗4 → V such that (V, [ , ], ( , , )) is an Akivis algebra and the following identities hold: ([a, b], c, d) − [a, (b, c, d)] + [b, (a, c, d)] = {a, b, c, d} − {b, a, c, d}, (19) (a, [b, c], d) − [b, (a, c, d)] + [c, (a, b, d)] = [|a, b, c, d|] − [|a, c, b, d|],(20) [b, (a, c, d)] − [b, (a, d, c)] − (a, b, [c, d]) = {a, b, c, d} − {a, b, d, c} −[|a, b, c, d|] + [|a, b, d, c|].

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Lemma 2.7 ([8]) BTQQ algebras are equivalent to Sabinin algebras of degree 4. If we consider Hom-Sabinin algebras of degree 3, one gets Hom-Akivis algebra, as mentioned in [6]. Consider the identity (5) on the definition of a Hom-Sabinin algebra, applied to three arbitrary elements a, b, c and x = 1, one gets the HomAkivis identity. Hom-Sabinin algebras of degree 4 leads to Hom-BTQQ algebras that are defined as Definition 2.8 A Hom-BTQQ algebra is a vector space V endowed with four linear maps [ , ] : V ⊗2 → V , ( , , ) : V ⊗3 → V , { , , , } : V ⊗4 → V and [| , , , |] : V ⊗4 → V ; and an endomorphism α : V → V s, such that the following equations hold:

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

[a, b] + [b, a] = 0,

35

(22)

[[a, b], α(c)] + [[b, c], α(a)] + [[c, a], α(b)] = (a, b, c) − (a, c, b) − (b, a, c) +(b, c, a) + (c, a, b) − (c, b, a), (23) ([a, b], α(c), α(d)) − [α2 (a), (b, c, d)] + [α2 (b), (a, c, d)] = {a, b, c, d} − {b, a, c, d}, (α(a), [b, c], α(d)) − [α2 (b), (a, c, d)] + [α2 (c), (a, b, d)] =

(24)

[|a, b, c, d|] − [|a, c, b, d|] [α2 (b), (a, c, d)] − [α2 (b), (a, d, c)] − (α(a), α(b), [c, d]) =

(25)

{a, b, c, d} − {a, b, d, c} − [|a, b, c, d|] + [|a, b, d, c|].

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Theorem 2.9 Hom-BTQQ algebras are equivalent to Hom-Sabinin algebras of degree 4. Proof The proof given in [8] for the non Hom-type case can be copied for the Homtype case; using the Main Theorem.  Corollary 2.10 Let (V, [ , ], ( , , ), { , , , }, [| , , , |], α) be a Hom-BTQQ algebra and β : V → V be an endomorphism of the Hom-algebra V . Then (V, β[ , ], β 2 ( , , ), β 3 { , , , }, β 3 [| , , , |], β ◦ α) is a Hom-BTQQ algebra. Proof Is a consequence of the same result for Hom-Sabinin algebras and the fact that Hom-BTQQ algebras are Hom-Sabinin algebras of degree 4; using the Main Theorem.  This procedure does not always give nice examples as it may seem. We present in the following an example where it leads to a trivial Hom-Akivis algebra. Example 2.11 Consider the following Hom-algebra V over C generated by x, y

with a trilinear operation (x, x, x) = 0 = (x, y, x) = (y, x, y) = (y, y, y), (x, x, y) = −2x − 4y = −(y, x, x) = 2(y, y, x) = −2(x, y, y). Then this algebra is an Akivis algebra by considering [a, b] = 0. Set α(x) = −2α(y) = −2x, which provides an Akivis algebra morphism; leading to a Hom-Akivis algebra defined by (x, x, x) = 0 = (x, y, x) = (y, x, y) = (y, y, y), (x, x, y) = −2α2 (x) − 4α2 (y) = −(y, x, x) = 2(y, y, x) = −2(x, y, y). Since −2α(x) − 4α(y) = 0, then the resulting Hom-Akivis algebra is trivial.

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Example 2.12 Consider now any Lie algebra (L , μ) and a Lie algebra morphism α. Then the operations [ , ]μ , the commutator of μ, and ( , , )α , the Hom-associator of μ, form a Hom-Akivis algebra. In particular, the operations are: [a, b]μ = 2μ(a, b), (a, b, c)α = μ(α(b), μ(c, a)). A more specific example would be sl2 (C) as the Lie algebra and α(h) = −h, α(x) = y and α(y) = x. The formulas for the non zero new operations in this particular case are [x, y] = 2h, [h, x] = 4x, [h, y] = −4y (x, x, y) = −(y, x, x) = −2y, (x, x, h) = −(h, x, x) = −2h, (x, y, y) = −(y, y, x) = 2x, (h, y, y) = −(y, y, h) = 2h, (h, h, x) = −(x, h, h) = 4x, (h, h, y) = −(y, h, h) = 4y. The same constructions can be considered to form BTQQ Hom-algebras from any Hom-algebra using the Hom-type formulas of those given in [8].

2.3 Hom-Power Associative Algebras and Hom-Type Teichmüller Identity There are several consequences of the Main Theorem already proven in the literature, such as the example of Hom-power associative algebras. Theorem 2.13 ([18, Corollary 5.2]) Let (A, μ, α) be a multiplicative Hom-algebra. Then the following statements are equivalent. 1. 2. 3. 4.

A satisfies x 2 α(x) = α(x)x 2 and x 4 = α(x 2 )α(x 2 ) for all x ∈ A. A satisfies (x, x, x)α = 0 = (x 2 , α(x), α(x))α for all x ∈ A. A is up to fourth Hom-power associative. A is Hom-power associative.

Proof Firstly, the equations on 1 and 2 are the same equations writen in two different ways, hence 1 ⇔ 2. It is trivial that 4 ⇒ 3 ⇒ 2. The fact 2 ⇒ 4 is true for power associative algebras as proven in [1]. Also, the free power associative algebras are N+ -graded since their defining relations are homogeneous polynomials. Hence, by application of the Main Theorem, it follows that the result is true since they are the Hom-type identities of the power associative case. 

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In the following, we recover the Hom-Type Teichmüller identity [19]. Theorem 2.14 ([19, Lemma 7.3]) Let (A, μ, α) be a multiplicative Hom-algebra. Then for any w, x, y, z ∈ A the following equation holds: (wx, α(y), α(z))α − (α(w), x y, α(z))α + (α(w), α(x), yz)α −α2 (w)(x, y, z)α − (w, x, y)α α2 (z) = 0. Proof The Teichmüller identity holds for any algebra. The previous equation is its Hom-type identity, hence it holds by application of the Main Theorem. 

3 Towards Universal Enveloping Algebras of Hom-Sabinin Algebras We aim to discuss universal enveloping algebras of Hom-Sabinin algebras. We consider the functor U : Sab Alg → Bi Algcc ; from Sabinin algebras to cocommutative, coassociative and magmatic bialgebras. We show that it is adjoint to the functor Y I I I , defined in [15], that gives a Sabinin algebra structure to any algebra. This construction is, in fact, an equivalence between Sabinin algebras and connected cocommutative bialgebras. In this section, we will construct the functor Y I I Ihom . α Definition 3.1 Consider the operations qn,m that make the equation

(αv−1 [u]α , αu−1 [v]α , αu+v−2 z)α = q α (u, v, z)  αu 2 +v2 (αv1 −1 ([u (1) ]α )αu 1 −1 ([v(1) ]α ))αu 1 +v1 −1 (q α (u (2) , v(2) , z)), + u 1 +v1 >0

hold. Then we define Y I I Ihom (A, μ, α) as < u; a, b >= −q α (u, a, b) + q α (u, b, a), < a, b >= −[a, b], and also  1 α q n,m (u, v) = (u σ(1) . . . u σ(n) , vτ (1) . . . vτ (m−1) , vτ (m) ). n!m! n,m−1 σ∈S ,τ ∈S n

m

In this definition, we consider that [x1 ⊗ · · · ⊗ xn ]α = (x1 ⊗ · · · ⊗ xn )ατ where τ is the tree of right-normed words: (x1 ⊗ · · · ⊗ xn )τ = (. . . ((x1 x2 )x3 ) . . . )xn . Note that in case α = id, we recover the original functor from [15]. Example 3.2 In case (A, μ, α) is Hom-associative, then Y I I Ihom (A, μ, α) is the corresponding Hom-Lie algebra A− .

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Example 3.3 In case (A, μ, α) is Hom-alternative, then Y I I Ihom (A, μ, α) is the corresponding Hom-Malcev algebra A− ; as −q(c, a, b) + q(c, b, a) = 2(c, b, a)α = −Jα (a, b, c) − 4(c, b, a)α . 1 Concluding that < c; a, b >= − Jα (a, b, c). 3 Example 3.4 Here are the first computations of the q α operations: α (x; y; z) = (x, y, z)α . • In case n = 1, m = 1, one gets q1,1 • In case n = 2, m = 1, one gets α (x, y; t; z) = (x y, α(t), α(z))α − α2 (x)(y, t, z)α − α2 (y)(x, t, z)α . q2,1

• In case n = 1, m = 2, one gets α (x; y, t; z) = (α(x), yt, α(z))α − α2 (y)(x, t, z)α − α2 (t)(x, y, z)α . q1,2

3.1 Hom-Bialgebras and Their Primitive Elements In this section, we define and study (nonassociative) Hom-bialgebras and some properties of their primitive elements. Definition 3.5 A coalgebra is a pair (A, ) where A is a K-vector space and  : A → A ⊗ A is a linear map. Definition 3.6 Let (A, μ, α) be a Hom-algebra, with a binary operation. We say that A is unitary if there is a linear map u : K → A, such that μ(u(1), x) = μ(x, u(1)) = α(x) for any x ∈ A, and α ◦ u = u. Definition 3.7 A Hom-coalgebra is a tuple (A, , α), where (A, ) is a coalgebra and α : A → A is a coalgebra morphism. We say that a Hom-coalgebra is counitary if there is a map  : A → K such that 

(x(1) ) ⊗ x(2) ) = 1 ⊗ x and



x(1) ⊗ (x(2) ) = x ⊗ 1.

Definition 3.8 A Hom-bialgebra is a sextuple (A, μ, u, , , α), where (A, μ, u, α) is a unitary Hom-algebra, (A, , , α) is a counitary Hom-coalgebra and the maps  : A → K and  : A → A ⊗ A are  Hom-algebra morphisms. As a consequence, the relation x(1) u((x(2) )) = u((x(1) ))x(2) = α(x) follows; equivalently μ ◦ (id ⊗ (u ◦ )) ◦  = μ ◦ ((u ◦ ) ⊗ id) ◦  = α.

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Note that in case α is the identity map, one recovers the definition of a unitary and counitary bialgebra. Remark 3.9 Notice that this definition is different from the usual one where the multiplication is assumed to be Hom-associative and the comultiplication is Homcoassociative [11]. Lemma 3.10 Let B be a free unitary Hom-algebra generated by a set G. Define α ( p) = u(1) ⊗ p + p ⊗ u(1) for any p ∈ G and extend it to B as an algebra morphism. In particular, α (u(1)) = u(1) ⊗ u(1). Define also ( p) = 0 for p ∈ G and (u(1)) = 1. Then B = (B, μ, u, α , , α) is a coassociative, cocommutative, counitary and unitary Hom-bialgebra. Proof The proof is straightforward.



Note that the difference between the Hom-bialgebra structure of a free Homalgebra and the bialgebra structure of a free algebra is mainly the definition of the adjoint map of the unit element, that is, a group-like element of the coalgebra structure. Since all bialgebras we are going to consider are cocommutative, we shall rewrite a ⊗ b + b ⊗ a as a ◦ b so that our equations are half the size. Note that a ◦ b = b ◦ a. Lemma 3.11 If p is a primitive element of a Hom-bialgebra (A, μ, u, , , α), then α( p) is also primitive. Proof α (α( p)) = α⊗2 (α ( p)) = u(1) ⊗ α( p) + α( p) ⊗ u(1).



Theorem 3.12 Let p be a primitive element of (B, μ, 1, , ε), the free algebra generated by a set G with its natural bialgebra structure. Then p α , the Hom-type element obtained by our procedure in Definition 1.4 applied to p, is a primitive element of (B, μ, u, α , , α), the free Hom-algebra generated by (K G , α) with its natural Hom-bialgebra structure.  Proof Consider p = λi m i some element in a free algebra B where m i are nonwords, p is a non-associative associative words on the generators andλi ∈ K. In other polynomial. If p is primitive, then λi α (m i ) = λi (1 ⊗ m i + m i ⊗ 1). Let m i = (xi,1 ⊗ · · · ⊗ xi,ni )τi ; then α (m i ) = (α (xi,1 ⊗ · · · ⊗ xi,ni ))τi . In conclusion, one gets the following equation on B ⊗ B: 

λi ((1 ◦ xi,1 ) ⊗ · · · ⊗ (1 ◦ xi,ni ))τi  = λi (((1 ⊗ xi,1 ) ⊗ · · · ⊗ (1 ⊗ xi,ni ))τi + ((xi,1 ⊗ 1) ⊗ · · · ⊗ (xi,ni ⊗ 1))τi ),

where a ◦ b = a ⊗ b + b ⊗ a. By the Main Theorem; since F ⊗ F is free, the Hom-type equation obtained by applying our procedure to the one above is also true for a free Hom-algebra. Indeed

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λi ((1 ◦ xi,1 ) ⊗ · · · ⊗ (1 ◦ xi,ni ))ατi  = λi (((1 ⊗ xi,1 ) ⊗ · · · ⊗ (1 ⊗ xi,ni ))ατi + ((xi,1 ⊗ 1) ⊗ · · · ⊗ (xi,ni ⊗ 1))ατi ). Undoing the coproducts, the equation translates to 

λi α ((xi,1 ⊗ · · · ⊗ i xni )ατi )  = λi (u(1) ⊗ (xi,1 ⊗ · · · ⊗ xi,ni )ατi ) + ((xi,1 ⊗ · · · ⊗ xi,ni )ατi ⊗ u(1)).

Therefore, the Hom-type monomial of p = bialgebra.



λi m i is primitive for the Hom

Corollary 3.13 The operations q α : B ⊗n ⊗ B ⊗m ⊗ B → B defined in Definition 3.1 are primitive in the free Hom-algebra B generated by G = {x1 , . . . , xn , . . .} ∪ {y1 , . . . , yn , . . .} ∪ {z} and an endomorphism α : K G → K G . Corollary 3.14 The operations q α : C ⊗n ⊗ C ⊗m ⊗ C → C are primitive in the Hom-bialgebra (C, μ, u, δ, , α). Proof Let a1 , . . . , an , b1 , . . . , bm , c be primitive elements for (C, δ) and define φ(xi ) = ai , φ(yi ) = bi , φ(z) = c and φ(u) = 0 for any other u ∈ G. Also, take any β : K G → K G such that φ ◦ β = α ◦ φ. Then extend φ : B → C as a Hom-algebra morphism. Since φ(G) are primitive elements of (C, δ), it follows that (φ ⊗ φ) ◦  = δ ◦ φ on G; but then it is true for every element since G generates B as an algebra and they are algebra morphisms. Finally, let’s compute δ(q α (a, b, c)) δ(q α (a, b, c)) = δ(q α (φ(x), φ(y), φ(z))) = δ(φ(q β (x, y, z))) = ((φ ⊗ φ) ◦ )(q β (x, y, z)) = 1 ◦ φ((q β (x, y, z))) = 1 ◦ (q α (a, b, c)).  The combinatorics in relation with α in the coproduct of non-associative words are very interesting and has deep consequences in the structure of Hom-bialgebras. Here are some properties Consider a right-normed word u = ((. . . (x1 x2 ) . . . xn−1 )xn ). The question is: How do α appear in a particular summand of (u)? First, consider only three generators (x y)z, it follows then that ((x y)z) = (x y)z ◦ 1 + α⊗2 (x y ◦ z) + α(y)z ◦ α2 (x) + α(x)z ◦ α2 (y). We realize that α keeps some record of the order of multiplications of the elements. In every summand, the left and right elements of the tensor are right-normed words and the instances of α appearing in each side of the tensor indicate where there are elements missing from the original word. Here is an algorithm to compute these elements for every type of non-associative monomial using trees:

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Lemma 3.15 Let m = (x1 ⊗ · · · ⊗ xn )τ be a non-associative monomial on a free algebra and consider P1 ∪ P2 a partition of the variables x1 , . . . , xn . Then the summand of (m) associated to the partition P1 ∪ P2 is given by the following procedure: 1. In the tree τ , label every leaf with 1 or −1 depending on the partition of the variables. 2. Next, go down the tree associating to every internal node a label 0, 1 or −1 if both branches come from a node with that label; but associate 0 if the branches come from nodes with different labels. 3. Finally, going up from the root, and considering all the internal nodes do the following: • For nodes with labels 1 or −1 do nothing. • For nodes with label 0 that came from two 0 labelled branches do nothing. • For nodes with label 0 that came from at least one non zero node, apply α to the leaves of the other branch with opposite label. Proof We only need to check the three cases of the lemma and show how α is applied. If we have the case (a ⊗ 1)(b ⊗ 1), it means that a, b are in the same part of the partition and in this case (ab ⊗ 1) is the solution where there are no α applications to any variable. This is case (1) in our list. Also, there is the possibility (a ⊗ b)(c ⊗ d); where there is no α application either. This is case (2) in our list. If we have the case (a ⊗ b)(c ⊗ 1), it means that one side comes from a 0 node and the other comes from either 1 or −1. In this case, we have (ac ⊗ α(b), which is the application of α to every variable in the side opposite to c. It may also happened that (1 ⊗ b)(c ⊗ 1) = α(c) ⊗ α(b); which is the same case applied to both branches. This is case (3) in our list.  Consider now (A, μ, α) a free Hom-associative unitary algebra, it should have a natural coalgebra structure to be able to consider its primitive elements. In the case of the tensor algebra, which is the free associative algebra, the coproduct that appears naturally is called shuffle coproduct. It is the one that we consider in this context, but with the exception that our algebra is not associative, but Hom-associative. Hence, since u · x = α(x), the coproduct is no longer the shuffle coproduct. The coproduct on the free Hom-associative algebra will be called the α-shuffle coproduct; and in case α = id, it is the shuffle coproduct on the tensor algebra.

3.2 Universal Enveloping Algebras In this section, we aim to find an appropriate universal enveloping algebra to any Hom-Sabinin algebra.

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Proposition 3.16 Let (A, α) be a Hom-algebra. Then Y I I Ihom (A, α) defined in Definition 3.1 is a Hom-Sabinin algebra. Proof In any algebra; the q operations give a Sabinin algebra. By the Main Theorem, the q α operations give rise to a Hom-Sabinin algebra.  The previous proposition shows that we have a functor Y I I Ihom : Hom-Alg → Hom-SabAlg, from the category of Hom-algebras with one binary multiplication to the category of Hom-Sabinin algebras. This functor has a left adjoint functor that we shall denote by Uhom : Hom-SabAlg → Hom-Alg. The image Uhom (S, α) can be defined as the quotient of the free Hom-algebra (K{S}, α) ˆ generated by S; by the ideal I generated by {< u; a, b > +q α (u, a, b) − α q(u, b, a)| u ∈ T (S); a, b ∈ S} ∪ {[a, b]+ < a, b > | a, b ∈ S} ∪ {(u, v) − 1 q α (u, v)| u, v ∈ T (S)}. The endomorphism is n!m! αU ((s1 . . . sn )τ ) = (α(s1 ) . . . α(sn ))τ for si ∈ S and τ a tree of n leaves. ˆ ) ⊂ I for αˆ the endomorphism of the This functor Uhom is well defined since α(I free algebra K{S}. We call this functor the universal enveloping Hom-algebra; in view of the following result: Proposition 3.17 The functor Uhom is left adjoint to Y I Ihom . Proof We have to show that H om Hom-Alg (Uhom (S, α), (A, β)) ∼ = H om Hom-SabAlg ((S, α), Y I I Ihom (A, β)), where the first maps are Hom-algebra morphisms and the second maps are HomSabinin algebra morphisms. Consider f : (S, α) → Y I I Ihom (A, β) a Hom-Sabinin algebra morphism. Then it is a linear map f : S → A where f ◦ α = β ◦ f and defines F : K{S, α} → (A, β) a Hom-algebra morphism in a unique way where F| S = f . F(< u; a, b > +q β (u, a, b) − q β (u, b, a)) = f (< u; a, b >) +q β ( f (u 1 ) . . . f (u n ), f (a), f (b)) − q β ( f (u 1 ) . . . f (u n ), f (b), f (a)) =< f (u 1 ) . . . f (u n ); f (a), f (b) > +q β ( f (u 1 ) . . . f (u n ), f (a), f (b)) −q β ( f (u 1 ) . . . f (u n ), f (b), f (a)) = 0. F([a, b]+ < a, b >) = [ f a, f b] + f < a, b >= [ f a, f b]+ < f a, f b >= 0.

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 1 q β (u, v)) n!m!  1 = f ((u, v)) − q β ( f (u 1 ) . . . f (u n ), f (v1 ) . . . f (vm−1 ), f (u m ))) n!m! = ( f (u 1 ) . . . f (u n ), f (v1 ) . . . f (vm−1 ), f (vm ))  1 q β ( f (u 1 ) . . . f (u n ), f (v1 ) . . . f (vm−1 ), f (u m ))). − n!m! F((u, v) −

 : Uhom (S) → A as F(x  + I ) = f (x) is a HomTherefore, I ⊆ ker (F) and F algebra morphism. This is a functor from H om((S, α), Y I I Ihom (A, β)) to H om (Uhom (S, α), (A, β)). Consider g : Uhom (S) → A, then Y I I Ihom (g) : Y I I Ihom (Uhom (S, α)) → Y I I Ihom (A, β) is a Hom-Sabinin algebra morphism; and also is π : S → Y I I Ihom (Uhom (S, α)) defined as π(s) = s + I . In conclusion, Y I I Ihom (g) ◦ π : (S, α) → Y I I Ihom (A, β) is a Hom-Sabinin algebra morphism and this is a functor from H om(Uhom (S, α), (A, β)) to H om ((S, α), Y I I Ihom (A, β)). This two functors are inverses of each other, proving our claim.  The question of whether π is injective or not is still open. We conjecture that in general π might not be injective depending on the properties of α. The proof in case of Sabinin algebras (α = id) can be read in [13].

3.3 Universal Envelopes as Hom-Bialgebras We consider in this section Hom-Hopf algebra, which are (α, id)-Hom-Hopf algebra with respect to the definition given in [7], where the structure maps for the product and the coproduct may be different. We show that the universal enveloping algebra of a Hom-Lie algebra is endowed with such a structure while the universal enveloping algebra of a Hom-Sabinin algebra is endowed with a Hom-bialgebra structure. Theorem 3.18 For any Hom-Sabinin algebra (S, α); its universal enveloping algebra Uhom (S, α) is a Hom-bialgebra. Proof We only need to check that the coproduct of the free algebra generated by S can be taken to the quotient. This is true because the ideal is generated by primitive elements and  is an algebra morphism. Since  is a morphism, (I ) is an ideal on K{S} ⊗ K{S}. Also, for any generator p ∈ I , ( p) = 1 ⊗ p + p ⊗ 1 ∈ B ⊗ I + I ⊗ B. Hence (I ) ⊆ B ⊗ I + I ⊗ B; by ideals properties. 

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Definition 3.19 A Hom-Hopf algebra is a unitary and counitary Hom-bialgebra (H, μ, u, , , α) which is Hom-associative, coassociative and such that there is a map S : H → H satisfying μ ◦ (id ⊗ S) ◦  = u ◦ , μ ◦ (S ⊗ id) ◦  = u ◦  and S ◦ α = α ◦ S. The map S is called an antipode. This definition corresponds to that of (α, id)-Hom-Hopf algebra given in [7]. Proposition 3.20 Let (H, μ, u, , , α) be a Hom-associative, coassociative Hombialgebra. If H admits an antipode, then it is unique up to im(α); i.e., α ◦ S = α ◦ S  if S, S  are two antipodes for H . Proof α ◦ S = μ ◦ (α ◦ S ⊗ [μ ◦ (id ⊗ S  ) ◦ ]) ◦  = μ ◦ (α ⊗ μ) ◦ (S ⊗ id ⊗ S  ) ◦ (id ⊗ ) ◦  = μ ◦ (μ ⊗ α) ◦ (S ⊗ id ⊗ S  ) ◦ ( ⊗ id) ◦  = μ ◦ ([μ ◦ (S ⊗ id) ◦ ] ⊗ α ◦ S  ) ◦  = α ◦ S  .  Definition 3.21 be a finite set and X i denote a copie of that set for i ∈  Let X  N. Define α : X i → X i as α(xi ) = xi+1 and consider Fα (X ) the free algebra generated by X i ; and extend α as an endomorphism. Finally, make a quotient of that free algebra by the ideal generated by I = α(a)(bc) − (ab)α(c)| a, b, c ∈ F(X ) . We call that quotient the free Homassociative algebra generated by X and we denote it by Fα,ass (X ). Note that this is a free construction from the category Set, or FinVec, the category of finite- dimensional vector spaces; and not from the category of Hom-K-modules. The difference lies in the fact that, in this case, α is freely defined and in the latter case, α is already given and is only extended. Proposition 3.22 The map α is injective on Fα,ass (X ). Proof First note that α : Fα,ass (X ) → Fα,ass (X ) is well defined as α(I ) ⊆ I . Consider α([s]) = 0 for some [s] ∈ Fα,ass (X ); then α(s) ∈ I considered as α : Fα (X ) → Fα (X ). Hence α(s)(bc) ∈ I and also (sb)α(c) ∈ I for all b, c ∈ Fα (X ). By the definition of I and the fact that the algebra is free, it is only possible that sb ∈ I . By the same argument as before, we have s ∈ I . Therefore, [s] = 0 and α is injective.  Proposition 3.23 Fα,ass (X ) is a Hom-Hopf algebra. Proof Define (xi ) = 1 ⊗ xi + xi ⊗ 1 for xi ∈ X i and extend it as an algebra morphism. Also, consider (xi ) = 0 and (1) = 1 and extend them as algebra morphisms.

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Set also S(xi ) = −x i , S(1) = 1 and S(u v) = S(v)S(u) and extend it linearly. We need to show that u (1) S(u (2) ) = (u). The proof is trivial for the case of u = 1, xi . Assume now that |u| > 1. Then u = a b, where |a|, |b| ≥ 1, and    (a(1) b(1) )(S(b(2) )S(a(2) )) u (1) S(u (2) ) = α   ((a(1) b(1) )α(S(b(2) )))α2 (S(a(2) )) = α(a(1) b(1) )(α(S(b(2) ))α(S(a(2) ))) =   α(a(1) )α2 (S(a(2) ))(b) = 0. = (α(a(1) )(b(1) S(b(2) )))α2 (S(a(2) )) = α



By injectivity, it follows that



u (1) S(u (2) ) = (u).



Corollary 3.24 Let (L , β) be a Hom-Lie algebra. Then its universal enveloping algebra Uhom (L , β) as defined in [16] is a Hom-Hopf algebra. Proof Since Fα,ass (L) is a Hom-Hopf algebra and Uhom (L , β) is defined as a quotient and generated by primitive elements. This quotient preserves the coproduct. In particular, the ideal is generated by {[x, y] − x y + yx| x, y ∈ L} ∪ {xi −  β i−1 (x1 )| xi ∈ X i , i > 1}.  Here are some examples of the equation u (1) S(u (2) ) = (u), for small |u| cases.  Example 3.25 • If u = x then u (1) S(u (2) ) = x 1 + 1S(x) = α(x) − α(x) = 0. • If u = (x y) then 

u (1) S(u (2) ) = x y 1 + 1 S(x y) + α(x)S(α(y)) + α(y)S(α(x))

= α(x y) + α(yx) − α(x y) − α(yx) = 0. • If u = (x y)z then 

u (1) S(u (2) ) = (x y)z 1 + 1 S((x y)z) + α2 (x)S(α(y)z) + α2 (y)S(α(x)z)+

α(x y)S(α(z)) + α(z)S(α(x y)) + α(y)zS(α2 (x)) + α(x)zS(α2 (y)) = α((x y)z) − α(z(yx)) + α2 (x)(zα(y)) + α2 (y)(zα(x)) − α((x y)z)+ α(z)(α(yx)) − (α(y)z)α2 (x) − (α(x)z)α2 (y) = α2 (x)(zα(y)) + α2 (y)(zα(x)) − (α(y)z)α2 (x) − (α(x)z)α2 (y) = 0.

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• If u = (a(bc))d then 

u (1) S(u (2) ) = (a(bc))d 1 + 1 S((a(bc))d) + α(a(bc))S(α(d))

+ α(d)S(α(a(bc))) + α2 (bc)S(α(a)d) + (α(a)d)S(α2 (bc)) + α3 (b)S((aα(c))d) + ((aα(c))d)S(α3 (b)) + α3 (c)S((aα(b))d) + ((aα(b))d)S(α3 (c)) + α2 (a)S(α(bc)d) + (α(bc)d)S(α2 (a)) + α(aα(b))S(α2 (c)d) + (α2 (c)d)S(α(aα(b))) + α(aα(c))S(α2 (b)d) + (α2 (b)d)S(α(aα(c))) = α((a(bc))d) + α(d((cb)a)) − α((a(bc))d) − α(d((cb)a))+ α2 (bc)(dα(a)) + (α(a)d)α2 (cb) − α3 (b)(d(α(c)a)) − ((aα(c))d)α3 (b) − α3 (c)(d(α(b)a)) − ((aα(b))d)α3 (c) − α2 (a)(dα(cb)) − (α(bc)d)α2 (a) + α(aα(b))(dα2 (c)) + (α2 (c)d)(α(α(b)a)) + α(aα(c))(dα2 (b)) + (α2 (b)d)(α(α(c)a)) = 0. Note that this last universal enveloping algebra may be different from our definition since it is adjoint to Y I I Ihom : Hom-Ass → Hom-Lie, which considers only the Hom-associative algebras and not all Hom-algebras where Y I I Ihom (A, α) is a Hom-Lie; called Hom-Lie admissible algebra.

3.4 An Example: Antisymmetric Algebras As an example, we will show the relation between algebras with an antisymmetric bilinear product, and connected (0, id)-Hom-Hopf algebras. Proposition 3.26 There is an equivalence of categories between Hom-Lie algebras where α = 0 and algebras with an antisymmetric bilinear product. Proof Let (L , [ , ], 0) be a Hom-Lie algebra. Then [ , ] is an antisymmetric bilinear product. Let (L , ·) be an algebra with an antisymmetric bilinear product. Then the HomJacobiator Jα (x, y, z) = 0 for this product in case α = 0. Hence (L , ·, 0) is a HomLie algebra where α = 0. Let f : L 1 → L 2 be a morphism of Hom-Lie algebras. Then f is a morphism of the corresponding antisymmetric algebras. Consider finally, f : L 1 → L 2 a morphism of antisymmetric algebras. Then f ◦ 0 = 0 = 0 ◦ f . Therefore, f is a morphism of the corresponding Hom-Lie algebras  where α1 = α2 = 0. Proposition 3.27 There is an equivalence of categories between Hom-associative algebras with α = 0 and magmatic algebras; i.e., algebras with one binary operation.

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Proof The Hom-associativity is trivially satisfied for any algebra in case α = 0.  Corollary 3.28 The category of (0, id)-H om-H op f -algebras is equivalent to the category of algebras which are magmatic algebras; i.e., algebras with just one product, and that product is bilinear; with an element u such that u H = H u = 0. Since K u is an ideal one can also consider only the algebra H/K u and the element u can be recovered to define the structure of a (0, id)-H om-H op f -algebra. Proof Let H be a (0, id)-Hom-Hopf algebra. Then it is a magmatic algebra with an element u as specified; by forgetting the coalgebra structure. Let’s denote this construction as the F functor. Consider any algebra (H, ·) with an element u ∈ H such that u H = H u = 0. Define a complement P ⊕ K u = H and a coproduct as (u) = u ⊗ u and (x) = x ⊗ u + u ⊗ x for x ∈ P. Also, a counit as (u) = 1 and (x) = 0 for any x ∈ P. Then (H, ·, u, , ) is a (0, id)-Hom-Hopf algebra. Let’s denote this construction as the G functor This two constructions are inverse to one another. It is obvious that F ◦ G = I D since the product of the algebra is not modified by G; and the coalgebra construction of G is forgotten by F. Let H be a (0, id)-Hom-Hopf algebra. First, let’s prove that Prim H is a subalgebra of H : (x y) = (x)(y) = (x ⊗ u + u ⊗ x)(y ⊗ u + u ⊗ y) = x y ⊗ u + u ⊗ x y. Since (Prim H )∗ ⊕ H0∗ generates Gr (H ∗ ), see for instance [12, Lemma 5.6.6], it follows that H = H0 ⊕ Prim H . The coproduct is then fixed as (u) = u ⊗ u and (x) = x ⊗ u + u ⊗ x for x ∈ Prim H . Therefore, G ◦ F = I D.  Proposition 3.29 The Hom-associative universal enveloping algebra of a Hom-Lie algebra L where α = 0 is given by Uα (L) = K{L}/ x y − yx − [x, y]| x, y ∈ L . Proof Since α = 0 by the previous proposition, any algebra is Hom-associative in this case, and the result follows.  Corollary 3.30 We have π(L) ∼ = L as vector spaces, where π : K{L} → Uα (L) is the canonical quotient map.  Proof The enveloping algebra U (L) has a filtration given by Fn = π( nk=1 L ⊗k ) and the associated graded algebra is the free commutative algebra generated by L. We have the following result.  Theorem 3.31 The map π : (L , [ , ], 0) → Y I I Iα (Uα (L)) is an injective morphism of Hom-Lie algebras; even more, the Hom-Lie algebra L can be recovered from the (0, id)-Hom-Hopf algebra U (L) as F1 /F0 , where Fn is the associated filtration defined previously.

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Theorem 3.32 Let H be a (0, id)-Hom-Hopf algebra with a filtration where Gr (H ), the graded algebra associated to this filtration, is free commutative generated by H1 . Then H1 < Y I I Iα (H ) and H ∼ = Uα (H1 /H0 , [ , ]). Proof Since Gr (H ) is commutative, it follows that [H1 , H1 ] ⊆ H1 . Hence, H1 is closed by the commutator product. Also, H0 · H = 0, and hence H1 /H0 is a quotient of (H1 , [ , ]). Also, there is a morphism f : H1 /H0 → Y I I I0 (H/H0 ) given by the inclusion of H1 into H . By the universal property, then there is F : Uα (H1 /H0 ) → H/H0 a (0, id)-Hom-Hopf algebra morphism. This map is given by F([h 1 . . . h n ]) = h1 . . . hn . The map F also respects the filtrations; since F(Fn ) ⊆ Hn /H0 . Then there is a map Gr F : K{H1 /H0 }/ x y − yx| x, y ∈ H1 /H0 → Gr (H/H0 ) ⊕ H0 that is an  isomorphism, hence F is an isomorphism when identifying F(F0 ) = H0 . We have found then an equivalence between this two categories, but the class of (0, id)-Hom-Hopf algebras H with a filtration where Gr (H ), the graded algebra associated to this filtration, is free commutative generated by H1 ; should be studied in more detail. This problem remains open whether our hypothesis on the filtration are necessary or less conditions already imply them. Remember that every connected Hopf algebra has a coradical filtration such that Gr (H ) is free in the class of commutative and associative algebras; and is generated by H1 . Acknowledgements FPU Grant from the Ministerio de Educación, Cultura y Deporte in Spain. Research project MTM2013-45588-C3-3-P.

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Eigenvalues of Positive Pseudo-Hermitian Matrices Jacques Faraut

Abstract We consider positive pseudo-Hermitian matrices and study properties of their eigenvalues. Our results are analogous to classical results about the eigenvalues of Hermitian matrices: Cauchy interlacing property, Laplace transform of orbital measures, Horn’s convexity theorem. We consider also Horn’s problem in the setting of pseudo-Hermitian matrices. Keywords Eigenvalue · Cauchy interlacing theorem · Baryshnikov’s formula · Orbital measure · Laplace transform · Horn’s convexity theorem · Horn’s problem Mathematics Subject Classification 15A18 · 15A42 · 44A10 Pseudo-Hermitian matrices are square complex matrices which are self-adjoint with respect to an indefinite inner product. In general, the eigenvalues of a pseudoHermitian matrix are not real, but for a positive pseudo-Hermitian matrix, the eigenvalues are real. If the inner product is of signature ( p, q), p eigenvalues are positive and q ones are negative. The pseudo-unitary group U ( p, q) acts on the cone of positive pseudo-Hermitian matrices, preserving the eigenvalues. We study in this paper the properties of the eigenvalues of these matrices and integral formulas related to these eigenvalues. These properties and formulas are analogous to classical properties and formulas related to the eigenvalues of Hermitian matrices, with differences due to the fact that the pseudo-unitary group U ( p, q) is not compact for p and q ≥ 1. In Sect. 2, we consider the projection of an n × n pseudo-Hermitian matrix onto the space of (n − 1) × (n − 1) pseudo-Hermitian matrices. In this setting, there is an interlacing property similar to the Cauchy interlacing one [3]. This interlacing property has been established in [5]. Related to this interlacing property, we obtain an integration formula analogous to Baryshnikov’s formula [1]. In Sect. 3, we conTo the memory of Takaaki Nomura. J. Faraut (B) Institut de Mathématiques de Jussieu, Sorbonne Université, 4 place Jussieu, case 247, 75252 Cedex 05 Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_4

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sider orbits for the action of the pseudo-unitary group U ( p, q), and orbital measures. By using the integration formula established in Sect. 2, we determine the Laplace transform of an orbital measure. The result is similar to Itzykson–Zuber’s formula [10]. In Sect. 4, we look at the projection of orbits on the space of diagonal matrices, and determine the projection of orbital measures. There is in this setting a convexity theorem, analogous to Horn’s convexity theorem [8]. But there is an important difference in the geometry for these projections due to the fact that the pseudo-unitary group U ( p, q) is not compact for p and q ≥ 1. The orbits are not compact, and orbital measures are unbounded. In last section, we consider Horn’s problem for the eigenvalues of positive pseudo-Hermitian matrices. We determine a measure whose support should be Horn’s set. But we could not determine Horn’s set. It would be natural to consider the case of real pseudo-symmetric matrices. The interlacing property of the eigenvalues holds in this setting, but we have no result about the projection of orbital measures for the action of O( p, q) on the space of diagonal matrices. In fact, the proof of our results in case of pseudo-Hermitian matrices are based on explicit integral formulas, and we don’t know such formulas in case of pseudo-symmetric matrices. Note that the letter C denotes a constant which can be different from one place to another, which may depend on the signature ( p, q) and the choice of the Haar measure of U ( p, q).

1 Pseudo-Hermitian Matrices We fix p > 0, q ≥ 0, p + q = n, and consider on Cn the sesquilinear form [x, y] = x1 y1 + · · · + x p y p − x p+1 y p+1 − · · · − x p+q y p+q = (I p,q x|y), for x, y ∈ Cn , with

 I p,q =

Ip 0 0 −Iq



The pseudo-adjoint A# of a square matrix A ∈ M(n, C) is defined by [Ax, y] = [x, A# y] (x, y ∈ Cn ), or A# = I p,q A∗ I p,q , where A∗ is the usual adjoint of A. The matrix A is said to be pseudo-Hermitian if A# = A. We denote by H p,q the real vector space of pseudoHermitian matrices. A matrix U ∈ M(n, C) belongs to the pseudo-unitary group U ( p, q) if [U x, U y] = [x, y] (x, y ∈ Cn ), or U −1 = U # , or U I p,q U ∗ = I p,q . The group U ( p, q) acts on the space H p,q by the transformations

Eigenvalues of Positive Pseudo-Hermitian Matrices

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A → U AU −1 . We say that a pseudo-Hermitian matrix A is positive if [Ax, x] > 0 ∀x ∈ Cn , x = 0. The set  p,q of positive pseudo-Hermitian matrices is a convex cone. This cone is linearly isomorphic to the cone n of positive definite Hermitian matrices. In fact the map A → AI p,q is a bijection from n onto  p,q . The cone  p,q is invariant under the action of the group U ( p, q). The eigenvalues of a Hermitian matrix are real, i.e. for q = 0. But if q ≥ 1, in general, the eigenvalues of a pseudo-Hermitian matrix are not real. For instance, if  A=

0 B0 −B0∗ 0

 B0 ∈ M( p, q, C)

the eigenvalues or A are pure imaginary. We will see that the eigenvalues of a positive pseudo-Hermitian matrix are real, and in this paper, we will only consider positive pseudo-Hermitian matrices. We start with a preliminary result. Proposition 1.1 A matrix g ∈ G L(n, C) admits the following decomposition g = U DV, with U ∈ U ( p, q), V = U (n), and D diagonal with positive diagonal entries. Proof The matrix A = g −1 I p,q g ∗ −1 is Hermitian with signature ( p, q). By the classical polar decomposition, A = V1 D1 V1∗ with V1 ∈ U (n) and D1 diagonal with p positive diagonal entries and q negative ones. Put D1 = D2 I p,q D2 with D2 diagonal with positive diagonal entries. Hence g −1 I p,q g ∗ −1 = V1 D2 I p,q D2 V1∗ or (gV1 D2 )I p,q (gV1 D2 )∗ = I p,q . Therefore

U = gV1 D2 ∈ U ( p, q), or g = U D2−1 V1−1 . 

Proposition 1.2 (a) The eigenvalues of a positive pseudo-Hermitian matrix are real, with p positive eigenvalues and q negative ones. (b) A positive pseudo-Hermitian matrix A admits the following pseudo polar decomposition A = U DU −1 , with U ∈ U ( p, q) and D diagonal with p positive diagonal entries and q negative ones.

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Note that (b) implies (a). However, we will give a simple proof of (a). Proof (a) Let λ be an eigenvalue of A, and x ∈ Cn , x = 0, an associated eigenvector: Ax = λx. Then [Ax, x] = λ[x, x]. Since [Ax, x] > 0, then λ = 0, [x, x] = 0, and since [x, x] is real, then λ is real. (b) Let A be a positive pseudo-Hermitian matrix. The matrix A1 = AI p,q is a positive definite Hermitian matrix, which can be written A1 = gg ∗ with g ∈ G L(n, C). By Proposition 1.1, g can be decomposed as g = U D1 V with U ∈ U ( p, q), V ∈ U (n) and D1 diagonal with positive diagonal entries. Hence AI p,q = A1 = gg ∗ = U D1 V V ∗ D1 U ∗ = U D12 U ∗ = U D12 I p,q U # I p,q . Therefore, A = U DU −1 with D = D12 I p,q .



2 Interlacing Property and Analogue of Baryshnikov’s Formula By Proposition 1.2 an orbit of U ( p, q) contained in the cone  p,q can be written O A = {U AU −1 | U ∈ U ( p, q)}, where A is a diagonal matrix with p positive diagonal entries and q negative ones. More precisely, we can take A = diag(α1 , . . . , αn ) with α1 ≥ · · · ≥ α p > 0 > α p+1 ≥ · · · ≥ α p+q . All matrices X ∈ O A have the same eigenvalues αi , moreover   O A = X ∈  p,q | spectrum(X ) = {α1 , . . . , αn } . Therefore, we will denote the orbit O A by Oα . We assume q ≥ 1. For X ∈ Oα consider the projection Y = p(X ) onto the subspace of H p,q with zeros on the last row and last column, which can be identified with H p,q−1 . The matrix Y belongs to  p,q−1 . The eigenvalues μ1 , . . . , μn−1 of Y are ordered as follows μ1 ≥ · · · ≥ μ p > 0 > μ p+1 ≥ · · · ≥ μ p+q−1 . Theorem 2.1 The eigenvalues μ1 , . . . , μn−1 of Y interlace the eigenvalues α1 , . . . , αn of X in the following way

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μ1 ≥ α1 ≥ μ2 ≥ · · · ≥ μ p ≥ α p > 0 > α p+1 ≥ μ p+1 ≥ · · · ≥ α p+q−1 ≥ μ p+q−1 ≥ α p+q . The proof is given in [5] (Theorem 3.2). We will denote by Iα this interlacing set Iα = {t ∈ Rn−1 |t1 ≥ α1 ≥ t2 ≥ · · · ≥ t p ≥ α p > 0 > α p+1 ≥ t p+1 ≥ · · · ≥ α p+q−1 ≥ t p+q−1 ≥ α p+q }.

One introduces the rational function R(z) = [(z In − X )−1 ]n,n , the lower right entry of the inverse (z In − X )−1 , and establish two formulas. By observing that det(z In−1 − Y ) , R(z) = det(z In − X ) one obtains

n−1 (z − μi ) R(z) = i=1 . n i=1 (z − αi )

Also R(z) =

n  j=1

wj , z − αi

with w j ≤ 0 if 1 ≤ j ≤ p, wi ≥ 0 if p + 1 ≤ j ≤ p + q, and w1 + · · · + wn = 1. This second formula is the key for proving the interlacing property. The eigenvalues of X are the poles of R(z), and the eigenvalues of Y are the zeros. To prove this formula one observes that, if X = U AU # with U ∈ U ( p, q) (z I − X )−1 = U (z I − A)−1 U # , n  1 R(z) = Un j U #jn . z − α j j=1 Moreover, U #jn = −Un j if 1 ≤ j ≤ p, and U #jn = Un j if p + 1 ≤ j ≤ p + q. Therefore, p p+q   |Un j |2 |Un j |2 R(z) = − + . z − αj z − αj j=1 j= p+1 Furthermore

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J. Faraut n 

Un j U #jn = 1, and −

j=1

p+1 

|Un j |2 +

j=1

p+q 

|Un j |2 = 1.

j= p+1

Proposition 2.2 The isotropy subgroup U ( p, q) A = {U ∈ U ( p, q) | U AU −1 = A} is compact. Proof Decompose the matrices A and U in blocs according to n = p + q:   U11 U12 , U= U21 U22

 A=

 A1 0 , 0 −A2

with A1 = diag(α1 , . . . , α p ), A2 = diag(−α p+1 , . . . , −α p+q ). If U ∈ U ( p, q) A : U AU # = A, then ∗ ∗ ∗ ∗ + U12 A1 U12 = A1 , U21 A1 U21 + U22 A2 U22 = A2 . U11 A1 U11

Together with the relation U −1 = U # , these relations imply that U ( p, q) A is compact (Fig. 1).  Since the isotropy subgroup U ( p, q) A is compact (Proposition 2.2), the orbit Oα carries a positive measure which is invariant under the unimodular group U ( p, q). We fix a (left and right) Haar measure ω on U ( p, q) and define the orbital measure μα on the orbit Oα by 

 Oα

f (X )μα (d X ) =

U ( p,q)

f (U AU −1 )ω(dU ).

From the polar decomposition (Proposition 1.2), it follows that if μ is a positive measure on  p,q which is U ( p, q) invariant, there is a positive measure ν, the pseudo Fig. 1 Graph of the rational function R(z), p = 2, q = 2. The four eigenvalues of X are the black dots, and the three eigenvalues of Y are the white dots

0

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57

radial part of μ, on the cone D p,q = {A = diag(α1 , . . . , αn ) | α1 ≥ · · · α p > 0 > α p+1 ≥ · · · ≥ α p+q } such that 

 ( p,q)

f (X )μ(d X ) =

D p,q



f (U AU −1 )ω(dU ) ν(dα).

U ( p,q)

We consider the projection μ(αp,q−1) of the orbital measure μα on H p,q−1 : for a function f on H p,q−1 ,  H p,q−1

f (Y )μα( p,q−1) (dY ) =

 Oα

 f p(X ) μα (d X ).

The existence of this projection will follow from the next Theorem. Theorem 2.3 The pseudo radial part να( p,q−1) of the projection μ(αp,q−1) of the orbital measure μα is the measure on D p,q−1 supported by the interlacing set Iα with density C

Vn−1 (t1 , . . . , tn−1 ) , Vn (α1 , . . . , αn )

where Vn is the Vandermonde polynomial

Vn (x1 , . . . , xn ) =

(xi − x j ).

1≤i< j≤n

In other words, for a function f on Rn−1 , 

C f (t)να( p,q−1) (dt) = n−1  α1  α p−1Vn (α1 , α. .p+1. , αn )  R∞ dt1 dt2 . . . dt p dt p+1 . . . α1

α2

αp

α p+2

αn−1

αn

dtn−1 Vn−1 (t) f (t).

Theorem 2.3 is proven in [5]. We will give here a different proof. This formula is a non-compact analogue of Baryshnikov’s formula ([1], Proposition 4.2). Proof (a) The coset space U ( p, q)/U ( p, q − 1) can be identified with the hyperboloid in Cn with equation −|z 1 |2 − · · · − |z p |2 + |z p+1 |2 + · · · + |z p+q |2 = 1. Let β be a positive measure on this hyperboloid which is invariant under U ( p, q). Consider the convex set

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 p,q = {w ∈ Rn | wi ≤ 0 if 1 ≤ i ≤ p, wi ≥ 0 if p + 1 ≤ i ≤ p + q, w1 + · · · + wn = 1},

and the map  : U ( p, q)/U ( p, q − 1) →  p,q , z → w, with wi = −|z i |2 if 1 ≤ i ≤ p, and wi = |z i |2 if p + 1 ≤ i ≤ p + q. The image under  of the measure β is, up to a constant factor, the Lebesgue measure of the hyperplane w1 + · · · + wn = 1, restricted to  p,q , defined by the differential form λ = dw1 ∧ · · · ∧ dwn−1 . In case of p = 0, q ≥ 0, 0,q is a compact simplex and this property of  is classical. The proof for p ≥ 1 is analogous. (b) Let μ1 , . . . , μn−1 satisfying the interlacing condition of Theorem 3.1. Consider the poles wi of the rational function R(z). Then wi ≤ 0 if 1 ≤ i ≤ p + 1, and wi ≥ 0 if p + 1 ≤ i ≤ p + q. Furthermore, since R(z) ∼

1 as z → ∞, z

we get w1 + · · · + wn = 1. In this way, we obtain a map  : μ → w, Iα → D p,q . As in the case of q = 0 (Proposition 1.2 in [4]), one proves that ∗ (λ) =

Vn−1 (μ1 , . . . , μn−1 ) dμ1 ∧ · · · ∧ dμn−1 . Vn (α1 , . . . , αn )

Theorem 2.3 is obtained by composing the maps  and :  ◦  : U ( p, q)/U ( p, q − 1) → Iα , z → μ.



3 The Laplace Transform of an Orbital Measure For A ∈  p,q , A = diag(α1 , . . . , αn ), we consider the Laplace transform of the orbital measure μα   −1 L(μα )(Z ) = e−tr(Z X ) μα (d X ) = e−tr(ZU AU ) ω(dU ). Oα

U ( p,q)

The function L(μα ) is holomorphic in the tube ( p, q) + iH p,q and invariant under U ( p, q). It can be computed explicitly. The formula we will give below is an analogue

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59

of the Itzykson–Zuber formula which corresponds to the case n = p, q = 0 ([10], formulae (3.2) and (3.4)). Theorem 3.1 Let A = diag(α1 , . . . , αn ) ∈ D p,q and Z = diag(z 1 , . . . , z n ) with Re(z i ) > 0 if 1 ≤ i ≤ p, and Re(z i ) < 0 if p + 1 ≤ i ≤ p + q. Define  E p,q (z, α) =

Oα

e−tr(Z X ) μα (d X ).

Then E p,q (z, α) =

C det(e−αi z j )1≤i, j≤ p det(e−αi z j ) p+1≤i, j≤ p+q . Vn (α)Vn (z)

This formula is essentially a special case of a formula given in a more general setting in [2] (Corollary 4.15). We give below a different proof using Theorem 2.3. Proof We will prove the formula recursively with respect to q. For q = 0, this is Itzykson–Zuber’s formula. We assume q ≥ 1 and that the formula holds for ( p, q − 1). (a) In the first step, we will show that E p,q (z, α) can be written as an integral with respect to the projection μ(αp,q−1) of the orbital measure μα on H p,q−1 . Let Y = p(X ) be the projection of X on H p,q−1 . By using the fact that tr X = tr A, we get, with Z  = p(Z ), tr(Z X ) = tr(Z  Y ) + z n tr(A − Y ). Therefore, E p,q (z, α) = e

−z n tr A

 Oα



e zn trY e−tr(Z Y ) μα (dY ).

The integrant only depends on Y = p(X ). Therefore, E p,q (z, α) = e−zn tr A





H p,q−1

e zn trY etr(Z Y ) μα( p,q−1) (dY ).

This integral can be written by using the pseudo radial part να( p,q−1) of μ(αp,q−1) : E p,q (z, α) = e−z n tr A

 Rn−1

= e−zn tr A

e z n (t1 +···+tn−1 )

 Rn−1

 U ( p,q−1)



e−tr(Z U T U

−1 )

ω p,q−1 (dU ) να( p,q−1) (dt)

E( p,q−1) (z  , t)e zn (t1 +···+tn−1 ) να( p,q−1) (dt).

This is a recursion formula from E p,q−1 to E p,q . (b) By using the recursion hypothesis,

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Vn−1 (z 1 , . . . , z n−1 )E p,q (z, α) = Ce−zn (α1 +···+αn ) A × B, with,

 A=



∞

dt1

α1

α1

α2

 dt2 . . .

α p−1

αp

dt p e zn (t1 +···+t p ) det(e−zi t j )1≤i, j≤ p ,

and  B=

α p+1 α p+2

 dt p+1 . . .

a p+q−1

α p+q

dt p+q−1 e zn (t p+1 +···+t p+q−1 ) det(e−zi t j ) p+1≤i, j≤ p+q−1 .

In case of q = 1, the second integral does not appear and B = 1. By integration, we obtain A = det(Ai j )1≤i j≤ p , 

with Ai1 =

∞ α1

e−(zi −zn )t1 dt1 =

1 e−(zi −zn )α1 , zn − zi

and, for 2 ≤ j ≤ p,  Ai j =

α j−1

αj

e−(zi −zn )t j dt j =

 −(zi −zn )α j 1 e − e−(zi −zn )α j−1 . (z i − z n )

Adding the first line to the second one, then the second one to the third one and so on, we get e zn (α1 +···+α p ) det(e−zi α j ]1≤i, j≤ p . A = p i=1 (z i − z n ) Similarly, if q ≥ 2, 

with Bi j =

αj α j+1

B = det(Bi j ) p+1≤i, j≤ p+q−1 , e−(zi −zn )t j dt j =

1  −(zi −zn )α j+1 e − e−(zi −zn )α j . zi − zn

By using the identity  det(a λj i )1≤i, j≤m = (a1 . . . am )λm det (a j+1 )λi −λm − (a j )λi −λm ) 1≤i, j≤m−1 , with a j = e−αq+ j , λi = z q+i , m = q, one gets e zn (α p+1 +···+α p+q ) B =  p+q det(e−zi α j ) p+1≤i, j≤ p+q . (z − z ) i n i= p+1

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Finally, we obtain e z n (α1 +···+α p ) Vn−1 (z 1 , . . . , z n−1 )E p,q (z, α) = Ce−z n (α1 +···+αn ) ×  p det(e−zi α j )1≤i, j≤ p i=1 (z i − z n )

e zn (α p+1 +···+α p+q ) ×  p+q det(e−zi α j ) p+1≤i, j≤ p+q . i= p+1 (z i − z n ) After simplification, we get the formula of Theorem 4.1.



4 Convexity Theorem and Analogue of Heckman’s Measures Let us first recall the classical Horn’s convexity theorem [8] which corresponds to the case p = n, q = 0. Consider the projection Proj : Hn → Dn of the space of Hermitian matrices onto the subspace Dn  Rn of real diagonal matrices. Recall that Oα denotes the orbit of A = diag(α1 , . . . , αn ) under the action of the unitary group U (n). Theorem 4.1 The projection of the orbit Oα is equal to the convex hull of the orbit of α under the action of the permutation group Sn :  Proj(Oα ) = Conv σ(α) | σ ∈ Sn . (σ(α) = (ασ(1) , . . . , ασ(n) ).) The projection Mα of the orbital measure μα will be called Heckman’s measure. (Such measures are introduced in [7]). It is a probability measure on Dn , invariant under Sn with compact support: supp(Mα ) = Conv(Sn · α). The Fourier–Laplace transform of Heckman’s measure Mα α (z) = M

 Rn

e−(z|x) Mα (d x),

α (z) = En (z, α), with is given by Itzykson–Zuber’s formula: M En (z, α) = δn !

 1 det e−zi α j 1≤i, j≤n . Vn (z)Vn (α)

(δn = (n − 1, . . . , 2, 1, 0), δn ! = (n − 1)! . . . 2!.)

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We turn now to the case of pseudo-Hermitian matrices (q ≥ 1), and consider the projection Proj :  p,q → D p,q , of the cone of positive pseudo-Hermitian matrices onto the cone of diagonal matrices with p positive diagonal entries and q negative ones. The group W p,q = S p × Sq acts on D p,q , S p permuting the positive entries, and Sq permuting the negative ones. p q Let C p,q be the convex cone in D p,q  R+ × R− generated by the vectors ei − e j with 1 ≤ i ≤ p, p + 1 ≤ j ≤ p + q. ({ei } denotes the canonical basis in Dn  Rn .) Recall that Oα denotes the orbit under the action of U ( p, q) of the diagonal matrix A = diag(α, . . . , αn ) ∈ D p,q . Next theorem is the non-compact analogue of Horn’s convexity theorem. Theorem 4.2 For α ∈ D p,q , Proj(Oα ) = Conv(W p,q · α) + C p,q . This result is proven by [11] (Sect. III) in a slightly different form. We will give an alternative proof using the Laplace transform. Proof Define the measure G p,q on D p,q by, for a continuous function f with compact support, 

 D p,q

f (x)G p,q (d x) =

pq

R+



f

ti j (ei − e j )

1≤i≤ p, p+1≤ j≤ p+q



dti j .

1≤i≤ p, p+1≤ j≤ p+q

The support of G p,q is equal to the cone C p,q . The measure G p,q has a density with respect to the Lebesgue measure of the hyperplane with equation x1 + · · · + xn = 0, which is piecewise polynomial and homogeneous. Lemma 4.3 The Laplace transform of the measure G p,q ,  G p,q (z) =

 D p,q

is given by

e−(z|x) G p,q (d x),

 G p,q (z) =

1≤i≤ p, p+1≤ j≤ p+q

1 . zi − z j

Proof The measure G p,q is the convolution product of the pq Heaviside measures Yi j given by  ∞  f t (ei − e j ) dt. Yi j , f  = 0

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The Laplace transform of G p,q is the product of the Laplace transform of the Heaviside measures Yi j ,  ∞ 1  e−t (zi −z j ) dt = . Yi j (z) = z − zj i 0  

We denote by Mα and Mα the Heckman’s measures with α = (α1 , . . . , α p ) and α = (α p+1 , . . . , α p+q ) supported by the convex sets Conv(σ(α ) | σ ∈ S p ) ⊂ R+ , Conv(σ(α ) | σ ∈ Sq ) ⊂ R− . p

q

Theorem 4.4 The projection Mα of the orbital measure μα on D p,q is equal to the following convolution product Mα = C G p,q ∗ (Mα × Mα ). Proof The Laplace transform of the projection Mα on D pq is equal to the restriction to D p,q of the Laplace transform L(μα ) of μα :  Mα (z) = E p,q (z, α). We decompose the formula for E p,q (z, α) in Theorem 3.1 in three factors E p,q (z, α) = C

1≤i≤ p, p+1≤ j≤ j

1 zi − z j

 1 det e−zi α j 1≤i, j≤ p Vp p (z 1 , . . . , z p )  1 det e−zi a j p+1≤i, j≤ p+q . × Vq (α )Vq (z p+1 , . . . , z p+q ) ×

(α )V

The first factor is, up to the constant C, equal to the Laplace transform of the measure G p,q , the second one is the Laplace transform of Heckman’s measure Mα , and the third one is the Laplace transform of Heckman’s measure Mα . The theorem follows since the Laplace transform carries the convolution product onto the ordinary product and is injective (Fig. 2).  Alternative proof of Theorem 4.2 The support of the projection of the orbital measure μα is equal to the projection of the orbit Oα : supp(Mα ) = Proj(Oα ). Since the measures G p,q and Mα × Mα are positive  supp G p,q ∗ (Mα × Mα ) = supp(G p,q ) + supp(Mα × Mα ).

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x3

x2

C2,1

ε23

σ(α)

ε13 x1

α

Proj(Oα )

Fig. 2 Projection of the orbit, p = 2, q = 1. The projection of the orbit Oα is a domain in the plane x1 + x2 + x3 = α1 + α2 + α3 bounded by the segment [α, σ(α)] and two half-lines parallel to ε13 and to ε23 (εi j = ei − e j ), and σ is the symmetry (x1 , x2 , x3 ) → (x2 , x1 , x3 )

Furthermore supp(G p,q ) = C p,q , and supp(Mα × Mα ) = Conv(S p · α ) × Conv(Sq · α ) = Conv(W p,q · α). This proves that Proj(Oα ) = Conv(W p,q · α) + C p,q . 

Eigenvalues of Positive Pseudo-Hermitian Matrices

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5 Horn’s Problem for Pseudo-Hermitian Matrices The classical Horn’s problem is as follows: let A and B be n × n Hermitian matrices. Assume that the eigenvalues α1 , . . . , αn of A and β1 , . . . , βn of B are known. What are the possible eigenvalues of the sum A + B? [9]. There is also a probabilistic version of Horn’s problem. Consider the orbits Oα of A and Oβ of B for the action of the unitary group U (n). Assume that the random matrix X is uniformly distributed on Oα , i.e. whose law is the orbital measure μα , and Y uniformly distributed on Oβ . What is the distribution of the eigenvalues of the sum X + Y ? There is a numerous literature about Horn’s problem. In [6], we gave a formula for this distribution. The support of this distribution is the set Horn(α, β) of possible eigenvalues for the sum X + Y. It is possible to consider Horn’s problem for positive pseudo-Hermitian matrices in H p,q . But, if q ≥ 1, the orbital measures are unbounded, and there is no probabilistic version. Let X in H p,q , with eigenvalues α1 ≥ · · · ≥ α p > 0 > α p+1 ≥ · · · ≥ α p+q , and Y with eigenvalues β1 ≥ · · · ≥ β p > 0 > β p+1 ≥ · · · ≥ β p+q . The problem is to determine the set Horn(α, β) of possible eigenvalues for the sum Z = X + Y . Let A = diag(α1 , . . . , αn ), and B = diag(β1 , . . . , βn ). The orbital measures μα and μβ are contained in the convex cone  p,q . The convolution product μα ∗ μβ is well defined. The set Horn(α, β) is equal to the support of the pseudo radial part να,β of μα ∗ μβ . The measure να,β is determined by the relation  D p,q

E(z, t)να,β (dt) = Lμα (z)Lμβ (z).

The group W p,q = S p × Sq acts on the space of diagonal matrices D p,q , S p permuting the positive diagonal entries, and Sq permuting the negative ones. Define the discrete measure   1 ε(σ1 )δσ1 (α ) × ε(σ2 )δσ2 (α ) Vn (α) σ1 ∈S p σ2 ∈Sq  1 = ε(w)δwα . Vn (α) w∈W

ηα =

p,q

where ε(w) is the product of the signatures of σ1 and σ2 , if w = σ1 × σ2 . Recall that Mα is the projection of the orbital measure μα on the space Dn of diagonal matrices.

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Theorem 5.1 The radial part να,β of μα ∗ μβ , whose support is Horn’s set Horn (α, β), is given by  να,β = C Vn (t) ηα ∗ Mβ . Proof The proof follows the same lines as the one of Theorem 4.1 in [6]. The measure M = Mα ∗ ηβ is skew-symmetric with respect to W p,q = S p × Sq , and its Laplace  as well, transform M  = M(z) Mα (z)ηβ (z) = E p,q (α, z)

1 det(e−zi βi )1≤i, j≤ p det(e−zi β j ) p+1≤i, j≤ p+q . Vn (β)

Let us compute the integral  E p,q (z, t)Vn (t)M(dt)  C = det(e−zi t j )1≤i, j≤ p det(e−zi t j ) p+1≤i, j≤ p+q M(dt) Vn (z) Rn   C     = ε(σ1 )e−(σ1 (z )|t ) × ε(σ2 )e−(σ2 (z )|t ) M(dt), Vn (z) Rn

I (z) =

Rn

σ1 ∈S p

σ2 ∈Sq

with z  = (z 1 , . . . , z p ), z  = (z p+1 , . . . , z p+q ), and t  = (t1 , . . . , t p ), t  = (t p+1 , . . . , t p+q ). We go on the computation   C ε(w)e−(w(z)|t) M(dt) I (z) = Vn (z) Rn w∈W C   ε(w)M(w(z)) = Vn (z) w∈W C  = p!q!E p,q (z, α)E p,q (z, β). p!q!M(z) = Vn (z)



We don’t know how to deduce the support of να,β from this formula since ηα is an alternate sum of Dirac measures and cancellations are possible in the sum να,β = C Vn (t)



 ε(w)δwα ∗ Mβ .

w∈W

References 1. Y. Baryshnikov, GUEs and queues. Probab. Theory Relat. Fields 119, 256–274 (2001) 2. S. Ben Said, B. Orsted, Bessel functions for root systems via the trigonometric setting. Int. Math. Res. Not. 9, 551–585 (2005)

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3. A.-L. Cauchy, Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des mouvements des planètes. Œuvres complètes, série 2, tome 9 (1829), pp. 174–195 4. J. Faraut, Rayleigh theorem, projection of orbital measures, and spline functions. Adv. Pure Appl. Math. 4, 261–283 (2015) 5. J. Faraut, Projections of orbital measures for the action of a pseudo-unitary group. Banach Cent. Publ. 113 (2017) 6. J. Faraut, Horn’s problem and Fourier analysis. Tunis. J. Math. 1 (2019) 7. G. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups. Invent. math. 67, 333–356 (1982) 8. A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix. Am. J. Math 76, 620–630 (1954) 9. A. Horn, Eigenvalues of sums of Hermitian matrices. Pac. J. Math. 12, 225–241 (1962) 10. C. Itzykson, J.-B. Zuber, The planar approximation II. J. Math. Phys. 21, 411–421 (1980) 11. S.M. Paneitz, Invariant cones and causality in semisimple Lie algebras and groups. J. Funct. Anal. 43, 313–359 (1981)

On the Subgroup Lattices of Lie Groups with Finitely Many Connected Components Hatem Hamrouni and Zouhour Jlali

Dedicated to the memory of Professor Takaaki Nomura (1953–2020)

Abstract Let G be a locally compact group. The lattice formed by all closed subgroups of G will be denoted by LAT (G) and will be called the closed subgroup lattice of the group G. The purpose of this paper is to show that if LAT (G) is isomorphic to a closed subgroup lattice of a Lie group with finitely many connected components, then G itself is a Lie group with finitely many connected components. Moreover, we establish that G is finite if and only if LAT (G) is finite. Keywords Locally compact group · Almost connected group · Totally disconnected group · Subgroup lattice · Projectivity · Chabauty topology 2010 Mathematics Subject Classification Primary 22D05

1 Introduction and Main Results The set LAT (G) of all closed subgroups of a locally compact topological group G is a lattice with respect to the operations of intersection A ∩ B and taking the smallest closed subgroup A ∨ B = A, B containing subgroups A and B in LAT (G). The lattice LAT (G) will be called the closed subgroup lattice of the group G. The H. Hamrouni (B) · Z. Jlali Faculty of Sciences at Sfax. Department of Mathematics., Sfax University, B.P. 1171. 3000, Sfax, Tunisia e-mail: [email protected] Z. Jlali e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_5

69

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investigation of connections between the structure of a locally compact group and the structure of its lattice of closed subgroups is one of the main general approaches in the study of locally compact groups. There are many interesting results in this field, but also many important unsolved problems (a survey of the principal results can be found in [20], see also [5, 15–17]). There exists a natural topology on LAT (G), the Chabauty topology ([4]), which is generated by open sets of the form O1 (K ) = {H ∈ LAT (G) | H ∩ K = ∅} , O2 (V ) = {H ∈ LAT (G) | H ∩ V = ∅} , where V and K run, respectively, over all open and compact subsets of G. The lattice LAT (G) endowed with the Chabauty topology is denoted by SUB (G). The space SUB (G) is compact (See Théorème 1, page 181, in [3]. For more information about this space; see Chap. 1 in [9]). In this paper, we shall prove the following. Theorem A (Proposition 2.3 below) For a locally compact group G, the following three conditions are equivalent: (1) G is finite. (2) LAT (G) is finite (3) SUB (G) is discrete. Remark 1.1 We note that in the case when G is compact and abelian, the above result was proved by Morris (Lemma 1 in [14]). Following Suzuki [24], we define a projectivity. Definition 1.2 (Projectivity) Let G and H two locally compact groups. (1) An isomorphic mapping of the subgroup lattice LAT (G) of G onto the subgroup lattice LAT (H ) of H is called a projectivity from G onto H . (2) H is called a projective image of G if there is a projectivity from G onto H . Remark 1.3 (1) If A is a subgroup of G and ϕ is a projectivity from G onto H , then the restriction ϕ A of ϕ to LAT (A) clearly is a projectivity from A onto ϕ(A); we say that ϕ induces the projectivity ϕ A in A. (2) If ϕ is a projectivity from G to H and N a closed normal subgroup of G such that ϕ(N ) is normal in H , then the map ϕ N : LAT (G/N ) → LAT (H/ϕ(N )) , A/N → ϕ(A)/ϕ(N ) is a projectivity from G/N to H/ϕ(N ); we call ϕ N the projectivity induced by ϕ in G/N . Remark 1.4 The role of the projectivities is fundamental in the lattice theory and in the structure of groups (for example, see Sects. 2.5 and 2.6 in [21]). They play an important role in the classification of topologically modular locally compact groups and topologically quasihamiltonian locally compact groups (see [10] and [9]).

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If X is a class of locally compact groups, let X be the class of all locally compact groups H for which there exists G ∈ X such that LAT (G) is isomorphic to LAT (H ). We call a class X of locally compact groups invariant under projectivities, if X = X. One of the main problems in the theory of subgroup lattices is to find classes of locally compact groups invariant under projectivities: Question 1.5 Which classes of locally compact groups X satisfy X = X? The most important classes of locally compact groups to be considered are: • [C] is the class of all connected locally compact groups. • [K] is the class of compact groups. • [Lie] is the class of Lie groups. The following example, taken from [5, page 707], shows that the classes [Lie], [K] and the class [D] of discrete groups are not invariant under projectivities. Example 1.6 Let P be the set ofall prime numbers. For p ∈ P, let Z( p) be the cyclic group of order p. Let G = p∈P Z( p) be the cartesian product  of the discrete groups Z( p) endowed with the product topology and let H = p∈P Z( p) be the direct sum of the discrete groups Z( p) endowed with the discrete topology. Then G is an infinite profinite group and H an infinite discrete group (hence, in particular, a Lie group). The map ϕ : LAT (H ) → LAT (G) , S → S is a projectivity from H onto G with T → T ∩ H as inverse ([5, page 708]). The identity component of a topological group G, denoted by G 0 , is the connected component of the identity in G. We recall that a topological group G is said to be almost connected if the factor group G/G 0 of G modulo the identity component G 0 is compact. A Lie group is almost connected if and only if it has only finitely many connected components. The following is the main result of this paper. Theorem B (Theorem 3.13 below) The class of almost connected Lie groups is invariant under projectivities.

2 Locally Compact Groups Possessing Only a Finite Number of Closed Subgroups Lemma 2.1 (Proposition 2.5 in [8]) Let G be a locally compact group. If LAT (G) is finite then G is a Lie group. Proof Let E = {e} be the trivial subgroup of G. By our assumption, SUB (G) is discrete and so E is isolated in SUB (G). Let K be a compact subset of G such that

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O1 (K ) = {E}. Since K ∩ E = ∅, there is a compact neighborhood V of e such that V ∩ K = ∅. It is clear that the trivial subgroup E is the unique closed subgroup of G contained in V , and so G has no small subgroups. Consequently, G is a Lie group. Proposition 2.2 Let G be an infinite non-discrete locally compact abelian group. The following conditions are equivalent: (1) G is topologically isomorphic with T, where T is the circle group; (2) every proper closed subgroup is finite; (3) every proper closed subgroup is of the form {g ∈ G | g n = e}, where n is any non-negative integer and e denotes the identity element. Proof This follows from Theorem 1.2 and Proposition 1.16 in [1]. In the following, we show that a locally compact group with only a finite number of closed subgroups must be a finite group. Proposition 2.3 (Groups with only a finite number of closed subgroups) For a locally compact group G, the following three conditions are equivalent: (1) G is finite; (2) LAT (G) is finite; (3) SUB (G) is discrete. Proof Clearly (1) implies (2). Suppose that (2) holds. We have to show that G is finite. Since the infinite cyclic group Z has infinitely many subgroups, any element of G is compact; that is, for any g ∈ G, the subgroup generated by g is relatively compact (see Weil’s Lemma, Proposition 7.43 in [12]). Thus, G is a finite union of compact subgroups and so it is compact. Further, since LAT (G) is finite, by Lemma 2.1, G is a Lie group. On the other hand, G 0 does not have the circle group, T, as a subgroup as T has an infinite number of closed subgroups. Thus, by the Maximal Torus Theorem (Theorem 6.30 in [12]), G 0 is trivial. Then G is discrete, and therefore, it is finite. Hence, (2) implies (1). The equivalence between (2) and (3) follows from the fact that the Chabauty topology is Hausdorff and compact. Remark 2.4 If A is a subgroup of G and ϕ is a projectivity from G onto H , then the restriction ϕ A of ϕ to LAT (A) is a projectivity from A onto ϕ(A). Proposition 2.3 shows that if A is finite, so is ϕ(A) since it is the image of A under the projectivity ϕ A . Therefore, we have the following: Any projectivity from G onto H establishes a one-to-one correspondence between the sets of finite subgroups of G and H. Combining Propositions 2.3 and 2.2 we get Corollary 2.5 Let G be an infinite non-discrete locally compact abelian group. The following conditions are equivalent: (1) G is topologically isomorphic with T; (2) every proper closed subgroup has only a finite number of closed subgroups.

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Remark 2.6 A more general version of Proposition 2.2 can be found in [14]. Example 2.7

•

LAT (G) =

if and only if G is cyclic of prime order. • •

LAT (G) =

•

if and only if G is cyclic of order p 2 for a prime p.

•

• LAT (G) = •

•

if and only if G is cyclic of order pq (with primes p = q)

•

3 Projectivities and Almost Connected Lie Groups Proposition 3.1 The class [C] is invariant under projectivities. Proof See [16] (see also [5, page 705]). Remark 3.2 Let A be a closed subgroup of G and ϕ a projectivity from G onto H . Proposition 3.1 shows that if A ∈ [C], so is ϕ(A) since it is the image of A under the projectivity ϕ A induced by ϕ in A. Therefore, we have the following: Any projectivity from G onto H establishes a one-to-one correspondence between the sets of connected closed subgroups of G and H. A topological group G is called totally disconnected if G 0 is trivial. Let [TD] be the class of totally disconnected locally compact groups. Corollary 3.3 If ϕ is a projectivity from a locally compact group G onto a locally compact group H , then ϕ(G 0 ) = H0 . In particular, the class [TD] is invariant under projectivities. Proof By Proposition 3.1, the closed subgroup ϕ(G 0 ) is connected and so ϕ(G 0 ) ⊆ H0 . Similarly, since ϕ−1 is a projectivity from H onto G, we have ϕ−1 (H0 ) ⊆ G 0 . Finally, we obtain ϕ(G 0 ) = H0 . The second statement follows from the fact that the identity subgroup of G corresponds under projectivity to the identity subgroup of H . If G is a locally compact topological group and g ∈ G, then g will be called compact if it is an element of a compact subgroup of G. The set of all compact elements of G is denoted by comp (G). It is exactly the union of all compact subgroups of G.

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Lemma 3.4 Let ϕ be a projectivity from a locally compact group G onto a locally compact group H . If comp (G) is a subgroup, then ϕ(comp (G)) = comp (H ). Proof See Lemma in [17]. Corollary 3.5 The class [C] ∩ [K] is invariant under projectivities. Proof Let ϕ be a projectivity from a locally compact group G onto a locally compact group H . We assume that G ∈ [C] ∩ [K]. Since G is compact, comp (G) = G and so, by Lemma 3.4, comp (H ) = H . Further, by Proposition 3.1, H is connected. Using Theorem 1 of [7], we deduce that H is compact. Consequently H ∈ [C] ∩ [K]. We recall the following basic facts from [19]. (F1) A locally compact group G is finite dimensional if and only if every strictly decreasing chain of closed connected subgroups of G is of finite length. (F2) A locally compact group G is a Lie group if and only if G is finite dimensional and every closed totally disconnected subgroup of G is discrete. Corollary 3.6 The class [Lie] ∩ [K] of compact Lie groups is invariant under projectivities. Proof Let G ∈ [Lie] ∩ [K], and let ϕ be a projectivity from G onto a locally compact group H . We will prove that H is a compact Lie group. First, we show that H ∈ [Lie]. Let (Hi )i∈I be a strictly decreasing chain of subgroups of H . Since  closed connected  ϕ−1 is a projectivity, By Proposition 3.1, ϕ−1 (Hi ) i∈I is a strictly decreasing chain of closed connected subgroups of H , and so I is finite. Thus, H is finite dimensional. Now, let L be a totally disconnected closed subgroup of H . By Corollary 3.3, ϕ−1 (L) is a totally disconnected closed subgroup of G, and therefore, ϕ−1 (L) is discrete. Since G is compact, ϕ−1 (L) is finite and so, by Remark 2.4, L is finite. Thus L is discrete. By (F1) and (F2), H ∈ [Lie]. It remains to show that H is compact. By Proposition 3.4, comp (H ) = H and therefore H0 is compact. On the other hand, by Corollary 3.3, let ϕG 0 be the projectivity induced by ϕ in G/G 0 . Since G/G 0 is finite, H/H0 is finite. Consequently, both H0 and H/H0 are compact, and so is H . This finishes the proof of the lemma. The following two lemmas are used to prove the main result of this section. The first is an apparent generalization of Theorem 3 in [7]. This result can be extracted from [11] (see Proposition 5.8 and Lemma 12.39). Lemma 3.7 Let G be an almost connected locally compact group and H a closed subgroup of G such that H ⊆ comp (G). Then H is compact. Proof First, we suppose that G is a Lie group. As H ∩ G 0 ⊆ comp (G) ∩ G 0 = comp (G 0 ), by Theorem 3 in [7], H ∩ G 0 is compact. On the other hand, since G 0 is open in G, the groups H/H ∩ G 0 and H G 0 /G 0 are topologically isomorphic, and therefore, H/H ∩ G 0 is compact. Consequently, H ∩ G 0 and H/H ∩ G 0 are both compact, and so is H . To prove the general case, let N be a compact normal subgroup

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of G such that G/N is a Lie group. Let π : G → G/N be the canonical projection. Since π is proper, we obtain π(H ) ⊆ π (comp (G)) = comp (π(G)), and therefore, in view of the first case, π(H ) is compact. Thus, H N is compact and so H is compact. In every totally disconnected locally compact group, the compact open subgroups form a basis of the neighborhood of the identity element (van Dantzig’s theorem). If G is a locally compact group, then G/G 0 is a totally disconnected locally compact, and so there is an open subgroup H of G containing G 0 such that H/G 0 is compact. On the other hand, we have G 0 = H0 . Therefore, we have the following (Lemma 6 in [6]): Every locally compact group contains an open almost connected subgroup. Lemma 3.8 (Theorem 32.6 in [22]) A locally compact group G is a Lie group if, and only if, every compact subgroup of G is a Lie group. Proof The “only if” part follows from the fact that the class of Lie groups is stable under taking closed subgroups. For the “if” part, let H be an open almost connected subgroup of G. By a famous theorem of Yamabe ([25]), every identity neighborhood of H contains a compact normal subgroup N of H such that H/N is a Lie group. Since the class of Lie groups is closed with respect to extensions (Theorem 7 in [13]), H is a Lie group, and G is a Lie group as well, since H is open in G. A topological group G is called compact-free if it has no compact subgroup except the trivial one, or equivalently, if comp (G) = {e}. Let [CF] be the class of locally compact compact-free groups. As an immediate consequence of Lemma 3.8 we get the following corollary. Corollary 3.9 We have [CF] ⊆ [Lie] . The following proposition is a consequence of Lemma 3.4. Proposition 3.10 The class [CF] is invariant under projectivities. Proof Let G ∈ [CF] and let ϕ be a projectivity from G onto a locally compact group H . We will prove that H ∈ [CF]. We have comp (G) = {e} and therefore, by Lemma 3.4, comp (H ) = ϕ({e}). Since ϕ maps the identity subgroup of G to the identity subgroup of H , ϕ({e}) = {e} and so comp (H ) = {e}. Thus H ∈ [CF]. The class of monothetic groups is not invariant under projectivities (see Example 1.6). The following proposition is interesting its own right. Proposition 3.11 (Projectivities and infinite discrete cyclic groups) If there is a projectivity from the additive group of integers Z onto a locally compact group G, then G is topologically isomorphic to Z. Proof Let ϕ be a projectivity from Z onto G. By Corollary 3.3, ϕ(Z) = G is totally disconnected. On the other hand, since Z is compact-free, G is also compact-free and so it is a Lie group, by Corollary 3.9. Thus, G is discrete, and therefore, the result follows from Theorem 3.2 in [2] (see also Theorem 1.2.10, page 15, in [21]).

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Remark 3.12 Let A be a subgroup of G and ϕ a projectivity from G onto H . Proposition 3.11 shows that if A is infinite discrete cyclic, so is ϕ(A) since it is the image of A under the projectivity ϕ A induced by ϕ in A. The following is the main result of this paper. Theorem 3.13 The class of almost connected Lie groups is invariant under projectivities. Proof Let G be an almost connected Lie group, and let ϕ be a projectivity from G onto a locally compact group H . We will prove that H is an almost connected Lie group. First, we prove that H is a Lie group. In view of Lemma 3.8, it suffices to prove that any compact subgroup K of H is a Lie group. We have ϕ−1 (K ) = ϕ−1 (comp (K ))   = comp ϕ−1 (K ) ⊆ comp (G) .

(by Lemma 3.4)

From Lemma 3.7, we deduce that ϕ−1 (K ) is a compact subgroup of G. Thus, ϕ−1 (K ) ∈ [Lie] ∩ [K] and so, by Corollary 3.6, K = ϕ(ϕ−1 (K )) ∈ [Lie] ∩ [K]. Consequently, H ∈ [Lie]. We now show that H is almost connected. By Corollary 3.3, we have that ϕ(G 0 ) = H0 , and therefore, let ϕG 0 : LAT (G/G 0 ) → LAT (H/H0 ) be the projectivity induced by ϕ in G/G 0 (see Remark 1.3 (2)). As G/G 0 ∈ [Lie] ∩ [K], H/H0 ∈ [Lie] ∩ [K], which achieves the proof of the Theorem. Acknowledgements It is a pleasure to thank both referees for their careful reading of the manuscript and for their useful comments in improving the form of the paper.

References 1. D.L. Armacost, The Structure of Locally Compact Abelian Groups, Monographs and Textbooks in Pure and Applied Mathematics (Marcel Dekker, New York, 1981) 2. R. Baer, The significance of the system of subgroups for the structure of the group. Am. J. Math. 61, 1–44 (1939) 3. N. Bourbaki, Éléments de Mathématique, Intégration, chapitres 7–8 (Springer, Berlin, 2007) 4. C. Chabauty, Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78, 143– 151 (1950) 5. V.S. Charin, I.V. Protasov, Projections of topological groups, Mathematical Notes of the Academy of Sciences of the USSR 24 (1978), 705–708. Translated from Matematicheskie Zametki 24(3), 383–389 (1978) 6. J. Cleary, S.A. Morris, Trinity ...A Tale of Three Cardinals, Miniconference on Operator Theory and Partial Differential Equations, 117–127 (Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1986)

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7. D.Z. Djocovic, The union of compact subgroups of a connected locally compact group. Math. Z. 158, 99–105 (1978) 8. H. Hamrouni, A. Omri, Discrete subgroups of a locally compact group with jointly discrete Chabauty neighborhoods. J. Lie Theory 29, 89–93 (2019) 9. Herfort W., Hofmann K. H., Russo F. G., Periodic Locally Compact Groups. De Gruyter Studies in Mathematics, vol. 71 (De Gruyter, Berlin, 2019). A Study of a Class of Totally Disconnected Topological Groups 10. W. Herfort, K.H. Hofmann, F.G. Russo, When is the sum of two closed subgroups closed in a locally compact abelian group? Topol. Appl. 270 (2020) 11. K.H. Hofmann, S.A. Morris, The Lie Theory of Connected Pro-lie Groups (European Mathematical Society Publishing House, Zürich, 2007) 12. K.H. Hofmann, S.A. Morris, The Structure of Compact Groups. De Gruyter Studies in Mathematics, vol. 25 (De Gruyter, Berlin). A primer for the student - a handbook for the expert, 4th revised and expanded edition, 2020 13. K. Iwasawa, On some types of topological groups. Ann. Math. 50, 507–558 (1949) 14. S.A. Morris, The circle group. Bull. Austral. Math. Soc. 36, 279–282 (1987) 15. Y.N. Mukhin, Locally compact groups with a distributive structure of closed subgroups. Sib. Mat. Zh., 8, 268–274 (1967). Translated from Sibirskii Mathematicheskii Zhurnal, Vol. 8, No. 2, pp. 366–375, March-April, 1967 16. Y.N. Mukhin, S.P. Khomenko, Monothetic groups and the subgroup lattice. Matem. Zap. Ural’skogo Un-ta 6, 67–79 (1967) 17. Y.N. Mukhin, I.V. Protasov, Projection of Abelian Topological Groups, Translated form ukrainskii matematicheskii zhurnal (1979) 18. V.P. Platonov, Periodic and compact subgroups of topological groups. Siberian Math. J. Springer 7, 681–697 (1966) 19. N.W. Rickert, Locally compact topologies for groups. Trans. Am. Math. Soc. 126, 225–235 (1967) 20. L.E. Sadovskii, Some lattice-theoretical problems in the theory of groups. Uspekhi Mat. Nauk 23, 123–157. Russian Math. Surv. 23 (1968), 125–156 (1968) 21. R. Schmidt, Subgroup Lattices of Groups. Expositions in Math., vol. 14 (de Gruyter, 1994), xv+572 pp 22. M. Stroppel, Locally Compact Groups (European Mathematical Society (EMS), Zürich, EMS Textbk. Math., 2006) 23. M. Suzuki, On the lattice of subgroups of finite groups. Trans. Am. Math. Soc. 70, 345–371 (1951) 24. M. Suzuki, Structure of a Group and the Structure of its Lattice of Subgroups (Springer, Berlin, 1956) 25. H. Yamabe, On the conjecture of Iwasawa and Gleason. Ann. Math. 58, 48–54 (1953)

Semi-invariant Vectors Associated with Holomorphically Induced Representations of Exponential Lie Groups Junko Inoue

Abstract We study Frobenius reciprocity in distribution sense for holomorphically induced representations ρ = ρ( f, h) of exponential groups G with Lie algebras g, from complex subalgebras h of gC and real linear forms f . We treat some examples of groups G = R  Rn , and give a new example ρ = ρ( f, h) such that the space H(h, f, δ) of ρ is non-zero, and the reciprocity does not hold in the following sense. In our example, the set a of irreducible representations of G, such that the spaces of semi-invariant generalized vectors are non-zero does not coincide with the support of the Plancherel measure associated with the decomposition of ρ. Keywords Holomorphically induced representation · Solvable Lie group · Plancherel formula

1 Introduction As a sequel of our previous works in [9, 10], we study Frobenius reciprocity in distribution sense associated with Penney’s abstract Plancherel theorem for holomorphically induced representations. First, we recall some definitions and preliminaries in [10]. Let G = exp g be an exponential Lie group with Lie algebra g and f ∈ g∗ be a real linear form, which is extended to gC by complex linearity. We denote by B f the skew symmetric bilinear form on gC defined by B f (X, Y ) := f ([X, Y ]), X, Y ∈ gC . Let h ⊂ gC be an isotropic complex subalgebra for B f , that is, h satisfies f ([h, h]) = {0}.

This work was supported by JSPS KAKENHI Grant Number JP17K05280 J. Inoue (B) Center for Data Science Education, Organization for Educational Support and International Affairs, Tottori University, Tottori 680-8550, Japan e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_6

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Let d := g ∩ h, D = exp d be the corresponding subgroup, χ f be the unitary character of D defined by χ f (exp X ) = ei f (X ) , X ∈ d. We denote E := (h + h) ∩ g noting that E is not necessarily a subalgebra. Take δ ∈ E ∗ such that δ|d =

1 tr ad g/d and δ|E∩n = 0, 2

(1.1)

where n is the nilradical of g, and extend it to EC by complex linearity. From a triple (h, f, δ), we define a holomorphically induced representation ρ = ρ(h, f, δ), which is a subrepresentation of ind GD χ f , by the following Definition 1. Let us recall that on the space of continuous functions ψ on G with compact support modulo D satisfying ψ(x y) =

 D (y) ψ(x) ∀y ∈ D, ∀x ∈ G, G (y)

where G and  D are the modular functions of G and D, respectively, there exists a positive left invariant linear functional  ψ(g) dμG,D

ψ → G/D

uniquely up to a positive constant factor. (See [2, Chap.V], [5, Chap.3].) Definition 1 Let C ∞ (h, f, δ) be the space of C ∞ functions φ on G such that 1. φ(gy) = χ f (y)

−1



 D (y) G (y)

1/2 φ(g), ∀g ∈ G, ∀y ∈ D,

(1.2)



2.

φ :=

|φ(g)|2 dμG,D (g) < ∞,

2

G/D

3. R(X )φ = (−i f (X ) + δ(X ))φ, ∀X ∈ h,

(1.3)

where R denotes the action as the left invariant vector field: R(X )φ(g) :=

 d φ(g exp(t X ))t=0 , dt

X ∈ g,

which is extended to gC by complex linearity. Let H(h, f, δ) be the completion of C ∞ (h, f, δ) and define a representation ρ of G on H(h, f, δ) by the left translation ρ(g)φ(x) := φ(g −1 x), g, x ∈ G.

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for φ ∈ H(h, f, δ). Let us remark that the property (1.1) of the linear form δ ∈ E ∗ makes (1.2) compatible with (1.3). Suppose H(h, f, δ) = {0}, and denote the space of C ∞ -vectors by H(h, f, δ)∞ and its anti-dual space by H(h, f, δ)−∞ . Elements of the antidual space H(h, f, δ)−∞ are called generalized vectors. Then the anti-linear form aρ ∈ H(h, f, δ)−∞ defined by H(h, f, δ)∞ φ → aρ , φ := φ(e), where e is the unit element of G, satisfies the semi-invariance ρ(X )aρ = (i f (X ) + δ(X ))aρ ,

X ∈ h.

⊕ Suppose ρ is decomposed into a direct integral G m(π )π dμ(π ) of irreducible  on Hilbert spaces Hπ , where dμ is a Borel measure on G  and representations π ∈ G m(π ) is the multiplicity of π . Then by Penney’s abstract Plancherel theorem [13], aρ is decomposed into a direct integral of generalized vectors aπ of π , and each aπ satisfies the semi-invariance π(X )aπ = (i f (X ) + δ(X ))aπ , Let

X ∈ h.

(Hπ−∞ )h, f,δ := {a ∈ Hπ−∞ ; π(X )a = (i f (X ) + δ(X ))a,

Then we have

X ∈ h}.

dim(Hπ−∞ )h, f,δ ≥ m(π )

for μ-almost all π [13]. Our problem of ‘reciprocity’ is as follows: Problem 1 (See [5, Chap.10]) Does the equality dim(Hπ−∞ )h, f,δ = m(π )

(1.4)

hold for μ-almost all π ? Many affirmative results have been obtained with various groups and representations, including the cases of monomial representations. (e.g., [1, 3–9, 11, 12, 14].) On the other hand, in [10], we found an example that the space (Hπ−∞ )h, f,δ = {0}  where ν is the Plancherel measure for the monomial reprefor ν-almost all π ∈ G, G sentation ind D χ f , although ρ(h, f, δ) vanishes. In this article, we show a new example that ρ does not vanish nor does the reciprocity (1.4) hold. We first study the direct product of ax + b group and R, and give an example of non-zero ρ that the reciprocity (1.4) does not hold. Next, we study semidirect product groups R  Rn defined in Sect. 3 and improve our previous results in [9] by discussing a necessary and sufficient condition that ρ is non-vanishing. In

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order to construct intertwining operators, we apply the idea of using matrix elements associated with semi-invariant vectors developed by Penney [14] and Magneron [11, 12].

2 Direct Product of ax + b Group and R Let G be the connected and simply connected Lie group with Lie algebra g = R-Span{T, X, Z } = R-Span{T, X } × R defined by [T, X ] = X. Let V = X + Z + i T ∈ gC , h = CV,

f ∈ g∗ , δ = δ(T )T ∗ ,

(2.1)

where {T ∗ , X ∗ , Z ∗ } is the dual basis in g∗ . Remark that E = RT + R(X + Z ) generates g. We study the holomorphically induced representation ρ := ρ(h, f, δ) of G. Since h ∩ h = {0}, the representation ρ is a subrepresentation of the regular representation of G. For  = t ∗ T ∗ + x ∗ X ∗ + z ∗ Z ∗ ∈ g∗ (t ∗ , x ∗ , z ∗ ∈ R), the coadjoint orbits are Ad∗ (G) = RT ∗ + R+ X ∗ + z ∗ Z ∗ := {τ T ∗ + ε X ∗ + z ∗ Z ∗ , ε > 0, τ ∈ R}

if

x ∗ > 0,

Ad∗ (G) = RT ∗ + R− X ∗ + z ∗ Z ∗ := {τ T ∗ + ε X ∗ + z ∗ Z ∗ , ε < 0, τ ∈ R}

if

x ∗ < 0,

Ad∗ (G) = {}

if

x ∗ = 0.

Let + : = { ∈ g∗ , (X ) > 0}, ∗

− : = { ∈ g , (X ) < 0},  : = + ∪ − ,

+ : = X ∗ + RZ ∗ ,

(2.2)

− : = −X ∗ + RZ ∗ ,  : = + ∪ − .

Then the set  is open dense in g∗ and consists of 2-dimensional orbits. We parametrize / Ad∗ (G) by .  with the space of coadjoint By the orbit method, we identify the unitary dual G  and we have that orbits g∗ / Ad∗ (G) by the Kirillov-Bernat mapping  : g∗ → G  corresponding to the regular representation is supported on the generic subset of G / Ad∗ (G), which is identified with ; the Plancherel measure μ is obtained as the ˜ by  of a finite measure μ˜ on g∗ equivalent to the Lebesgue measure. image ∗ (μ) (See [5, Theorem 8.1.9].) We also regard μ as the measure on  which is defined by the image of μ˜ by the quotient map g∗ → g∗ / Ad∗ (G). Let  = ε X ∗ + z ∗ Z ∗ , ε = ±1, z ∗ ∈ R. We realize the irreducible representation π corresponding to Ad∗ (G) by the Kirillov-Bernat mapping as follows: Taking a Pukanszky polarization b = RX + RZ , we have π = ind GB χ , where χ is the

Semi-invariant Vectors Associated …

83

character of B = exp b with χ (exp Y ) = ei(Y ) for Y ∈ b. By the diffeomorphism R3 (t, x, z) → g = exp(t T ) exp(x X ) exp(z Z ) ∈ G, we transfer a Lebesgue measure dtd xdz on R3 to a left invariant measure dg on G. Identifying G/B with R by R s → exp(sT )B ∈ G/B, we realize π on Hπ = L 2 (R) by π (exp(t T ) exp(x X ) exp(z Z ))φ(s) = exp(i z ∗ z + iεxet−s )φ(s − t) for t, x, z, s ∈ R, φ ∈ L 2 (R). We also obtain dφ , ds π (X )φ(s) = iεe−s φ(s), π (Z ) = i z ∗ . π (T )φ(s) = −

(2.3)

We first compute the semi-invariant distributions a := aπ for generic orbits  ∈ ± . By the semi-invariance π (V )a = (i f (V ) + δ(V ))a , we have   d εe−s + z ∗ + a (s) = ( f (X + Z − i T ) − δ(T ))a (s) ds and

a (s) = exp(εe−s + (−z ∗ + f (X + Z ) − δ(T ))s − i f (T )s)

up to a constant factor. That is, a is described by 

a , φ =

∞

eεe

−s

+(−z ∗ + f (X +Z )−δ(T ))s−i f (T )s

−∞

φ(s) ds,

)h, f,δ ≤ 1. up to a constant factor, and dim(Hπ−∞  Proposition 1 Under the assumptions above, we have 1. (Hπ−∞ )h, f,δ = Ca and a ∈ Hπ if ε = −1 and f (X + Z ) − δ(T ) < z ∗ .  −∞ h, f,δ = {0} if ε = 1 or f (X + Z ) − δ(T ) ≥ z ∗ . 2. (Hπ )

(2.4)

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Proof Suppose ε = −1 and f (X + Z ) − δ(T ) < z ∗ . Then we have that the function a (s) is square integrable and gives a non-zero generalized vector by (2.3) and (2.4). We next suppose ε = −1 and f (X + Z ) − δ(T ) = z ∗ . Let ψ(s) be a C ∞ function 1 such that supp ψ ⊂ [0, ∞) and ψ(s) = for s > 1. Then by (2.3), ψ is a C ∞ -vector s but a , ψ = ∞ since a (s) → 1 (s → ∞). Similarly, we have a , ψ = ∞ for the case of ε = −1 and f (X + Z ) − δ(T ) > z ∗ . Hence, (Hπ−∞ )h, f,δ = {0} for the case  ∗ of ε = −1 and f (X + Z ) − δ(T ) ≥ z . 2 Finally, suppose ε = 1. Let ψ(s) = e−s . Then ψ is a C ∞ vector and a , ψ = ∞ a (s) → 1 (s → −∞). Thus, (Hπ−∞ )h, f,δ = {0}.   since  exp(e−s ) )h, f,δ ≤ 1 for  ∈  and (Hπ−∞ )h, f,δ = {0} for  ∈ + , we have Now, since dim(Hπ−∞   ⊕ that ρ is decomposed into a subrepresentation of the direct integral − π dμ() on − with multiplicity free. Let R be an intertwining operator  R : H( f, h, δ) → L 2 (− × R) 

⊕

−

Hπ dμ().

Suppose  ∈ H( f, h, δ)∞ be a C ∞ -vector of ρ. Denoting φ(, s) := (R)(, s) for (, s) ∈ − × R, let us define a function σφ = σφ (g) on G by  σφ (g) : =

−

π (g)−1 φ(, ·), a dμ().

Let  ∈ H(h, f, δ)∞ ⊂ L 2 (G), and take a compactly supported smooth function F ∈ Cc∞ (G) on G arbitrarily. Then Penney’s Plancherel theorem concerning the decomposition of the Dirac measure aρ for ρ reads that

F,  L 2 (G) = ρ(F)aρ ,  L 2 (G)

 F(g)ρ(g)aρ dg,  = 

G −1

(2.5) L 2 (G)



F(g) aρ , ρ(g)  dg = F(g) ρ(g)−1 , aρ  dg G   F(g)

π (g)−1 φ(, ·), a dμ() dg = F, σφ  L 2 (G) . =

=

G

G

−

That is, σφ is necessarily square integrable on G. Let F be a smooth function on G. We say that F is of the Schwartz class, denoting by F ∈ S (exp g), if the function

Semi-invariant Vectors Associated …

85

R3 (t, x, z) → F(exp(t T ) exp(x X ) exp(z Z )) is a Schwartz class function on R3 . For F ∈ S (exp g), we define the partial Euclidean X Z ∈ S (R3 ) by Fourier transform F X Z (t, v, w) : = F

 ei(vx+wz) F(exp(t T ) exp(x X ) exp(z Z )) d xdz, (t, v, w) ∈ R3 .

R2

Let F ∈ S (exp g) and φ ∈ Cc∞ (− × R). For  = −X ∗ + z ∗ Z ∗ , s, z ∗ ∈ R, writing φ(, s) = φ(z ∗ , s), a = az ∗ , dμ() = dμ(z ∗ ), we have  σφ (g) =

−

 =

−

π (g)−1 φ(, ·), a dμ()  ∗ dμ(z ) exp(−i z ∗ z + i xe−s )φ(z ∗ , s + t)az ∗ (s) ds, R



σφ , F :=

σφ (g)F(g) dg G  exp(−i z ∗ z + i xe−s )φ(z ∗ , s + t)az ∗ (s) ds dμ(z ∗ ) = R3

−

R

F(exp(t T ) exp(x X ) exp(z Z )) dtd xdz    X Z (t, e−s , −z ∗ ) dsdμ(z ∗ )dt = φ(z ∗ , s + t)az ∗ (s) F R  R   − s s X Z (t, e−s , −z ∗ ) dsdμ(z ∗ )dt φ(z ∗ , s + t)az ∗ (s)e 2 e− 2 F = R  R   − s φ(z ∗ , s + t)az ∗ (s)e 2 N F (t, s, z ∗ ) dsdμ(z ∗ )dt, (2.6) = R

where

−

R

s X Z (t, e−s , −z ∗ ). N F (t, s, z ∗ ) = e− 2 F

Since μ is the image of μ, ˜ which is equivalent to the Lebesgue measure, we may assume dμ(z ∗ ) = c(z ∗ )dz ∗ , where dz ∗ is the Lebesgue measure on −  R and c(z ∗ ) is a positive, continuous function on R. X Z is contained in X Z of F Suppose that the support supp F R × R+ × R := {(t, u, z ∗ ); u > 0, t, u, z ∗ ∈ R}. Then

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X Z (t, e−s , −z ∗ )|2 dtdsdz ∗ e−s | F   ∞ ∗ X Z (t, u, −z ∗ )|2 du = dtdz |F R2 0   ∞ ∗ X Z (t, u, −z ∗ )|2 du = dtdz |F

N F 2 =

R3

R2

−∞

= (2π )2 F 2 . X Z ⊂ R × R+ × R} is dense in L 2 (R3 ). In The space {N F ; F ∈ S (exp g), supp F fact, for N (t, s, z ∗ ) ∈ Cc∞ (R3 ), let 

u − 2 N (t, − log u, −z ∗ ) (u > 0) F0 (t, u, z ) := , (t, u, z ∗ ) ∈ R3 . 0 (u ≤ 0) 1

∗

Then we have supp F0 ⊂ R × R+ × R and F0 ∈ Cc∞ (R3 ), and there exists a function X Z = F0 . Thus we have F = F(exp(t T ) exp(s X ) exp(z ∗ Z )) ∈ S (exp g) such that F X Z (t, e−s , −z ∗ ) N F (t, s, z ∗ ) = e− 2 F s

= e− 2 F0 (t, e−s , −z ∗ ) = e 2 e− 2 N (t, − log(e−s ), z ∗ ) = N (t, s, z ∗ ). s

s

s

Now, Suppose that σφ is square integrable on G. Then for functions F ∈ S (exp g) X Z ⊂ R × R+ × R, we have such that supp F    σφ , F ≤ σφ

F = (2π )−1 σφ

N F = (2π )−1 σφ

N

F and by the calculation (2.6),       σφ , F =   R

 −

  φ(z , s + t)a (s)e N F (t, s, z ) dsc(z )dz dt  ∗

R

s 2

z∗

∗

∗

∗

≤ (2π )−1 σφ

N F = (2π )−1 σφ

N F . Since such functions N F are dense in L 2 (R3 ), we have that the function s φ(z ∗ , s + t)az ∗ (s)e 2 c(z ∗ ) is square integrable on R3 .  ∞>

|φ(z ∗ , s + t)az ∗ (s)e 2 c(z ∗ )|2 dtdsdz ∗  = |φ(z ∗ , t)az ∗ (s)|2 es c(z ∗ )2 dtdsdz ∗ s

R3

R3

Thus, for dz ∗ -almost all z ∗ ∈ R, we have

Semi-invariant Vectors Associated …

 ∞>

R2

87

|φ(z ∗ , t)az ∗ (s)|2 es dtds,

which implies that  R =

R

|φ(z ∗ , t)az ∗ (s)|2 es ds |φ(z ∗ , t)|2 e2(−e

−s

+(−z ∗ + f (X +Z )−δ(T )+ 21 )s )

ds < ∞

for almost all (z ∗ , t) ∈ R2 . Letting β := f (X + Z ) − δ(T ) + 21 we have that the −s ∗ function h(s) = e−e +(−z +β)s is square integrable on R if and only if −z ∗ + β < 0. Thus, we have

supp (φ(z ∗ , t)) ⊂ (z ∗ , t) ∈ R2 ; z ∗ ≥ β and denoting   1 , β := −X ∗ + z ∗ Z ∗ ; z ∗ > f (X + Z ) − δ(T ) + 2 we have that R(H( f, h, δ)) ⊂ L 2 (β × R).

(2.7)

Now we are ready to describe the decomposition of ρ: Theorem 1 Let G = exp g be the connected and simply connected Lie group with Lie algebra g = R-Span{T, X, Z } defined by [T, X ] = X, and let V = X + Z + i T , h = CV , f ∈ g∗ , δ = δ(T )T ∗ be as (2.1), and , − be as (2.2). Denote a := { = −X ∗ + z ∗ Z ∗ ; z ∗ > f (X + Z ) − δ(T )},   1 ∗ ∗ ∗ ∗ . β :=  = −X + z Z ; z > f (X + Z ) − δ(T ) + 2 Then we have the following: 1. If  ∈ a , then we have dim(Hπ−∞ )h, f,δ = 1 and semi-invariant generalized vec tors are elements of Hπ . )h, f,δ = {0}. 2. If  ∈  \ a , then we have dim(Hπ−∞  3. The space H(h, f, δ) = {0}. The representation ρ = ρ(h, f, δ) decomposes into a multiplicity-free direct integral over β :

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 ρ=

⊕

β

πz ∗ dz ∗ ,

(2.8)

where πz ∗ = π with  = −X ∗ + z ∗ Z ∗ , and dz ∗ is a Lebesgue measure on the open subset β of − . Proof  ⊕ By (2.7), we have proved that ρ is a subrepresentation of the direct integral πz ∗ dz ∗ . Now, we construct an intertwining operator as follows: β

For z ∗ > f (X + Z ) − δ(T ) + 21 , let 

∗

A(z ) :=

R

s 2

|a (s)e | ds z∗

2

 21

,

which is finite. For φ ∈ Cc∞ (β × R), define

 σ˜ φ (g) :=

β

az ∗ π (g) φ(, ·), dz ∗ , A(z ∗ ) −1

(2.9)

where dz ∗ is a Lebesgue measure on β ⊂ − . Then for F ∈ Cc∞ (G), by similar computations as (2.6), we have         ∗ (s) s a z  ∗ ∗ ∗  σ˜ φ , F L 2 (G)  =  2 N (t, s, z ) dsdz dt  . e φ(z , s + t) F  R β R  A(z ∗ ) Here we have 2     az ∗ (s) s   ∗ 2  dsdz ∗ dt = e φ(z , s + t) A(z ∗ )  R β R  R β   =

 

β

R

2     ∗ az ∗ (s) 2s  e φ(z , t)  dsdz ∗ dt A(z ∗ )  R  ∗ 2 ∗ φ(z , t) dz dt = φ 2 β ×R

and  

N F 2β ×R =

R

β

  =  ≤

R

R3

β

 R

X Z (t, e−s , −z ∗ )|2 dtdsdz ∗ e−s | F

dtdz ∗



∞

X Z (t, u, −z ∗ )|2 du |F

0

X Z (t, u, −z ∗ )|2 dudtdz ∗ |F

= (2π )2 F 2L 2 (G) .

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89

Thus, we have    σ˜ φ , F L 2 (G)  ≤ (2π )2 φ 2

2 β ×R F L 2 (G)

and the mapping φ → σ˜ φ defined by (2.9) extends to a bounded intertwining operator   on L 2 (β × R) to H( f, h, δ). This implies (2.8). Remark 1 If  = −X ∗ + z ∗ Z ∗ satisfies 1 f (X + Z ) − δ(T ) < z ∗ < f (X + Z ) − δ(T ) + , 2 )h, f,δ = 1, but the representation π does not appear in the then we have dim(Hπ−∞  decomposition of ρ.

3 Semidirect Product Group R  Rn Let n := Rn be an n-dimensional real vector space and L be a non-zero linear endomorphism of n. Let g = R  Rn be the semidirect product of abelian Lie algebra R and Rn defined by g = R-span{X, n}, [X, Y ] = L(Y ), Y ∈ n. Extending L to nC , suppose L has no purely imaginary eigenvalues. Then g is exponential solvable Lie algebra. Let  be the set of eigenvalues of L, r =  ∩ R be the set of real eigenvalues of L, and c =  \ r . We denote by λ M := max r and λm := min r the maximum element and the minimum element of real eigenvalues r , respectively. We may assume λ M ≥ 0 by considering −L if necessary. Furthermore, we assume the following: (P1) There exists Y ∈ n such that E := RX + RY generates g. (P2) For all complex eigenvalues λ ∈ c , the real part Re(λ) satisfy Re(λ) < λ M and Re(λ) > λm . Let G = exp g be the connected and simply connected Lie group with Lie algebra g. Let U = X + iY and h = CU . Take f ∈ g∗ arbitrarily, and δ := δ(X )X ∗ , and define the holomorphically induced representation ρ(h, f, δ) of G. In [9], we gave a sufficient condition of δ so that ρ(h, f, δ) is non-vanishing and the reciprocity (1.4) holds, under the additional assumption (P3) λm = 0. In this paper, we treat the non-nilpotent cases of λ M > 0 without the assumption (P3), where the reciprocity (1.4) may not hold with non-zero H(h, f, δ). We discuss a necessary and sufficient condition of non-vanishing of ρ and improve our results in [9].

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Here we remark that the nilpotent case λ M = 0 was studied by Magneron in [12].

3.1 Preliminaries for R  Rn Let us recall some preliminary computations in [9]. By the Jordan’s theory and our assumption of E generating g, we take a basis of n as follows: Let  be the set of all eigenvalues of ad X , and r :=  ∩ R = {λ j ; j ∈ Jr }, c :=  \ r = {λ j , λ j , ; j ∈ Jc }, where Jr and Jc are sets of indices, respectively. Denote J := Jr ∪ Jc and λ0 = 0 if 0 ∈ , for the sake of convenience. Let nλ j = {V ∈ nC ; ( ad X − λ j I )d V = 0 for some d > 0} and d j = dim nλ j for j ∈ J . Then we have a basis of n of the form B = {Y js , 1 ≤ s ≤ d j , j ∈ Jr } ∪ {Yks , Wks , 1 ≤ s ≤ dk , k ∈ Jc } satisfying Y =



Y j1 ,

ad X (V js ) = λ j V js + V js+1 (1 ≤ s ≤ d j − 1),

j∈Jr ∪Jc d

d

ad X (V j j ) = λ j V j j ,  Y js for j ∈ Jr , s where V j = Y js + i W js for j ∈ Jc (1 ≤ s ≤ d j − 1). Using the basis B, we take the Euclidean measure d Z on n, Z ∈ n = Rn , such that the unit cube has volume 1. By the diffeomorphism G with R × Rn defined by R × Rn (x, Z ) → exp(x X ) exp(Z ) ∈ exp(RX ) exp(n) = G, we transfer the Euclidean measure d xd Z to the left Haar measure on G. For the basis {X } ∪ B of g, we denote by q∗

q∗

{X ∗ , Y js ∗ , Yk , Wk ; 1 ≤ s ≤ d j , j ∈ Jr , 1 ≤ q ≤ dk , k ∈ Jc } the dual basis of g∗ . Concerning coadjoint orbits for  ∈ g∗ , we have Ad∗ (G) = RX ∗ + Ad∗ (exp(RX )) Ad∗ (G) = {}

if if

([X, n]) = {0}, ([X, n) = {0}.

Semi-invariant Vectors Associated …

91

3.2 Semi-invariant Generalized Vectors of Irreducible Representations of G Let  ∈ g∗ be an element of a two-dimensional orbit. We realize the irreducible  corresponding to the orbit of  ∈ g∗ by the Kirillovunitary representation π ∈ G Bernat mapping as follows: Noting that n is a Pukanszky polarization at , we have π = ind GN χ , where N = exp n and χ (exp(Z )) = ei(Z ) for Z ∈ n. Identifying G/N with R by R t → exp(t X )N ∈ G/N , we realize π on L 2 (R) by π (exp(x X ) exp(Z ))φ(t) = χ (exp( Ad exp((−t + x)X )(Z )))φ(t − x) for x ∈ R, Z ∈ n, φ ∈ L 2 (R). We also obtain dφ , dt π (Z )φ(t) = i( Ad exp(−t X )(Z ))φ(t).

π (X )φ(t) = −

Suppose that a generalized vector a := aπ satisfies the semi-invariance π (U )a = (i f (U ) + δ(U ))a . Then we have   d − + ( Ad exp(−t X )(Y )) a (t) = ( f (Y ) + δ(X ) + i f (X ))a (t) dt and a (t) = exp(β (t) − ( f (Y ) + δ(X ) + i f (X ))t) up to a constant factor, where 

t

β (t) :=

( Ad exp(−s X )(Y )) ds.

0

That is, a is described by 

a , φ =

∞ −∞

eβ (t)−( f (Y )+δ(X )+i f (X ))t φ(t) dt,

(3.1)

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J. Inoue

up to a constant factor, and dim(Hπ−∞ )h, f,δ ≤ 1. Here we give an explicit description  of π (Z ) and β associated with the basis B recalling some calculations in [9, Sect. 2.2]. Denoting  js := (V js )

for j ∈ Jr ∪ Jc , 1 ≤ s ≤ d j .

we have π (Y js )φ(t) = i

dj  (−t)k−s Re(e−λ j t  jk )φ(t), (k − s)! k=s

π (W js )φ(t) = i

dj  (−t)k−s Im(e−λ j t  jk )φ(t) (k − s)! k=s

(3.2)

for φ ∈ Cc∞ (R) and j ∈ Jr ∪ Jc , 1 ≤ s ≤ d j , where Im(e−λ j t  jk ) denotes the imaginary part of e−λ j t  jk . Letting ⎛

⎞ dj  (−1)s  js ⎠

b jk (t) := Re ⎝e−iIm(λ j )t

s=k+1

we have β (t) =



k!λs−k j

for λ j = 0, 0 ≤ k ≤ d j − 1,

d j −1

e

−Re(λ j )t

j∈J, j=0

 k=0

b jk (t)t k +

d0  s=1

0s

(−1)s−1 t s . s!

When 0 ∈ / , we regard 0s = 0 for all s. We remark that b jk (t) is a smooth bounded function on t ∈ R. For real eigenvalue λ j , we have that b jk (t) = b jk is constant. Suppose  ∈ g∗ is an element of a two-dimensional coadjoint orbit such that  Md M = 0 and mdm = 0. We denote (−1)d M  Md M (λ M = 0) (d M − 1)!λ M (−1)dm mdm bm := bmdm −1 = (λm = 0) (dm − 1)!λm (−1)d0 −1 . b0 := 0d0 d0 !

b M := b Md M −1 =

Then for γ ∈ C, we have the following: β (t) + γ t → 1 (t → −∞) if λ M = 0. e−λ M t b M t d M −1

(3.3)

Semi-invariant Vectors Associated …

93

β (t) + γ t −λ e m t bm t dm −1

→ 1 (t → ∞) if λm < 0.

β (t) + γ t → 1 (t → ∞) if λm > 0, γt

(3.4)

(3.5)

where we assume γ = 0 in (3.5). β (t) + γ t → 1 (t → ∞) if λm = λ0 = 0 and d0 ≥ 2. b0 t d0

(3.6)

β (t) + γ t → 1 (t → ∞) if λm = λ0 = 0 and d0 = 1, (l01 + γ )t

(3.7)

where we assume 01 + γ = 0 in (3.7). Now we obtain the following lemma, which is a refinement of [9, Lemma 2.3]. Lemma 1 For γ ∈ C, let a,γ (t) := eβ (t)+γ t and define a distribution a,γ on Cc∞ (R) by  ∞

a,γ , φ = a,γ (t)φ(t) dt, φ ∈ Cc∞ (R). −∞

Then we have the following: 1. Suppose λm < 0 < λ M , then we have that (1.1) a,γ (t) ∈ L 2 (R) if either the condition (i) or (ii) is satisfied. (i)  Md M > 0, mdm > 0 and dm is even. (ii)  Md M > 0, mdm < 0 and dm is odd. (1.2) a,γ does not extend as a C −∞ -vector if one of the following conditions (iii) or (iv) or (v) is satisfied. (iii)  Md M < 0. (iv) mdm < 0 and dm is even. (v) mdm > 0 and dm is odd. 2. Suppose 0 < λm ≤ λ M , then we have that (2.1) a,γ ∈ L 2 (R) if the conditions  Md M > 0 and Re(γ ) < 0 are satisfied. (2.2) a,γ does not extend as a C −∞ -vector if either the condition  Md M < 0 or Re(γ ) ≥ 0

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is satisfied. 3. Suppose λm = λ0 = 0 < λ M and d0 ≥ 2. Then (3.1) a,γ (t) ∈ L 2 (R) if either the condition (i  ) or (ii  ) is satisfied. (i  )  Md M > 0, 0d0 > 0 and d0 is even. (ii  )  Md M > 0, 0d0 < 0 and d0 is odd. (3.2) a,γ does not extend as a C −∞ -vector if one of the following conditions (iii) or (iv  ) or (v  ) is satisfied. (iii)  Md M < 0. (iv  ) 0d0 < 0 and d0 is even. (v  ) 0d0 > 0 and d0 is odd. 4. Suppose λm = λ0 = 0 < λ M , and d0 = 1. Then we have that (4.1) a,γ ∈ L 2 (R) if the conditions  Md M > 0 and 01 + Re(γ ) < 0 are satisfied. (4.2) a,γ does not extend as a C −∞ -vector if either the condition  Md M < 0 or 01 + Re(γ ) ≥ 0 is satisfied. Proof By the description (3.2) of the action of g and (3.3), we have that a,γ does not extend as a C −∞ -vector if  Md M < 0. We also have (1.1), (1.2), (2.1), (3.1) and (4.1) considering (3.4), (3.5), (3.6), (3.7). For (2.2) and (4.2), Suppose Re(γ ) ≥ 0 (for the case (2.2)) or 01 + Re(γ ) ≥ 0 (for the case (4.2)). Let φ(t) be a smooth non-negative function such that supp φ ⊂ [0, ∞) and φ(t) = 1t for t > 1. Then φ(t) is a C ∞ -vector, but a , φ = ∞. Thus,   a,γ does not extend as an C −∞ -vector.

3.3 Conditions for H(h, f, δ)  = 0 and the Decompositions of ρ Now we assume λ M > 0. Let  := { ∈ g∗ ; (Y Md M )(Ymdm ) = 0}. Then the set  is open dense in g∗ and consists of 2-dimensional orbits. Let

(3.8)

Semi-invariant Vectors Associated …

95

+M : = { ∈ ; (Y Md M ) > 0},

+ : = { ∈ g∗ ; (X ) = 0, (Y Md M ) = 1},

−M : = { ∈ ; (Y Md M ) < 0},

− : = { ∈ g∗ ; (X ) = 0, (Y Md M ) = −1},

 := (+ ∩ ) ∪ (− ∩ ).

(3.9)

We parametrize / Ad∗ (G) by  through θ˜ :  → Ad∗ (exp(x()X ) exp(y()Y Md M )) ∈ , 1 (X ) x() := . log |(Y Md M )|, y() := − λM λ M (Y Md M )  with the space of coadjoint orbits g∗ / Ad∗ (G) by We identify the unitary dual G  and we have that the Plancherel measure the Kirillov-Bernat mapping  : g∗ → G ˜ by  of a finite μ for the regular representation is obtained as the image ∗ (μ) measure μ˜ on g∗ equivalent to the Lebesgue measure. (See [5, Theorem 8.1.9].)  corresponding to / Ad∗ (G), which Thus, μ is supported on the generic subset of G is identified with . We also regard μ as the measure on  defined by the image ˜ of μ. ˜ Let θ˜∗ (μ) dm + m := { ∈ ; (Ym ) > 0},

⎧ + M ⎪ ⎪ ⎪ ⎪ ⎪ +M ⎪ ⎪ ⎪ ⎨+ M a := + ⎪  ⎪ M ⎪ ⎪ ⎪ ⎪+M ⎪ ⎪ ⎩ + M

dm − m := { ∈ ; (Ym ) < 0},

∩ + m if λm < 0 and dm is even ∩ − m if λm < 0 and dm is odd if 0 < λm ≤ λ M and δ(X ) > − f (Y ) ∩ + 0 if λm = λ0 = 0, d0 ≥ 2, and d0 is even ∩ − 0 if λm = λ0 = 0, d0 ≥ 3, and d0 is odd if λm = λ0 = 0, d0 = 1, and δ(X ) > 01 − f (Y ).

a :=  ∩ a .

(3.10)

Proposition 2 Under the assumptions above, we have )h, f,δ = Ca for  ∈ a , (Hπ−∞  )h, f,δ = {0} for  ∈  \ a . (Hπ−∞  Proof Let γ = −( f (Y ) + δ(X ) + i f (X )) and apply Lemma 1 to a = a,γ . Then we can verify the conclusion.    ⊕Thus we have that ρ is decomposed into a subrepresentation of the direct integral π dμ() on a , which is an open subset of the affine space + , with multiplicity a

free. Suppose H( f, h, δ) = {0} and let R be an intertwining operator:

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 R : H( f, h, δ) → L 2 (a × R) 

⊕

a

Hπ dμ().

Suppose  ∈ H( f, h, δ)∞ be a C ∞ -vector of ρ. Denoting φ(, s) := (R)(, s) for (, s) ∈ a × R, let us define a function σφ = σφ (g) on G by  σφ (g) : =

a

π (g)−1 φ(, ·), a dμ().

Let  ∈ H(h, f, δ)∞ ⊂ L 2 (G), and take F ∈ Cc∞ (G) arbitrarily. Then by some calculation similar to (2.5) in Sect. 2, Penney’s Plancherel theorem concerning the decomposition of the Dirac measure aρ for ρ reads that

F,  L 2 (G) = ρ(F)aρ ,  L 2 (G)   = F(g)

π (g)−1 φ(, ·), a dμ() dg = F, σφ  L 2 (G) . a

G

That is, σφ is necessarily square integrable on G. Let F be a smooth function on G. We say that F is of the Schwartz class, F ∈ S (G), if the function g = R × n (t, Z ) → F(exp(t X ) exp(Z )) is a Schwartz class function on g = Rn+1 . For F ∈ S (G), we define the partial n on n by Euclidean Fourier transform F n (t, ) : = F

 n

ei(Z ) F(exp(t X ) exp(Z )) d Z , (t, ) ∈ R × n∗ .

Let F ∈ S (G) and φ ∈ Cc∞ (a × R). For g = exp(x X ) exp Z , x ∈ R, Z ∈ n, we have  σφ (g) =

π (g)−1 φ(, ·), a dμ() a   = dμ() exp(−i( Ad exp(−t X )(Z )))φ(, x + t)a (t) dt, a

R

Semi-invariant Vectors Associated …

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σφ , F :=

G

=

σφ (g)F(g) dg   exp(−i( Ad exp(−t X )Z )φ(, x + t)a (t) dt dμ()

Rn+1 a R

  =



R a R

  =

R a R

  =



F(exp(x X ) exp(Z )) d xd Z n (x, − Ad∗ exp(t X )|n ) d xdμ()dt φ(, x + t)a (t) F 1



R a R

1

n (x, − Ad∗ exp(t X )|n ) d xdμ()dt φ(, x + t)a (t)ε(t) 2 ε(t)− 2 F 1

φ(, x + t)a (t)ε(t) 2 P F (x, t, ) d xdμ()dt,

(3.11)

where 1 t tr ad X n (x, − Ad∗ exp(t X )|n ), ε(t) := λ−1 , P F (x, t, ) = ε(t)− 2 F M e

(x, t, ) ∈ R × R × a . Now, let q∗

q∗

B ∗ : = {Y js ∗ , Yk , Wk ; 1 ≤ s ≤ d j , j ∈ Jr , 1 ≤ q ≤ dk , k ∈ Jc } be the dual basis for B of n∗ . We take the Euclidean measure dm on n∗ (m ∈ n∗ ) such that the unit cube spanned by B has volume 1. Let d  () be the Euclidean measure on + associated with the description =

 j∈Jr ,1≤s≤d j , ( j,s)=(M,d M )

∗

 js Y js ∗ + Y Md M +



js Y js + js W js ∈ + ,

j∈Jc ,1≤s≤d j

where  js , js , js ∈ R, such that the unit cube has volume 1. We identify R × + with n∗ by the map ∗

R × + (t, ) → m = (t − 1)Y Md M + |n ∈ n∗ with Euclidean measures dm = dtd  (). ˜ of μ, ˜ which is equivalent to the Lebesgue measure, Since μ is the image θ˜∗ (μ) we may assume dμ() = c()d  (), where c() is a positive, continuous function on + . Let Q : R × + → n∗ defined by Q(s, ) := Ad∗ exp(s X )|n , (s, ) ∈ R × + .

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Then the image of R × a by the map Q is an open set Q(R × a ) = a |n := { =  |n ;  ∈ a } in n∗ , and Q is a diffeomorphism with the absolute value of the Jacobian | det(dQ(x, ))| = λ M e−x tr ad X . n is contained in n of F Suppose that the support supp F R × (−a |n ) := {(x, u); x ∈ R, −u ∈ a |n }. Then 

P F 2L 2 (R×R×a )

= 

R×R×a



R×R×a



R×a |n

= = =

R×n∗

n (x, − Ad∗ exp(t X )|n )|2 d xdtd   ε(t)−1 | F n (x, −Q(t, ))|2 d xdtd   λ M e−t tr ad X | F

n (x, −q)|2 d xdq |F

n (x, q)|2 d xdq |F

= (2π )n F 2L 2 (R×n) . Thus, by (3.11) with dμ() = c()d  , we have      R



a

R

  1 φ(, x + t)a (t)ε(t) 2 P F (x, t, ) c()d xd  dt  = | σφ , F L 2 (G) |

≤ (2π )− 2 σφ

P F L 2 (R×R×a ) . n

(3.12)

n ⊂ R × (−a |n )} is On the other hand, the space {P = P F ; F ∈ S (G), supp F dense in L 2 (R × R × a ). In fact, for P ∈ Cc∞ (R × R × a ), let F0 be a function on R × n∗ defined by  F0 (x, ζ ) :=

1

ε(u) 2 P(x, Q−1 (−ζ )) (−ζ ∈ a |n = Q(R × ξa ), Q−1 (−ζ ) = (u, )) , 0 (−ζ ∈ / a |n )

(x, ζ ) ∈ R × n∗ .

Then we have supp F0 ⊂ R × (−a |n ) and F0 ∈ Cc∞ (R × n∗ ), and there exists n = F0 . Thus we have F = F(exp(x T ) exp(Z )) ∈ S (G) such that F n (x, − Ad∗ exp(t X )|n ) = ε(t)− 2 F0 (x, − Ad∗ exp(t X )|n ) P F (x, t, ) = ε(t)− 2 F 1

1

= ε(t)− 2 ε(t) 2 P(x, Q−1 ( Ad∗ exp(t X )|n )) = P(x, t, ). 1

1

Semi-invariant Vectors Associated …

99

Thus, (3.12) implies that   ∞>

  2 1   φ(, x + t)a (t)ε(t) 2 c() d xd  dt R a R     2 1   = φ(, x)a (t)ε(t) 2 c() d xd  dt, R

a

R

and for d  -almost all , the function φ(, x)a (t)ε(t) 2 is square integrable on R × R. 1

−1

1

For  ∈ a , the function a (t)ε(t) 2 = a,γ + 21 tr ad X (t)λ M 2 is square integrable if and only if  ∈ ρ , where ⎧ ⎪  ⎪ ⎪ a ⎪ ⎪ ⎪ ⎨a ρ := ∅ ⎪ ⎪ ⎪ a ⎪ ⎪ ⎪ ⎩ { ∈ a ; δ(X ) > 01 − f (Y ) + 21 tr ad X }

if λm < 0 if λm > 0 and δ(X ) > − f (Y ) + 21 tr ad X

if λm > 0 and δ(X ) ≤ − f (Y ) + 21 tr ad X if λm = λ0 = 0 and d0 ≥ 2 if λm = λ0 = 0 and d0 = 1,

(3.13) which is an open subset of g∗ . Thus, the support of φ(, x) is contained in the closure of ( ∩ ρ ) × R, which implies that the image of the intertwining operator R is contained in L 2 (( ∩ ρ ) × R). Theorem 2 Let g = R  n be a Lie algebra defined by Sect. 3.1 with assumptions (P1), (P2) and λ M > 0, and G = exp g. Let U = X + iY and h = CU . Take f ∈ g∗ arbitrarily, and δ := δ(X )X ∗ , and define the holomorphically induced representation ρ(h, f, δ) of G. Let , , a and ρ be defined by (3.8), (3.9), (3.10) and (3.13), respectively, and let ρ =  ∩ ρ . Then we have the following: 1. For  ∈ a , we have  h, f,δ dim Hπ−∞ = 1.  Furthermore, we have that semi-invariant generalized vectors a are elements of h, f,δ  Hπ , that is, Hπ−∞ ⊂ Hπ for all  ∈ a .  2. For  ∈  \ a , we have  −∞ h, f,δ Hπ = {0}. 3. If ρ = ∅, then the space H(h, f, δ) = {0}. The representation ρ = ρ(h, f, δ) decomposes into a multiplicity-free direct integral over ρ :

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 ρ=

⊕ ρ

π d,

where d is a Lebesgue measure on the open subset ρ of . 4. If ρ = ∅, then the space H(h, f, δ) = {0}. Proof We have obtained that if H(h, f, δ) = {0}, then ρ is a subrepresentation of a multiplicity-free direct integral over ρ × R. If ρ = ∅, then we can construct an intertwining operator from L 2 (ρ × R) to H(h, f, δ) as follows: For  ∈ ρ , let a be defined by (3.1), and  J () :=

R

|a (t)|2 et tr ad X dt.

For φ = φ(, x) ∈ Cc∞ (ρ × R) we define  σφ (g) : =

ρ

1

2

π (g)−1 φ(, ·), a J ()− 2 λ M d  (), g ∈ G. 1

Then for F ∈ S (G), by some similar computations to those in (3.11), we have       t 1   tr ad X −  ¯ φ(, x + t)a (t)e 2 J () 2 P F (x, t, ) d xd ()dt  | σφ , F L 2 (G) | =   R ρ R  n

≤ (2π ) 2 φ L 2 (ρ ×R) F L 2 (G) . Thus, σφ (g) is square integrable on G and σφ ∈ H(h, f, δ), and the mapping φ → σφ extends to a bounded intertwining operator on L 2 (ρ × R) to H( f, h, δ).   Remark 2 1. In the following cases, we have that a = ρ , and the reciprocity holds: (case 1) (case 2) (case 3) (case 4)

λm < 0. λm > 0 and δ(X ) > − f (Y ) + 21 tr ad X . λm = 0 and d0 ≥ 2. λm > 0 and δ(X ) ≤ − f (Y ). In this case, a = ρ = ∅, and H(h, f, δ) = {0}.

In our previous work [9, Theorem 2.4], we obtained the result concerning the case 1 and the case 2. 2. Suppose λm > 0 and − f (Y ) < δ(X ) ≤ − f (Y ) + 21 tr ad X . Then ρ = ∅, thus h, f,δ  = 1 for  ∈ a . H(h, f, δ) = {0}, but dim Hπ−∞  3. Suppose λm = 0 and d0 = 1. Then representations π corresponding to  satisfying 1  ∈ a , f (Y ) + δ(X ) − tr ad X < (Y01 ) < f (Y ) + δ(X ) 2

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 h, f,δ do not appear in the decomposition of ρ but dim Hπ−∞ = 1.  Acknowledgements The author would like to thank the referee for many valuable comments to improve the final form of the paper.

References 1. Y. Benoist, Multiplicité un pour les espaces symétriques exponentiels, Mém. Soc. Math. France (N.S.) 15, 1–37 (1984) 2. P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Rais, P. Renouard, M. Vergne, Représentations des groupes de Lie résolubles, Monographies de la Société Mathématique de France, No. 4. Dunod, Paris (1972) 3. H. Fujiwara, Représentations monomiales des groupes de Lie résolubles exponentiels, in The orbit method in representation theory (Birkhäuser, 1990), pp. 61–84 4. H. Fujiwara, Une réciprocité de Frobenius, in Infinite Dimensional Harmonic Analysis III (World Scientific Publishing, Hackensack, 2005), pp. 17–35 5. H. Fujiwara, J. Ludwig, Harmonic Analysis on Exponential Solvable Lie Groups, Springer Monographs in Mathematics (Springer, Tokyo, 2015) 6. H. Fujiwara, S. Yamagami, Certaines représentations monomiales d’un groupe de Lie résoluble exponentiel, in Representations of Lie Groups, Kyoto, Hiroshima, 1986. Adv. Stud. Pure Math., vol. 14 (Academic Press, Boston), pp. 153–190 7. J. Inoue, Semi-invariant vectors associated to decompositions of monomial representations of exponential Lie groups. J. Math. Soc. Japan 49(4), 647–661 (1997) 8. J. Inoue, Holomorphically induced representations of some solvable Lie groups. J. Funct. Anal. 186(2), 269–328 (2001) 9. J. Inoue, Holomorphically induced representations of exponential solvable semi-direct product groups R  Rn . Adv. Pure Appl. Math. 6(2), 113–123 (2015) 10. J. Inoue, An example of holomorphically induced representations of exponential solvable Lie groups, in Geometric and Harmonic Analysis on Homogeneous Spaces. Springer Proc. Math. Stat., vol. 290 (Springer, Cham, 2019), pp. 111–120 11. B. Magneron, Représentations induites holomorphes des groupes de Lie nilpotents et involutions complexes. C. R. Acad. Sci. Paris, Ser. I Math. 317, 37–42 (1993) 12. B. Magneron, Involutions complexes et vecteurs sphériques associés pour les groupes de Lie nilpotents réels. Astérisque 253 (1999) 13. R. Penney, Abstract Plancherel theorems and a Frobenius reciprocity theorem. J. Funct. Anal. 18, 177–190 (1975) 14. R. Penney, Holomorphically induced representations of exponential Lie groups. J. Funct. Anal. 64, 1–18 (1985)

Singular Integral Operators of Convolution Type on Jacobi Hypergroup Takeshi Kawazoe

To the Memory of Takaaki Nomura

Abstract We shall define a Calderón-Zygmund class CZ p (α,β ) on the Jacobi hypergroup (R+ , α,β , ∗) such that, if a function g on R+ belongs to CZ p (α,β ), then the convolution operator g∗ is bounded from L q (α,β ) to itself for p ≤ q ≤ 2. Actually, we shall obtain a relation between the L p norms of g and the Abel transform Ag and a transference principle between the L p operator norms of g∗ and 2 the Euclidean operator φ, where φ(x) = e( p −1)ρx Ag(x). Therefore, to define the p Calderón-Zygmund class CZ (α,β ), we shall obtain some conditions on g under which φ belongs to CZ(R). Then, φ is bounded on L q (R) and, by the transference principle, g∗ is bounded on L q (α,β ). Keywords Singular integrals · Jacobi hypergroup · Calderón-Zygmund operators

1 Introduction Let φ be a locally integrable function on R − {0} and Tφ = φ the singular integral operator of convolution type with the kernel φ:  Tφ ( f )(x) = φ  f (x) = lim

→0 |x−y|>

φ(x − y) f (y)dy.

(1)

T. Kawazoe (B) Department of Mathematics, Keio University at SFC, Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_7

103

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T. Kawazoe

We say that φ belongs to the Calderón-Zygmund class CZ(R) if φ satisfies the following conditions:  ∈ L ∞ (R), (a) φ (b) there exists a constant c0 such that for all y = 0,  |φ(x − y) − φ(x)|d x ≤ c0 , |x|≥2|y|

 is the Fourier transform of φ. Then, it is known that Tφ exists in L p (R), where φ 1 < p < ∞, and it is bounded from L p (R) to itself. Moreover, Tφ satisfies the weak type L 1 estimate (see Theorem 5.4 in [11], p. 289, Theorem 3 in [10], p. 19 and its proof). The integral in (1) is ensured by the condition (a), which follows from the following additional conditions (see [11], p. 288, Lemma 5.3); (c) there exists a constant c1 such that for all 0 <  < N ,     φ(x)d x  ≤ c1 ,  ≤|x|≤N

(d) there exists a constant c2 such that for all R > 0,  |x||φ(x)|d x ≤ c2 R. |x|≤R

The aim of this paper is to generalize this result to the convolution operators on the Jacobi hypergroup (R+ , , ∗) with the weight function  = α,β and the convolution ∗ for α ≥ β ≥ − 21 . We shall obtain a class CZ p () on R+ such that, if a function g on R+ belongs to CZ p (), then the Jacobi convolution operator Tg = g∗ is bounded on L p (), the  weighted L p space on R+ . Ionescu [4] already treated singular integrals of convolution type on the symmetric spaces G/K of real rank one. Hence, if we apply his results to K -invariant functions on G/K , we can obtain some results corresponding to the group cases, that is, the cases that α, β are suitable integers or half integers based on the root structure of G. The key in his paper is a transference 2 principle between the L p operator norms of Tg and Tφ , where φ(x) = e p ρx g(x), x ≥ 0, in our setting. On the other hand, in this paper, we shall obtain a transference 2 principle between the L p operator norms of Tg and Tφ , where φ(x) = e( p −1)ρx Ag(x) and A is the Abel transform on R+ . Since the Abel transform changes the Jacobi convolution ∗ to the Euclidean convolution  such as A( f ∗ g) = A f  Ag, our transference principle is natural and our approach is pretty straightforward. As mentioned above we focus on a transference principle of convolution operators. Some important singular integrals are not of Jacobi convolution type and excluded from our transference principle. For example, the Hilbert transform defined by Ben Salem and Samaali [1], which is bounded on L p () for 1 < p < ∞, is not of Jacobi convolution type. Therefore, we expect wider and more extensive transference principle in future.

Singular Integral Operators of Convolution Type on Jacobi Hypergroup

105

This paper is organized as follows. In Sect. 2, we recall some basic properties of the Jacobi hypergroup. In Sect. 3, applying the inversion formula of the Abel transform F = A f in [5], we shall obtain a relation between the L p norms of a function f on R+ and F = A f (see Theorem 3.5). A noting point is that, when α + 21 < 1, this equivalence relation does not hold, due to the shape of the HarishChandra’s C-function. Hence, in the following arguments, we have to treat the case of α + 21 < 1 separately. In Sect. 4, we shall obtain a transference principle between the L p operator norms of Tg and Tφ (see Theorem 4.1). Especially, if α + 21 ≥ 1 and φ belongs to CZ(R), then Tg is bounded from L p () to itself (see Corollary 4.2). In Sect. 5, we shall define CZ p () on the Jacobi hypergroup (see Definition 5.1) such g of g is bounded on the tube that, if g belongs to CZ p (), then the Jacobi transform  ( 2p −1)ρx 2 domain | (λ)| ≤ ( p − 1)ρ and φ(x) = e Ag(x) belongs to CZ(R). Then, by the transference theorem, Tg is bounded from L q () to itself for p ≤ q ≤ 2 (see Theorem 5.3). In Sect. 6, we shall consider some additional conditions on CZ p (), which correspond to (c) and (d) in the Euclidean case (see Proposition 6.2).

2 Notations We recall some basic properties of the Jacobi hypergroup (R+ , , ∗). We refer to [3] and [7] for the details. Let α, β ∈ R and α ≥ β ≥ − 21 , (α, β) = (− 21 , − 21 ). We denote (x) = α,β (x) = 22(α + β+1) (shx)2α+1 (chx)2β+1 and put ρ = α + β + 1. Then (x) ∼ (thx)2α+1 e2ρx . Let w(x) be a positive measurable function on R+ . We denote by L p (R+ , w) the space of measurable functions f on R+ with finite L p norm  f  L p (R+ ,w) with respect to w(x)d x. When w = , L p (R+ , ) is abbreviated by L p (). For λ ∈ C, the solutions of the differential equation (x)−1

du  d  (x) = −(λ2 + ρ2 )u(x) dx dx

with u(0) = 1 and u (0) = 0 is given by the Jacobi functions of the first kind with order (α, β): (α,β)

φλ (x) = φλ

 (x) = 2 F1

1 1 (ρ + iλ), (ρ − iλ); α + 1; −(shx)2 , 2 2

where 2 F1 denotes the Gauss hypergeometric function. For f ∈ L 1 (), the Jacobi transform  f (λ), λ ∈ R, is defined by

106

T. Kawazoe 1

 f (λ) =

22 (α + 1)



∞

f (x)φλ (x)(x)d x.

(2)

0

The Paley-Wiener theorem asserts that the map f →  f is a bijection from C0∞ (R), the space of infinitely differentiable even functions on R with compact support, onto the space of entire holomorphic even functions of exponential type, and the inverse transform is given by 1

f (x) =

22 (α + 1)



∞

 f (λ)φλ (x)|C(λ)|−2 dλ,

0

where C(λ) = Cα,β (λ) is Harish-Chandra’s C-function. The Plancherel theorem asserts that the map f →  f extends to an isometry from L 2 () onto L 2 (R+ , |C(λ)|−2 dλ): 

∞



∞

| f (x)| (x)d x = 2

0

| f (λ)|2 |C(λ)|−2 dλ.

(3)

0

For suitable functions f, g on R+ , the convolution product f ∗ g is defined by 1

f ∗ g(x) =

22 (α + 1)



∞

f (y)Tx g(y)(y)dy,

0

where Tx is the generalized translation operator defined by 1

Tx f (y) =

22 (α + 1)



∞

 f (λ)φλ (x)φλ (y)|C(λ)|−2 dλ.

0

Especially, f ∗ g is well-defined for f, g ∈ L 1 () and (L 1 (), ∗) is a commutative    Banach algebra. Moreover, since (T x f )(λ) = f (λ)φλ (x), it follows that f ∗ g = 1  f · g . As a function of λ, φλ (s) is the Fourier Cosine transform of an L function A(·, s) on R+ , which is supported on [0, s]: 1 1 (s)φλ (s) = √ (α + 1) π



s

cos λx A(x, s)d x.

(4)

0

We recall from [7], (2.19) that A(x, s) is a constant multiple of 1

1

sh2s · (chs)β− 2 (chs − chx)α− 2 2 F1



1 2

+ β, 21 − β; α + 21 ;

chs − chx  . 2chs

Since chs − chx ∼ es th(s 2 − x 2 ) for s > x > 0, we see the following estimate. Lemma 1 Let 0 < x < s. Then

(5)

Singular Integral Operators of Convolution Type on Jacobi Hypergroup

107

A(x, s) ∼ eρs ths(th(s 2 − x 2 ))α− 2 . 1

The Abel transform A f on R+ of f is defined by 

∞

A f (x) =

f (y)A(x, y)dy.

x

We extend A f on R+ as an even function on R. It follows from (2) and (4) that

f (λ),  f (λ) = A

(6)

where ∼ is the classical Fourier transform. Since ( f ∗ g)(λ) =  f (λ) g (λ), it follows that 1 A( f ∗ g) = √ A f  Ag, 2π

(7)

where  denotes the Euclidean convolution. 3 The inversion formula of the Abel transform of F = A f is given as f = 2−3α− 2 2 1 σ W−(β+ 1 ) ◦ W−(α−β) (F), where W−μ is the Weyl type fractional differential operator 2 σ on the Jacobi hypergroup (see [7], Corollary 3.3). We can replace W−μ with the R classical Weyl type fractional differential operator W−μ on R+ and obtain a version of this formula. Actually, by letting ν = β + 21 and ν = α − β in [5], Theorem 3.6 or by replacing F with eρx F in [5], Corollary 3.7, we can deduce the following formula. Let β + 21 = n + μ and α − β = n + μ , where n, n ∈ Z and 0 ≤ μ, μ < 1, and let 0 = {k + μ + μ |k ∈ Z, 1n+n ≤ k ≤ n + n }, 1 = {k, k + μ, k + μ |k ∈ Z, 1n+n ≤ k ≤ n + n }, where 1n = 1 if n ≥ 1 and 1n = 0 if n = 0. Then, there exist constants cγ , γ ∈ 0 , and functions Aγ (x, s), γ ∈ 1 , such that for x > 0, f (x) ∼

eρx   R cγ (thx)γ W−γ (F)(x) (x) γ∈ 0  ∞   γ R (thx) W−γ (F)(s)Aγ (x, s)ds , + γ∈1

(8)

x

where Aγ (x, s) is of the form Aγ (x, s) = Q γ (x, s)Z γ (s − x) and Aγ , Q γ , Z γ satisfy the following estimates: For γ ∈ 1 , there exists 0 < ξγ < 1 such that

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T. Kawazoe

(i) |Z γ (u)| ≤ c(thu)u −(1+ξγ ) for u > 0, (thx)ξγ for s > x, (ii) |Q σγ (x, s)| ≤ c (ths)  s (iii) |Aγ (x, s)|d x ≤ c for all s > 0,  0∞ (iv) |Aγ (x, s)|ds ≤ c for all x > 0. x R , γ ∈ 0 ∪ 1 , In what follows we denote by γα the maximal order of differentials W−γ in (8), that is, 1 γα = α + . 2

3 L p Norms of the Abel Transform We shall obtain a relation between the L p norms of a function f on the hypergroup and its Abel transform A f on R+ . According to (8), we can characterize the L 1 norm of f in terms of the Abel transform A f (see [5], Theorem 3.6 and [6], Proposition 4):  f  L 1 () ∼



R W−γ A( f ) L 1 (R+ ,(thx)γ eρx ) .

(9)

γ∈0 ∪1

How about the case of p > 1. First we note the following lemma. R Lemma 3.1 Let 1 ≤ p < 2 and 0 ≤ γ < γα . For a function F on R+ , if W−γ (F) α R belongs to L p (R+ , (thx)(2− p)γα e(2− p)ρx ), then W−γ (F) belongs to L p (R+ , (thx)(2− p)γ e(2− p)ρx ) and R R (F) L p ((R+ ,(thx)(2− p)γ e(2− p)ρx ) ≤ cW−γ (F) L p ((R+ ,(thx)(2− p)γα e(2− p)ρx ) . W−γ α R (F) Proof We suppose that 0 ≤ γ1 < γ2 ≤ γα , γ2 = γ1 + μ for 0 < μ < 1, and W−γ 2 R belongs to L p ((R+ , (thx)(2− p)γ2 e(2− p)ρx ). We put F0 = W−γ (F) and we shall obtain 2 R R a relation between L p norms of W−γ (F) = W (F ) and F 0 0 . Since 1 ≤ p < 2, we μ 1 can choose ν > 0 such that (2 − p)ρ − ν > 0. Then, Wμ (F0 )(x) is dominated by

 ∞ x

γ (2− p) p2 ((2− p)ρ−ν) sp

|F0 (s)|(s − x)μ−1 ds ≤ cF0 (s)(ths)

e

γ −(2− p) p2 −((2− p)ρ−ν) sp

× (ths) ≤cF0

(μ−1) 1p

(s − x)

γ (2− p) p2 ((2− p)ρ−ν) sp (s)(ths) e

(s −

(μ−1) 1p x)

e

L p x

(μ−1) q1

(s − x)

γ −(2− p) p2 −((2− p)ρ−ν) xp  L p (thx) e x

,

L q x

Singular Integral Operators of Convolution Type on Jacobi Hypergroup

where

1 p

+

1 q

= 1 and h L xp = γ2





∞

|h(s)| p ds

1p

109

. Here we used that

x

(ths)−(2− p) p e−((2− p)ρ−ν) p (s − x)(μ−1) q  L q x  ∞ γ q −(2− p) 2p −((2− p)ρ−ν) sqp (μ−1) = (ths) e (s − x) ds x  ∞ γ2 q sq e−((2− p)ρ−ν) p (s − x)(μ−1) ds ≤(thx)−(2− p) p x  ∞ γ2 q sq −(2− p) p −((2− p)ρ−ν) xqp e e−((2− p)ρ−ν) p s (μ−1) ds. =(thx) s

1

q

0

Hence, we see that  ∞ R |W−γ (F)(x)| p (thx)(2− p)γ1 e(2− p)ρx d x 1 0  ∞ ∞  ≤c |F0 (s)| p (ths)(2− p)γ2 e((2− p)ρ−ν)s (s − x)μ−1 ds (thx)−(2− p)μ eνx d x 0 ∞ x  s  p (2− p)γ2 ((2− p)ρ−ν)s |F0 (s)| (ths) e (s − x)μ−1 (thx)−(2− p)μ eνx d x ds =c 0 0 ∞ p (2− p)γ2 (2− p)ρs ≤c |F0 (s)| (ths) ·e (ths)( p−1)μ ds 0 ∞ R ≤c |W−γ (F)(s)| p (ths)(2− p)γ2 e(2− p)ρs ds. 2 0

We take a sequence γ = ξ0 < ξ1 < ξ2 < · · · < ξn = γα such that ξi+1 − ξi < 1. Then, applying the above inequality first for γ1 = ξn−1 < γ2 = ξn = γα and repeating sequentially, we can deduce the desired one.  Proposition 3.2 Let F = A( f ) for f ∈ L p (). (i) If γα ≥ 1 and 1 ≤ p ≤ 2, then R (F) L p (R+ ,(thx)(2− p)γα e(2− p)ρx ) ≤ c f  L p () . W−γ α

(10)

(ii) If γα < 1 and 1 ≤ p < 2 satisfies (2 − p)γα > p − 1 and 2γα − (β + 21 ) p > 3( p − 1), then R (F) L p (R+ ,(thx)(2− p)γα e(2− p)ρx ) ≤ c f  L p ((x)(thx)3(1− p) ) . W−γ α

Proof (i) When p = 1, (10) follows from [6], Proposition 4 as R (F) L 1 (R+ ,(thx)γα eρx ) ≤ c f  L 1 (R+ ,(thx)2γα e2ρx ) ≤ c f  L 1 () . W−γ α

(11)

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T. Kawazoe

When p = 2, (10) follows from the Plancherel formula. Actually, we recall that R (F)(λ) = |λ|γα dγα (λ) |C(λ)|−1 ∼ |thλ|(1 + |λ|)γ0 (cf. [2], Theorem 2 (iii)) and W−γ α γ π γ π  F(λ), where dγα (λ) = cos α2 − isgn(λ) sin α2 (cf. [8], Lemma 3.4). Since γα ≥ 1 R   and |C(λ)| (F)(λ)| ≤ c|C(λ)|−1 | F(λ)|. Since F and |dγα (λ)| ≤ 1, we see that |W−γ α are even, the Euclidean Plancherel formula, (3) and (6) yield that R R (F)2L 2 (R+ ) ≤ W−γ (F)2L 2 (R) W−γ α α  ∞ 2  ≤c | F(λ)| |C(λ)|−2 dλ −∞  ∞ | f (λ)|2 |C(λ)|−2 dλ ∼  f 2L 2 () . ≤c 0

Hence, by interpolation, (10) holds for 1 < p < 2. (ii) When p = 1, (11) follows as in (i). We apply the argument used in the proof of [6], Proposition 4. We suppose that 1 < p < 2. First prove the following lemma. Lemma 3.3 Let 1 < p < 2 and 0 < γ < 1. (i) Let δ > 0 and  satisfy  > p − 1. Then R ◦ Wγσ ( f ) L p (R+ ,(thx) eδx ) ≤ c f  L p (R+ ,(thx)+γ p− p+1 e(σγ p+δ)x ) . W−γ

(ii) Let −σγ p + δ > 0 and  satisfy  > (γ + 1) p − 1. Then σ ◦ WγR ( f ) L p (R+ ,(thx) eδx ) ≤ c f  L p (R+ ,(thx)−γ p− p+1 e(−σγ p+δ)x ) . W−γ

Proof (i) According to the proof of [6], Lemma 3.2 (i), R (Wγσ ( f ))(x) =(σshσx)γ f (x) + W−γ



∞ x

σ γ,2 f (t)(W Bγ )(x, t)dt

σ γ,2 and |(W Bγ )(x, t)| ≤ c(tht)γ (thx)−1 eσγt . Therefore, it follows that



∞

0 ≤c 0

R |W−γ (Wγσ ( f ))(x)| p (thx) eδx d x

∞

| f (x)| p (thx)+γ p e(σγ p+δ)x d x (12)  ∞ ∞ p +c | f (t)|(tht)γ eγσt dt (thx)− p eδx d x. 0

x

Under the assumptions on δ, , we can choose a sufficiently small ν > 0 such that ν − δp < 0 and μ > 0 such that  > pμ > p − 1. Since − p + μ < 0 and − p + μ p > −1, it follows that

Singular Integral Operators of Convolution Type on Jacobi Hypergroup



∞

111

| f (t)|(tht)γ eγσt dt

x 

δ





δ





δ



δ

≤c f (t)(tht)γ+ p −μ e(γσ+ p −ν)t  L xp (tht)− p +μ e(ν− p )t  L qx δ

≤c f (t)(tht)γ+ p −μ e(γσ+ p −ν)t  L xp (thx)− p +μ e(ν− p )t  L qx δ

≤c f (t)(tht)γ+ p −μ e(γσ+ p −ν)t  L xp (thx)− p +μ e(ν− p )x and therefore, the second term in (12) is dominated as 



 | f (t)| p (tht)γ p+−μ p e(γσ p+δ−ν p)t dt (thx)− p+μ p eν px d x 0 x  ∞  t  p γ p+−μ p (γσ p+δ−ν p)t = | f (t)| (tht) e (thx)− p+μ p eν px d x dt 0 0 ∞ p γ p+− p+1 (γσ p+δ)t ≤c | f (t)| (tht) e dt ∞

∞

0

(ii) It follows from the proof of [6], Lemma 3.2 (ii) that σ (WγR ( W−γ

f ))(x) = (σshσx)

−γ



∞

f (x) + x

R γ,2 f (s)(W Cγ )(x, t)ds

R γ,2 Cγ )(x, t)| ≤ c(thx)−1−γ e−σγx . Therefore, since  − (γ + 1) p > −1 and and |(W −σγ p + δ > 0, it follows that



∞

 0

≤c 0

σ |W−γ (WγR ( f ))(x)| p (thx) eδx d x

∞

| f (x)|(thx)−γ p e(−σγ p+δ)x d x (13)  ∞ ∞ p +c | f (t)|dt (thx)−γ p− p e(−σγ p+δ)x d x. 0

x

Under the assumptions on δ, , we can choose a sufficiently small ν > 0 such that σγ − δp + ν < 0 and μ > 0 such that  − γ p > pμ > p − 1. Since − p + γ + μ < 0 and μ p − p > −1, it follows that 

∞ x



δ



δ



δ

| f (t)|dt =c f (t)(tht) p −γ−μ e(−σγ+ p −ν)t  L xp (tht)− p +γ+μ e(σγ− p +ν)t  L qx 

δ

=c f (t)(tht) p −γ−μ e(−σγ+ p −ν)t  L xp (thx)− p +γ+μ e(σγ− p +ν)x , and therefore, the second term in (13) is dominated as

112

T. Kawazoe



∞

0 =c

| f (t)| p (tht)−γ p−μ p e(−σγ p+δ−ν p)t

∞



t

 (thx)μ p− p eν px d x dt

0

| f (t)| (tht) p

−γ p− p+1 (−σγ p+δ)t

e

.

0

 Now we recall that R R R 1 2 W−(α+ ◦ W+ ( f ) = W−(α+ (f) 1 (F) =W 1 ◦ Wα−β ◦ W −(α+ 1 ) β+ 1 ) ) 2

2

2

2

R 1 1 R R 2 ) ◦ (W−(β+ ) ◦ (W−(β+ )( f ) =(W−(α+ 1 W 1 W 1 W ) α+ 1 ) β+ 1 ) β+ 1 2

2

2

2

2

2

=A ◦ B ◦ C( f ). Since the assumption on p implies that (2 − p)γα > p − 1, 2γα + (1 − p) > (β + 3 ) p − 1 and 2γα − (β + 21 ) p + 2(1 − p) > p − 1, we can apply Lemma 3.3 repeat2 edly and deduce that R (F) L p ((thx)(2− p)γα e(2− p)ρx ) W−γ α

=A ◦ B ◦ C( f ) L p ((thx)(2− p)γα e(2− p)ρx ) ≤cB ◦ C( f ) L p ((thx)(2− p)γα +γα p+(1− p) e((2− p)ρ+γα )x ) ≤cC( f ) L p ((thx)2γα −(β+ 21 ) p+2(1− p) e((2− p)ρ+ pγα −(β+ 21 ) px ) ≤c f  L p ((thx)2γα +3(1− p) e2ρx ) = c f  L p ((x)(thx)3(1− p) ) . This completes the proof of Proposition 3.2.



Remark 3.4 When γα < 1, if p > 1 is sufficiently close to 1, then the conditions (2 − p)γα > p − 1 and 2γα − (β + 21 ) p > 3( p − 1) hold and thus, (11) holds. In what follows we say that p is sufficiently close to 1 if p > 1 satisfies these conditions. Theorem 3.5 Let 1 ≤ p < 2 and γα ≥ 1. Then, there exist positive constants c1 , c2 such that for all F = A( f ), f ∈ L p (), R F L p (R+ ,(thx)(2− p)γα e(2− p)ρx ) ≤ c2  f  L p () . c1  f  L p () ≤ W−γ α

(14)

If γα < 1 and p is sufficiently close to 1, then (14) holds provided that the right-hand side is replaced by c2  f  L p (R+ ,(x)(thx)3(1− p) ) . Proof The second inequality was obtained in Proposition 3.2. We shall prove the first R one. It follows from (9) and Lemma 3.1 that  f  L 1 () ≤ W−γ F L 1 (R+ ,(thx)γα eρx ) . α On the other hand, the Plancherel formula yields that R F L 2 (R+ ) )  f  L 2 () ≤ c(F L 2 (R+ ) + W−γ α

(see the proof of Proposition 3.2). Then, by using the J -method of the interpolation and Lemma 3.1, we can deduce that for 1 < p < 2,

Singular Integral Operators of Convolution Type on Jacobi Hypergroup

113

R  f  L p () ≤ c(F L p (R+ ,e(2− p)ρx ) + W−γ F L p (R+ ,(thx)(2− p)γα e(2− p)ρx ) ) α R ≤ cW−γ F L p (R+ ,(thx)(2− p)γα e(2− p)ρx ) . α



Hence, (14) follows from Lemma 3.3 and Proposition 3.2.

Proposition 3.6 Let 1 ≤ p < ∞ and w be an even positive function on R. For γ ≥ 0, there exists a positive constant cγ such that R R (F) L p (R,w) ≤ cγ W−γ (F) L p (R+ ,w) W−γ

for all even functions F on R. Proof We suppose that γ = n + μ, where n is an integer and 0 ≤ μ ≤ 1. We put R N (μ) = inf W−μ (F) L p (R+ ,w) , F

R where F moves even or odd functions on R with W−μ (F) L p (R,w) = 1. Then 1 N (0) = N (1) = 2 . We shall prove that there exists  > 0 such that N (μ) ≥  for 0 ≤ μ ≤ 1. Actually, if N (μ) = 0, then there exists an even or odd function F on R R R (F) L p (R+ ,w) = 0, that is, W−μ (F) is supported on (−∞, 0]. Then such that W−μ R R F = Wμ (W−μ (F)) is also supported on (−∞, 0] by the definition of the fractional integral operator. Hence F = 0, because F is even or odd. This is in contradiction R (F) L p (R,w) = 1.  with W−μ

Remark 3.7 Let 1 ≤ p < 2 and

1 2

−

< α. Let F = A( f ) for f ∈ L p (). Then

1 p

F L p (R+ ,e(2− p)ρx ) ≤ c f  L p () .

(15)

When γα ≥ 1 or p = 1, Lemma 3.1 and Theorem 3.5 yield the desired result. Therefore, we may suppose that 21 − 1p < α < 21 and 1 < p < 2. We choose a sufficiently small  > 0 such that 1 − 2p +  < 0. Then, it follows from Lemma 1 that 

∞

|F(x)| ≤ ≤ ≤

x ∞ x ∞

| f (s)|A(x, s)ds | f (s)|eρs (ths)(th(s 2 − x 2 ))α− 2 ds 1

| f (s)|e( p −)ρs (ths)α+ 2 e(1− p +)ρs (th(s − x))α− 2 ds 2

1

2

1

x

≤c f (s)e( p −)ρs (ths)(α+ 2 ) (th(s − x))(α− 2 )  L xp e(1− p +)ρs  L qx 2

1

2

1

≤ce(1− p +)ρx  f (s)e( p −)ρs (ths)(α+ 2 ) (th(s − x))(α− 2 )  L xp . 2

2

1

1

Therefore, since (α − 21 ) p > −1 and 2α p + 1 > p − 1 > 0, it follows that

114

T. Kawazoe



∞



∞

0

≤

0

|F(x)| p e(2− p)ρx d x | f (s)| p e(2− p)ρs (ths)(α+ 2 ) p 1

∞

≤c



s

 1 e pρx (th(s − x))(α− 2 ) p d x ds

0

| f (s)| p e2ρs (ths)2α p+1 ds ≤ c f  L p () .

0

4 A Transference Formula Let g be a locally integrable function on R+ and Tg = g∗ the corresponding Jacobi convolution operator defined similarly as (1). We shall obtain a relation between the L p operator norms of Tg = g∗ and the Euclidean convolution operator Tg, p = 2 e( p −1)ρx Ag. Theorem 4.1 (i) Let γα ≥ 1 and 1 ≤ p < 2. If Tg, p is bounded from L p (R, |thx|(2− p)γα ) to itself, then Tg is bounded from L p () to itself. (ii) Let γα < 1 and p be sufficiently close to 1. If Tg, p is bounded from L p (R, |thx|(2− p)γα ) to itself, then Tg is bounded from L p (R+ , (x)(thx)3(1− p) ) to L p (). Proof (i) When γα ≥ 1, it follows from (7), Theorem 3.5 and Proposition 3.6 that p

R (A(g)  A( f )) L p (R+ ,(thx)(2− p)γα e(2− p)ρx ) g ∗ f  L p () ≤cW−γ α R ≤c(e( p −1)ρ(·) Ag)  (e( p −1)ρ(·) W−γ (A f )) L p (R,|thx|(2− p)γα ) α 2

2

R ≤c(e( p −1)ρ(·) W−γ (A f ) L p (R,|thx|(2− p)γα ) α 2

R ≤cW−γ (A f ) L p (R,|thx|(2− p)γα e(2− p)ρ|x| ) α p

R ≤ccγα W−γ (A f ) L p (R+ ,(thx)(2− p)γα e(2− p)ρx ) ≤ c f  L p () . α

(ii) When γα < 1 and p is sufficiently close to 1, the right-hand side of the last inequality is replaced by c f  L p (R+ ,(x)(thx)3(1− p) ) by Theorem 3.5.  Next we suppose that e( p −1)ρx Ag(x) belongs to CZ(R) (see §1). Then Tg, p = 2 e( p −1)ρ(·) Ag is bounded from L q (R) to itself for 1 < q < ∞. Here, we note that the maximal function of |thx|(2−q)γα is dominated by the same function. Hence it follows from [9], Theorem 8 that Tg, p is bounded from L q (R, |thx|(2−q)γα ) to itself. 2

Corollary 4.2 Let 1 < p < 2 and p ≤ q ≤ 2. We suppose that e( p −1)ρx Ag(x) belong to CZ(R). (i) If γα ≥ 1, then Tg is bounded from L q () to itself. (ii) If γα < 1 and p > 1 is sufficiently close to 1, then Tg is bounded from L q ((x)(thx)3(1−q) ) to L q (). 2

Singular Integral Operators of Convolution Type on Jacobi Hypergroup

115

Proof The argument before Corollary 4.2 and Theorem 4.1 yield the case of q = 2 p. Since e( p −1)ρx Ag(x) belong to CZ(R),  g (λ + ( 2p − 1)ρi) (see (6)) is bounded. Especially,  g is bounded on R and thus, Tg is bounded on L 2 (). Hence, the desired results follow by interpolation. 

5 A CZ Class on the Hypergroup (R+ , , ∗) We shall obtain some conditions on even functions g on R under which e( p −1)ρx Ag(x) belongs to CZ(R) in Sect. 1. In what follows, we suppose that g is of the form 2

g(x) = e− p ρ|x| h(x), 2

where h is an even function on R. We define a Calderón-Zygmund class CZ p () on the Jacobi hypergroup (R+ , , ∗). Definition 5.1 Let 1 ≤ p < 2. We denote by CZ p () the set of even functions 2 g(x) = e− p ρ|x| h(x) on R satisfies the following conditions. (A)  g (λ) has a holomorphic extension on the tube domain | λ| < ( 2p − 1)ρ and has a bounded extension on the boundary. (B) There exists a constant c2 such that for all y = 0,  |s|≥2|y|

|h(s − y) − h(s)||ths|2α+1 ds ≤ c2 .

(C) There exists a constant c3 such that |h(x)| ≤ c3 |thx|−2(α+1) . We note that for 1 ≤ p < q < 2, g(x) = e− p ρ|x| h(x) = e− q ρ|x| e( q − p )ρ|x| h(x) = e− q ρ|x| h (x) 2

2

2

2

2

and h (x) = O(|thx|−2(α+1) e( q − p )ρ|x| ). We can easily deduce that 2

2

CZ p () ⊂ CZq ().

Theorem 5.2 Let 1 ≤ p < 2. If an even function g on R belongs to CZ p (), then 2 e( p −1)ρx A(g)(x) belongs to CZ(R). Proof We shall prove that e( p −1)ρx A(g)(x) satisfies the conditions (a) and (b) of 2 CZ(R) in §1. (a) follows from (A), because the Fourier transform of e( p −1)ρx A(g)(x) 2

116

T. Kawazoe

is given by the Jacobi transform  g (λ + ( 2p − 1)ρi) (see (6)). As for (b), first we note that, if 0 < 2y ≤ x, then e( p −1)ρ(x−y) A(g)(x − y) − e( p −1)ρx A(g)(x)  ∞  ∞ 2 2 =e( p −1)ρ(x−y) g(s)A(x − y, s)ds − e( p −1)ρx g(s)A(x, s)ds x−y x  ∞  ∞ 2 2 g(s − y)A(x − y, s − y)ds − e( p −1)ρx g(s)A(x, s)ds =e( p −1)ρ(x−y) x x  ∞  ∞ h(s − y)B(x − y, s − y)ds − h(s)B(x, s)ds = x x ∞ (h(s − y) − h(s))B(x − y, s − y)ds = x  ∞ h(s)(B(x − y, s − y) − B(x, s))ds + 2

2

x

=I1 (x, y) + I2 (x, y), where B(x, s) = e( p −1)ρx e− p ρs A(x, s). Since B(x, s) is positive and B(x, s) ≤ 2 1 ce( p −1)ρ(x−s) (ths)(th(s 2 − x 2 ))α− 2 by Lemma 1, it follows that for 2y ≤ x < s 2

2

B(x − y, s − y) ≤ ce−( p −1)ρ(s−x) (th(s − y))(th((s − x)(s + x − 2y)))α− 2 2

1

≤ ce−( p −1)ρ(s−x) (th(s − x))α− 2 (ths)α+ 2 . 2

1

1

Therefore, it follows from (B) that 

∞

 |I1 (x, y)|d x =

2y

∞

|h(s − y) − h(s)|



2y

 ≤c

s

 B(x − y, s − y)d x ds

2y ∞

|h(s − y) − h(s)||ths|2α+1 ds ≤ cc2 .

2y

To estimate I2 (x, y), first we note from (5) that B(x, s) = e( p −1)ρx e− p ρs A(x, s) 2

2

∼ ce−( p −1)ρ(s−x) · ths · (th(s − x))α− 2 · (th(s + x))α− 2   × 2 F1 21 + β, 21 − β; α + 21 ; 21 − 21 ch(s − x) + 21 sh(s − x) · ths . 2

1

1

Here, we used that chx = ch(s+(x−s)) = ch(s − x) − sh(s − x) · ths. Then, by using chs chs the Taylor expansion, we see that for each 2y ≤ x ≤ s, there exist 0 < s − y < ξ < s and 0 < x − y < ξ < x such that

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|B(x − y, s − y) − B(x, s)| ≤cye−( p −1)ρ(s−x) (th(s − x))α− 2  1 3 × (chξ)−2 · (th(ξ + ξ ))α− 2 + thξ · (th(ξ + ξ ))α− 2 (ch(ξ + ξ ))−2  1 + thξ · (th(ξ + ξ ))α− 2 sh(s − x)(chξ)−2 2

1

≤cye−( p −1)ρ(s−x) e−(s−y) (th(s − x))α− 2 (ths)α− 2 , 2

because

s 2

1

1

< s − y < ξ < s and s ≤ x + s − 2y < ξ + ξ < x + s ≤ 2s. Hence 

s

|B(x − y, s − y) − B(x, s)|d x ≤ cye−(s−y) (ths)2α .

2y

Then, it follows from (C) that 

∞

2y





 s |h(s)| |B(x − y, s − y) − B(x, s)|d x ds 2y 2y  ∞ (ths)−2 e−(s−y) ds ≤ cy(thy)−1 e−y ≤ c. =cc3 y ·

|I2 (x, y)|d x =

∞

2y

Similarly, we can treat the case of −x ≤ −2y < 0 and we can deduce (b).



Theorem 5.3 Let 1 < p < 2 and p ≤ q ≤ 2 or p = 1 and 1 < q ≤ 2. Let g belong to CZ p (). Then (i) If γα ≥ 1, then Tg is bounded from L q () to itself. (ii) If γα < 1 and p is sufficiently close to 1, then Tg is bounded from L q ((x) (thx)3(1−q) ) to L q (). Proof When 1 < p < 2, the desired results follow from Theorem 5.2 and Corollary 4.2. When p = 1, we note that CZ1 () ⊂ CZq () for 1 < q < 2. Therefore, the desired results follow similarly. 

6 Additional Conditions The case (ii) of Theorem 5.3 is improved by an additional condition. Theorem 6.1 Let γα < 1. Let 1 ≤ p < 2 be sufficiently close to 1 and p < q ≤ 2. If g belongs to CZ p () and satisfies (D) there exists a constant c4 such that  0

1

|h(x)||thx|2α+1 d x ≤ c4 ,

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then Tg is bounded from L q () to itself. Proof We recall that g(x) = e− p ρ|x| h(x) = e− q ρ|x| h (x) and h (x) = O(|thx|−2(α+1) e2( q − p )|x| ). 2

2

1

1

Let g0 = g · χ1 and g1 = g · χc1 , where χ1 and χc1 are the characteristic functions of |x| ≤ 1 and |x| ≥ 1, respectively. Clearly, g1 ∈ L q () and g0 ∈ L 1 () by (D). Let f be in L q (R) and f = f 0 + f 1 be the above decomposition. Then by Theorem 5.3, g ∗ f 1  L q () ≤  f 1  L q ((x)(thx)3(1−q) ) ≤ c f 1  L q () . As for f 0 , g1 ∗ f 0  L q () ≤ g1  L q ()  f 0  L 1 () ≤ c f 0  L q () and g0 ∗ f 0  L q () ≤ g0  L 1 ()  f 0  L q () ≤ c   f 0  L q () . Hence, Tg is bounded from L q () to itself. We shall obtain some conditions from which (A) follows. We note that 

∞

−∞

 ∞ 2  2 e( p −1)ρx e− p ρs h(s)A(|x|, s)ds d x −∞ |x|  ∞  s 2  2 = h(s) e( p −1)ρx e− p ρs A(|x|, s)d x ds 0 −s  ∞ = h(s)W (s)ds. (16)

e( p −1)ρx A(g)(x)d x = 2



∞

0

W (s) is a positive bounded function on R+ . Let g,N (x) = g(x)χ≤|x|≤N (x) for 0 <  < N . Then, by the Paley-Wiener theorem,  g,N satisfies the condition (A) in Definition 5.1. Hence, if | g,N (λ + i( 2p − 1)ρ)|, λ ∈ R, is bounded by a constant independent of , N , then (A) follows. Since g,N (λ + i( 2p − 1)ρ) coincides with the Fourier transform of e( p −1)ρx A(g,N )(x), if e( p −1)ρx A(g,N )(x) satisfies (b), (c), (d) in §1, then (A) follows. 2

2

Proposition 6.2 We suppose that an even function g(x) = e− p ρ|x| h(x) satisfies (B) and (C) of Definition 5.1 and the additional conditions; (A1) there exists a constant c5 such that for all 0 < N , 2

  

N

  h(x)W (x)d x  ≤ c5 ,

0

(A2) there exists a constant c6 such that for all R > 0,  |x|≤R

|x||h(x)||thx|2α+1 d x ≤ c6 R.

Then (A) follows. Proof We note that h ,N = hχ≤|x|≤N also satisfies (B), (C), (A1), (A2) (see [11], p. 287, Lemma 5.2). Hence, as in the proof of Theorem 5.2, (B), (C) yield that

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e( p −1)ρx A(g,N )(x) satisfies (b). Since A(g,N )(x) is supported on [0, N ], (16) for g = g,N and (A1) yield that there exists a constant c such that for all N , 2

  

|x| 0. Therefore, we see that there exists a constant c such that for all R > 0,  2 |x||e( p −1)ρx A(g,N )(x)|d x ≤ c R |x|≤R

and thus, e( p −1)ρx A(g,N )(x) satisfies (d). By using the argument in [11], p. 288, 2 Lemma 5.3, we can deduce that the Fourier transform of e( p −1)ρx A(g,N ) is bounded by a constant independent of , N . Hence (A) follows.  2

Proposition 6.3 Let α > 21 . If the condition (C) is replaced by (C ) |h(x)| ≤ c|x|−2 |thx|−2α , then the fact that e( p −1)ρx A(g)(x) satisfies (b) follows from (C ) without (B). 2

Proof Since   2 d ( 2p −1)ρx d  ( 2p −1)ρx ∞ e e A(g)(x) = h(s)e− p ρs A(|x|, s)ds dx dx |x|  ∞  ∞   2 2 2 d ( p−1)ρx − p ρs 2 A(|x|, s)ds ( p − 1)ρ h(s)e A(|x|, s)ds + h(s)e− p ρs =e dx |x| |x| d  1 3   and  A(|x|, s) ≤ ceρs (ths)α+ 2 (th(s − x))α− 2 , it easily follows that dx d 2  c   ( −1)ρx A(g)(x) ≤ .  e p dx |x|2 Then, by using the mean value theorem, it follows from (C ) that for each 0 < 2|y| ≤ |x|, there exists |x − y| < |ξ| < |x| such that |e( p −1)ρ(x−y) A(g)(x − y) − e( p −1)ρx A(g)(x)| d  2   c|y|   e( p −1)ρx A(g) (ξ) · |y| ≤ . = dx |x − y|2 2

2

Hence, it follows that  2 2 |e( p −1)ρ(x−y) A(g)(x − y) − e( p −1)ρx A(g)(x)|d x ≤ c. |x|≥2|y|

Therefore, (b) follows from (C ) without (B).



Acknowledgements The author would like to thank the anonymous referee who provided useful and detailed comments on a previous version of the manuscript.

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References 1. N. Ben Salem, T. Samaali, Hilbert transform and related topics associated with the differential Jacobi operator on (0, +∞). Positivity 15, 221–240 (2011) 2. M. Flensted-Jensen, Paley-Wiener type theorems for a differential operator connected with symmetric spaces. Ark. Mat. 10, 143–162 (1972) 3. M. Flensted-Jensen, T. Koornwinder, The convolution structure for Jacobi function expansions. Ark. Mat. 11, 245–262 (1973) 4. A.D. Ionescu, Singular integrals on symmetric spaces of real rank one. Duke Math. J. 114, 101–122 (2002) 5. T. Kawazoe, H 1 -estimates of the Littlewood-Paley and Lusin functions for Jacobi analysis. Anal. Theory Appl. 25, 201–229 (2009) 6. T. Kawazoe, Applications of an inverse Abel transform for Jacobi analysis: weak-L 1 estimates and the Kunze-Stein phenomenon. Tokyo J. Math. 41, 77–112 (2018) 7. T. Koornwinder, A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, 145–159 (1975) 8. Y. Luchko, H. Matrìnez, J. Trujillo, Fractional Fourier transform and some of its applications. Fract. Calc. Appl. Anal. 11, 457–470 (2008) 9. Y. Rakotondratsimba, A remark on Feffermann-Stein’s inequalities collect. Math. 49, 1–8 (1998) 10. E.M. Stein, Harmonic Analysis Real-variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, 1993) 11. A. Torchinsky, Real-variable Methods in Harmonic Analysis. Pure and Applied Mathematics, vol. 123 (Academic, Orlando, 1986)

The Compression Semigroup of the Dual Vinberg Cone Hideyuki Ishi and Khalid Koufany

Dedicated to the memory of Professor Takaaki Nomura.

Abstract We investigate the semigroup associated with the dual Vinberg cone and prove its triple and Ol’shanski˘ı polar decompositions. Moreover, we show that the semigroup does not have the contraction property with respect to the canonical Riemannian metric on the cone. Keywords Dual Vinberg cone · Compression semigroup · Triple and Ol’shanski˘ı polar decompositions 2000 Mathematics Subject Classification Primary 20M20 · Secondary 22E1

1 Introduction and Preliminaries Semigroups of transformations leaving invariant a given set is a well-known tool in various fields of mathematics, for example, invariant convex cone theory and geometric control theory. In Lie group setting, probably the most important compression semigroups come from the Ol’shanski˘ı semigroups, i.e., compression semigroups of symmetric spaces G C /G, where G is a Hermitian Lie group. One extremely useH. Ishi Osaka City University, 3-3-138 Sugimoto-cho, Sumiyoshi-ku, Osaka 558-8585, Japan e-mail: [email protected] K. Koufany (B) CNRS, IECL, University of Lorraine, F-54000 Nancy, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_8

123

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ful structure property of such semigroups is the existence and uniqueness of the Ol’shanski˘ı polar decomposition G exp(iC), where C is a convex cone in the Lie algebra of G which is invariant under the adjoint action of G. This decomposition has many applications to representations theory, see for example [4, 11, 12]. A compression semigroup associated naturally to an Euclidean Jordan algebra E was introduced in [6]. It is the compression semigroup of a symmetric cone C (the open cone of invertible squares in E),  := {g ∈ Co(E) | g(C) ⊂ C}, where Co(E) is the conformal group of E. This semigroup  satisfies the Ol’shanski˘ı polar decomposition and, in addition,  admits a triple decomposition. Furthermore, elements of  are proved to be contractions for the invariant Riemannian metric on C [6, 7] and also for the Hilbert metric [8] and the Finsler metric [10]. The contraction property has many applications, for example, in Kalman Filtering theory (for the Hamiltonian semigroup) [1]. The purpose of this article is to study the compression semigroup of a homogeneous non-symmetric convex cone, which gives a new example of Lie semigroups admitting both the Ol’shanski˘ı polar decomposition and a triple decomposition, but does not have the contraction property with respect to the canonical metric. More precisely, the homogeneous cone  is given by    := x ∈ R5 | x1 > 0, x2 > 0, x1 x2 x3 − x1 x52 − x2 x42 > 0 , which is called the dual Vinberg cone [13, 14]. Let us first summarize some well-known facts about the real symplectic group and the symplectic semigroup, which will be utilized frequently in the investigation of the cone . Let Sym(3, R) denote the space of 3 × 3 real symmetric matrices, and Sym+ (3, R) (resp. Sym++ (3, R)) the subset of positive (resp. positive definite) matrices. Then, Sym++ (3, R) is a symmetric cone in the Euclidean Jordan algebra Sym(3, R) with the inner product given by (x|y) = tr x y. Denote by 1 , 2 and 3 the principal minors of matrices in Sym(3, R). For a matrix M, denote by M T its transpose, and if M is invertible, M −T will denote (M T )−1 . GL(6, R) | g J g T = J } with J =  Recall  the symplectic group Sp(6, R) ={g ∈  0 −I A B . In a block form, an element g = ∈ GL(6, R) with A, B, C, D ∈ I 0 C D Mat(3, R) belongs to Sp(6, R) if and only if A T C , D T B ∈ Sym(3, R), D T A − B T C = I,

(1.1)

or equivalently B A T , C D T ∈ Sym(3, R), AD T − BC T = I.  Lemma 1.1 An element g =

(1.2)

 A B ∈ Sp(6, R) has a unique triple decomposition C D

The Compression Semigroup of the Dual Vinberg Cone

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     A B I v L 0 I 0 = C D 0I 0 L −T u I

(1.3)

with u, v ∈ Sym(3, R) and L ∈ G L(3, R) if and only if D is invertible, and in this case L = D −T = A − B D −1 C, v = B D −1 and u = D −1 C. (1.4) It is well known that the symplectic group Sp(6, R) acts on the Siegel upper half space TSym++ (3,R) := Sym(3, R) + i Sym++ (3, R) by linear fractional transformations, that is,   A B −1 ∈ Sp(6, R), z ∈ TSym++ (3,R) g · z = (Az + B)(C z + D) , where g = C D which induces an isomorphism from Sp(6, R)/{±I } onto the holomorphic automorphism group G(TSym++ (3,R) ) of TSym++ (3,R) . Since Sym(3, R) is the Šilov boundary of TSym++ (3,R) , the action of Sp(6, R) is extended on Sym(3, R) (precisely, one should consider a conformal compacitification of Sym(3, R) on which the actions of all the elements g ∈ Sp(6, R) are well-defined [5]). In this action, we consider the compression semigroup (called also the symplectic semigroup) of the symmetric cone Sym++ (3, R),   Sp := g ∈ Sp(6, R) | g · Sym++ (3, R) ⊂ Sym++ (3, R) which is a closed subsemigroup of Sp(6, R). It was proved in [6] that Sp can be given by  Sp =

  A B T T + ∈ Sp(6, R) | D ∈ GL(3, R), C D , D B ∈ Sym (3, R) (1.5) C D

+ − and has a triple decomposition Sp = Sp G(3, R)Sp , where

  I B | B ∈ Sym+ (3, R) , 0 I    A 0 | A ∈ GL(3, R) , G(3, R) := 0 A−T    I 0 − Sp := | C ∈ Sym+ (3, R) . C I + := Sp



It was also proved that the symplectic semigroup satisfies the following Ol’shanski˘ı polar decomposition Sp = G(3, R) exp(CSp ), where CSp is the closed convex cone  CSp :=

0 B C 0



+



∈ sp(6, R) | B, C ∈ Sym (3, C) ,

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and sp(6, R) is the Lie algebra of Sp(6, R), that is, sp(6, R) = {X ∈ M(3, R) | X J + J X T = 0}    A B | A ∈ Mat(3, R), B, C ∈ Sym(3, R) . = C −A T Now we turn to the dual Vinberg cone . Let V be the subspace of Sym(3, R) defined by ⎧ ⎫ ⎛ ⎞ x1 0 x4 ⎨ ⎬ V := x = ⎝ 0 x2 x5 ⎠ , x1 , . . . , x5 ∈ R . ⎩ ⎭ x4 x5 x3 Then  is naturally identified with the intersection Sym++ (3, R) ∩ V , that is,  = {x ∈ V | 1 (x) > 0, 2 (x) > 0, 3 (x) > 0} = {x ∈ V | x is positive definite}. Let T := V + i ⊂ VC be the tube domain over , G(T ) the identity component of the holomorphic automorphism group on the tube domain T , and  the compression semigroup  := {g ∈ G(T ) | g ·  ⊂ }

(1.6)

of . This semigroup  is a main object of the present work. Here, we explain the organization of this paper. In Sect. 2, we describe the group G(T ) as a subgroup of Sp(6, R). Then, we give a characterization of  as a subset of G(T ) using the triple decomposition in Sect. 3. In Sect. 4, we show that  also admits an Ol’shanski˘ı polar decomposition. Finally, in Sect. 5, we show that  does not have a contraction property with respect to the canonical Riemannian metric on . We should like to thank the anonymous referee for the careful reading of this work. The second author is grateful to Institute Elie Cartan of Lorraine for providing him a visiting professor position in March 2011, where the present research began. This work was partially supported by KAKENHI 20K03657 and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).

2 The Holomorphic Automorphism Group of T First we shall determine the linear automorphism group G() := {g ∈ G L(V ) | g ·  = } of the cone . Define

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⎧ ⎨

⎫ ⎞ ⎛ a1 ⎬ H := A = ⎝ 0 a2 ⎠ | a1 , . . . , a5 ∈ R, a1 a2 = 0, a3 > 0 , ⎩ ⎭ a4 a5 a3 and let H + be the subset of H consisting of diagonal matrices with positive entries. Then H forms a Lie group, and H + is its identity component. Let ρ : H → G L(V ) be the representation of H given by ρ(A)x := Ax A T (A ∈ H, x ∈ V ). Then H + as well as H acts transitively on the cone  ⊂ V by ρ. In other words, we have ρ(H ) ⊂ G() and  = ρ(H + )I3 . For a parameter (s1 , s2 , s3 ) ∈ C3 , let (s1 ,s2 ,s3 ) be the function on  given by (s1 ,s2 ,s3 ) (x) := 1 (x)s1 −s2 2 (x)s2 −s3 3 (x)s3 = x1s1 −s3 x2s2 −s3 (det x)s3 (x ∈ ). The function (s1 ,s2 ,s3 ) is relatively invariant under the action of H + : (s1 ,s2 ,s3 ) (ρ(A)x) = a12s1 a22s2 a32s3 (s1 ,s2 ,s3 ) (x) (x ∈ , A ∈ H + ).

(2.1)

Indeed, this equality characterizes the function (s1 ,s2 ,s3 ) up to a constant multiple.   Let ∗ ⊂ V be the dual cone of . Namely, ∗ := ξ ∈ V | (x|ξ) > 0 for all x ∈  \ {0} . The Köcher–Vinberg characteristic function ϕ of  is defined by ϕ (x) :=  so-called −(x|ξ) e dξ for x ∈ . It is known (see [3, Proposition I.3.1]) that, for any ∗  g ∈ G(), we have ϕ (gx) = | Det g|−1 ϕ (x) (x ∈ ).

(2.2)

For A = diag(a1 , a2 , a3 ) ∈ H and x ∈ V , we observe that ⎛

⎞ a12 x1 0 a1 a3 x 4 a22 x2 a2 a3 x5 ⎠ , ρ(A)x = ⎝ 0 a1 a3 x4 a2 a3 x5 a32 x3 so that Det ρ(A) = a13 a23 a34 . For a general A ∈ H + , because of the factorization A = diag(a1 , a2 , a3 )A with a unipotent A ∈ H + , we have again Det ρ(A) = a13 a23 a34 . Therefore, comparing (2.1) and (2.2), we conclude that there exists a constant C > 0 for which 1/2 1/2 (2.3) ϕ (x) = C(−3/2,−3/2,−2) (x) = C x1 x2 (det x)−2 . Let G() I3 be the isotropy subgroup of G() at I3 ∈ , and take g ∈ G() I3 . In general, for a function F on , we denote by g ∗ F the pullback F ◦ g. Since G() I3 is a compact group, we have | Det g| = 1, so that g ∗ ϕ2 = ϕ2 thanks to (2.2). Thus, by the uniqueness of irreducible factorization of the rational function ϕ2 , we have g ∗ x1 = C1 x1 , g ∗ x2 = C2 x2 , g ∗ det x = C3 det x

(2.4)

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H. Ishi and K. Koufany

g ∗ x1 = C1 x2 , g ∗ x2 = C2 x1 , g ∗ det x = C3 det x

(2.5)

with C1 C2 C3 = 1. On the other hand, since g · I3 = I3 , we have C1 = C2 = C3 = 1. Let us consider the case (2.4). We have g ∗ det x = det x, which means that x1 x2 (g ∗ x3 ) − x1 (g ∗ x5 )2 − x2 (g ∗ x4 )2 = x1 x2 x3 − x1 x52 − x2 x42 .

(2.6)

From this equality, we deduce g ∗ x5 = ±x5 + αx2 with some α ∈ R. In fact, if g ∗ x5 would contain other terms, for instance γx3 , then the left-hand side should contain the term of x1 x3 x5 , which does not appear in the right-hand side. By the same argument, we have g ∗ x4 = ±x4 + βx1 with some β ∈ R. Actually, we have (2.6) in this case with g ∗ x3 = x3 + β 2 x1 + α2 x2 ± 2βx4 ± 2αx5 . On the other hand, since g · I3 = I3 , we have α = β = 0. Therefore we conclude that g = ρ(diag(±1, ±1, 1)). ⎛ ⎞ 010 Let us turn to the case (2.5). Put σ = ⎝1 0 0⎠. Then ρ(σ) : x = (x1 , . . . , x5 ) → 001 (x2 , x1 , x3 , x5 , x4 ) belongs to G() I3 satisfying (2.5). Furthermore, if g ∈ G() I3 satisfies (2.5), then g ◦ ρ(σ) satisfies (2.4). Now we conclude that Lemma 2.1 The isotropy subgroup G() I3 is a finite group of order 8 generated by ρ(diag(−1, 1, 1)), ρ(diag(1, −1, 1)), and ρ(σ). Corollary 2.1 One has G() = ρ(H + )  G() I3 . We extend the action of G() to T by g(z) = g(x) + ig(y), z = x + i y. The translation tv : z → z + v by v ∈ V is a holomorphic automorphism of T , and the group N + of all such translations is an Abelian group isomorphic to V . The rational map s on T defined by ⎛

− z11 0 s : T  z → ⎝ 0 − z12 z4 z1

z5 z2

z4 ⎞ z1 z5 ⎠ z2 det z z1 z2

∈ T

belongs to G(T ). Note that s 2 = ρ(diag(−1, −1, 1)) = Id, so that s is not an involution, but s 4 = I d. Let V  be the subspace of V defined by ⎧ ⎨

⎫ ⎞ u1 0 0 ⎬ V  := u = ⎝ 0 u 2 0⎠ | u 1 , u 2 ∈ R . ⎩ ⎭ 0 0 0 ⎛

For any u ∈ V  , let t˜u = s ◦ tu ◦ s and denote by N − the subgroup of G(T ) of these transformations.

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Keeping in mind the inclusion T ⊂ TSym++ (3,R) , we shall realize the group G(T ) as a subgroup of G(TSym++ (3,R) ) (see Theorem 2.2). In other words, we shall see that any g ∈ G(T ) can be described by an element of  For A ∈ H , the corre Sp(6, R). A 0 ∈ G(3, R) ⊂ Sp(6, R). sponding ρ(A) ∈ G() is induced by the matrix 0 A−T For v∈ V , we identify the translation tv : T  z → z + v ∈ T with the matrix  I v ∈ Sp(6, R). In this way, we regard G 0 := ρ(H ) and N + as subgroups of 0I Sp(6, R). On the other hand, from a straightforward calculation, we see that the map s corresponds to the matrix ⎞ ⎛ 0 −1 ⎜ 0 −1 ⎟ ⎟ ⎜ ⎜ 1 0⎟ ⎟ ∈ Sp(6, R). ⎜ ⎟ ⎜1 0 ⎟ ⎜ ⎝ 1 0 ⎠ 0 1 −1 − Then, an easy  matrix  calculation tells us that the transform t˜u = stu s ∈ N correI 0 sponds to ∈ Sp(6, R). −u I Put p0 = i I ∈ T and let K = {g ∈ G(T ) | g · p0 = p0 } be the isotropy subgroup of G(T ) at the point p0 .

Lemma 2.2 (cf.[2, Lemma 4.1]) One has  K = kθ,φ where

⎛ Cθ,φ = ⎝

   Cθ,φ −Sθ,φ | θ, φ ∈ [0, 2π) , = Sθ,φ Cθ,φ

cos θ

⎞ cos φ ⎠ , Sθ,φ 1

⎛ ⎞ sin θ sin φ ⎠ . =⎝ 0

Theorem 2.1 The group G(T ) is generated by G 0 , N + and s. Proof Let us take any g ∈ G(T ) and put z = g · p0 . Since y = z ∈ , we can find A ∈ H for which ρ(A) · I = y. Putting x = z ∈ V , we have g · p0 = z = tx ρ(A) · p0 , so that k = ρ(A)−1 tx−1 g belongs to K . Since g = tx ρ(A)k, it is enough to show that k is generated by N + , G 0 and s. By Lemma 2.2, we have k = kθ,φ with some θ, φ ∈ [0, 2π). First we consider the , then det Cθ,θ = cos2 θ = 0, and we have case θ = φ. If θ = π2 , 3π 2 kθ,θ = tv ρ(A)t˜−u ∈ N + G 0 N − with

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A = (Cθ,θ )−T ∈ H, u = (Cθ,θ )−1 Sθ,θ ∈ V  , v = −Sθ,θ (Cθ,θ )−1 ∈ V thanks to Lemma 1.1. Thus kθ,θ is generated by N + , G 0 and s in this case. For the case that θ = π2 and θ = 3π , the element kθ,θ equals s and s −1 , respectively, so that 2 the claim holds for these cases, too. Now we consider a general kθ,φ . We can take an appropriate α ∈ R for which det Cθ+α, φ+α = 0. Similarly to the argument above, we see from Lemma 1.1 that kθ+α,φ+α ∈ N + G 0 N − . Finally, we have kθ,φ = k−α,−α kθ+α,φ+α , which completes the proof.  We remark that G 0 is not equal to the whole group G(), but is a subgroup of G() of index 2. Indeed, ρ(σ) ∈ G() \ G 0 , and ρ(σ) is a holomorphic automorphism on T but not an element of G(T ). We also note that G 0 is not connected. Its identity component is ρ(H + ). Let us give another explicit description of the group G(T ) as a subgroup of Sp(6, R). We set ⎧⎛ ⎫ ⎞ ⎨ x1 0 x6 ⎬ W : = ⎝ 0 x2 x7 ⎠ | x1 , . . . , x7 ∈ R , ⎩ ⎭ x4 x5 x3 ⎧⎛ ⎫ ⎞ ⎨ a1 0 0 ⎬ H  := ⎝ 0 a2 0 ⎠ | a1 , . . . , a5 ∈ R, a3 > 0 , ⎩ ⎭ a4 a5 a3 Then, we have A, B ∈ H  ⇒ AB ∈ H  , 

(2.7)

A ∈ H , w ∈ W ⇒ Aw, w A ∈ W, 

and

T





(2.8)

A ∈ H , u ∈ V ⇒ u A, A u ∈ V ,

(2.9)

A ∈ H  , u ∈ V  , w ∈ W ⇒ A + wu ∈ H  .

(2.10)

T

Define   A B   T  ∈ Sp(6, R) | A ∈ H , B ∈ W, C ∈ V , D ∈ H . C D

 G :=

(2.11)

  A B Let us check that G is a subgroup of Sp(6, R). For two elements g = and C D    A B g = of G, we have C  D

The Compression Semigroup of the Dual Vinberg Cone

gg  =

131

 A A + BC  AB  + B D  . C A + DC  C B  + D D 



Then, we see from (2.7) to (2.10) that A A + BC  ∈ H  ,

AB  + B D  ∈ W, C A + DC  ∈ V  , (C B  + D D  )T ∈H  ,

so that gg  ∈ G. On the other hand, since G ⊂ Sp(6, R), we have g

−1



    T  0 I D −B T T 0 −I = g = , −I 0 −C T A T I 0

 for g =

 A B ∈ G, whence we see that g −1 ∈ G. C D

Theorem 2.2 The linear fractional action of Sp(6, R) on the Siegel upper half plane TSym++ (3,R) induces an isomorphism from G onto G(T ).   A B Proof For g = ∈ G and z ∈ T , we obtain Az + B ∈ WC by (2.8) and C D  T (C z + D) ∈ HC by (2.10), so that we have g · z ∈ WC by (2.8). On the other hand, since g ∈ Sp(6, R) and z ∈ TSym++ (3,R) , we have g · z ∈ TSym++ (3,R) . Thus, g · z ∈ T = TSym++ (3,R) ∩ WC , and we have a group homomorphism from G into G(T ). Thanks to Theorem 2.1, the map is surjective because G contains the matrices corresponding to tv ∈ N + (v ∈ V ), ρ(A) ∈ G 0 (A ∈ H ) and s. Let us show the injectivity. Take g ∈ G such that  g · z = zfor all z ∈ T . Then, g · p0 = p0 together A −B with g ∈ Sp(6, R) implies g = with A + i B ∈ U (3). Since g ∈ G, we B A  T   have A ∈ H , A ∈ H and B ∈ V . Thus, we get A = Cθ,φ and B = Sθ,φ with some θ, φ ∈ [0, 2π). Let us consider z ∈ T with z 1 = z 2 = i. Then ⎛

⎞ ⎞ ⎛ i 0 eiθ z 4 i 0 z4 ⎠. i eiφ z 5 g · ⎝ 0 i z5⎠ = ⎝ 0 2 iθ iθ iφ z4 z5 z3 e z 4 e z 5 z 3 + z 4 e sin θ + z 52 eiφ sin φ Thus, g · z = z implies θ = φ = 0, so that g = I .



We see from Theorem 2.2 that each g ∈ G(T ) is uniquely extended to a linear fractional transform on TSym++ (3,R) . Let us present one more description of the group G: Proposition 2.1 One has  G=

  A B  T  T T  ∈ Sp(6, R) | A ∈ H , D ∈ H , D B ∈ V, C D ∈ V . C D

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Proof Let G  be the right-hand side. By (1.1), (2.8), and (2.9), we have G ⊂ G  . To show the converse inclusion, we take g ∈ G  with det D = 0. By (2.8) and (2.9) again, we get B = D −T (D T B) ∈ W and C = (C D T )D −T ∈ V  . Thus g ∈ G. Since both G and G  are closed subset of Mat(6, R), we obtain G  ⊂ G by a closure argument.  Let ϒ ⊂ G(T ) be the set consisting of g ∈ G(T ) such that there exist v ∈ V, A ∈ H and u ∈ V  for which g = tv ρ(A)t˜u . Identifying G(T ) with G by Theorem 2.2, we get an explicit description of the set ϒ. Proposition 2.2 One has  ϒ=

  A B ∈ G | det D = 0 . C D

Therefore, ϒ is an open dense subset of G(T ).  Proof Let ϒ  be the right-hand side. The inclusion  ϒ ⊂ ϒ follows from Lemma A B 1.1. To show the converse inclusion, we take g = ∈ ϒ  . Then, we have the C D equality (1.3) and (1.4). In particular, v = B D −1 ∈ Sym(3, R) belongs to W by (2.8), so that we get v ∈ V = Sym(3, R) ∩ W . On the other hand, we have u = D −1 C ∈ V  by (2.9) and L = D −T ∈ H . Thus, g = tv ρ(L)t˜−u ∈ ϒ and the assertion is verified. 

3 The Triple Decomposition of  We shall investigate decomposition structures of the compression semigroup  defined by (1.6). More precisely, we will prove that any element g of the semigroup  admits a triple decomposition g = tv ρ(A)t˜−u , which is unique by Lemma 1.1. Consider the following two closed subsemigroups of    I v |v∈ , 0I      I 0 −  ˜ |u ∈∩V ,  := t−u | u ∈  = u I   := tv | v ∈  = +





and 

+0

−

0



  I v := {tv | v ∈ } = |v∈ , 0I      I 0  ˜ := t−u | u ∈  = |u ∈∩V . u I

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The latest are two subsemigroups of the interior  0 of . Now we state our first main theorem. Theorem 3.1 The semigroup  is contained in ϒ. Moreover, one has  =  + G 0  − .   A B Proof First we observe that for g = ∈ G with C D ⎞ ⎛ a1 A = ⎝ 0 a2 ⎠ , a4 a5 a3

⎛

⎞ ⎛ ⎞ b1 0 b6 c1 0 0 B = ⎝ 0 b2 b7 ⎠ , C = ⎝ 0 c2 0⎠ , b4 b5 b3 0 0 0

⎞ ⎛ d1 0 d4 D = ⎝ d2 d5 ⎠ , d3

the equality AD T − BC T = I implies ak dk − bk ck = 1

(k = 1, 2).

(3.1)

Moreover, if z  = g · z ∈ VC with z ∈ VC , then z k =

ak z k + bk ck z k + dk

(k = 1, 2).

(3.2)

Now we suppose g ∈ / ϒ, which means det D = 0 by Proposition 2.2. Since D T ∈ H  , we have d1 = 0 or d2 = 0. If d1 = 0, we have c1 = − b11 = 0 by (3.1). Let us consider the case ⎛ ⎞ x1 0 0 z = ⎝ 0 1 0⎠ ∈ V. 0 01 b2

By (3.2), we have z 1 = −a1 b1 − x11 , so that we can take x1 > 0 for which z 1 < 0. / , which imply that g ∈ / . Similarly, we can show Then z ∈  and z  = g · z ∈ g∈ /  if d2 = 0. Therefore, we conclude that  ⊂ ϒ. Now take g ∈  and let g = tv ρ(A)t˜−u (u ∈ V  , A ∈ H, v ∈ V ) be a triple decomposition. Let {xn }∞ n=1 ⊂  be a sequence converging to 0. Then,   g · x n = v + ρ(A)t˜−u xn → v as n → ∞, so that we get v ∈ . Thanks to (3.2), we have ak2 z k (k = 1, 2) for z  = g · z ∈ VC with z ∈ VC . In particular, if u 1 < 0, z k = vk + zk +u k then g · x is not defined for ⎛

⎞ −u 1 0 0 x = ⎝ 0 1 0⎠ ∈ , 0 01 which contradicts g ∈ . Therefore u 1 ≥ 0. We see that u 2 ≥ 0 similarly, which completes the proof of the theorem.  As a consequence, we have

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(3.3)

Let us describe  in a matrix block form. Proposition 3.1 One has  =

  A B T T  ∈ G | det (D) = 0, D B ∈ , C D ∈  ∩ V . C D

Proof Let g = tv ρ(L)t˜−u be the triple decomposition of g ∈ . Thanks to (1.4), we have u = D −1 C = D −1 (C D T )D −T and v = B D −1 = D −T (D T B)D −1 . Therefore, the assertion follows from Theorem 3.1. 

4 The Ol’shanski˘ı Polar Decomposition of  We see from (2.11) that the Lie algebra g of G equals the subalgebra of sp(6, R) given by    A v  | A ∈ h, v ∈ V, u ∈ V , g= u −A T where h is the Lie algebra of H ⊂ GL(3, R), that is, ⎧⎛ ⎫ ⎞ ⎨ a1 ⎬ h = ⎝ 0 a2 ⎠ | a1 , . . . , a5 ∈ R . ⎩ ⎭ a4 a5 a3 

 I /2 0 ∈ g. Namely, if gk := Then, g is graded by ad(Z 0 ) with Z 0 := 0 −I /2 { X ∈ g | [Z 0 , X ] = k X }, then g = g−1 ⊕ g0 ⊕ g1 with 

g−1 =

00 u 0



| u ∈ V





, g0 =



Let C :=

0v u0

A 0 0 −A T



    0v |v∈V . | A ∈ h , g1 = 00

 | v ∈ , u ∈  ∩ V



 .

Then C is an Ad(G())-invariant closed convex cone in g−1 ⊕ g1 which is proper (C ∩ −C = {0}) and generating (C 0 = ∅). Its interior C 0 is the set of matrices with v ∈  and u ∈  ∩ V  . Theorem 4.1 The compression semigroup  has the following Ol’shanski˘ı polar decomposition  = G 0 exp(C)

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with  0 = G 0 exp(C 0 ) as interior. Proof Let us denote S := G 0 exp(C) and prove that  = S. First by [9, 12] it follows that S is a closed subsemigroup of G. Further, it is clear ¯  − = exp( ¯ ∩ V  ) and G 0 are closed subsemigroups of S. Thus, that  + = exp(), we have  ⊂ S by Theorem 3.1. On the other hand, since G 0 and exp(C) are subsemigroups of Sp , we see that G 0 exp(C) ⊂ Sp . In addition, G 0 ⊂ G and exp(C) ⊂ G, so that S = G 0 exp(C) is  contained in both G and Sp . Therefore, S ⊂  thanks to (3.3).

5 A Counter-Example to the Contraction Property of  On a proper open convex cone C ⊂ Rn , the second derivative of the logarithm of the Köcher–Vinberg characteristic function ϕC of C gives a canonical Riemannian metric (see [3, Sect. I.4], [14, Chap. I, Sect. 3]): (v|v  )x := Dv Dv log ϕC (x)

(v, v  ∈ Rn , x ∈ C),

where Dv denotes the directional derivative in v. Thanks to the relative invariance of ϕC under the action of the linear automorphism group G(C) (see (2.2)), the canonical metric is G(C)-invariant. In particular, if C is a symmetric cone, the metric makes C a Riemannian symmetric space. For example, if C = Sym++ (3, R), then the metric is given by the formula (v|v  )x = 2tr (x −1 vx −1 v  )

(v, v  ∈ Sym(3, R), x ∈ Sym++ (3, R)).

(5.1)

It is proved in [6, Sect. 5] that, if C is symmetric, the compression semigroup C of C has the contraction property with respect to the canonical metric, that is, (J (g, x)v|J (g, x)v)g(x) ≤ (v|v)x

(g ∈ C , x ∈ C, v ∈ Rn ),

where J (g, x) stands for the Jacobi matrix of g at x. We shall see that it is no longer the case when C is the dual Vinberg cone . Recalling (2.3), we see that the canonical Riemannian metric on  is given by (v|v  )x = −

1  v1 v1 v2 v   + 2 2 + 2 tr (x −1 vx −1 v  ) 2 2 x1 x2

(v, v  ∈ V, x ∈ ). (5.2)

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⎛

⎞ 101 Now we consider the case v = v  = ⎝0 0 0⎠ and x = I3 . Then (v|v)x = − 21 + 2 × 101 ⎛ ⎞ 1 0 −1 4 = 7.5. In view of Theorem 3.1, we put g0 := tv0 ∈  with v0 := ⎝ 0 1 0 ⎠ ∈ −1 0 1.01 ⎛ ⎞ 2 0 −1 . Then g0 (x) = In + v0 = ⎝ 0 2 0 ⎠ ∈  and J (g0 , x)v = v since g0 is a −1 0 2.01 translation. We observe  6.01 2 1 (J (g0 , x)v|J (g0 , x)v)g0 (x) = − + 2 = 7.795 · · · > 7.5 = (v|v)x . 8 3.02 This phenomenon is caused by a behavior of ‘the extra term’ − 12 (

v1 v1 x12

+

v2 v2 ) in (5.2), x22 −1 −1 

compared with (5.1). Actually, the decrease of the second term 2 tr(x vx v ) is  2 6.01 little (from 8 to 2 3.02 = 7.920 · · · ), while the extra term increases from −1/2 to −1/8.

References 1. Ph. Bougerol, Kalman filtering with random coefficients and contractions, geometric methods in representation theory. SIAM J. Control Optim. 32, 942–942 (1993) 2. L. Geatti, Holomorphic automorphisms of some tube domains over nonselfadjoint cones. Rend. Circ. Mat. Palermo 36, 281–331 (1987) 3. J. Faraut, A. Korányi, Analysis on Symmetric Cones (Clarendon Press, Oxford, 1994) 4. J. Hilgert, K.-H. Neeb, Basic Theory of Lie Semigroups and Applications, Springer Lecture Notes in Mathematics, vol. 1552 (Springer, Berlin, 1993) 5. S. Kaneyuki, On the Causal Structures of the Šilov Boundaries of Symmetric Bounded Domains, pp. 127–159, Lecture Notes in Mathematics, vol. 1468 (Springer, Berlin, 1991) 6. K. Koufany, Semi-groupe de Lie associé à un cône symétrique. Annales de l’Institut Fourier 45, 1–29 (1995) 7. K. Koufany, Contractions of angles in symmetric cones. Publ. Res. Inst. Math. Sci. 38, 227–243 (2002) 8. K. Koufany, Application of Hilbert’s projective metric on symmetric cones. Acta Math. Sin. 22, 1467–1472 (2006) 9. J. Lawson, Polar and Ol’shanski decompositions. J. Reine Angew. Math. 448, 191–219 (1994) 10. Y. Lim, Finsler metrics on symmetric cones. Math. Ann. 316, 379–389 (2000) 11. K.-H. Neeb, On the complex and convex geometry of Ol’shanski˘ı semigroups. Annales de l’Institut Fourier 48, 149–203 (1998) 12. G.I. Ol’shanski, Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series. Funct. Anal. Appl. 15, 275–285 (1982) 13. E.B. Vinberg, Homogeneous cones. Soviet Math. Dokl. 1, 787–790 (1960) 14. E.B. Vinberg, The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12, 340– 403 (1963)

The Universal Pre-Lie–Rinehart Algebras of Aromatic Trees Gunnar Fløystad, Dominique Manchon, and Hans Z. Munthe-Kaas

Abstract We organize colored aromatic trees into a pre-Lie–Rinehart algebra (i.e., a flat torsion-free Lie–Rinehart algebra) endowed with a natural trace map, and show the freeness of this object among pre-Lie–Rinehart algebras with trace. This yields the algebraic foundations of aromatic B-series. Keywords Free pre-Lie algebra · Lie–Rinehart algebra · Aromatic tree · Trace map 2000 Mathematics Subject Classification Primary: 17A50 · 17D25

1 Introduction In the analysis of structure preserving discretisation of differential equations, series developments indexed by trees are fundamental tools. The relationship between algebraic and geometric properties of such series has been extensively developed in recent years. The mother of all these series is B-series, introduced the seminal works of John Butcher in the 1960s [2, 3]. However, the fundamental idea of denoting analytical forms of differential calculus with trees was conceived already a century earlier by Cayley [4]. This work was partially supported by the project Pure Mathematics in Norway, funded by Trond Mohn Foundation and Tromsø Research Foundation. G. Fløystad · H. Z. Munthe-Kaas Matematisk Institutt, Universitetet i Bergen, Bergen, Norway e-mail: [email protected] H. Z. Munthe-Kaas e-mail: [email protected] D. Manchon (B) Laboratoire de Mathématiques Blaise Pascal, CNRS and Université Clermont-Auvergne, Aubière, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_9

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A modern understanding of B-series stems from the algebra of flat and torsion-free connections naturally associated with locally Euclidean geometries. The vector fields on Rd form a pre-Lie algebra L with product given by the connection  in (9). The free pre-Lie algebra is the vector space spanned by rooted trees with tree grafting as the product [5]. A B-series can be defined as an element Ba in the graded completion of the free pre-Lie algebra, yielding infinite series of trees with coefficients a(t) ∈ R for each tree t. By the universal property, a mapping → f ∈ L, sending the single node tree to a vector field, extends uniquely to a mapping Ba → Ba ( f ), where Ba ( f ) is an infinite series of vector fields Ba ( f ) = a( ) f + a( ) f  f + a( )( f  f )  f + a(

  ) f  ( f  f) − ( f  f)  f + ··· .

On the geometric side, it has recently been shown [11] that B-series are intimately connected with (strongly) affine equivariant families of mappings of vector fields on Euclidean spaces. An infinite family of smooth mappings n : X Rn → X Rn for n ∈ N has a unique B-series expansion Ba if and only if the family respects all affine linear mappings ϕ(x) = Ax + b : Rm → Rn . This means that f ∈ X Rn being ϕrelated to g ∈ X Rm implies n ( f ) being ϕ-related to m (g). Subject to convergence of the formal series we have n ( f ) = Ba ( f ). Aromatic B-series is a generalization which was introduced for the study of volume preserving integration algorithms [6, 10], more recently studied in [1, 12]. The divergence of a tree is represented as a sum of “aromas”, graphs obtained by joining the tree root to any of the tree’s nodes. Aromas are connected directed graphs where each node has one outgoing edge. They consist of one cyclic sub-graph with trees attached to the nodes in the cycle. Aromatic B-series are indexed by aromatic trees, defined as a tree multiplied by a number of aromas. The geometric significance of aromatic B-series is established in [12]. Consider a smooth local mapping of vector fields on a finite-dimensional vector space,  : X Rd → X Rd . “Local” means that the support is non-increasing, supp (( f )) ⊂ supp ( f ). Such a mapping can be expanded in an aromatic B-series if and only if it is equivariant with respect to all affine (invertible) diffeomorphisms ϕ(x) = Ax + b : Rd → Rd . An equivalent formulation of this result is in terms of the preLie algebra L = (X Rd , ) defined in the Canonical example of Sect. 2.4. The isomorphisms of L are exactly the pullback of vector fields by affine diffeomorphisms ξ( f ) = A−1 f ◦ ϕ, hence, Theorem 1.1 Let L be the canonical pre-Lie algebra of vector fields on a finitedimensional euclidean space. A smooth local mapping  : L → L can be expanded in an aromatic B-series if and only if  ◦ ξ = ξ ◦  for all pre-Lie isomorphisms ξ : L → L. This result shows that aromatic B-series have a fundamental geometric significance. The question to be addressed in this paper is to understand their algebraic foundations. In what sense can aromatic B-series be defined as a free object in some

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category? Trees represent vector fields and aromas represent scalar functions on a domain. The derivation of a scalar field by a vector field is modeled by grafting the tree on the aromas. A suitable geometric model for this is pre-Lie algebroids, defined as Lie algebroids with a flat and torsion-free connection [13]. Lie algebroids are vector bundles on a domain together with an “anchor map”, associating sections of the vector bundle with derivations of the ring of smooth scalar functions. The algebraic structure of Lie algebroids is captured through the notion of Lie– Rinehart algebras; the aromatic trees form a module over the commutative ring of aromas, acting as derivations of the aromas through the anchor map given by grafting. However, it turns out that the operations of divergence of trees and the grafting anchor map are not sufficient to generate all aromas. Instead, a sufficient set of operations to generate everything is obtained by the graph versions of taking covariant exterior derivatives of vector fields and taking compositions and traces of the corresponding endomorphisms. These operations are well defined on any finite-dimensional pre-Lie algebroid. However, for the Lie–Rinehart algebra of aromas and trees the trace must be defined more carefully, since, e.g., the identity endomorphism on aromatic trees does not have a well-defined trace. In this paper, we define the notion of tracial pre-Lie-Rinehart algebras and show that the aromatic B-series arise from the free object in this category.

2 Lie–Rinehart and Pre-Lie–Rinehart Algebras Lie–Rinehart algebras were introduced by George S. Rinehart in 1963 [14]. They have been thoroughly studied by several authors since then, in particular by J. Hübschmann who emphasized their important applications in Poisson geometry [8]. After a brief reminder on these structures, we introduce pre-Lie–Rinehart algebras which are Lie–Rinehart algebras endowed with a flat and torsion-free connection. We also introduce the mild condition of traciality for Lie–Rinehart algebras. The main fact (Corollary 2.8) states the traciality of any finite-dimensional Lie algebroid over a smooth manifold.

2.1 Reminder on Lie–Rinehart Algebras Let k be a field, and let R be a unital commutative k-algebra. Recall that a Lie– Rinehart algebra over R consists of an R-module L and an R-linear map ρ : L → Derk (R, R) (the anchor map ), such that • L is a k-bilinear Lie algebra with bracket [[−, −]],

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• The anchor map ρ is a homomorphism of Lie algebras, • For f ∈ R and X, Y ∈ L the Leibniz rule holds:   [[X, f Y ]] = ρ(X ) f Y + f [[X, Y ]].

(1)

Remark 2.1 In the original article [14], G.Rinehart does not state that the anchor map should be a Lie algebra homomorphism. However, all articles on Lie-Rinehart algebras from the last two decades seem to require this. In the much-cited article by J. Hübschmann [8] from 1990 it is not quite clear whether this is required, but again in later articles like [9] from 1998 and onwards, he explicitly requires the anchor map to be a Lie algebra homomorphism. If one does not require the anchor map to be a Lie algebra homomorphism, then if Ann(L) is the annihilator of L in R, it is easy to see that L will be a Lie–Rinehart algebra over R/ Ann(L), with the anchor map being a Lie algebra homomorphism. In particular, for Lie algebroids (see Sect. 2.2), then Ann(L) = 0, and the anchor map will automatically be a Lie algebra homomorphism. A homomorphism (α, γ) : (L , R) → (K , S) of Lie–Rinehart algebras consists of a Lie k-algebra homomorphism α and k-algebra homomorphism γ: α : L → K, γ : R → S such that for f ∈ R and X ∈ L: • α( f X ) = γ( f )α(X ), • γ((ρ L (X ) · f ) = ρ K (α(X )) · γ( f ). A connection on a R-module N is a R-linear map ∇ : L −→ Endk (N ) X −→ ∇ X such that

  ∇ X ( f Y ) = ρ(X ). f Y + f ∇ X Y.

The curvature of the connection is given by R(X, Y ) := [∇ X , ∇Y ] − ∇[[X,Y ]] . If N = L, the torsion of the connection is given by T (X, Y ) := ∇ X Y − ∇Y X − [[X, Y ]]. The curvature vanishes if and only if N is a module over the Lie algebra L (via ∇). In that case, N is called a module over the Lie–Rinehart algebra (L , R). This is equivalent to the map ∇ being a homomorphism of Lie algebras where Endk (N ) is

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endowed with the commutator as the Lie bracket. In particular, the k-algebra R is a module over the Lie–Rinehart algebra (L , R). Let N be a R-module endowed with a connection ∇ with respect to the Lie– Rinehart algebra (L , R). The R-module Hom R (N , N ) can be equipped with the connection defined by (where u ∈ Hom R (N , N ), X ∈ L and Y ∈ N )   (∇ X u)(Y ) := ∇ X u(Y ) − u(∇ X Y ).

(2)

This connection verifies the Leibniz rule ∇ X (u ◦ v) = ∇ X u ◦ v + u ◦ ∇ X v,

(3)

as can be immediately checked. Proposition 2.2 If the connection ∇ on N is flat, the corresponding connection ∇ on Hom R (N , N ) given by (2) is also flat. Proof If ∇ is flat on N , it is well known that the corresponding L-module structure on N yields a L-module structure on Hom R (N , N ) via (2), hence a flat connection. To be concrete, a direct computation using (2) yields     ([∇ X , ∇Y ] − ∇[[X,Y ]] u)(Z ) = ∇ X ∇Y u(Z ) − u(∇Y Z ) − ∇Y (u∇ X Z ) + u(∇Y ∇ X Z )     −∇Y ∇ X u(Z ) − u(∇ X Z ) + ∇ X (u∇Y Z ) − u(∇ X ∇Y Z )   −∇[[X,Y ]] u(z) + u(∇[[X,Y ]] Z )     = ([∇ X , ∇Y ] − ∇[[X,Y ]] ) u(Z ) − u ([∇ X , ∇Y ] − ∇[[X,Y ]] )(Z ) .

 Definition 2.3 Let (L , R) be a Lie–Rinehart algebra. An R-module N is tracial if there exists a connection ∇ on N and a R-linear map τ : Hom R (N , N ) → R such that • τ (α ◦ β) = τ (β ◦ α) (trace property), • τ is compatible with the connection and the anchor, i.e., for any X ∈ L and α ∈ Hom R (N , N ) we have τ (∇ X α) = ρ(X ).τ (α).

(4)

If N is a module over (L , R), this means that τ is a homomorphism of (L , R)modules.

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2.2 Aside on Manifolds and Lie Algebroids Recall that a Lie algebroid on a smooth manifold M is a Lie–Rinehart algebra over the C-algebra of smooth C-valued functions on M. It is given by the smooth sections of a vector bundle E, and the anchor map comes from a vector bundle morphism from E to the tangent bundle T M. The terminology “anchor map” and the notation ρ are often used for the bundle morphism in the literature on Lie algebroids. Theorem 2.4 Let M be a finite-dimensional smooth manifold, and let V be a Lie algebroid on M. Any finite-dimensional vector bundle W endowed with a V connection is tracial, i.e., the C ∞ (M)-module N of smooth sections of W is tracial with respect to the Lie–Rinehart algebra L of sections of V . Proof We can consider the fiberwise trace on the algebra HomC ∞ (M) (N , N ) of smooth sections of the endomorphism bundle End W : it is given fiber by fiber by the ordinary trace of an endomorphism of a finite-dimensional vector space. The trace property is obviously verified. To prove the invariance property (4), choose two V -connections ∇ 1 and ∇ 2 on W . It is well known (and easily verified) that c X := ∇ X2 − ∇ X1 belongs to HomC ∞ (M) (N , N ), hence is a section of the vector bundle End(W ). Now for any section ϕ of End(W ) we have for any X ∈ L and α ∈ N : 

     ∇ X2 ϕ (α) = ∇ X2 ϕ(α) − ϕ ∇ X2 (α)     = (∇ X1 + c X ) ϕ(α) − ϕ ∇ X1 (α) + c X (α)   = ∇ X1 ϕ (α) + [c X , ϕ](α).

The trace of a commutator vanishes, hence we get Tr(∇ X2 ϕ) = Tr(∇ X1 ϕ).

(5)

In other words, the trace of ∇ X ϕ does not depend on the choice of the connection. We can locally (i.e., on any open chart of M trivializing the vector bundle W ) choose the canonical flat connection with respect to a coordinate system, namely   ∇ X0 α := ρ(X )α1 , . . . , ρ(X )α p for which (4) is obviously verified (here p is the dimension of the fiber bundle W ). Hence, from (5), we get that (4) is verified for any choice of connection ∇. 

2.3 Tracial Lie–Rinehart Algebras When the module N is the Lie–Rinehart algebra itself, it may be convenient to restrict the algebra on which the trace is defined:

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Definition 2.5 Suppose that N = L, and let us introduce the k-linear operator d : L −→ Hom R (L , L) defined by d X (Z ) := ∇ Z X.

(6)

Let the algebra of elementary R -module endomorphisms be the R-module subalgebra of Hom R (L , L) generated by {∇Y1 · · · ∇Yn d X : X, Y1 , . . . , Yn ∈ L}. It will be denoted by E R (L , L). Remark 2.6 The Leibniz rule (3) implies that the (L , R)-module structure on Hom R (L , L) (via the connection ∇) restricts to E R (L , L), making this an (L , R)submodule of Hom R (L , L). Definition 2.7 A Lie–Rinehart algebra L over the unital commutative k-algebra R is tracial if there exists a connection ∇ on L and a R-linear map τ : E R (L) → R such that • τ (α ◦ β) = τ (β ◦ α) (trace property), • τ is a homomorphism of L-modules, i.e., for any X ∈ L and α ∈ E R (L) we have τ (∇ X α) = ρ(X ).τ (α).

(7)

In this case, the divergence on L is the composition Div = τ ◦ d of d

τ

L −→ E R (L) −→ R. Corollary 2.8 Any finite-dimensional Lie algebroid is tracial for its natural canonical trace map. Proof It is an immediate consequence of Theorem 2.4.



We also have a an analog for the differential of a function, the first term in the De Rham complex: d : R → Hom R (L , R),

f → (X → ρ(X )( f )).

Given an element Y in L, we get a map in Hom R (L , L) denoted d f · X : X → ρ(X )( f ) · Y.

(8)

2.4 Pre-Lie–Rinehart Algebras Definition 2.9 A pre-Lie–Rinehart algebra is a Lie–Rinehart algebra L endowed with a flat torsion-free connection

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∇ : L → Endk (L , L). We have then, with the notation X  Y := ∇ X Y : • [[X, Y ]] = X  Y − Y  X , • X  (Y  Z ) − (X  Y )  Z = Y  (X  Z ) − (Y  X )  Z (left pre-Lie relation). A module over the pre-Lie–Rinehart algebra is the same as a module over the underlying Lie–Rinehart algebra. If N is a module and n and element, we write X  n for ∇ X n. In particular, for f ∈ R, the action of the anchor map ρ(X ). f is written X  f . Canonical example [4]: Let k = R, let R = C ∞ (Rd ) and let L be the space of smooth vector fields on Rd . Let X, Y ∈ L, which are written in coordinates: X=

d 

f i ∂i ,

Y =

i=1

X Y =

 d d   j=1

d i=1

i=1

d i=1

 f i (∂i g j ) ∂ j .

(9)

i=1

f i ∂i , the endomorphism d X sends

d 

In particular ∂ j →

g j∂j.

j=1

Then

For a vector field X =

d 

g j ∂ j →

d 

g j ∂ j ( f i )∂i .

i, j=1

∂ j ( f i )∂i , so the trace of d X is the divergence

d i=1

∂i ( f i ).

Proposition 2.10 In a pre-Lie–Rinehart algebra L, the algebra E R (L) of elementary module homomorphisms is generated by {d X : X ∈ L}. Proof In view of Definition 2.5, we first show that for any X, Y ∈ L, the endomorphism ∇Y (d X ) is obtained by linear combinations of products of endomorphisms of the form d Z , Z ∈ L. It derives immediately from the left pre-Lie relation, via the following computation (recall (2)):   (∇Y d X )(Z ) = ∇Y d X (Z ) − d X (∇Y Z ) = ∇Y (Z  X ) − (∇Y Z )  X = Y  (Z  X ) − (Y  Z )  X = Z  (Y  X ) − (Z  Y )  X   = d(Y  X ) − d X ◦ dY (Z ).

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To then show further that ∇Y1 ∇Y2 d X is a linear combination of products of endomorphisms, we use the Leibniz rule ∇Y1 (d X ◦ dY2 ) = (∇Y1 d X ) ◦ dY2 + d X ◦ ∇Y1 dY2 . 

In this way we may continue. Proposition 2.11 Let L be a pre-Lie–Rinehart algebra. a. For any X, Y ∈ L:

X  dY = d(X  Y ) − dY ◦ d X.

b. For f ∈ R (recall (8) for d f · Y ): X  (d f · Y ) = d f · (X  Y ) + d(X  f ) · Y − (d f · Y ) ◦ d X. Proof a. Using (2): (X  dY )(Z ) = X  (Z  Y ) − (X  Z )  Y = Z  (X  Y ) − (Z  X )  Y = d(X  Y )(Z ) − dY ◦ d X (Z ) b. Again using (2) this map sends Z to 

   X  (d f · Y ) (Z ) = X  (Z  f )Y − (d f · Y )(X  Z )     (connection property) = X  (Z  f ) Y + (Z  f )(X  Y ) − (X  Z )  f Y = (Z  (X  f ))Y − ((Z  X )  f )Y + (Z  f )(X  Y ). This is the map: d(X  f ) · Y − (d f · Y ) ◦ d X + d f · (X  Y ).  Definition 2.12 Let (L , R) and (K , S) be tracial pre-Lie–Rinehart algebras, and (α, γ) : (L , R) → (K , S) a homomorphism of pre-Lie-Rinehart algebras. For each X ∈ L there is a commutative diagram: L

dX

α

K

L α

dα(X )

K.

An elementary endomorphism φ : L → L is an R-linear combination of compositions d X 1 ◦ · · · ◦ d X r . It induces an elementary endomorphism ψ : K → K which

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is the corresponding R-linear combination of compositions of dα(X i )’s. Note: This ψ may not be unique since expressing φ as an R-linear combination of compositions may not be done uniquely. For instance it could be that d X is the zero map, while dα(X ) is not the zero map. The homomorphism (α, γ) is a homomorphism of tracial pre-Lie algebras if for each elementary endomorphism φ the trace of φ maps to the trace of ψ: γ(τ φ) = τ ψ. (This is regardless of which ψ that corresponds to φ.)

3 Aromatic Trees In this section we define rooted trees, aromas and aromatic trees, the latter being the relevant combinatorial objects for building up the free pre-Lie–Rinehart algebra.

3.1 Rooted Trees and Aromas Let C be a finite set, whose elements we shall think of as colors. We introduce some notation: Definition 3.1 TC is the vector space freely generated by rooted trees whose vertices are colored with elements of C. We denote by VC the vector space freely generated by pairs (v, t) where t is a C-colored tree and v is a vertex of t. There is an injective map VC → Endk (TC ), (v, t) → (s → s v t) where v is grafting the root of s on the vertex v. The composition β ◦ α of maps β, α in Endk (TC ) induces a multiplication (composition) ◦ on VC given by (v, t) ◦ (u, s) = (u, s v t). We may then identify VC as a k-subalgebra of Endk (TC ). Then d : TC → Endk (TC ), t →

 (v, t).

(10)

v∈t

Definition 3.2 A connected directed graph with vertices colored by C and where each vertex has precisely one outgoing edge is called a C-colored aroma or just an aroma since we will only consider this situation. It consists of a central cycle with trees attached to the vertices of this cycle. The arrows of each tree are oriented towards the cycle, which will be oriented counterclockwise by convention when an aroma is drawn in the two-dimensional plane. We let AC be the vector space freely generated by C-colored aromas. See Fig. 1 where the first four connected graphs are aromas.

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Now consider the linear map τ : VC −→ AC

(11)

which maps the pair (v, t) to an aroma by joining the root of t to the vertex v. Lemma 3.3 The vector space AC spanned by the aromas can be naturally identified with the quotient VC /[VC , VC ], where [VC , VC ] is the vector space spanned by the commutators in VC , so that the map τ becomes the natural projection from VC onto VC /[VC , VC ]. Proof An aroma has a unique interior cycle. Let r1 , · · · rn be the vertices on this cycle. At r j−1 there is a tree t j with root r j−1 . Let t j be this tree with r j−1 grafted onto r j , so r j is the root of t j . We may the write the aroma as   a = τ (r1 , t1 ) ◦ · · · ◦ (rn , tn ) .

(12)

and this is invariant under any cyclic permutation of the elements (ri , ti ). On the other hand, any tree t with marked point v admits the decomposition: (v, t) = (v1 , t1 ) ◦ · · · ◦ (v j , t j )

(13)

where v1 (resp. v j ) is the root of t (resp. the marked vertex v) and (v1 , v2 , . . . , v j ) is the path from the root to v in t. Each vertex vi of this path is the root of the tree ti . Now if (v , t ) = (v j+1 , t j+1 ) ◦ · · · ◦ (v j+k , t j+k ) is another tree with marked vertex, we have (v, t) ◦ (v , t ) = (v1 , t1 ) ◦ · · · ◦ (v j+k , t j+k ) and (v , t ) ◦ (v, t) = (v j+1 , t j+1 ) ◦ · · · ◦ (v j+k , t j+k ) ◦ (v1 , t1 ) ◦ · · · ◦ (v j , t j ).     The trace property τ (v, t) ◦ (v , t ) = τ (v , t ) ◦ (v, t) is then obvious by cyclic invariance of the decomposition of an aroma. Now any aroma is the image by τ of at most n trees with marked points, where n is the length of the cycle. It is clear that two such trees admit the same decomposition as above modulo cyclic permutation, which implies that they differ by a commutator. Now, we have to prove that any element T ∈ VC with τ (T ) = 0 is a linear combination of commutators. Decomposing T in the basis of trees with one marked points: T =

 (v,t)

=

α(v,t) (v, t)





a aroma (v,t), τ (v,t)=a

α(v,t) (v, t),

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from τ (T ) = 0 we get (v,t), τ (v,t)=a α(v,t) = 0 for any aroma a. Hence, the sum (v,t), τ (v,t)=a α(v,t) (v, t) is a sum of commutators for any aroma a, which proves that T is also a sum of commutators.  The canonical embedding of C into TC is given by c → •c . It is well-known [5, 7] that TC with grafting of trees as the operation  is the free pre-Lie algebra on the set C. Then (TC , k) becomes a pre-Lie–Rinehart algebra, with anchor map zero. Lemma 3.4 The algebra Ek (TC ) of elementary module morphisms of Definition 2.5, for the pre-Lie–Rinehart algebra (TC , k), coincides with the algebra VC of trees with one marked point. Proof Since dt = v∈t (v, t) by (10) we need only to show that each marked tree (v, t) is in Ek (TC ). Let tv be the subtree of t which has v as root. If v is not the root of t, it is attached to a node w. Take tv away from t and let t be the resulting tree. Then (v, t) = (w, t ) ◦ (v, tv ). We now show by induction on the number of nodes of (i) |t| and (ii) |tv |, that (v, t) is in Ek (TC ). (i) If the marked tree is (•c , •c ), then it is d(•c ) and is in Ek (TC ). (ii) If v is a root, then  (w, t). d(t) = (v, t) + w=v

The left term is in Ek (TC ), and the right term also by induction. (iii) If v is not a root then both (w, t ) and (v, tv ) are in Ek (TC ) by induction, and so also (v, t).  As a result VC is a TC -module by grafting: s  (v, t) = (v, s  t), where the latter is a sum of pairs (v, ti ) coming from that s  t is a sum of trees ti . The aromas AC also form a TC -module by grafting the trees on all vertices in an aroma. Lastly, the map τ : VC → AC is a TC -module map.

3.2 The Free Pre-Lie Algebra Let L be a pre-Lie algebra over the field k. The pre-Lie algebra TC has the universal property that given any map C → L there is a unique morphism of pre-Lie algebras TC → L such that the diagram below commutes: C

TC

α

L.

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Theorem 3.5 Let L be a tracial pre-Lie–Rinehart algebra over the k-algebra R, τ with trace map End R (L , L) −→ R. α

a. Given a set map ψ : C → L, the unique pre-Lie algebra homomorphism TC −→ L such that α(•c ) = ψ(c) for any c ∈ C induces a unique morphism of associative β

algebras VC −→ E R (L) such that the following diagram commutes: d

TC −−−−→ ⏐ ⏐ α

VC ⏐ ⏐ β

(14)

d

L −−−−→ E R (L). b. Moreover, there is a unique linear map γ extending the diagram (14) to a commutative diagram d

TC −−−−→ ⏐ ⏐ α d

VC ⏐ ⏐ β

τ

−−−−→ AC ⏐ ⏐ γ τ

L −−−−→ E R (L) −−−−→ R. c. These maps fulfill the following for an aroma a, tree t, and φ ∈ VC ⊆ Endk (TC ): i. β(φ)(α(t)) = α(φ(t)), ii. β(t  φ) = α(t)  β(φ), iii. γ(t  a) = α(t)  γ(a). Proof Part a. Any tree (v, t) with one marked point different from the root can be written as (v, t) = (v

, t

) ◦ (v, t ), (15) where the associative product on VC has been described in Sect. 3.1. Here, t is any upper sub tree containing the marked vertex v, and t

is the remaining tree, on which the marked vertex v

comes from the vertex immediately below the root of t . We then proceed by induction on the number of vertices: if t is reduced to the vertex v colored by c ∈ C, we obviously have β(v, t)(x) = dα(•c )(x) = x  α(•c )

(16)

for any x ∈ L. Suppose that the map β has been defined for any tree up to n vertices. Now if t has n + 1 vertices and one marked vertex v different from the root, we must have:   (17) β(v, t) = β (v

, t

) ◦ (v, t ) = β(v

, t

) ◦ β(v, t ). It is easily seen that this does not depend on the choice of the decomposition. Indeed, if v is not the root of t , then (v, t ) = (v , s )(v, s) where s is the subtree with root v,

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and s is the remaining tree inside t . The vertex v comes from the vertex immediately below v in t . We have then t) ◦ β(v, s) β(v, t) = β(v

, t

) ◦ β(v , s ) ◦ β(v, s) = β(v , where t is obtained by grafting s on t

at vertex v

. Hence any decomposition boils down to the unique one with minimal upper tree, for which the marked vertex is the root. Now if t has n + 1 vertices and if the marked vertex is the root, we must define β(v, t) as follows:  β(v , t). (18) β(v, t) = dα(t) − v =v

Part b. The map β is an algebra morphism, hence induces a map β : VC /[VC , VC ] → Hom R (L , L)/[Hom R (L , L), Hom R (L , L)]. The map τ of the bottom line of the diagram being a trace, it induces a map τ : Hom R (L , L)/[Hom R (L , L), Hom R (L , L)] → L. In view of Lemma 3.3, the map γ := τ ◦ β then makes Diagram (14) commute. Part c. We prove first ii. Let t, s be trees and first consider φ = ds. Recall by Proposition 2.11a: t  ds = d(t  s) − ds ◦ dt β(t  ds) = βd(t  s) − βd(s) ◦ βd(t) = dα(t  s) − dα(s) ◦ dα(t) = d(α(t)  α(s)) − dα(s) ◦ dα(t) Again by Proposition 2.11a this equals: α(t)  dα(s) = α(t)  β(ds). Now if part ii holds for φ1 and φ2 it is an immediate computation to verify that   β t  (φ1 ◦ φ2 ) = α(t)  β(φ1 ◦ φ2 ), thus showing part ii. Part i is shown in a similar way. First consider φ = du. Then β(du)(α(t)) = dα(u)(α(t)) = α(t)  α(u) = α(t  u) = α(du(t)).

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Lastly one may show that if i holds for φ1 and φ2 , it holds for their composition. For Part iii, the aroma a is an image τ φ for a marked tree φ. Hence, γ(t  a) =

γ(t  τ φ)

(trace is an L-homomorphism) = =

γτ (t  φ) τ β(t  φ)

(use Part ii.) = τ (α(t)  β(φ)) (trace is an L-homomorphism) = α(t)  τ β(φ) = α(t)  γτ (φ) = α(t)  γ(a). 

3.3 The Pre-Lie–Rinehart Algebra of Aromatic Trees Definition 3.6 Let RC be the vector space freely generated by C-colored directed graphs (not necessarily connected) where each vertex has precisely one outgoing edge. Such a directed graph is a multiset of aromas, and we call it a multi-aroma. The vector space RC has a commutative unital k-algebra structure coming from the monoid structure on multisets of aromas. Note that RC is the symmetric algebra Symk (AC ) on the vector space of C-colored aromas. Remark 3.7 Denote [n] = {1, 2, . . . , n}. In the case of one color, a multi-aroma on n vertices is simply a map f : [n] → [n]. More precisely the multi-aromas identify as orbits of such maps by the action of the symmetric group Sn . Definition 3.8 Denote RC ⊗k TC by L C . As a vector space it has as basis all expressions r ⊗k t where r is a multi-aroma and t is a tree. For short we write this as r t and call it an aromatic tree, [12]. See Fig. 1. On L C we have the product L C × L C −→ L C (qs, r t) −→ ∇qs r t = qs  r t given by grafting the root of the tree s on any vertex of the aromatic tree r t and summing up. Similarly, we can graft an aromatic tree on an aroma. From this, we get induced maps: ∇ : L C → Homk (L C , L C ) r t  → (qs  → r t  qs)

ρ : L C → Derk (RC , RC ) r t  → (q  → r t  q)

d : L C → Hom RC (L C , L C ) r t  → (qs  → qs  r t)

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Fig. 1 An example of aromatic tree made of four aromas and a rooted tree

Proposition 3.9 L C is a pre-Lie-Rinehart algebra over the commutative algebra RC spanned by multi-aromas, with anchor map ρ and connection ∇ defined above. Proof Checking the Leibniz rule for the anchor map and the left pre-Lie relation for  is an easy exercise left to the reader. 

3.4 The Algebra of Marked Aromatic Trees Definition 3.10 Let Eˆ C be the free RC -module spanned by all pairs (v, r t) where r t is an aromatic tree and v is a vertex of r t. It identifies naturally as an RC -submodule of End RC (L C ) by (v, r t) → (u → u v r t). In fact by composition ◦ it is an RC -submodule subalgebra. Extending the map τ of (11) we get an RC -linear map τ : Eˆ C → RC . It maps an element (v, r t) to the multi-aroma we get by joining the root of t to the vertex v. Lemma 3.11 a. Eˆ C is an L C -submodule of End RC (L C ), b. τ is an L C -module map, c. τ is a trace map, i.e., it vanishes on commutators. Proof a. For an aromatic tree r t and a marked aromatic tree α = (v, qs), one obtains r t  α by grafting the root of t on any vertex of qs and by summing up all possibilities, keeping of course the marked vertex v in each term. b. Consider the operations:

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• First graft t on each vertex of qs and then attach the root of s to v. This gives τ (r t  (v, qs)). • First attach the root of s on v and then graft the root of t to any vertex of the resulting aroma. This gives r t  τ (v, qs). We see these operations give the same result, and so τ (r t  (v, qs)) = r t  τ (v, qs). c. Look at τ ((w, r t) ◦ (v, qs)). We get this by 1. Composition ◦: Graft the root of s on w. 2. Map τ : Graft the root of t on v. But the order of these two operations can be switched without changing the result. Hence τ is a trace map. 

3.5 The Algebra of Elementary Endomorphisms Let BC be the vector space freely generated by pairs (v, a) where a is an aroma in AC and v a vertex of a. There is an injection BC → Homk (TC , AC ), (v, a) → (s → s v a), where the tree s is grafted on the vertex v in a. The TC -module structure on AC gives by adjunction an injective linear map d : AC → Homk (TC , AC ), a → (t → t  a). Its image lies in the image of BC . Thus AC may be considered a subspace of BC , identifying da = v∈a (v, a). Let DC be the k-vector space generated by expressions da · t, where a is an aroma and t a tree. It identifies as d AC ⊗k TC . We have compositions: VC ◦ VC → VC

:

DC ◦ VC

:

(v, s) ◦ (w, t) =(w, t w s)  (da · t) ◦ (v, s) = (v, s u a) · t u∈a

(19) VC ◦ DC → DC D C ◦ D C → R C ⊗k D C

: :

(v, s) ◦ (da · t) →da · (t v s) (da · t) ◦ (db · s) →(s  a) · db · t

Proposition 3.12 The RC -submodule E C of Eˆ C generated by DC , VC and their composition DC ◦ VC , is the algebra of elementary endomorphisms E RC (L C ).

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It decomposes as a free RC -module:       E C = RC ⊗k VC R C ⊗k D C RC ⊗k (DC ◦ VC ) .

(20)

Proof By the relations (19) above, we see that E C is an RC -module subalgebra of Eˆ C . Inclusion E RC (L C ) ⊆ E C : Considering the map d: d(r t) = dr · t + r · dt. Looking at the right side of this, the first term is in DC , and the second term in RC ⊗k VC . Inclusion E C ⊆ E RC (L C ): By Lemma 3.4, VC ⊆ E RC (L C ). Since d(at) − adt = da · t, also DC ⊆ E RC (L C ). Free decomposition: An element of RC ⊗k VC has its marks on trees, and so cannot be an R-linear combination of the two other parts. Any element of RC ⊗k DC must have a term with a mark on the interior cycle of an aroma and so cannot be a sum of  terms in DC ◦ VC . By Lemma 3.11 and Proposition 3.12 above we have Corollary 3.13 The pre-Lie–Rinehart algebra L C of aromatic trees is tracial.

4 The Universal Tracial Pre-Lie–Rinehart Algebra We show that the pair (L C , RC ) is a universal tracial pre-Lie-Rinehart algebra. Remark 4.1 Originally we aimed to show that the pair (L C , RC ) was a universal pre-Lie–Rinehart algebra. However from a given map of sets C→L we could not extend this to maps LC → L ,

RC → R.

The problem is that one cannot generate all of L C or RC by starting from C and using the operations Div = τ ◦ d and  applied on the algebra L C , either between aromatic trees s  t or on an aroma s  a. In particular, one cannot generate all of the multi-aromas RC .

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To remedy this we have introduced the subalgebra VC of Endk (TC ) generated by the image of TC → Endk (TC ) together with its trace map τ . From this subalgebra, one can get all aromas by applying the trace map. Furthermore, VC is “fattened up” to the subalgebra EC of End RC (L C ) over RC . To get the universality property, we have therefore introduced the class of tracial pre-Lie–Rinehart algebras. The map γ of Theorem 3.5 extends to γˆ : RC → R given by γ(a ˆ 1 · · · a p ) := γ(a1 ) · · · γ(a p ). The map α of Theorem 3.5 extends to αˆ : L C → L given by ˆ 1 · · · a p )α(t) = γ(a1 ) · · · γ(ai )α(t) α(a ˆ 1 · · · ai t) := γ(a for any a1 , . . . , ai ∈ AC and t ∈ TC ,. Theorem 4.2 (Universality property) Let (L , R) be a tracial pre-Lie–Rinehart algebra, and C → L a map of sets. a. This extends to a unique homomorphism of tracial pre-Lie–Rinehart algebras: (α, ˆ γ) ˆ : (L C , RC ) → (L , R). b. The map β of Theorem 3.5 extends to a homomorphism βˆ of associative algebras giving a commutative diagram d

L C −−−−→ ⏐ ⏐ αˆ 

EC ⏐ ⏐ βˆ 

τ

−−−−→ RC ⏐ ⏐ γˆ 

d

(21)

τ

L −−−−→ E R (L) −−−−→ R. c. It fulfills the following for u ∈ L C and φ ∈ E C : ˆ i. β(φ)( α(u)) ˆ = α(φ(u)), ˆ ˆ  φ) = α(u) ˆ ii. β(u ˆ  β(φ). Proof Part a. We show ai. γˆ is a k-algebra homomorphism, aii. For an aromatic tree r t and a multi-aroma q: γ(r ˆ t  q) = α(r ˆ t)  γ(q). ˆ Note: It is to establish this property that we require the trace map τ to be an L-module homomorphism. aiii. αˆ is a homomorphism of pre-Lie algebras, aiv. For a multi-aroma q and an aromatic tree r t: α(q ˆ · r t) = γ(q) ˆ · α(r ˆ t). av. Uniqueness of αˆ and γ. ˆ

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Property ai is by construction since RC is a free commutative algebra. Property aiv is by definition of α. ˆ Since the action of  of L C on RC is a derivation (the anchor map), it is enough for Property aii to show for an aroma a and tree t that: γ(t  a) = α(t)  γ(a) and this is done in Theorem 3.5. For Property aiii, we have for multi-aromas r, q and trees t, s that   α(r ˆ t  qs) = αˆ r (t  q)s + rq(t  s) ,   = γˆ r (t  q) α(s) + γ(rq)α(t ˆ  s) + γ(r ˆ )γ(q)(α(t) ˆ  α(s)). = γ(r ˆ )(α(t)  γ(q))α(s) ˆ This again equals: α(r ˆ t)  α(qs) ˆ = γ(r ˆ )α(t)  γ(q)α(s). ˆ The uniqueness, Property av, of αˆ is by L C being the free pre-Lie algebra. As for γˆ it is determined by its restriction AC → R. By the requirement of Definition 2.12 and the uniqueness of γ for making a commutative diagram in Theorem 3.5 we see that the γˆ restricted to AC must equal γ. ˆ E C decomposes as a free RC -module (20). We let β(r ˆ φ) = Part b. Definition of β: ˆ when φ is a basis element for these free modules. On VC we let βˆ be given γ(r ˆ )βφ by β. On DC we define ˆ β(da · t) = dα(t) · α(t). Lastly consider the map DC ⊗k VC → DC ◦ VC , where by the latter we mean the vector space spanned by all compositions. This map is a bijection. To see this, consider (19). Note that an element ω in DC ◦ VC has no marked point on the interior cycle of the aroma. Let then v be a marked point in a term of the element ω which has minimal distance from v to the interior cycle. Following the path from v to the interior cycle, the vertex attached to the interior cycle (but not on the cycle), must be the root of a tree s with v ∈ s, which is grafted onto an aroma a. Thus, we have reconstructed a, (v, s) and t and can subtract the image of a multiple of (da · t) ◦ (v, s) from ω. In this way, we may continue and get ω as the image of a unique element in DC ⊗k VC . We may then define βˆ on DC ⊗k VC by   βˆ (da · t) ◦ (v, s) = (dγ(a) · α(t)) ◦ β(v, s).

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Now we show the homomorphism property of βˆ . It respects composition of VC since β does. It respects compositions DC ◦ VC by the above definition. It respects composition DC ◦ DC by     βˆ (da · t) ◦ (db · s) = βˆ (s  a)db · t ˆ = γ(s  a)β(db · t) = γ(s  a)dγ(b) · α(t) = (α(s)  γ(a))dγ(b) · α(t) = (dγ(a) · α(t)) ◦ (dγ(b) · α(s)) ˆ ˆ = β(da · t) ◦ β(db · s). Applying βˆ to the composition VC ◦ DC   βˆ da · (t v s) dγ(a) · α(t v s)   (use Part i. of Theorem 3.5) = dγ(a) · β (v, s))(α(t) .   βˆ (v, s) ◦ (da · t) = =

This map sends u ∈ L C to (u  γ(a)) · β((v, s))(α(t)), and so does the map ˆ ˆ ˆ β((v, s)) ◦ dγ(a) · α(t) = β((v, s)) ◦ β(da · t). So these maps are equal and βˆ respects composition on DC ◦ VC . Part c. i. For φ in VC , this follows easily from Part i in Theorem 3.5. For φ in DC , it is an easy computation. Since βˆ respects compositions, we then derive it for general φ. ii. When φ is in VC this is by Part ii in Theorem 3.5. When φ is in DC we have the following computation using Proposition 2.11b :     βˆ t  (da · s) = βˆ da · (t  s) + d(t  a) · s − (da · s) ◦ dt = dγ(a) · α(t  s) + d(γ(t  a)) · α(s) − dγ(a) · α(s) ◦ βd(t) = dγ(a) · (α(t)  α(s)) + d(α(t)  γ(a)) · α(s) − (dγ(a) · α(s)) ◦ dα(t) (use Proposition 2.11b) = α(t)  (dγ(a) · α(s)) ˆ = α(t)  β(da · s).

Now ii follows by the easily checked fact that it holds for compositions if it holds for each factor. 

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5 Remarks on Equivariance We finally return to some remarks on Theorem 1.1 in the light of the universal diagram (21). Consider the canonical example (L , ) of vector fields on Rd , where R = C ∞ (Rd ). Let C = { } and choose a mapping → f ∈ L inducing the universal ˆ γˆ in (21). Any affine diffeomorphism ξ(x) = Ax + b on Rd induces arrows α, ˆ β, isomorphisms on L, End R (L , L) and R by pullback of tensors: ξ · f := A−1 f ◦ ξ (ξ · G)( f ) := ξ · (G(ξ · f )) ξ · r := r ◦ ξ for f ∈ L, G ∈ End R (L , L) and r ∈ R. Given three finite series B L ∈ L C , B E ∈ EC , B R ∈ RC we obtain three mappings ˆ L): L → L  L ( f ) := α(B ˆ E ) : L → End R (L , L)  E ( f ) := β(B ˆ R ) : L → R.  R ( f ) := β(B It is straightforward to check that these are all with respectto the action  equivariant   (ξ · f ) = ξ ·  ( f ) ,  (ξ( f )) = ξ ·  E ( f ) and of affine diffeomorphisms:  L E   L  R (ξ( f )) = ξ ·  R ( f ) . Theorem 1.1 states that any smooth local affine equivariant mapping  : L → L has an aromatic B-series B L ∈ L C , where the overline denotes the graded completion, i.e., the space of formal infinite series. The proof technique [12], seems to work also for smooth local mappings between different tensor bundles. Hence, we claim: Claim 5.1 A smooth, local mapping  E : L → End R (L , L) has an aromatic series B E ∈ EC if and only it is affinely equivariant. A smooth, local mapping  R : L → R has an aromatic series B R ∈ RC if and only it is affinely equivariant. Subject to convergence, the mappings are represented by their aromatic B-series. Acknowledgements The second author thanks Universitetet i Bergen for the warm welcome and stimulating atmosphere during his two visits in September 2018 and May 2019, and Trond Mohn Foundation for support. He also thanks Camille Laurent-Gengoux for an illuminating e-mail discussion on Lie algebroids.

References 1. Geir Bogfjellmo, Algebraic structure of aromatic B-series. Journal of Computational Dynamics 6(2), 199 (2019) 2. J.C. Butcher, Coefficients for the study of Runge-Kutta integration processes. J. Aust. Math. Soc. 3(2), 185–201 (1963)

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3. J.C. Butcher, An algebraic theory of integration methods. Math. Comput. 26(117), 79–106 (1972) 4. A. Cayley, On the theory of the analytical forms called trees. Lon. Edinburgh Dublin Philosoph. Mag. J. Sci. 13(85), 172–176 (1857) 5. F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 2001(8), 395–408 (2001) 6. P. Chartier, A. Murua, Preserving first integrals and volume forms of additively split systems. IMA J. Numer. Anal. 27(2), 381–405 (2007) 7. A. Dzhumadil’daev, C. Löfwall, Trees, free right-symmetric algebras, free Novikov algebras and identities. Homol. Homot. Appl. 4(2), 165–190 (2002) 8. J. Huebschmann, Poisson cohomology and quantization. J. f.d. Reine u. Angew. Math. 408, 57–113 (1990) 9. J. Huebschmann, Twilled Lie-Rinehart algebras and differential Batalin-Vilkovisky algebras (1998). arXiv:math/9811069 10. A. Iserles, G.R.W. Quispel, P.S.P. Tse, B-series methods cannot be volume-preserving. BIT Num. Math. 47(2), 351–378 (2007) 11. R.I. McLachlan, K. Modin, H. Munthe-Kaas, O. Verdier, B-series methods are exactly the affine equivariant methods. Numerische Mathematik 133(3), 599–622 (2016) 12. H. Munthe-Kaas, O. Verdier, Aromatic Butcher series. Found. Comput. Math. 16(1), 183–215 (2016) 13. H.Z. Munthe-Kaas, A. Stern, O. Verdier, Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds. SIAM J. Appl. Algebra Geom. 4(1), 49–68 (2020) 14. G.S. Rinehart, Differential forms on general commutative algebras. Trans. Amer. Math. Soc. 108(2), 195–222 (1963)

Variants of Confluent q-Hypergeometric Equations Ryuya Matsunawa, Tomoki Sato, and Kouichi Takemura

Abstract Variants of the q-hypergeometric equation were introduced in our previous paper with Hatano. In this paper, we consider degenerations of the variant of the q-hypergeometric equation, which is a q-analogue of confluence of singularities in the setting of the differential equation. We also consider degenerations of solutions to the q-difference equations. Keywords q-hypergeometric equation · Confluence of singularities · Degeneration · Series solution · Hypergeometric function 2010 Mathematics Subject Classification. 33D15 · 39A13

1 Introduction The special functions have rich mathematical structures, and some of them have been applied to physics. Gauss’ hypergeometric function 2 F1 (α, β; γ; z)

=1+

α(α + 1)β(β + 1) 2 αβ (α)n (β)n n z+ z + ··· + z + ··· γ 2! γ(γ + 1) n!(γ)n (1.1)

is a typical example of the special functions. Here (λ)n = λ(λ + 1) . . . (λ + n − 1). It is essentially characterized by Gauss’ hypergeometric equation

R. Matsunawa · T. Sato Department of Mathematics, Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan K. Takemura (B) Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_10

161

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z(1 − z)

d2 y dy − αβ y = 0. + (γ − (α + β + 1)z) dz 2 dz

(1.2)

It is a second-order Fuchsian differential equation with three singularities {0, 1, ∞}. Here, the Fuchsian differential equation is a linear differential equation whose singularity on the Riemann sphere is always the regular singularity. However, several special functions are related with the differential equation which may not be Fuchsian. For example, Kummer’s confluent hypergeometric function 1 F1 (α; γ; z)

=1+

α(α + 1) 2 α (α)n n z+ z + ··· + z + .... γ 2! γ(γ + 1) n!(γ)n

(1.3)

satisfies the differential equation z

d2 y dy − αy = 0, + (γ − z) dz 2 dz

(1.4)

which has an irregular singularity at z = ∞. Equation (1.4) is called Kummer’s differential equation or the confluent hypergeometric differential equation. It is widely known that Kummer’s function and Kummer’s differential equation are obtained by confluence of the singularity of Gauss’ hypergeometric equation. Namely, we replace the variable z in Gauss’ hypergeometric equation with z/β and take the limit β → ∞. Then, the singularity z = 1 of Gauss’ hypergeometric equation merges into the singularity z = ∞ and we obtain the confluent equation. We can also consider the confluence process from Kummer’s differential equation. We set z = u 1 x + u 2 , u 2 = u 21 /2, γ = u 2 and α = −λ/2, and take the limit u 1 → ∞. Then, we obtain the Hermite-Weber equation; dy d2 y + λy = 0. − 2x dx2 dx

(1.5)

By setting y = e x /2 u, it is transformed to the differential equation −u  + x 2 u = (1 + λ)u, which is related with the quantum harmonic oscillator. The q-analogue of the hypergeometric functions has been studied well from the nineteenth century. Heine’s basic hypergeometric series was introduced as 2

2 φ1 (a, b; c; x) =

∞  (a; q)n (b; q)n n=0

(q; q)n (c; q)n

x n , (λ; q)n =

n−1 

(1 − λq i ).

(1.6)

i=0

It satisfies the equation (x − q) f (x/q) − ((a + b)x − q − c) f (x) + (abx − c) f (q x) = 0.

(1.7)

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It is known that Gauss’ hypergeometric differential equation (1.2) is obtained from Eq. (1.7) by the limit q → 1, and the differential equation has singularities at x = 0, 1, ∞. In this paper, we investigate a q-analogue of confluence processes which is related with the hypergeometric equations. In particular, we consider the confluence processes from the variant of q-hypergeometric equation of degree two, which was introduced in [3] as (x − q h 1 +1/2 t1 )(x − q h 2 +1/2 t2 )g(x/q) +q

α1 +α2

− [(q

α1

(x − q α2

l1 −1/2

t1 )(x − q

l2 −1/2

+ q )x + E x + p(q 2

p = q (h 1 +h 2 +l1 +l2 +α1 +α2 )/2 ,

1/2

(1.8)

t2 )g(q x)

+ q −1/2 )t1 t2 ]g(x) = 0,

E = − p{(q −h 2 + q −l2 )t1 + (q −h 1 + q −l1 )t2 }.

By taking the limit q → 1, we essentially obtain the second-order Fuchsian differential equation with three singularities {t1 , t2 , ∞} (see [3]). Recall that Hahn [2] introduced a q-difference analogue of Heun’s differential equation of the form {a2 x 2 + a1 x + a0 }g(x/q) − {b2 x 2 + b1 x + b0 }g(x)

(1.9)

+ {c2 x + c1 x + c0 }g(xq) = 0, 2

with the condition a2 a0 c2 c0 = 0, and it was rediscovered in [6] by considering degenerations of the Ruijsenaars-van Diejen system. We call Eq. (1.9) the q-Heun equation. Then, Eq. (1.8) is a specialization of the q-Heun equation, and it is characterized by the condition that the difference of the exponents at the origin x = 0 is one and the singularity x = 0 is apparent (see [3, 7]). We consider the degeneration of the variant of q-hypergeometric equation of degree two such that the polynomial q α1 +α2 (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 ) tends to a linear polynomial. We take the limit q α2 → 0 formally in Eq. (1.8) with the condition that α2 + l2 = h 1 + h 2 − l1 − α1 + 1 − 2λ is fixed. Then, we have q h 1 +h 2 −l1 −2λ+1/2 t2 (q l1 −1/2 t1 − x)g(q x) + (x − q α1 2

h 1 +1/2

− [q x − q

t1 )(x − q

h 2 +1/2

h 1 +h 2 −λ+1/2

(q

−h 2

(1.10)

t2 )g(x/q) t1 + q −h 1 t2 + q −l1 t2 )x

+ q h 1 +h 2 −λ (q + 1)t1 t2 ]g(x) = 0. We may regard the parameter λ in Eq. (1.10) to be independent from the other parameters h 1 , h 2 , l1 , α1 , t1 and t2 . We call Eq. (1.10) a variant of the singly confluent q-hypergeometric equation or the variant of the confluent q-hypergeometric equation of type (1, 2). By taking the limit q → 1, we essentially obtain Kummer’s confluent hypergeometric equation although the position of the regular singularity is deformed to x = t1 (see Sect. 7 for details).

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We consider further degeneration. We take the limit q −l1 → 0 formally in Eq. (1.10). Then we have the following equation; g(q x) + q 2λ+1 (1 − q −h 1 −1/2 t1−1 x)(1 − q −h 2 −1/2 t2−1 x)g(x/q) −

[q α1 +2λ−h 1 −h 2 t1−1 t2−1 x 2

−q

λ+1/2

(q −h 1 t1−1

+

q −h 2 t2−1 )x

(1.11) λ

+ q (q + 1)]g(x) = 0.

We call Eq. (1.11) a variant of the biconfluent q-hypergeometric equation or the variant of the confluent q-hypergeometric equation of type (0, 2). By taking the limit q → 1, we essentially obtain the Hermite-Weber differential equation (see Sect. 7). Next we investigate solutions of the q-difference equations. It was discovered in [3] that the variant of q-hypergeometric equation of degree two (Eq. (1.8)) has several explicit formal solutions, and we describe them in Proposition 2.1. For example, the function g(x) = x λ

∞ 

 cn

x q l1 −1/2 t1

n=0

;q

 n

, cn =

(q λ+α1 ; q)n (q λ+α2 ; q)n q n (q h 1 −l1 +1 ; q)n (q h 2 −l1 +1 t2 /t1 ; q)n (q; q)n (1.12)

is a formal solution of Eq. (1.8), where λ = (h 1 + h 2 − l1 − l2 − α1 − α2 + 1)/2. Here, the formal solution means that the coefficients of the solution (e.g., cn in Eq. (1.12)) are determined recursively. On the degeneration to the variant of the singly confluent q-hypergeometric equation (Eq. (1.10)) as q α2 → 0, we can also take the limit of the solutions of Eq. (1.8), and we obtain several explicit formal solutions of Eq. (1.10) (see theorems in Sect. 3). For example, the function g(x) = x λ

∞  n=0

 cn

x q l1 −1/2 t1

;q

 n

, cn =

(q λ+α1 ; q)n q n (q h 1 −l1 +1 ; q)n (q h 2 −l1 +1 t2 /t1 ; q)n (q; q)n (1.13)

is a solution of Eq. (1.10), in which the term (q λ+α2 ; q)n in Eq. (1.12) is replaced with 1 by the limit q α2 → 0. We can also several explicit formal solutions of Eq. (1.11) by the limit q −l1 → 0 in Eq. (1.10) (see theorems in Sect. 4). On this research, we eventually found new solutions of Eq. (1.8), which we note in Theorem 2.2. We can also consider the limits of the functions in Theorem 2.2 and we obtain several explicit formal solutions of Eq. (1.10) and those of Eq. (1.11). Recall that the coefficient of g(q x) in the variant of the singly confluent qhypergeometric equation (i.e., Eq. (1.10)) is a linear polynomial. We can also consider the degeneration that the coefficient of g(x/q) tends to a linear polynomial. We perform it in Sect. 5, and we obtain another variant of the singly confluent qhypergeometric equation (see Eq. (5.1)) together with several formal solutions of that. We also obtain another variant of the biconfluent q-hypergeometric equation (see Eq. (5.8)) and several formal solutions. It is seen in Sect. 6 that another variant of the singly confluent q-hypergeometric equation (resp. another variant of the bicon-

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fluent q-hypergeometric equation) is transformed to Eq. (1.10) (resp. Eq. (1.11)) by multiplying a scalar function (a gauge factor). Then, we obtain a solution of Eq. (1.10) (resp. Eq. (1.11)) by multiplying a solution of another variant of the singly confluent q-hypergeometric equation (resp. another variant of the biconfluent q-hypergeometric equation) by the gauge factor. This paper is organized as follows. In Sect. 2, we review some solutions of the variant of q-hypergeometric equation of degree two (Eq. (1.8)) and we obtain new solutions. In Sect. 3, we obtain some solutions of the variant of the singly confluent q-hypergeometric equation (Eq. (1.10)). In Sect. 4, we obtain some solutions of the variant of the singly confluent q-hypergeometric equation (Eq. (1.11)). In Sect. 5, we discuss other variants of the singly confluent q-hypergeometric equation and the biconfluent q-hypergeometric equation. In Sect. 6, we discuss relationships between another variant of the singly confluent q-hypergeometric equation (resp. another variant of the biconfluent q-hypergeometric equation) and Eq. (1.10) (resp. Eq. (1.11)), and we give applications to the solutions. In Sect. 7, we consider the limits to the singly confluent differential equation of Kummer and the biconfluent differential equation of Hermite-Weber as q → 1. In Sect. 8, we give concluding remarks.

2 Solutions to the Variant of q-Hypergeometric Equation of Degree Two Recall that the variant of q-hypergeometric equation of degree two was given in Eq. (1.8), i.e., (x − q h 1 +1/2 t1 )(x − q h 2 +1/2 t2 )g(x/q) + q α1 +α2 (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 )g(q x) − [(q α1 + q α2 )x 2 + E x + p(q 1/2 + q −1/2 )t1 t2 ]g(x) = 0, p = q (h 1 +h 2 +l1 +l2 +α1 +α2 )/2 ,

E = − p{(q −h 2 + q −l2 )t1 + (q −h 1 + q −l1 )t2 }.

In this paper, we do not consider convergence of solutions of the q-difference equations, and we treat solutions formally. If the infinite summation in the formal solution terminates as a finite summation, then it is the exact solution. Several solutions of the variant of q-hypergeometric equation of degree two were obtained in [3] as follows: Proposition 2.1 ([3]) Let λ = (h 1 + h 2 − l1 − l2 − α1 − α2 + 1)/2. (i) The function g(x) =x −α1

∞  n=0

·

(q 1/2 x −1 )n

(q λ+α1 ; q)n (q α1 −α2 +1 ; q)n

n  (q λ+α1 −h 2 +l2 ; q)k (q λ+α1 −h 1 +l1 ; q)n−k k=0

(q; q)k (q; q)n−k

(2.1) (q l1 t1 )k (q l2 t2 )n−k

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is a solution of the variant of q-hypergeometric equation of degree two (i.e., Eq. (1.8)). (ii) Set (i, i  ) = (1, 2) or (2, 1). Then, the function ∞  



(q λ+α1 ; q)n (q λ+α2 ; q)n qn h i  −li +1 t  /t ; q) (q; q) (q n i i n n n=0    x = x λ 3 φ2 q λ+α1 , q λ+α2 , l −1/2 ; q h i −li +1 , q h i  −li +1 ti  /ti ; q, q qi ti

g(x) = x λ

x

q li −1/2 ti

;q

(2.2)

n (q h i −li +1 ; q)

is a solution of Eq. (1.8). (iii) Set (i, i  ) = (1, 2) or (2, 1). Then the function g(x) = x −α1

∞  h i +1/2  q t

i

n=0

·

n  k=0

x

;q



(q λ+α1 ; q)n qn h n (q i −li  +1 ti /ti  ; q)n

(2.3)

(q λ−h i  +li  +α1 ; q)k k(k+1)/2 h i −li  k ) q (−q t /t i i (q h i −li +1 ; q)k (q; q)k (q; q)n−k

is a solution of Eq. (1.8). Note that the functions which are obtained by replacing α1 with α2 are also solutions of the variant of q-hypergeometric equation of degree two. We discovered new solutions of Eq. (1.8) as follows, which can be confirmed similarly to Proposition 2.1 established in [3]. Theorem 2.2 Let λ = (h 1 + h 2 − l1 − l2 − α1 − α2 + 1)/2. Set (i, i  ) = (1, 2) or (2, 1). (i) The function g(x) = x λ

∞ 

(q λ+α1 ; q)n (q λ+α2 ; q)n h −l +1 i i (q ; q)n (q h i −li  +1 ti /ti  ; q)n (q; q)n n=0

 q h i +1/2 t

i

x

;q

 

x

n

q h i  −1/2 ti 

n

   q h i +1/2 ti h i −li +1 h i −l  +1 x i = x λ 3 φ2 q λ+α1 , q λ+α2 , ;q ,q ti /ti  ; q, h  −1/2 x q i ti 

(2.4)

is a solution of Eq. (1.8). (ii) The function g(x) = x α1

∞ 

cn

n=0

cn =

n  k=0

q −nk+k

2

 x (q λ+α1 ; q)n 1+h −l  (q i i ti  /ti ; q)n q li −1/2 ti

/2

;q

  q −λ−α1 +h i  +1/2 t  n i , (2.5) n x

 q −λ−α1 +h i +1/2 t k (q λ+α1 −h i  +li  ; q)k i − (q h i −li +1 ; q)k (q; q)n−k (q; q)k q li  ti 

is a solution of Eq. (1.8).

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167

The functions which are obtained by replacing α1 with α2 are also solutions of the variant of q-hypergeometric equation of degree two. Note that solutions in Theorem 2.2 correspond to the ones obtained by replacing the role of q with q −1 in Proposition 2.1. If λ + α1 is a negative integer and set N = −λ − α1 , then the summation of each solution is finite, and the solution is a product of x λ and the polynomial of the variable x of degree N − 1. The polynomial essentially coincides with the Big q-Jacobi polynomial (cf. [1, 4]).

3 Singly Confluent Limit In the introduction, we introduced a variant of the singly confluent q-hypergeometric equation or the variant of the confluent q-hypergeometric equation of type (1, 2) as Eq. (1.10), i.e., q h 1 +h 2 −l1 −2λ+1/2 t2 (q l1 −1/2 t1 − x)g(q x) + (x − q h 1 +1/2 t1 )(x − q h 2 +1/2 t2 )g(x/q) − [q α1 x 2 − q h 1 +h 2 −λ+1/2 (q −h 2 t1 + q −h 1 t2 + q −l1 t2 )x + q h 1 +h 2 −λ (q + 1)t1 t2 ]g(x) = 0. By taking the limit q → 1, we essentially obtain the second-order linear differential equation with one regular singularity x = t1 and one irregular singularity x = ∞ (see Eq. (7.4)). Recall that Eq. (1.10) was obtained by the limit q α2 → 0 from the variant of qhypergeometric equation of degree two (i.e., Eq. (1.8)). We can obtain solutions of a variant of singly confluent q-hypergeometric equation by considering the limit of the solutions of the variant of q-hypergeometric equation. By taking the limit q α2 → 0 in Proposition 2.1, we may obtain solutions of Eq. (1.10). Theorem 3.1 (i) The function g(x) = x −α1

∞  (q −λ−α1 +h 1 +1/2 t1 x −1 )n (q λ+α1 ; q)n

(3.1)

n=0

·

n  (q λ+α1 −h 1 +l1 ; q) =0

is a solution of Eq. (1.10).

(q; q)n− (q; q)

q −(2n−−1)/2 (−q −λ−α1 −l1 +h 2 t2 /t1 )

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(ii) The function g(x) = x λ

∞ 

qn

n=0

(q λ+α1 ; q)n h −l +1 1 1 (q ; q)n (q h 2 −l1 +1 t2 /t1 ; q)n (q; q)n

  = x λ 3 φ2 q λ+α1 , 0,

x

q l1 −1/2 t1  ; q h 1 −l1 +1 , q h 2 −l1 +1 t2 /t1 ; q, q

x q l1 −1/2 t



 ;q

(3.2) n

1

is a solution of Eq. (1.10). (iii) The function g(x) = x −α1

∞  h 1 +1/2  q t n=0

x

1

;q

 n

(q λ+α1 ; q)n q n

n  k=0

q k (q λ+α1 +h 1 −h 2 t1 /t2 )k (q h 1 −l1 +1 ; q)k (q; q)k (q; q)n−k 2

(3.3) is a solution of Eq. (1.10). (iv) The function g(x) = x −α1

∞  h 2 +1/2  q t

x

n=0

·

2

;q



q n (q λ+α1 ; q)n n (q h 2 −l1 +1 t2 /t1 ; q)n

n  (q λ−h 1 +l1 +α1 ; q)k k=0

(q; q)k (q; q)n−k

q k(k+1)/2 (−q h 2 −l1 t2 /t1 )k

(3.4)

is a solution of Eq. (1.10). Proof We show (ii). It was shown in [3] that the function g(x) = x −α1

∞  n=0

 an

x q l1 −1/2 t1

 ;q

(3.5) n

is a solution of Eq. (1.8), if the coefficient an satisfies (1 − q h 1 −l1 +n+1 )(1 − q h 2 −l1 +n+1 t2 /t1 )(1 − q n+1 )an+1 − q(1 − q λ+α1 +n )(1 − q λ+α2 +n )an − q 3 (1 + q −1 )(1 − q h 1 −l1 +n )(1 − q h 2 −l1 +n t2 /t1 )(1 − q n )an + q 4 (1 + q −1 )(1 − q λ+α1 +n−1 )(1 − q λ+α2 +n−1 )an−1 + q 5 (1 − q h 1 −l1 +n−1 )(1 − q h 2 −l1 +n−1 t2 /t1 )(1 − q n−1 )an−1 − q 6 (1 − q λ+α1 +n−2 )(1 − q λ+α2 +n−2 )an−2 = 0 for all integer n, where an = 0 for n < 0, and the sequence

(3.6)

Variants of Confluent q-Hypergeometric Equations

an = q n

169

(q λ+α1 ; q)n (q λ+α2 ; q)n (n ≥ 0) (q h 1 −l1 +1 ; q)n (q h 2 −l1 +1 t2 /t1 ; q)n (q; q)n

(3.7)

satisfies the recursive relation. Thus, we recover Proposition 2.1 (ii) for the case (i, i  ) = (1, 2). By taking the limit q α2 → 0, the function g(x) of the form in Eq. (3.5) is a solution of Eq. (1.10), if the coefficient an satisfies (1 − q h 1 −l1 +n+1 )(1 − q h 2 −l1 +n+1 t2 /t1 )(1 − q n+1 )an+1 − q(1 − q

λ+α1 +n

)an − q (1 + q 3

+ q (1 + q

−1

+ q (1 − q

h 1 −l1 +n−1

4 5

)(1 − q

λ+α1 +n−1

)(1 − q

−1

)(1 − q

h 1 −l1 +n

)(1 − q

(3.8) h 2 −l1 +n

t2 /t1 )(1 − q )an n

)an−1

h 2 −l1 +n−1

t2 /t1 )(1 − q n−1 )an−1

− q 6 (1 − q λ+α1 +n−2 )an−2 = 0 for all integer n, where an = 0 for n < 0, and it is shown by taking the limit q α2 → 0 that the sequence an = q n

(q λ+α1 ; q)n (n ≥ 0) (q h 1 −l1 +1 ; q)n (q h 2 −l1 +1 t2 /t1 ; q)n (q; q)n

(3.9)

satisfies the recursive relation. Namely, we obtained (ii) by taking the limit of the function in Proposition 2.1 (ii) for the case (i, i  ) = (1, 2) as q α2 → 0. We can obtain Theorem (i) by taking the limit of the function in Proposition 2.1 (i) as q α2 → 0. We can also obtain (iii) (resp. (iv)) by taking the limit of the function in Proposition 2.1 (iii) for the case (i, i  ) = (1, 2) (resp. for the case (i, i  ) = (2, 1))  as q α2 → 0. By considering the limit of the functions in Theorem 2.2 (i) and Theorem 2.2 (ii) for the case (i, i  ) = (1, 2), we may guess the solutions of Eq. (1.10) and we can actually establish the following theorem. Theorem 3.2 (i) Set (i, i  ) = (1, 2) or (2, 1). Then, the function    q h i +1/2 t n (q λ+α1 ; q)n x i ; q n q h i  −1/2 ti  (q h i −l1 +1 ti /t1 ; q)n (q; q)n x n=0    q h i +1/2 ti h i −l1 +1 x ;q = x λ 2 φ1 q λ+α1 , ti /t1 ; q, h  −1/2 x q i ti 

g(x) = x λ

∞ 

(3.10)

is a solution of Eq. (1.10). (ii) The function g(x) = x −α1

∞  n=0

cn

   q −λ−α1 +h 2 +1/2 t n (q λ+α1 ; q)n x 2 ;q , l −1/2 1 n /t ; q) q t x 2 1 n 1

(q 1+h 2 −l1 t

(3.11)

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cn =

n 

q −nk+k

2

k=0

(q h 1 −l1 +1 ; q)

1 (q h 1 −h 2 t1 /t2 )k k (q; q)n−k (q; q)k

is a solution of Eq. (1.10). Note that, by the limit of the function in Theorem 2.2 (ii) for the case (i, i  ) = (2, 1) as q α2 → 0, we obtain Eq. (3.1). If λ + α1 is a negative integer and set N = −λ − α1 , then the summation of each solution is finite, and the solution is a product of x λ and the polynomial of the variable x of degree N − 1. The polynomial essentially coincides with the Big q-Laguerre polynomial (cf. [4]).

4 Biconfluent Limit In the introduction, we introduced a variant of the biconfluent q-hypergeometric equation or the variant of the confluent q-hypergeometric equation of type (0, 2) as Eq. (1.11), i.e., g(q x) + q 2λ+1 (1 − q −h 1 −1/2 t1−1 x)(1 − q −h 2 −1/2 t2−1 x)g(x/q) − [q α1 +2λ−h 1 −h 2 t1−1 t2−1 x 2 − q λ+1/2 (q −h 1 t1−1 + q −h 2 t2−1 )x + q λ (q + 1)]g(x) = 0. The equation has the symmetry of replacing (t1 , h 1 ) with (t2 , h 2 ). By taking the limit q → 1, we obtain the differential equation in Eq. (7.9). Equation (1.11) was obtained by the limit q −l1 → 0 from a variant of singly confluent q-hypergeometric equation (i.e., Eq. (1.10)). We can obtain solutions of the variant of biconfluent q-hypergeometric equation by considering the limit of the solutions of the variant of singly confluent q-hypergeometric equation. Theorem 4.1 (i) The function x −α1

∞ n   q −(n−) (q h 1 t1 )n− (q h 2 t2 ) (q −λ−α1 +1/2 x −1 )n (q λ+α1 ; q)n (q; q)n− (q; q) n=0 =0

(4.1)

is a solution of Eq. (1.11). (ii) Set (i, i  ) = (1, 2) or (2, 1). Then, the function x −α1

∞  h i +1/2  q t

i

n=0

x

;q

 n

(q λ+α1 ; q)n q n

2 n  q k (q λ+α1 +h i −h i  ti /ti  )k

k=0

is a solution of Eq. (1.11). (iii) Set (i, i  ) = (1, 2) or (2, 1). Then, the function

(q; q)k (q; q)n−k

(4.2)

Variants of Confluent q-Hypergeometric Equations

171

 (q λ+α1 ; q)  n x n n x (q; q)n q h i  −1/2 ti  n=0    x q h i +1/2 ti ; 0; q, h  −1/2 = x λ 2 φ1 q λ+α1 , x q i ti 

g(x) = x λ

∞  h i +1/2  ti q

;q

(4.3)

is a solution of Eq. (1.11). Note that the function in Theorem 4.1 (i) (resp. (ii), (iii)) is obtained by taking the limit of the function in Eq. (3.1) (resp. Eq. (3.3) or Eq. (3.4), Eq. (3.10)) as q −l1 → 0. If λ + α1 is a negative integer and set N = −λ − α1 , then the summation of each solution is finite, and the solution is a product of x λ and the polynomial of the variable x of degree N − 1. The polynomial essentially coincides with the Al-Salam-Carlitz I polynomial (cf. [4]).

5 Other Confluences We consider other confluence processes such that the degree of the polynomial on the coefficient of g(x/q) decreases. We take the limit q −α2 → 0 in Eq. (1.8) with the condition that the value h 2 − α2 = −h 1 + l1 + l2 + α1 − 1 + 2λ is fixed. Then we have (x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 )g(q x) −q

−h 1 +l1 +l2 +2λ−1/2

−[q

−α1 2

+q

x −q

l1 +l2 +λ

(5.1)

h 1 +1/2

t2 (x − q t1 )g(x/q)

−l2  q t1 + q −h 1 + q −l1 t2 x

l1 +l2 +λ−1/2

(1 + q −1 )t1 t2 ]g(x) = 0.

We may regard the parameter λ to be independent from the other parameters h 1 , h 2 , l1 , α1 , t1 , and t2 . We call it another variant of the singly confluent q-hypergeometric equation or the variant of the confluent q-hypergeometric equation of type (2, 1). We can obtain solutions of the variant of the confluent q-hypergeometric equation of type (2, 1) by considering the limit of the solutions of the variant of qhypergeometric equation. Theorem 5.1 (i) Set (i, i  ) = (1, 2) or (2, 1). Then, the function xλ

∞  (q −λ−α1 +h 1 −li  +1 t1 /ti  )n n=0

  = x λ 2 φ1 q λ+α1 ,

x q li −1/2 ti

(q λ+α1 ; q)n h −l +1 (q 1 i t1 /ti ; q)n (q; q)n



x q li −1/2 ti

; q h 1 −li +1 t1 /ti ; q, q −λ−α1 +h 1 −li  +1 t1 /ti 

 ;q 

(5.2) n

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is a solution of Eq. (5.1). (ii) The function g(x) =x −α1

∞ n   (q λ+α1 −h 1 +l1 ; q)n−k l1 k l2 n−k (q 1/2 x −1 )n (q λ+α1 ; q)n (q t1 ) (q t2 ) (q; q)k (q; q)n−k n=0 k=0

(5.3) is a solution of Eq. (5.1). (iii) The function g(x) = x −α1

∞  (q h 1 +1/2 t1 /x; q)n n=0

(q λ+α1 ; q)n h (q 1 −l2 +1 t1 /t2 ; q)n

qn

n  k=0

q k(k+1)/2 (−q h 1 −l2 t1 /t2 )k k (q; q)k (q; q)n−k

(q h 1 −l1 +1 ; q)

(5.4) is a solution of Eq. (5.1). We can obtain the functions in Theorem 5.1 (i) (resp. (ii), (iii)) by taking the limit of the function in Proposition 2.1 (ii) (resp. Proposition 2.1 (i), Proposition 2.1 (iii) for the case (i, i  ) = (1, 2)) as q −α2 → 0. We can also obtain the functions in Theorem 5.1 (ii) by taking the limit of the function in Proposition 2.1 (iii) for the case (i, i  ) = (2, 1). Theorem 5.2 (i) The function g(x) = x λ 

∞  h 1 +1/2  t1 q n=0

x

;q

 (q λ+α1 ; q) q n(n−1)/2 (−q −λ−α1 −l1 −l2 +h 1 +3/2 x/t )n n 2 n (q h 1 −l1 +1 ; q)n (q h 1 −l2 +1 t1 /t2 ; q)n (q; q)n

(5.5)

  q h 1 +1/2 t1 h 1 −l1 +1 h 1 −l2 +1 = x λ 2 φ2 q λ+α1 , ;q ,q t1 /t2 ; q, q −λ−α1 −l1 −l2 +h 1 +3/2 x/t2 x

is a solution of Eq. (5.1). (ii) The function g(x) = x −α1

∞ 

 cn

n=0

cn =

n  k=0

q −nk+k

2

/2

x q l1 −1/2 t1

;q

(q h 1 −l1 +1 ; q)

is a solution of Eq. (5.1). (iii) The function

  n

−

q −λ−α1 +l1 t1 n λ+α1 2 (q ; q)n q −n /2 , x

 q −λ−α1 +h 1 +1/2 t k 1 1 − l2 t (q; q) (q; q) q k n−k k 2

(5.6)

Variants of Confluent q-Hypergeometric Equations

g(x) = x −α1

∞ 

 cn

n=0

cn =

n  k=0

q −nk

x q l2 −1/2 t2

;q

173

  q −λ−α1 +h 1 +1/2 t n (q λ+α1 ; q)n 1 , (5.7) n x (q 1+h 1 −l2 t1 /t2 ; q)n

(q λ+α1 −h 1 +l1 ; q)k  q −λ−α1 +l2 t2 k (q; q)n−k (q; q)k q l 1 t1

is a solution of Eq. (5.1). The function in Theorem 5.2 (i) (resp. (ii), (iii)) appears as the limit of the function in Theorem 2.2 (i) for the case (i, i  ) = (1, 2) (resp. Theorem 2.2 (ii) for the case (i, i  ) = (1, 2), Theorem 2.2 (ii) for the case (i, i  ) = (2, 1)) as q −α2 → 0. To obtain a variant of another biconfluent q-hypergeometric equation, we take the limit q −h 1 → 0 in Eq. (5.1). Then, we have q −2λ−1 (1 − q −l1 +1/2 t1−1 x)(1 − q −l2 +1/2 t2−1 x)g(q x) + g(x/q) − [q −α1 −l1 −l2 −2λ t1−1 t2−1 x 2

−q

−λ−1/2

(q −l1 t1−1

+ q −l2 t2−1 )x

+q

(5.8) −λ

(1 + q

−1

)]g(x) = 0.

We call it another variant of the biconfluent q-hypergeometric equation or the variant of the confluent q-hypergeometric equation of type (2, 0). The equation has the symmetry of replacing (t1 , l1 ) with (t2 , l2 ). By taking the limit q → 1, we essentially obtain the Hermite-Weber differential equation (see Sect. 7). We can obtain solutions of the variant of the confluent q-hypergeometric equation of type (2, 0) by considering the limit of the solutions of the variant of the confluent q-hypergeometric equation of type (2, 1). Theorem 5.3 (i) Set (i, i  ) = (1, 2) or (2, 1). Then, the function   ∞  x q −n(n−1)/2 (q λ+α1 ; q)n (−q −λ−α1 +li −li  ti /ti  )n ; q (q; q)n q li −1/2 ti n n=0    x = x λ 2 φ0 q λ+α1 , l −1/2 ; −; q, q −λ−α1 +li −li  ti /ti  qi ti xλ

(5.9)

is a solution of Eq. (5.8). (ii) The function g(x) =x −α1

∞ n   (q l1 t1 )k (q l2 t2 )n−k (q 1/2 x −1 )n (q λ+α1 ; q)n (q; q)k (q; q)n−k n=0 k=0

is a solution of Eq. (5.8). (iii) Set (i, i  ) = (1, 2) or (2, 1). Then, the function

(5.10)

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g(x) = x −α1

∞  n=0

cn =

n  k=0

q −nk

 cn

x q li −1/2 ti

;q

  n

−

q −λ−α1 +li ti n λ+α1 2 (q ; q)n q −n /2 , x

1

 q −λ−α1 +li −li  t k

(q; q)n−k (q; q)k

ti 

(5.11)

i

is a solution of Eq. (5.8). We can obtain the function in Theorem 5.3 (i) (resp. Theorem 5.3 (ii)) by taking the limit of the function in Eq. (5.2) (resp. Eq. (5.3)) as q −h 1 → 0. We can also obtain the function in Theorem 5.3 (iii) for the case (i.i  ) = (1, 2) (resp. the case (i.i  ) = (2, 1)) by taking the limit of the function in Eq. (5.6) (resp. Eq. (5.7)) as q −h 1 → 0.

6 Gauge Transformation We investigate the gauge transformation of linear difference equations. We apply it to the correspondence between the variant of the confluent q-hypergeometric equation ∞ (1 − q i αx). of type (i, j) and that of type ( j, i). We set (αx; q)∞ = i=0 Proposition 6.1 (i) If y(x) is a solution of the difference equation (1 − αx)a(x)g(x/q) + b(x)g(x) + c(x)g(q x) = 0,

(6.1)

then the function u(x) = (αq x; q)∞ y(x) satisfies a(x)g(x/q) + b(x)g(x) + (1 − αq x)c(x)g(q x) = 0.

(6.2)

(ii) If y(x) is a solution of the difference equation a(x)g(x/q) + b(x)g(x) + (1 − αx)c(x)g(q x) = 0,

(6.3)

then the function u(x) = y(x)/(αx; q)∞ satisfies (1 − αx/q)a(x)g(x/q) + b(x)g(x) + c(x)g(q x) = 0.

(6.4)

Proof It follows from the definition that (αx; q)∞ = (1 − αx)(αq x; q)∞ and (αq 2 x; q)∞ = (αq x; q)∞ /(1 − αq x). If the function y(x) satisfies Eq. (6.1) and u(x) = (αq x; q)∞ y(x), then we have a(x)u(x/q) + b(x)u(x) + (1 − αq x)c(x)u(q x) = (αq x; q)∞ {(1 − αx)a(x)y(x/q) + b(x)y(x) + c(x)y(q x)} = 0.

(6.5)

Variants of Confluent q-Hypergeometric Equations

175



Hence we obtain (i). We can show (ii) similarly.

By applying Proposition 6.1, we obtain correspondences among the variants of the confluent q-hypergeometric equation. A correspondence between Eq. (1.10) and Eq. (5.1) is given as follows. Proposition 6.2 Assume that function y(x) satisfies Eq. (1.10), which is a variant of the confluent q-hypergeometric equation of type (1, 2). Then, the function u(x) = (q −h 2 +1/2 t2−1 x; q)∞ y(x) is a solution of the equation ˜

˜

(x − q l1 −1/2 t1 )(x − q l2 −1/2 t2 )g(q x) −q

−h˜ 1 +l˜1 +l˜2 +2λ−1/2 ˜

t2 (x − q

˜

h˜ 1 +1/2

(6.6)

t1 )g(x/q)

˜

˜

˜

− [ q −α˜ 1 x 2 − q l1 +l2 +λ−1/2 {q −l2 t1 + (q −h 1 + q −l1 )t2 }x ˜

˜

+ q l1 +l2 +λ (q + 1)t1 t2 ]g(x) = 0, where l˜1 = l1 , l˜2 = h 2 , h˜ 1 = h 1 , α˜ 1 + α1 = h 1 − l1 + 1 − 2λ.

(6.7)

Note that Eq. (6.6) is a variant of the confluent q-hypergeometric equation of type (2, 1) (see Eq. (5.1)). Conversely, if the function u(x) satisfies Eq. (6.6), then the ˜ function y(x) = u(x)/(q −l2 +1/2 t2−1 x; q)∞ satisfies Eq. (1.10) where the relationship among the parameters is given by Eq. (6.7). Proof It follows from Eq. (1.10) that (1 − q −l1 +1/2 t1−1 x)g(q x) + q 2λ+1 (1 − q −h 1 −1/2 t1−1 x)(1 − q −h 2 −1/2 t2−1 x)g(x/q) − [q 2λ+α1 −h 1 −h 2 t1−1 t2−1 x 2 − q λ+1/2 (q −h 2 t2−1 + q −h 1 t1−1 + q −l1 t1−1 )x + q λ (q + 1)]g(x) = 0. We apply Proposition 6.1 (i). Set α = q −h 2 −1/2 t1−1 . Then, the function u(x) = (q −h 2 +1/2 t2−1 x; q)∞ y(x) satisfies (1 − q −l1 +1/2 t1−1 x)(1 − q −h 2 +1/2 t2−1 x)g(q x) + q 2λ+1 (1 − q −h 1 −1/2 t1−1 x)g(x/q) − [q 2λ+α1 −h 1 −h 2 t1−1 t2−1 x 2 − q λ+1/2 (q −h 2 t2−1 + q −h 1 t1−1 + q −l1 t1−1 )x + q λ (q + 1)]g(x) = 0. On the other hand, Eq. (6.6) is rewritten as ˜

˜

˜

(1 − q −l1 +1/2 t1−1 x)(1 − q −l2 +1/2 t2−1 x)g(q x) + q 2λ+1 (1 − q −h 1 −1/2 t1−1 x)g(x/q) ˜

˜

˜

˜

˜

− [q −α˜ 1 −l1 −l2 +1 t1−1 t2−1 x 2 − q λ+1/2 (q −l2 t2−1 + q −h 1 t1−1 + q −l1 t1−1 )x + q λ (q + 1)]g(x) = 0.

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By these expressions, we obtain the proposition.

As a consequence of Proposition 6.2, we obtain the following series solutions at x = ∞. Proposition 6.3 (i) The functions g1 (x) = x −α1 ·

∞ 

(q −λ−α1 +h 1 +1/2 t1 x −1 )n (q λ+α1 ; q)n

(6.8)

n=0 n  =0

(q λ+α1 −h 1 +l1 ; q) −(2n−−1)/2 q (−q −λ−α1 −l1 +h 2 t2 /t1 ) , (q; q)n− (q; q)

g2 (x) = ((q −h 2 +1/2 t2−1 x; q)∞ )−1 x 2λ+α1 −h 1 +l1 −1

∞ 

(q 1/2 x −1 )n

n=0

· (q −λ−α1 +h 1 −l1 +1 ; q)n

n  (q −λ−α1 +1 ; q)n−k k=0

(q; q)k (q; q)n−k

(q l1 t1 )k (q h 2 t2 )n−k

are solutions of Eq. (1.10). (ii) The functions g3 (x) = (q −l2 +1/2 t2−1 x; q)∞ x 2λ+α1 −h 1 +l1 −1

∞ 

(q λ+α1 +l1 −1/2 t1 x −1 )n

(6.9)

n=0

· (q −λ−α1 +h 1 −l1 +1 ; q)n

n  (q −λ−α1 +1 ; q) −(2n−−1)/2 q (−q λ+α1 −h 1 +l2 −1 t2 /t1 ) , (q; q)n− (q; q) =0

g4 (x) = x −α1

∞ 

(q 1/2 x −1 )n (q λ+α1 ; q)n

n=0

n  (q λ+α1 −h 1 +l1 ; q)n−k l1 k l2 n−k (q t1 ) (q t2 ) (q; q)k (q; q)n−k k=0

are solutions of Eq. (5.1). We can show similar properties for the biconfluent equations (Eq. (1.11) and Eq. (5.8)). Proposition 6.4 (i) Assume that function y(x) satisfies Eq. (1.11), which is a variant of the confluent q-hypergeometric equation of type (0, 2). Then, the function u(x) = (q −h 1 +1/2 t1−1 x; q)∞ (q −h 2 +1/2 t2−1 x; q)∞ y(x) is a solution of the equation q 2λ+1 g(x/q) + (1 − q −l1 +1/2 t1−1 x)(1 − q −l2 +1/2 t2−1 x)g(q x) −

[q −α˜ 1 −l1 −l2 +1 t1−1 t2−1 x 2

−

q λ+1/2 (q −l2 t2−1

+

q −l1 t1−1 )x

(6.10)

+ q λ (q + 1)]g(x) = 0

where l1 = h 1 , l2 = h 2 , α˜ 1 + α1 = 1 − 2λ.

(6.11)

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177

Note that Eq. (6.10) is a variant of the confluent q-hypergeometric equation of type (2, 0) (see Eq. (5.8)). Conversely, if the function u(x) satisfies Eq. (6.10), then the function y(x) = u(x)/((q −l1 +1/2 t1−1 x; q)∞ (q −l2 +1/2 t2−1 x; q)∞ ) satisfies Eq. (1.11) where the relationship among the parameters is given by Eq. (6.11). (ii) The functions g1 (x) = x −α1

∞ 

(q −λ−α1 +1/2 x −1 )n (q λ+α1 ; q)n

n  q −(n−) (q h 1 t1 )n− (q h 2 t2 ) =0

n=0

(q; q)n− (q; q)

,

(6.12) g2 (x)

= ((q −h 1 +1/2 t1−1 x; q)∞ (q −h 2 +1/2 t2−1 x; q)∞ )−1 x 2λ+α1 −1 ∞ n   (q h 1 t1 )k (q h 2 t2 )n−k · (q 1/2 x −1 )n (q −λ−α1 −1 ; q)n (q; q)k (q; q)n−k n=0 k=0

are solutions of Eq. (1.11). (iii) The functions g3 (x) = (q −l1 +1/2 t1−1 x; q)∞ (q −l2 +1/2 t2−1 x; q)∞ x 2λ+α1 −1 ·

∞ 

(q λ+α1 −1/2 x −1 )n (q λ+α1 ; q)n

=0

n=0

g4 (x) = x −α1

n 

∞ 

(q 1/2 x −1 )n (q λ+α1 ; q)n

n=0

(6.13)

q −(n−) (q l1 t1 )n− (q l2 t2 ) , (q; q)n− (q; q)

n  (q l1 t1 )k (q l2 t2 )n−k k=0

(q; q)k (q; q)n−k

are solutions of Eq. (5.8).

7 Limit to the Differential Equation We take the continuum limit q → 1 from the difference equations given in this paper. Recall that the variant of the confluent q-hypergeometric equation of type (1, 2) was given by q h 1 +h 2 −l1 −2λ+1/2 t2 (q l1 −1/2 t1 − x)g(q x) + (x − q α1 2

h 1 +1/2

− [q x − q

t1 )(x − q

h 2 +1/2

h 1 +h 2 −λ+1/2

(q

−h 2

(7.1)

t2 )g(x/q) t1 + q −h 1 t2 + q −l1 t2 )x

+ q h 1 +h 2 −λ (q + 1)t1 t2 ]g(x) = 0. Set q = 1 + ε, t2 = 1/(T ε) and consider the limit ε → 0, which is equivalent to q → 1. By using Taylor’s expansion

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g(x/q) = g(x) + (−ε + ε2 )xg  (x) + ε2 x 2 g  (x)/2 + O(ε3 ), 



(7.2)

g(q x) = g(x) + εxg (x) + ε x g (x)/2 + O(ε ), 2 2

3

we find the following limit as ε → 0: x 2 (x − t1 )g  (x) + x{T x(x − t1 ) + (h 1 − l1 + 1)x − 2λ(x − t1 )}g  (x)

(7.3)

+ {α1 T x + λ(t1 T − h 1 + l1 + λ)x − λ(λ + 1)t1 }g(x) = 0. 2

This equation has an irregular singularity at x = ∞ and regular singularities at x = 0, t1 , and we can show that the singularity x = 0 is apparent. Set f (x) = x λ g(x). Then, we essentially obtain Kummer’s confluent hypergeometric equation. (x − t1 ) f  (x) + {(x − t1 )T + h 1 − l1 + 1} f  (x) + (λ + α1 )T f (x) = 0.

(7.4)

By the limit to the differential equation, some solutions of Eq. (7.1) may converge to the solutions of Eq. (7.3). For example, the function x λ 3 φ2 (q λ+α1 , 0, x/(q l1 −1/2 t1 ); q h 1 −l1 +1 , q h 2 −l1 +1 t2 /t1 ; q, q)

(7.5)

in Eq. (3.2), which is a solution of Eq. (7.1), converges to the function ∞   x λ 1 F1 (λ + α1 , h 1 − l1 + 1; T (t1 − x)) = x λ n=0

 (λ + α1 )n T n (t1 − x)n (h 1 − l1 + 1)n n! (7.6)

for each component of the series as ε → 0 where q = 1 + ε and t2 = 1/(T ε). We can also obtain Eq. (7.3) from the variant of the confluent q-hypergeometric equation of type (2, 1). Namely, we can obtain Eq. (7.3) from Eq. (5.1) as ε → 0 by setting q = 1 + ε and t2 = −1/(T ε). We consider the limit from variants of the biconfluent q-hypergeometric equation. Recall that the variant of the confluent q-hypergeometric equation of type (0, 2) was given by g(q x) + q 2λ+1 (t1−1 q −h 1 −1/2 x − 1)(t2−1 q −h 2 −1/2 x − 1)g(x/q) −

[t1−1 t2−1 q α1 +2λ−h 1 −h 2 x 2

−q

λ+1/2

(q −h 2 t2−1

+

q −h 1 t1−1 )x

(7.7) λ

+ q (q + 1)]g(x) = 0.

Set q = 1 + ε, t1−1 = Bε1/2 , t2−1 = −Bε1/2 , and consider the limit ε → 0. Then we find the following limit as ε → 0: x 2 g  (x) + x(B 2 x 2 − 2λ)g  (x) + {α1 B 2 x 2 + λ(λ + 1)}g(x) = 0. Set f (x) = x λ g(x). Then we have

(7.8)

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179

f  (x) + B 2 x f  (x) + (α1 + λ)B 2 g(x) = 0,

(7.9)

which is essentially the Hermite-Weber differential equation. It seems that the solution of Eq. (7.7) given in Eq. (4.1) tends to the formal series x −α1

∞  (λ + α1 )2n n=0

n!B 2n

x −2n

(7.10)

for each component as ε → 0. This series is a formal solution of Eq. (7.8). We can obtain Eq. (7.8) from the variant of the confluent q-hypergeometric equation of type (2, 0). Namely, we obtain Eq. (7.8) from Eq. (5.8) as ε → 0 by setting q = 1 + ε, t1−1 = B(−ε)1/2 and t2−1 = −B(−ε)1/2 .

8 Concluding Remarks In this paper, we investigated degenerations of the variant of q-hypergeometric equation of degree two and obtained several formal solutions of the variant of singly confluent and biconfluent q-hypergeometric equations. Convergence or divergence of the formal solutions and the resummation should be clarified in a near future. We propose a problem for obtaining further degeneration of the variant of biconfluent qhypergeometric equation. We may define a variant of triconfluent q-hypergeometric equation by considering the degeneration such that the coefficient of g(x/q) in Eq. (1.11) is a linear polynomial, although we do not know how to obtain the limit to the differential equation (e.g., Airy’s differential equation). In [5], Ohyama proposed a coalescent diagram of q-special functions starting from Heine’s q-hypergeometric function, which is related with special solutions of q-difference Painlevé equations. Although the degenerations of the variant of q-hypergeometric equation in this paper are different objects from the ones in Ohyama’s paper, it would be interesting to find a unified theory. For example, the standard singly confluent q-hypergeometric equation (c − ax)u(q x) − (c + q − x)u(x) + qu(x/q) = 0 can be essentially obtained from the variant of the confluent q-hypergeometric equation of type (2, 1) (i.e., Eq. (5.1)) by the limit t1 → 0. However, it would not be simple for the biconfluent case. The monograph by KoekoekLesky-Swarttouw [4] might be useful for further studies. Acknowledgements The authors are grateful to the referee for valuable comments. The third author would like to thank Professor Yousuke Ohyama for discussions. He was supported by JSPS KAKENHI Grant Number JP18K03378.

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References 1. G. Gasper, M. Rahman, Basic Hypergeometric Series, vol. 96 (Cambridge university press, Cambridge, 2004) 2. W. Hahn, On linear geometric difference equations with accessory parameters. Funkcial. Ekvac. 14, 73–78 (1971) 3. N. Hatano, R. Matsunawa, T. Sato, K. Takemura, Variants of q-hypergeometric equation, to appear in Funkcial. Ekvac.. arXiv:1910.12560 4. R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics (Springer, Berlin, 2010) 5. Y. Ohyama, A unified approach to q-special functions of the Laplace type. arXiv:1103.5232 6. K. Takemura, Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations. J. Integr. Syst. 2, xyx008 (2017) 7. K. Takemura, On q-deformations of the Heun equation. SIGMA 14, paper 061 (2018)

Rapid Decay Property for Algebraic p-Adic Groups Sami Mustapha

Dedicated to the memory of Professor Takaaki Nomura.

Abstract For a locally compact group, the property of rapid decay (property (RD)) gives a control on the convolutor norm of any compactly supported function in terms of its L 2 -norm and the diameter of its support. We investigate in this paper the algebraic structure of compactly generated p-adic groups that have property (RD). We prove in particular that an algebraic group over Q p which is compactly generated as well as its radical has property (RD) if and only if it is reductive. Keywords Locally compact groups · p-adic groups · Length functions · Property (RD) · Volume growth · Random walks · Unimodularity · Amenability

1 Property (RD) Let G be a locally compact group which we assume compactly generated. We shall symmetric compact denote by dg the left Haar measure on G and by  = −1 some  neighborhood of the identity e in G which generates G, i.e., n n = G, where n =  · · ·  (n times), n = 0, 1, . . . with 0 = {e}. For g ∈ G the distance from e (or the length function induced by ) is defined by l (g) = inf{n = 0, 1, . . . such that g ∈ n }

(1)

S. Mustapha (B) Institut Mathématique de Jussieu, Sorbonne Université, Tour 25 5e étage Boite 247, 4 Place Jussieu, F-75252 CEDEX 05 Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_11

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and for R ≥ 1 we set

B R (e) = {g ∈ G : l (g) ≤ R}.

If 1 , 2 are two neighborhoods of e as above it is not difficult to check that there exists C > 0 such that (2) C −1 ≤ l2 (.)/l1 (.) ≤ C. We say that the group G has the rapid decay property (RD-property) if there exist two constants C, D > 0 such that for any R ≥ 1 and any continuous function ϕ ∈ Cc (G) with compact support contained in B R (e):  f ∗ ϕ L 2 (G) ≤ C R D ϕ L 2 (G)  f  L 2 (G) , f ∈ Cc (G),

R ≥ 1,

(3)

where L 2 (G) is defined with respect to the left Haar measure dg and the convolution product f ∗ ϕ is defined by  f ∗ ϕ(g) =

ϕ(h −1 g) f (h)dh, g ∈ G.

Observe that the left side of (3) relies on the group law and the algebraic structure of G while the right side relies on the large-scale geometry of G. There are several other equivalent definitions of the rapid decay property which give more flexibility for using this property. The most important is the definition used in [4] which says that G has RD-property if the rapid decay functions on G are contained in the reduced C ∗ -algebra of G: Hl∞ (G) ⊆ Cr∗ (G). For a length function l on G as in (1) (we drop  in the notation because of the equivalence (2)) the Sobolev space of order s (with respect to l) is the set Hls (G) of functions f satisfying f (1 + l)s ∈ L 2 (G). The space of rapidly decreasing functions on G (with respect to l) is the set Hl∞ (G) =



Hls (G).

s∈R

Equipped with the inner product  < ϕ, f >l,s =

ϕ(g) f (g)(1 + l(g))2s dg G

Hls (G) defines, for each s ∈ R, a Hilbert space and Hl∞ (G) is a Fréchet space for the projective limit topology induced by the inclusions of Hl∞ (G) in Hls (G) for each s ∈ R. Equivalently G has the RD-property if there exist constants C, s > 0 such that

Rapid Decay Property for Algebraic p-Adic Groups

λG (φ) L 2 (G)→L 2 (G) ≤ Cφ Hls (G) ,

183

(4)

where λG (φ) L 2 (G)→L 2 (G) denotes the operator norm of φ acting on L 2 (G) via its regular representation. The equivalence between the two formulations of property RD is well-known. It is easy to see that (4) with s implies (3) with the same s, and that (3) with D implies (4) with s = D + ,  > 0. Property (RD) appeared in the work of Haagerup [7] and was studied systematically by Jolissaint in [10]. Its usefulness in the theory of C ∗ -algebras was pointed out by Connes and Moscovosi (cf. [4] where this property is used to prove the Novikov conjecture) and by Lafforgue who used it in [12] to prove the Baum–Connes conjecture for some discrete groups having property (T). The validity of property (RD) was established for several classes of groups: free groups (Haagerup [7]), locally compact groups of polynomial volume growth and classical hyperbolic groups (Jolissaint [10]), Gromov hyperbolic groups (de la Harpe [5]), co-compacts discrete subgroups of S L 3 (Q p ) (Ramagge, Robertson and Steger [15]) and co-compacts discret subgroups of S L 3 (R) (Laforgue [11]). As examples of groups without the rapid decay property, one can cite amenable groups with exponential volume growth and S L n (Z), n ≥ 3 (see [10]). It is clear from its definition that property (RD) is inherited by open subgroups with the induced length. Having amenable open subgroup of super-polynomial growth therefore constitutes an obstruction to the rapid decay property. Actually S L n (Z), n ≥ 3, contains a subgroup with an exponential growth. For cocompact lattices in S L n (Q p ) or S L n (R), n ≥ 4, and more generally in semisimple Lie groups, the rapid decay property is open and a conjecture of Valette [17] is that cocompact lattices in a semisimple Lie group (real or p-adic) have the rapid decay property. In [3] Chatterji, Pittet and Saloff-Coste gave a precise description of those connected (real) Lie groups that have property (RD). They also proved that the group of k-points of a connected linear algebraic semisimple group defined over a local field k has property (RD) [3, Theorem 4.5]. The aim of this paper is to give a complete algebraic characterization of compactly generated p-adic groups that have property (RD). The main results of this paper were announced in [14]. Finally, we point out that for groups that have the rapid decay property very little is known about their RD degree. The rapid decay degree of G is defined as r d(G) = inf{s ∈ R such that Hls (G) ⊆ C  (G)}. Knowing the exact degree has interesting consequences, but unfortunately most methods for establishing the rapid decay property give a bad estimate for that degree. A problem that is completely open is to calculate r d(G) when G is a reductive p-adic group.

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p-Adic Algebraic Groups

For p a fixed prime number, let Q p denote the field of p-adic numbers and let G be an algebraic group over Q p . Let G o denote the connected component of the identity in G and R(G o ) (resp. Ru (G o )) the radical (resp. the unipotent radical) of G o . Recall that R(G o ) is the largest closed solvable connected normal subgroup of G o and that Ru (G o ) is the largest closed unipotent connected normal subgroup of G o . Let V = Ru (G o )/(Ru (G o ), Ru (G o )) and let ρ denote the representation of R(G o ) in the Lie algebra v of V induced by the adjoint representation of G. Theorem 2.1 Let G be an algebraic group over Q p as above. Assume that G is compactly generated. Suppose that G has property (RD). Then, either G is reductive, or v contains a subspace w = 0 stable under the action of R(G o ) and such that the image of R(G o ) in GL(w) is compact. Corollary 2.1 Let G be an algebraic group over Q p as above. Assume that both G and R(G) are compactly generated. Then G possesses RD-property if and only if G is reductive. The connected component of the identity in G being of finite index [1, Corollaire 6.4] we can assume, according to Lemma 3.3 of [3], that G is connected. Assume that G is reductive. It follows then that G is compactly generated (see [2, Paragraph 13]) and that is has property (RD). This is an easy consequence of [3, Theorem 4.5], [10, Corollary 3.1.8] and of [3, Lemma 3.1] (combined with structure theorems for reductive groups established in [2, Paragraph 1]). Assume now that G is not reductive, has property (RD) and that: (hypothesis (H)) v does not contain a subspace w = 0 stable under the action of R(G) and such that the image of R(G) in G L(w) by ρ is compact. It follows then (see [2, Paragraph 13]) that R(G) is compactly generated and of exponential volume growth (for the notion of volume growth of a compactly generated group see [6, 8]). Observe on the other hand that the group G is necessarily nonamenable. This follows from the fact that if G is an amenable group (see Leptin [13]) and has property (RD) then G is of polynomial volume growth [9]. Since R(G) is of exponential growth, the group G himself is of exponential growth [6, Theorem 1.2]. Observe also that according to [9, Theorem 2.2] the group G is unimodular. The key ingredient of the proof of Theorem 2.1 lies in the behavior of random walks on G as we will explain now. Let dμ(g) = (g)dg ∈ P(G) denote a probability measure on G. We assume that the measure μ is symmetric (i.e., dμ(g −1 ) = dμ(g)) and that the density (g) is a continuous compactly supported function on G. We shall also assume that there exists a symmetric compact neighborhood of the identity e ∈ 0 = −1 0 ⊂ G such that  n0 . (5) inf { (g), g ∈ 0 } > 0, G = n≥0

Rapid Decay Property for Algebraic p-Adic Groups

185

The last condition in (5) implies that supp(μ) generates the group G. Let dμ∗n (g) ∈ P(G), n = 1, 2, . . . denote the nth convolution power of μ and let n denote the corresponding density. It follows from the unimodularity of G that n is symmetric. We shall use the notation μ∗n to denote the convolution operator induced by the measure dμn∗ (g) ∈ P(G). Theorem 2.2 Let G, dμ(g) ∈ P(G) and n be as above. Then there exists C > 0 such that  1/3  −n , n = 1, 2, . . . . (6)  n 2L 2 (G) ≤ Cμ∗2n  L 2 (G)→L 2 (G) exp C The estimate (6) is of probabilistic nature. Indeed, due to the symmetry of the density n , the norm  n  L 2 (G) satisfies  n  L 2 (G) = [ 2n (e)]1/2 .

(7)

In other words  n  L 2 (G) can be interpreted as the square root of a return probability to the origin in the group G. An immediate corollary of Theorem 2.2 is that the group G cannot have (RD) (see [3], Sect. 7). Indeed the support condition satisfied by the density and (2) imply that μ∗2n  L 2 (G)→L 2 (G) = μ∗n 2L 2 (G)→L 2 (G) ≤ Cn 2D  n 2L 2 (G) ,

n = 1, 2, . . . . . .

where C and D are independent of n. Combining with (6) and letting n → ∞ lead to a contradiction which shows that the hypothesis (H) is absurd and completes the proof of Theorem 2.1. Corollary 2.1 is an immediate consequence of Theorem 2.1 and [2, Theorem 13.4].

3 Proof of Theorem 2.2 Let G = R σ S denote a Levi decomposition of the group G where S denotes a Levi semisimple subgroup and where σ denotes the action of S on R. Let us denote by dg = dr ds the Haar measure on G, where we use the coordinates g = (r, s) ∈ G = R σ S. Our starting point will be formula (7)  n 2L 2 (G) = 2n (e) = 2n−1 ∗ (e) =< μ∗(2n−1) , > .

(8)

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Here and throughout we use the notation < μ, ϕ > to denote the action of a measure μ on a function ϕ. Using the coordinates (8), we deduce   n 2L 2 (G) =

 ...



(g1 . . . g2n−1 )dμ(g1 ) . . . dμ(g2n−1 ) 

G

G

...

= G

(r1 .σ (s1 )r2 . . . σ (s1 . . . s2n−2 )r2n−1 , s1 . . . s2n−1 ) G

(r1 , s1 ) . . . (r2n−1 , s2n−1 )dr1 ds1 . . . dr2n−1 ds2n−1 . Using Fubini and the unimodularity of the changes of variable σ (s1 . . . si−1 )ri ↔ ri , i = 2, . . . , 2n − 1, which follows from the semi-simplicity of S, we deduce that  

  n 2L 2 (G)

=

  (r1r2 . . . r2n−1 , s1 . . . s2n−1 ) ...

... S

S

R

R

(r1 , s1 ) (σ (s1 )−1r2 , s2 ) . . . (σ (s1 . . . s2n−2 )−1r2n−1 , s2n−1 ) dr1 . . . dr2n−1 ds1 . . . ds2n−1 . Let us introduce the R-convolution product μ(s1 , . . . , s2n−1 ) = (., s1 ) ∗ R (σ (s1 )−1 (.), s2 ) ∗ R · · · ∗ R (σ (s1 . . . s2n−2 )−1 (.), s2n−1 ),

(9) s1 , . . . , s2n−1 ∈ S. We can rewrite   n 2L 2 (G) =

 ... S

μ(s1 , . . . , s2n−1 ), (., s1 . . . s2n−1 )ds1 . . . ds2n−1 . S

Observe that if one of the s j ’s in (9) is such that



(r, s j )dr = 0 then

μ(s1 , . . . , s2n−1 ) = 0 which allows us to limit ourselves in (10) to integration over the sets  {s j ∈ S;

(r, s j )dr > 0}, R

for j = 1, . . . , 2n − 1.

(10)

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187

Now suppose that 0 <  < 1 is fixed (this  will be chosen small enough later) and consider the sets S0 = S0 () ⊂ S, S1 = S1 () ⊂ S, defined by  S0 = {s ∈ S;

(r, s)dr ≥ },

(11)

R

 S1 = {s ∈ S; 0
0 (we used the fact that ϕ j (r )dr = 1, 1 ≤ j ≤ n 1 − 1). The next proposition is the heart of the proof. We follow the proof of estimate (2.8) in [18, Sect. 2]. Proposition 3.1 Let the notation be as above. Then there exists a positive constant C independent of n and k such that ϕn 1 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1  L 2 (R) ≤ Ce−

k 1/3 C

.

Proof Let  f (r ) = ϕn 1 +1 ∗ R . . . ∗ R ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 σ (s1 . . . sn 1 −1 )(r ) , r ∈ R.

(17) For each positive integer l we shall denote by vo(l) the Haar measure of V l = V · · · V (l times) and fl , the mean of the function f on r V l fl (r ) =

1 vo(l)

 rVl

f (y)dy, r ∈ R.

Following [16] we write  f  L 2 (R) ≤  f − fl  L 2 (R) +  fl  L 2 (R) ≤  f − fl  L 2 (R) + vo(l)−1/2  f  L 1 (R) .

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Using (14) and Proposition VII 3.2 of [19] we can then estimate  f 2L 2 (R) by  

χV 3 (h) 2 dhd x +  f 2L 1 (R) vo(3) vo(l)    2 C l 2  f 2L 1 (R) . ≤ ( f (xh) − f (x))2 ϕˇsn1 ∗ ϕsn1 (h)dhd x + 2 vo(l)

 f 2L 2 (R) ≤ Cl 2

( f (xh) − f (x))2

The Dirichlet expression Dn 1 ( f ) =

1 2

 

 ( f (xh) − f (x))2 ϕˇsn1 ∗ ϕsn1 (h)dhd x

can be rewritten as follows: Dn 1 ( f ) =  f 2L 2 (R) − ϕsn1 ∗ f 2L 2 (R)     2 r (sn 1 )y, sn 1 dy dh =  f 2L 2 (R) − f (y −1 h)

R (r, sn 1 )dr    2 y, sn 1 2 −1 −1 dy dh f (y r (sn 1 ) h)

=  f  L 2 (R) − R (r, sn 1 )dr    2 y, sn 1 2 −1

dy dh, f (y h) =  f  L 2 (R) − R (r, sn 1 )dr where we performed successively the two changes of variable y ↔ r (sn 1 )−1 y, h ↔ r (sn 1 )h and use the unimodularity of the normal subgroup R. In the same way  

f (y −1 h)

   2 y, sn 1 dy dh = f (σ (s1 . . . sn 1 −1 )−1 (y −1 h)) R (r, sn 1 )dr   2 σ (s1 . . . sn 1 −1 )−1 (y), sn 1

dy dh (r, s )dr n1   R = ϕn 1 +1 ∗ R . . . ∗ R ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R   2 σ (s1 . . . sn 1 −1 )−1 (y), sn 1

ϕ2n−1 ((y −1 h)) dy dh, R (r, sn 1 )dr

where we used (17) to obtain the last equality. The previous considerations combined with the fact that vo(l) ≥ cecl (recall that R is of exponential volume growth) imply that ϕn 1 +1 ∗ R . . . ∗ R ϕ2n−1 2 2

L (R)

  ≤ Cl 2 ϕn 1 +1 ∗ R . . . ∗ R ϕ2n−1 2 2 − ϕn 1 ∗ R ϕn 1 +1 ∗ R . . . ∗ R ϕ2n−1 2 2 L (R) L (R) +Ce−cl ϕn 1 +1 ∗ R . . . ∗ R ϕ2n−1  L 1 (R) .

Rapid Decay Property for Algebraic p-Adic Groups

191

Taking into account the fact that ϕn 1 +1 ∗ R . . . ∗ R ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1  L 1 (R) ≤ 1, we deduce that Cl 2 ϕn 1 ∗ R ϕn 1 +1 ∗ R . . . ∗ R ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) ≤ (Cl 2 − 1)ϕn 1 +1 ∗ R . . . ∗ R ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) + Ce−cl ≤ (Cl 2 − 1)ϕn 2 ∗ R . . . ∗ R ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) + Ce−cl , and then  ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) ≤ Cl 2 ϕn 2 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R)  −ϕn 1 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) + Ce−cl .

The same argument can be iterated and gives (with exactly the same constants):  ϕn j+1 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) ≤ Cl 2 ϕn j+1 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R)  −ϕn j ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) + Ce−cl ,

j = 1, . . . , k − 1.

From this it follows that for every j = 1, . . . , k − 1, we have 

  ϕn j+1 ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) − Ce−cl ≤ Cl 2 ϕn j+1 ∗ R . . . ∗ R ϕn k ∗ R

  . . . ∗ R ϕ2n−1 2L 2 (R) − Ce−cl − ϕn j ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) − Ce−cl ,

which leads, for every j = 1, . . . , k − 1, to the following upper bound for the difference ϕn j ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) − Ce−cl ,   1 k− j 1− 2 ϕn k ∗ R . . . ∗ R ϕn k ∗ R . . . ∗ R ϕ2n−1 2L 2 (R) . Cl In particular we obtain ϕn 1 ∗ R . . . ∗ R ϕn k ∗ R . . .

∗ R ϕ2n−1 2L 2 (R)

≤C





1 1− 2 Cl

k−1

+ Ce−cl ,

192

S. Mustapha

where the constants C, C  and c are independent of n, k and l. Optimizing over l (by  choosing l ≈ k 1/3 ) yields the desired upper estimate. Another important ingredient in the proof of estimate (6) is the following result. Proposition 3.2 Let the notation be as above. Then μ˜ ∗n  L 2 (S)→L 2 (S) ≤ μ∗n  L 2 (G)→L 2 (G) .

(18)

Proof Let us introduce  ψn (s) =

n (., s) dr, s ∈ S, n = 1, 2, . . . . R

Let ak , bk ∈ Cc (R) satisfying ak  L 2 (R) = bk  L 2 (R) = 1, k = 1, 2, . . . and  ak (rr1 )bk (r )dr → 1, k → ∞, R

uniformly. The existence of such functions is a consequence of the amenability of the subgroup R. Let c,c ∈ Cc (S) satisfying c L 2 (S) = c  L 2 (S) = 1. Let 



Ik =

(ak ⊗ c) ∗ μ∗n (g)(bk ⊗ c )(g)dg.

G

We have Ik =

      ak ∗ R n σ (s1−1 )(.), s −1 s (r )c(s1 )bk (r )c (s)ds1 dr ds S

R

S

       = ak ∗ R n σ (s1−1 )(.), s −1 s (r )bk (r )dr c(s1 )c (s)ds1 ds. S

S

R

Let us denote by Jk =

    ak ∗ R n σ (s1−1 )(.), s −1 s (r )bk (r )dr, k = 1, 2, . . . . R

We have   Jk =

R R

 n σ (s1−1 )(r1−1r ), s −1 s ak (r1 )bk (r )dr1 dr

 n σ (s1−1 )(r1 ), s −1 s ak (rr1 )bk (r )dr1 dr R   R   ak (rr1 )bk (r )dr n σ (s1−1 )(r1 ), s −1 s dr1 . =

=

R

R

Rapid Decay Property for Algebraic p-Adic Groups

193

 Letting k → ∞ we deduce that Jk → ψn s −1 s , from which it follows that Ik →



−1   S S ψn s s c(s1 )c (s)ds1 ds. This easily implies (18). Proof of Theorem 2.2 Fix (i 1 , . . . , i 2n−1 ) ∈ {0, 1}2n−1 a (2n − 1)-uplet satisfying i 1 + · · · + i 2n−1 = 2n − 1 − k and assume that k ≥ n + 1. Thanks to (16) and Proposition 3.1 we can then estimate 

 Si 1

...

Si 2n−1

 μ(s , . . . , s  2n−1   1 2n−1 ) , (., s . . . s ) (r, si )dr ds1 . . . ds2n−1

2n−1 1 2n−1 (r, si )dr i=1 i=1 R

(19) by 

n 1/3 C exp − C



 Si 1

...

Si 2n−1

χsupp(μ) ˜ (s1 . . . s2n−1 )

2n−1  

(r, si )dr ds1 . . . ds2n−1 .

i=1

(20) Consider the sum of all integrals (19), for k varying from n + 1 to 2n − 1, appearing in the decomposition (15). This sum can be dominated by a sum of integrals (20) extended to all possible indices k: Ce−

n 1/3 C



 ...

(i 1 ,i 2 ,...,i 2n−1 ) Si1 ∈{0,1}2n−1

Si2n−1

χsupp(μ) ˜ (s1 . . . s2n−1 )

2n−1  

(r, si )dr ds1 . . . ds2n−1 ,

i=1

which makes it possible to reconstitute for the considered sum an upper bound of the form  1/3    2n−1  n C exp − . . . χsupp(μ) (r, si )dr ds1 . . . ds2n−1 , ˜ (s1 . . . s2n−1 ) C S S i=1 hence, an upper estimate   1/3   n n 1/3  ∗(2n−1)  μ˜ μ˜ ∗2n  L 2 (S)→L 2 (S) , , χsupp(μ) C exp − ˜ ≤ C exp − C C 

where for the last inequality we used (5). Using Proposition 3.2 we deduce that the sum of all integrals (19) in (15) corresponding to indices k ≥ n + 1 can be bounded by C  e−

n 1/3 C

μ∗2n  L 2 (G)→L 2 (G) .

(21)

Integrals analogous to (19) which correspond to indices k smaller than n are much easier to handle. The easiest way is to consider them in the form

194

S. Mustapha









...

(r1 , s1 ) (σ (s1 )−1 r2 , s2 ) . . . (σ (s1 . . . s2n−2 )−1 r2n−1 , s2n−1 )

...

Si1

Si2n−1

R

R

(22)

(r1r2 . . . r2n−1 , s1 . . . s2n−1 )dr1 . . . dr2n−1 ds1 . . . ds2n−1 . Each integral (22) can be dominated by   ∞







... Si1

(r1 , s1 ) (σ (s1 )−1 r2 , s2 ) . . . (σ (s1 . . . s2n−2 )−1 r2n−1 , s2n−1 )

... Si2n−1

R

R

(23) dr1 . . . dr2n−1 ds1 . . . ds2n−1 .

The semi-simplicity of S allows to get rid of the action of σ on the R-variable in  each σ (s1 s j )−1 (.), s j , reducing (23) to   ∞



 Si 1

...

Si 2n−1

 R

...

R

(r1 , s1 ) (r2 , s2 ) . . . (r2n−1 , s2n−1 )dr1 . . . dr2n−1 ds1 . . . ds2n−1 .

(24)

Thanks to (13) and the fact that k ≤ n (24) admits the upper bound  ∞ (C1 )n .

(25)

Since there are at most 22n−1 terms of this type, it follows from (25) that the corresponding sum is dominated by  ∞ (4C1 )n .

(26)

Combining (15) and the bounds (21), (26) we deduce that  n 2L 2 (G) ≤ C  e−

n 1/3 C

μ∗2n  L 2 (G)→L 2 (G) +  ∞ (4C1 )n .

(27)

By choosing  sufficiently small in (13), (26) can be absorbed by (21) and we get in (27) the right upper estimate. This completes the proof of Theorem 2.2. Acknowledgements The author would like to express his gratitude to A. Baklouti and H. Ishi, the organizers of the 6th Tunisian-Japanese Conference of “Geometric and Harmonic Analysis on homogeneous spaces and Applications” in honor of Professor Takaaki Nomura, for their warm hospitality during the stimulating conference held in Djerba Island. He would like to thank warmly Professor Baklouti for the interest he has shown in the rapid decay property and for fruitful discussions.

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195

References 1. A. Borel, J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne. Comment. Math. Helv. 39, 111–164 (1964) 2. A. Borel, J. Tits, Groupes réductifs. Publ. Math. IHES 27, 55–150 (1965) 3. I. Chatterji, C. Pittet, L. Saloff-Coste, Connected Lie groups and property (RD). Duke Math. J. 137, 511–536 (2007) 4. A. Connes, H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29, 345–388 (1990) 5. P. de la Harpe, Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint. C. R. Acad. Sci. Paris Sér. I Math. 307, 771–774 (1988) 6. Y. Guivarc’h, Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101, 333–379 (1973) 7. U. Haagerup, An example of a nonnuclear C ∗ -algebra which has the metric approximation property. Invent. Math. 50, 279–293 (1979) 8. J.W. Jenkins, Growth of connected locally compact groups. J. Funct. Anal. 12, 113–127 (1973) 9. R. Ji, L.B. Schweitzer, Spectral invariance of smooth crossed products and rapid decay locally compact groups. K-Theory 10, 283–305 (1996) 10. P. Jolissaint, Rapidly decreasing functions in reduced C ∗ -algebras of groups. Trans. Amer. Math. Soc. 317, 167–196 (1990) 11. V. Lafforgue, A proof of property (RD) for cocompact lattices of S L(3, R) and S L(3, C). J. Lie Theory 10, 255–267 (2000) 12. V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. Invent. Math. 149, 1–95 (2002) 13. H. Leptin, On locally compact groups with invariant mean. Proc. Amer. Math. Soc. 19, 489–494 (1968) 14. S. Mustapha, La propriété (RD) pour les groupes algébriques p-adiques. C.R. Acad. Sci. Paris, Ser. I 348, 411–413 (2010) 15. J. Ramagge, G. Robertson, T. Steger, A Haagerup inequality for A˜ 1 × A˜ 1 and A˜ 2 buildings. Geom. Funct. Anal. 8, 702–731 (1998) 16. D. Robinson, Elliptic Operators and Lie Groups (Oxford University Press, Oxford, 1991) 17. A. Valette, Introduction to the Baum-Connes Conjecture. Lectures Math. (ETH Zürich, Birkhäuser, 2002) 18. NTh. Varopoulos, Hardy-Littlewood theory on unimodular groups. Ann. Inst. Henri Poincaré Probab. Stat. 31, 669–688 (1995) 19. N.Th. Varopoulos, L. Saloff-Coste, Th. Coulhon, Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 102 (1993)

Rings of Invariant Differential Operators on Homogeneous Cones and Capelli-Type Identities Hideto Nakashima

Abstract The aim of this paper is to study structures of rings of invariant differential operators on homogeneous cones, and give an explicit description of their generators. For the group actions, we consider split solvable Lie groups acting on the cones linearly and simply transitively. As an application, we present Capelli-type identities for generalized Vinberg cones. Keywords Invariant differential operators · Capelli-type identities · Homogeneous cones · Basic relative invariants 2010 Mathematics Subject Classification Primary 16S32 · Secondary 22E25, 22F30, 17D99 Introduction In the study of a homogeneous space M, it is fundamental to investigate the ring of invariant differential operators on M. In the case that M = G/K , where (G, K ) is a symmetric pair with G a separable unimodular Lie group and K a compact subgroup, it has been extensively studied from various points of view (cf. [1–6, 10]), and its fundamental theorem is the commutativity of the rings (see [3, p. 293]). Moreover, some of them admit Capelli-type identities which play a central role in invariant theory, non-commutative ring theory and representation theory (cf. Howe and Umeda [6]). Apart from symmetric cases, there are few studies on rings of invariant differential operators (cf. Zhong [12]), and one can ask questions how far the rings are away from being commutative or whether or not they admit Capelli-type identities. In this paper, we consider the rings of differential operators on homogeneous open convex cones  containing no entire lines (homogeneous cones for short) which are invariant under the action of split solvable Lie groups acting on  linearly and simply transitively. We give an explicit structure of the rings and describe its generators precisely. Moreover, Capelli-type identities are presented for generalized Vinberg cones, which are typical examples of non-symmetric homogeneous cones. H. Nakashima (B) The Institute of Statistical Mathematics, Midori-cho 10-3, Tachikawa, Tokyo 190-8562, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_12

197

198

H. Nakashima

We now describe the body of the paper in more detail. Let  be a homogeneous cone of rank r in an n-dimensional vector space V . We denote by H the solvable Lie group acting on  linearly and simply transitively. The dual vector space of V is n be a basis denoted by V ∗ , and write ξ(v) =  v | ξ  for ξ ∈ V ∗ and v ∈ V . Let {v i }i=1 ∗ n ∗ ∗ of V and {v i }i=1 ⊂ V its dual basis. Let F( × V ) be the space of functions P on  × V ∗ which are polynomial in ξ ∈ V ∗ (see (1.1) for detail). Then, any differential operator on  can be described as P(x, ∂x ) by using some P ∈ F( × V ∗ ) (cf. [3, Chap. II], see also (1.2)). We denote by ρ the representation of H on V by which H acts on  simply transitively, and by ρ ∗ its contragredient representation on V ∗ . ∗ of H if we Then, afunction P in F(  × V ) is said to be invariant under the action ∗ have P ρ(h)x, ρ (h)ξ = P(x, ξ ) for all h ∈ H , x ∈  and ξ ∈ V ∗ . A differential operator P(x, ∂x ) on  is said to be invariant under the action of H if P is H invariant. Let D() be the ring of all H -invariant differential operators on . Then, we see in Theorem 1.1 that it is isomorphic to the enveloping algebra U(h) of the Lie algebra h of H (cf. [5, Chap. II]). Let us take a reference point e, and fix it. For x ∈ , we introduce h x ∈ H by x = ρ(h x )e. For each v ∈ V , we introduce an H -invariant function Fv on  × V ∗ by Fv (x, ξ ) :=  ρ(h x )v | ξ  (x ∈ , ξ ∈ V ∗ ). Then, D() = {Fv (x, ∂x ); v ∈ V } equipped with the bracket product is a Lie algebra which is isomorphic to h (see Lemma 1.2). we see that D() can  As a corollary,  n is any basis of V (Corolbe generated by Fvi (x, ∂x ); i = 1, . . . , n where {v i }i=1 lary 1.3) and that the Euler operator E is in the center of D() (Corollary 1.4). In Sect. 2, we deal with concrete examples of homogeneous cones and show that they admit Capelli-type identities. Let V be a real vector space of dimension 2r + 1 defined by    x y V = v = (v1 , . . . , vr ); v j := 0 j , x0 , y j , z j ∈ R , yj z j and let  be a subset of V defined by  = {x = (x1 , . . . , xr ) ∈ V ; i (x) > 0 (i = 0, 1, . . . , r )} , where 0 (x) := x0 and  j (x)  := det x j for j = 1, . . . ,r . The set  is called a generalized Vinberg cone. Let X 0 , Y j , Z j ; j = 1, . . . , r be the basis  of V defined

by (2.3) and its dual basis is denoted by X 0∗ , Y j∗ , Z ∗j ; j = 1, . . . , r ⊂ V ∗ . Then, we see in Proposition 2.1 that D() is generated by

Rings of Invariant Differential Operators on Homogeneous Cones …

199

 r y 2j ∂ ∂ ∂ FX 0 (x, ∂x ) = x0 + + yj , ∂ x0 ∂yj x0 ∂z j j=1   x0 z j − y 2j  ∂ ∂ FY j (x, ∂x ) = + 2y j x0 ( j = 1, . . . , r ), x0 ∂yj ∂z j x0 z j − y 2j ∂ ( j = 1, . . . , r ). FZ j (x, ∂x ) = x0 ∂z j For ξ = ξ0 X 0∗ +

r j=1

∗0 (ξ )

  η j Y j∗ + ζ j Z ∗j ∈ V ∗ , we put

= ξ0 · ζ1 · · · ζr −

r

ζ1 · · · ζ j−1 ·

1

j=1

2

ηj

2

· ζ j+1 · · · ζr

and ∗j (ξ ) = ζ j for j = 1, . . . , r . When we substitute non-commutative variables in the argument of ∗0 , we use this order of the product. Set D(x) = FX 0 (x, ∂x )X 0∗ +

r 

 FY j (x, ∂x )Y j∗ + FZ j (x, ∂x )Z ∗j ∈ D() ⊗ V ∗

j=1 +1 and for m = (m 0 , m 1 , . . . , m r ) ∈ Zr +1 and n = (n 0 , n 1 , . . . , n r ) ∈ Zr≥0

m (x) :=

r 

n

i (x)m i (x ∈ ), ∗ (ξ ) :=

i=0

r 

i∗ (ξ )ni (ξ ∈ V ∗ ).

i=0

Then, Theorem 2.4 shows that  admits a Capelli-type identity of the form nj r  

  −1 n ∗j D(x) + S j,n j −k = nσ∗ σ (x)∗ (∂x ) (x ∈ )

j=0 k=1 +1 for n = (n 0 , n 1 , . . . , n r ) ∈ Zr≥0 . Here, the order of products of non-commutative variables is determined by (2.15), σ , σ∗ are square matrices defined by (2.12) and S jk ∈ V ∗ ( j = 1, . . . , r, k = 1, . . . , n j ) are elements in V ∗ given by

S0k :=

r 2

r  − k X 0∗ − (n i + k)Z i∗ ∈ V ∗ , S jk := −k Z ∗j ∈ V ∗ . i=1

Acknowledgment The author would like to express his sincere gratitude to Professor Takaaki Nomura for invaluable discussions on the topic of this article.

200

H. Nakashima

1 Rings of Invariant Differential Operators Let  be a homogeneous cone of rank r in an n-dimensional vector space V . The dual vector space of V is denoted by V ∗ , and we write ξ(v) =  v | ξ  for ξ ∈ V ∗ and n be a basis of V and {v ∗j }nj=1 ⊂ V ∗ its dual basis. Let F( × V ∗ ) = v ∈ V . Let {v i }i=1 ∞ C ()[ξ1 , . . . , ξn ] be the space of functions P on  × V ∗ which are polynomial in ξ ∈ V ∗ , that is, functions of the form P(x, ξ ) =



aα (x)ξ1α1 · · · ξnαn , x ∈ , ξ =

α∈Zn≥0

n

ξ j v ∗j ∈ V ∗ ,

(1.1)

j=1

where aα (x) ∈ C ∞ () are smooth functions on  which equal to zero except for finite α. Then, it is known (cf. [3, Chap. II]) that any linear differential operator on  can be described by using P ∈ F( × V ∗ ) as P(x, ∂x ) :=

α

aα (x)∂xα =

α

 aα (x)

∂ ∂ x1

α1

 ···

∂ ∂ xn

αn

.

(1.2)

Vinberg [11] tells us that, corresponding to , there exists a split solvable Lie group H acting on  linearly and simply transitively. Let ρ denote the representation of H on V by which H acts on  simply transitively. We take a reference point e ∈ , and fix it. Let φ(h) := ρ(h)e for h ∈ H . Since H acts on  simply transitively via ρ, we see that φ is a differmorphism from H onto . We denote by L(g) the left translation L(g)h = gh of H (g ∈ H ). Then, ρ(g) commutes L(g) with respect to φ for any g ∈ H , that is, φ ◦ L(g) = ρ(g) ◦ φ, φ −1 ◦ ρ(g) = L(g) ◦ φ −1 ,

(1.3)

where φ −1 is the inverse function of φ. In fact, we have for any h ∈ H   φ L(g)h = φ(gh) = ρ(gh)e = ρ(g)ρ(h)e = ρ(g)φ(h). Let ρ ∗ be the contragredient representation of ρ, that is, we have  ρ(h)v | ρ ∗ (h)ξ  =  v | ξ  for any h ∈ H , v ∈ V and ξ ∈ V ∗ . A function P ∈ F( × V ∗ ) is said to be invariant under the action of H if we have   P ρ(h)x, ρ ∗ (h)ξ = P(x, ξ ) (h ∈ H, x ∈ , ξ ∈ V ∗ ). A differential operator D on  is said to be invariant under the action of H if we have D ρ(h) ˆ = ρ(h)D ˆ for any h ∈ H , where we set ρ(h) ˆ f = f ◦ ρ(h −1 ) for h ∈ H and for a function f on . It is equivalent to the condition that the function P ∈ F( × V ∗ ) corresponding to D is H -invariant (cf. [1, Chap. XIV]). Let D()

Rings of Invariant Differential Operators on Homogeneous Cones …

201

denote the ring of all differential operators on  which are invariant under the action of H . Then, we have the following theorem (cf. Helgason [3, Chap. II, Sect. 4]). Theorem 1.1 The algebra D() is isomorphic to the enveloping algebra U(h) of the Lie algebra h corresponding to H . Proof Since H is a Lie group, its Lie algebra h is isomorphic to the Lie algebra consisting of all the left invariant vector fields on H and the ring of all left invariant differential operators on H is isomorphic to U(h) (cf. [5, Chap. II, Sect. 1, No. 4]). φ For a differential operator D H on H , we introduce a differential operator D H on   φ  by D H f = D H ( f ◦ φ) ◦ φ −1 for any smooth function f ∈ C ∞ () on  (cf. [3, Chap. II]). Note that for any two differential operators D H , D H , we have  φ φ φ −1 D H D H f = D H D H ( f ◦ φ) ◦ φ     = DH D H ( f ◦ φ) ◦ φ −1 ◦ φ φ −1 φ    = D H D H ( f ◦ φ) ◦ φ −1 = D H D H f, φ

φ

which implies D H D H = (D H D H )φ . A differential operator D H on H is said to be left −1 ˆ ˆ ˆ invariant if D H L(g) = L(g)D H for any g ∈ H , where we set L(g)q = q ◦ L(g ) for g ∈ H and for a function q on H . In what follows, we shall show that, if D H is left φ φ ˆ = invariant, then D H is an H -invariant differential operator on , that is, D H ρ(g) φ ρ(g)D ˆ for any g ∈ H . Let us assume that D is a left invariant differential operator H H on H . Then, for f ∈ C ∞ (), we have by (1.3)  φ     φ ˆ f = D H f ◦ ρ(g −1 ) ◦ φ ◦ φ −1 = D H f ◦ φ ◦ L(g −1 ◦ φ −1 D H ρ(g)    ˆ = D H L(g) f ◦ φ ◦ φ −1 . Since D H is left invariant, we proceed a calculation again by (1.3) as 

    −1 ˆ ˆ D H L(g) f ◦ φ ◦ φ −1 =  L(g)D H ( f ◦ φ) ◦ φ −1 =  D H ( f ◦ φ) ◦ L(g ) ◦ φ −1 = D H ( f ◦ φ) ◦ φ −1 ◦ ρ(g −1 ). φ

Recalling the definition of D H , we arrive at 

    φ D H ( f ◦ φ) ◦ φ −1 ◦ ρ(g −1 ) = ρ(g) ˆ ˆ D H ( f ◦ φ) ◦ φ −1 = ρ(g)D H f, φ

whence we have proved that D H is H -invariant. Similarly, we introduce an differenφ −1 tial operator D on H for any H -invariant differential operator D on , that is, for   φ −1 any smooth function q ∈ C ∞ (H ) on H , we set D q := D (q ◦ φ −1 ) ◦ φ. Then,  −1  φ −1  φ we see that D is a left invariant differential operator on H and D D = φ −1

φ −1

D D

 for any two H -invariant differential operators D , D on . The proof is

202

H. Nakashima

proceeded similarly, and so we omit it. Hence, we conclude that D() is isomorphic to the ring of left invariant differential operators on H , and hence to U(h).  Let us construct generators of D() explicitly. In what follows, we shall write h x = φ −1 (x) ∈ H (x ∈ ) for simplicity. Then, the latter equation in (1.3) is rewritten as h ρ(h)x = hh x (x ∈ ).

(1.4)

For v ∈ V , we define a function Fv :  × V ∗ → R by Fv (x, ξ ) :=  ρ(h x )v | ξ  (x ∈ , ξ ∈ V ∗ ). It is obvious that Fv (x, ξ ) is linear both in v ∈ V and ξ ∈ V ∗ . Moreover, Fv is H -invariant because (1.4) yields that     Fv (ρ(h)x, ρ ∗ (h)ξ ) = ρ(h ρ(h)x )v | ρ ∗ (h)ξ = ρ(hh x )v | ρ ∗ (h)ξ =  ρ(h x )v | ξ  = Fv (x, ξ ). Let be a V ∗ -valued function on  × V ∗ defined by (x, ξ ) :=

n

Fv j (x, ξ )v ∗j ∈ V ∗ (x ∈ , ξ ∈ V ∗ ).

(1.5)

j=1

  Since Fvi (x, ξ ) = v i | ρ ∗ (h −1 x )ξ (i = 1, . . . , n), the function is described as ∗ (x, ξ ) = ρ ∗ (h −1 x )ξ (x ∈ , ξ ∈ V ).

(1.6)

By construction, (x, ξ ) is invariant under the action of H . It will be used in Sect. 2. For each v ∈ V , we introduce an H -invariant differential operator Dv (x) on  by Dv (x) := Fv (x, ∂x ) (x ∈ ). v (x) := ρ(h x )v = We n note that Dv (x) is linear in v ∈ V . Moreover, by setting   v (x)v , it is a differential operator of the first order of the form i i=1 i Dv (x) =

n i=1

 vi (x)

∂ ∂ xi

(x ∈ ).

In other words, Dv (x) is a directional derivative along  v (x) ∈ V . Put D() := {Dv (x); v ∈ V }. Lemma 1.2 The set D() endowed with the product [Du (x), Dv (x)] := Du (x)Dv (x) − Dv (x)Du (x) (u, v ∈ V )

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203

is a Lie algebra which is isomorphic to h. Before giving a proof to this lemma, we recall an algebraic structure on homogeneous cones introduced by Vinberg [11]. By differentiating the orbit map H h → ρ(h)e ∈  at the unit element 1 H of H , we obtain a linear isomorphism h X → X.e := dρ(X )e ∈ V . Namely, if h(t) is a smooth curve in H such that h(0) = 1 H and h  (0) = X ∈ h, then we have     d  ρ h(t) e = dρ h  (0) e = dρ(X )e. t=0 dt Denoting its inverse map by L : V v → L v ∈ h, we introduce a bilinear product  in V by u  v := L u .v = dρ(L u )v (u, v ∈ V ). We know that the reference point e is a unit element of V with respect to this product (cf. [11, p. 363]). Moreover, the algebra (V, ) satisfies the following three conditions. (V1) (V2) (V3)

L u  v−v  u = L u L v − L v L u for any u, v ∈ V , there exists a linear form s ∈ V ∗ such that s(u  v) defines an inner product in V , the linear operator L v has only real eigenvalues for each v ∈ V .

The algebra (V, ) is called the Vinberg algebra, or clan. In particular, the condition (V1) implies that V endowed with the product [u  v] := u  v − v  u is a Lie algebra isomorphic to h. Note that the product  is neither commutative nor associative in general. Proof Let us begin a proof of Lemma 1.2. Since Du (x) is a directional derivative along  u (x) = ρ(h x )u, we have      d  v (x) = Du (x) ρ(h x )v = ρ(h x+tu (x) )v  . Du (x)  t=0 dt By (1.4), we obtain   h x+tu (x) = h ρ(h x )(e + tu) = h x h e+tu , where h(x) stands for h x . Set g(t) :=h e+tu ∈ H . It is a smooth curve in H such that g(0) = 1 H . By definition, we have ρ g(t) e = e + tu so that      d  d e + tu  = u = L u .e, dρ g  (0) e = ρ g(t) e = t=0 t=0 dt dt whence g  (0) = L u . Therefore, since ρ is a representation, we obtain

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H. Nakashima

       d  d d  ρ h x+tu (x) v  = ρ(h x )ρ g(t) v  = ρ(h x ) · ρ g(t) v  t=0 t=0 dt dt dt   t=0  = ρ(h x ) · dρ g  (0) v = ρ(h x ) u  v . Since Du (x) is a differential operator of the first order, we have   Du (x)Dv (x) = Du  v (x) + Fu (x, ξ )Fv (x, ξ )

ξ =∂x

(1.7)

and hence recalling that Dv (x) is linear in v ∈ V we obtain Du (x)Dv (x) − Dv (x)Du (x) = Du  v (x) − Dv  u (x) = Du  v−v  u (x) for any u, v ∈ V . This shows that the correspondence u → Du (x) induces an algebraic isomorphism from (V, [·  ·]) to (D(), [·, ·]). Since (V, [·  ·]) is isomorphic to h, the lemma is now proved.  The following corollary is given immediately from Lemma 1.2 and Theorem 1.1. n Corollary 1.3 Let {v i }i=1 be any basis of V . Then, D() is generated by Dvi (x) (i = 1, . . . , n).

Before closing this section, we consider the Euler operator E=

n i=1

xi

∂ =  x | ∂x  . ∂ xi

Since it can be described as E = De (x), the operator E is invariant under the action of H and E ∈ D() ⊂ D(). Combining Lemma 1.2 and the fact that e is the unit element in (V, ), we conclude the following. Corollary 1.4 The Euler operator E commutes to all elements in D().

2 Capelli-Type Identities for Generalized Vinberg Cones In this section, we present a concrete calculation of invariant differential operators for generalized Vinberg cones, and show that they admit Capelli-type identities. Let V be a real vector space of dimension 2r + 1 defined by    x y V = v = (v1 , . . . , vr ); v j := 0 j , x0 , y j , z j ∈ R (1 ≤ j ≤ r ) . yj z j Then, a generalized Vinberg cone  is defined by  = {x = (x1 , . . . , xr ) ∈ V ; i (x) > 0 (i = 0, 1, . . . , r )} ,

Rings of Invariant Differential Operators on Homogeneous Cones …

205

where 0 (x) := x0 ,  j (x) := det x j = x0 z j − y 2j ( j = 1, . . . , r ). If we set r = 2, then  is the original Vinberg cone (cf. Vinberg [11, p. 397]). The acting group H is given as  a 0 , a, c j > 0, b j ∈ R H = h = (h 1 , . . . , h r ); h j = bj cj 



(2.1)

with the action ρ on V being ρ(h)v := (h 1 v1 th 1 , . . . , h r vr th r ) (h ∈ H, v ∈ V ). Note that  j (x) are relatively invariant under the action of H , that is,     0 ρ(h)v = a 2 0 (v),  j ρ(h)v = a 2 c2j  j (v)

(2.2)

for any h ∈ H , v ∈ V and j = 1, . . . , r . We take a reference point e ∈  as  e = (e0 , . . . , e0 ) ∈ , e0 :=

 10 . 01

Then, the product  of the Vinberg algebra (V, ) is given by   v  v  = v1  v1 , . . . , vr  vr (v, v  ∈ V ), where v j  v j := =

     1  x0 y j x0 y j x yj 2 0 + y j z j y j z j 0 21 z j  1    x y + x0 y j + 21 y j z j x0 x0 2 0 j . 1 x y  + x0 y j + 21 y j z j 2y j y j + z j z j 2 0 j

1

x 0 2 0 y j 21 z j 



We equip V with an inner product  · | ·  defined by 

r    v | v  := x0 x0 + 2y j y j + z j z j (v, v  ∈ V ). j=1

We note that it satisfies the property (V2) of Vinberg algebras by the linear form s(v) =x0 + z 1 + · · · + zr (v ∈ V ). Let X 0 , Y j , Z j ; j = 1, . . . , r be the basis of V defined by

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H. Nakashima



   10 10 ,..., , 00 00

X0 = j

j

 ˇ   01 ,...,0 , Y j = 0, . . . , 10 

 ˇ    00 Z j = 0, . . . , ,...,0 01

(2.3)

for j = 1, . . . , r . Then, we have the following multiplication table.  X0

Yk

Zk

X0 X0

1 Y 2 k

0

Y j Y j 2δ jk Z j Zj 0

1 δ Y 2 jk j

(2.4)

0 δ jk Z j

Here, left factors of the products are placed in row entries, and right ones are placed in column

entries, and j, k = 1, . . . , r . Let X 0∗ , Y j∗ , Z ∗j ; 1 ≤ j ≤ r ⊂ V ∗ be the dual basis of {X 0 , Y j , Z j } with respect to the inner product  · | · . If we identify V ∗ with V through  · | · , then these can be written as X 0∗ = X 0 , Y j∗ =

1 Yj, 2

Z ∗j = Z j ( j = 1, . . . , r ).

An element ξ in V ∗ will be expressed as ξ = ξ0 X 0∗ +

r   η j Y j∗ + ζ j Z ∗j (ξ0 , η j , ζ j ∈ R).

(2.5)

j=1

For ξ ∈ V ∗ , we introduce irreducible polynomials i∗ (ξ ) (i = 0, 1, . . . , r ) on V ∗ by ∗0 (ξ ) := ξ0 · ζ1 · · · ζr − ∗j (ξ )

r

ζ1 · · · ζ j−1 ·

j=1

1 2

ηj

2

· ζ j+1 · · · ζr ,

(2.6)

:= ζ j ( j = 1, . . . , r ).

When we substitute non-commutative variables into the argument of ∗0 , we use this order of the product. We note that these polynomials i∗ (ξ ) (i = 0, 1, . . . , r ) are the basic relative invariants of the dual cone ∗ of . In fact, it is easily verified that     ∗ ∗0 ρ ∗ (h)ξ = a −2 c1−2 · · · cr−2 ∗0 (ξ ), ∗j ρ ∗ (h)ξ = c−2 j  j (ξ ) for j = 1, . . . , r , where h ∈ H is as in (2.1). Proposition 2.1 For x ∈ , ξ ∈ V ∗ and j = 1, . . . , r , one has

(2.7)

Rings of Invariant Differential Operators on Homogeneous Cones …

FX 0 (x, ξ ) = x0 ξ0 +  FY j (x, ξ ) =

r

yjηj +

j=1

x0 z j − y 2j  x0

y 2j x0

207

 ζj ,

x0 z j − y 2j  x0 η j + 2y j ζ j , FZ j (x, ξ ) = ζj. x0

Proof At first, we calculate  ρ(h)v | ξ  for v = X 0 , Y j , Z j with h ∈ H and ξ ∈ V ∗ . Let h = (h 1 , . . . , h r ) ∈ H as in (2.1). Then, we have   2  a abr a 2 ab1 ,..., , ρ(h)X 0 = 2 abr br2   ab1 b1  0 ac j , 0, . . . , 0 , ρ(h)Y j = 0, . . . , 0,  jcj   ac j 2b 0 0 , 0, . . . , 0 ρ(h)Z j = 0, . . . , 0, 0 c2j 

for j = 1, . . . , r . Here, the non-zero entries of ρ(h)Y j and ρ(h)Z j are on the jth position. Let ξ ∈ V ∗ be as in (2.5). Then, we obtain  ρ(h)X 0 | ξ  = a 2 ξ0 + 

r   ab j η j + b2j ζ j ,

j=1    ρ(h)Y j | ξ = ac j η j + 2b j c j ζ j , ρ(h)Z j | ξ = c2j ζ j .

Now we assume that h = h x for x = (x1 , . . . , xr ) ∈ , namely, ρ(h)e = x. Since x j = h j e0 th j = h j th j , we have  xj = and hence a=

√

x0 y j yj z j



 =

a 2 ab j ab j b2j + c2j

 ( j = 1, . . . , r ),

 yj x0 , b j = √ , c j = z j − b2j = x0



 j (x) . x0

This yields that FX 0 (x, ξ ) = x0 ξ0 +

r 

yjηj +

 j=1  2y j  j (x) FY j (x, ξ ) =  j (x)η j + ζj, x0 whence the proposition is now proved.

y 2j x0

 ζj ,

FZ j (x, ξ ) =

 j (x) ζj, x0 

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H. Nakashima

   Set ∂x := ∂∂x0 X 0∗ + rj=1 ∂∂y j Y j∗ + ∂z∂ j Z ∗j . Corollary 1.3 together with Proposition 2.1 tells us that D() is generated by the following invariant differential operators ( j = 1, . . . , r ).

 r y 2j ∂ ∂ ∂ D X 0 (x) = FX 0 (x, ∂x ) = x0 + + yj , ∂ x0 ∂yj x0 ∂z j j=1   x0 z j − y 2j  ∂ ∂ x0 , + 2y j DY j (x) = FY j (x, ∂x ) = x0 ∂yj ∂z j 2 x0 z j − y j ∂ D Z j (x) = FZ j (x, ∂x ) = . x0 ∂z j To go further, we prepare some notations. Let P ∈ F( × V ∗ ) be a function of the form (1.1). Let us denote by Du .P the action of Du (x) (u ∈ V ) on P defined by Du .P(x, ξ ) :=



 Du (x)aα (x) ξ1α1 · · · ξnαn (x ∈ ).

α∈Zn≥0

Then, (1.7) yields that Du .Fv (x, ∂x ) = Fu  v (x, ∂x ).

(2.8)

Moreover, if we set P(x, ξ ) = P1 (x, ξ ) · · · Pm (x, ξ ) with Pi ∈ F( × V ∗ ) for i = 1, . . . , m, then we have Dv .P(x, ξ ) =

m  (Dv .Pi )(x, ξ ) P j (x, ξ )

(2.9)

j=i

i=1

because Dv (x) is a differential operator of the first order. In particular, if Pi (x, ξ ) = Fvi (x, ξ ) (vi ∈ V ) in (2.9), then (2.8) yields that Du .P(x, ξ ) =

m i=1

Fu  vi (x, ξ )



Fv j (x, ξ ).

(2.10)

j=i

For P1 , . . . , Pm ∈ F( × V ∗ ), we denote by (P1 · · · Pm )(x, ∂x ) the differential operator obtained by substituting ∂x after expanding the product of polynomials, that is,   (P1 · · · Pm )(x, ∂x ) := P1 (x, ξ ) · · · Pm (x, ξ )

ξ =∂x

.

Then, since Dv (x) is a differential operator of the first order, we have   Dv (x) P(x, ∂x ) = Dv .P(x, ∂x ) + (Fv P)(x, ∂x ).

(2.11)

Rings of Invariant Differential Operators on Homogeneous Cones …

209

+1 For m = (m 0 , m 1 , . . . , m r ) ∈ Zr +1 and n = (n 0 , n 1 , . . . , n r ) ∈ Zr≥0 , we put

 (x) := m

r 

i (x)

mi

(x ∈ ),

n ∗ (ξ )

:=

i=0

r 

i∗ (ξ )ni (ξ ∈ V ∗ ).

i=0

Using a V ∗ -valued function on  × V ∗ defined in (1.5), we introduce H -invariant +1 by functions Rn on  × V ∗ for n ∈ Zr≥0  n Rn (x, ξ ) := ∗ (x, ξ ) (x ∈ , ξ ∈ V ∗ ). Since (x, ξ ) is H -invariant, so is Rn . Moreover, the formulas (1.6), (2.2) and (2.7) yield that (cf. see also [9, (3.5)]) −1

Rn (x, ξ ) = nσ∗ σ (x) ∗ (ξ ) (x ∈ , ξ ∈ V ∗ ), n

where σ , σ∗ are square matrices of size r + 1 defined, respectively, by    1 1 1 0 , σ∗ = . σ = t 1 Ir 0 Ir 

(2.12)

Here, Ir is the identity matrix of size r and 1 = (1, . . . , 1) ∈ Zr . Note that these matrices σ , σ∗ are called the multiplier matrices of  and ∗ , respectively (cf. [8, +1 , we see that Rn ∈ F( × V ∗ ) and (1.4)], see also [7, Theorem 2.2]). Since n ∈ Zr≥0 hence we can consider the differential operators Rn (x, ∂x ). Let e0 , e1 , . . . , er be the standard basis of Rr +1 , that is, e j has 1 on the position j and zeros elsewhere. Since   Rei (x, ξ ) = i (x, ξ ) (i = 0, 1, . . . , r ) by definition, (2.6) yields that r    Re0 (x, ∂x ) = FX 0 FZ 1 · · · FZ r − ( 21 FY j )2 FZ k (x, ∂x ) j=1

(2.13)

k= j

and Re j (x, ∂x ) = FZ j (x, ∂x ) ( j = 1, . . . , r ). For brevity, we set Dw = Dw (x) (w = X 0 , Y j , Z j ). Lemma 2.2 For j, k = 1, . . . , r , one has (i) D X 0 .Re0 (x, ∂x ) = Re0 (x, ∂x ) and D X 0 .Rek (x, ∂x ) = 0, (ii) D Z j .Re0 (x, ∂x ) = Re0 (x, ∂x ) and D Z j .Rek (x, ∂x ) = δ jk Re j (x, ∂x ), (iii) DY j .Re0 (x, ∂x ) = 0 and DY j .Rek (x, ∂x ) = 0. Proof The formula (2.8) together with the multiplication table (2.4) yields that

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D X 0 .Rek (x, ∂x ) = D X 0 .FZ k (x, ∂x ) = FX 0  Z k (x, ∂x ) = 0, D Z j .Rek (x, ∂x ) = D Z j .FZ k (x, ∂x ) = FZ j  Z k (x, ∂x ) = δ jk FZ j (x, ∂x ) = δ jk Re j (x, ∂x ) and DY j .Rek (x, ∂x ) = DY j .FZ k (x, ∂x ) = FY j  Z k (x, ∂x ) = 0 for j, k = 1, . . . , r . These equations indicate the latter assertions of (i)–(iii) in the lemma. Again by (2.8) and (2.4), we have (D  X 0 .FX20)(x, ∂x ) = FX 0  X 0 (x, ∂x ) = FX 0 (x, ∂2x ), D X 0 .FY j (x, ξ ) = 2FY j · FX 0  Y j (x, ξ ) = FY j (x, ξ ) for j = 1, . . . , r . Therefore, (2.10) shows that  

and

 D X 0 .(FX 0 FZ 1 · · · FZ r ) (x, ∂x ) = (FX 0 FZ 1 · · · FZ r )(x, ∂x )

      FZ k (x, ∂x ) = ( 21 FY j )2 FZ k (x, ∂x ), D X 0 . ( 21 FY j )2 k= j

k= j

whence we obtain D X 0 .Re0 (x, ∂x ) = Re0 (x, ∂x ). Similarly, we have by (2.10) and (2.4) D Z j .FX 0 (x, ∂x ) = FZ j  X 0 (x, ∂x ) = 0 and

  (D Z j .FY2k )(x, ∂x ) = 2FYk · FZ j  Yk (x, ∂x ) = δ jk FY2k (X, ∂x ).

(2.14)

  Hence, we see that the equation D Z j .Re0 (x, ∂x ) = Re0 (x, ∂x ) holds. Next, let us consider the action of DY j (x) for j = 1, . . . , r . Again by (2.8) and (2.4), we have for k = 1, . . . , r DY j .FX 0 (x, ∂x ) = FY j  X 0 (x, ∂x ) = FY j (x, ∂x ), DY j .FYk (x, ∂x ) = FY j  Yk (x, ∂x ) = 2δ jk FZ j (x, ∂x ), DY j .FZ k (x, ∂x ) = FY j  Z k (x, ∂x ) = 0 and by (2.10) 

     DY j .FY2k (x, ∂x ) = 2FYk · FY j  Yk (x, ∂x ) = 4δ jk FYk FZ k (x, ∂x ).

Therefore, (2.10) yields that 

    DY j . FX 0 FZ 1 · · · FZ r (x, ∂x ) = FY j FZ 1 · · · FZ r (x, ∂x )

Rings of Invariant Differential Operators on Homogeneous Cones …



and

211

     FZ l (x, ∂x ) = δ jk FYk FZ 1 · · · FZ r (x, ∂x ) DY j . ( 21 FYk )2 l=k

for k = 1, . . . , r . Hence, we obtain DY j .Re0 (x, ∂x ) = 0 and now the proof is completed.  +1 Lemma 2.3 Let n = (n 0 , n 1 , . . . , n r ) ∈ Zr≥0 . For j = 1, . . . , r , one has

  (i) D X 0  Rn (x, ∂x ) = (FX 0 Rn )(x, ∂x ) + n 0 Rn (x, ∂x ), (ii) D Z j Rn (x, ∂x ) = FZ j Rn (x, ∂x ) + (n 0 + n j )Rn (x, ∂x ),  2     (iii) 21 DY j Rn (x, ∂x ) = 41 FY2j Rn + 2FZ j Rn (x, ∂x ). Proof (i) By Lemma 2.2 and (2.9), we obtain D X 0 .Rn (x, ∂x ) =

r   n k Rn−ek · (D X 0 .Rek ) (x, ∂x ) k=0

= (n 0 Rn−e0 · Re0 )(x, ∂x ) = n 0 Rn (x, ∂x ) and

r   n k Rn−ei · (D Z j .Rek ) (x, ∂x ) D Z j .Rn (x, ∂x ) = k=0

= (n 0 Rn−e0 · Re0 )(x, ∂x ) + (n j Rn−e j · Re j )(x, ∂x ) = (n 0 + n j )Rn (x, ∂x ). By (2.11), this shows the assertions (i) and (ii). For the assertion (iii), we use again Lemma 2.2 and (2.9) so that DY j (Rn (x, ∂x )) = (FY j Rn )(x, ∂x ), and hence DY2 j (Rn (x, ∂x )) = DY j ((FY j Rn )(x, ∂x ))     =  FY j  Y j Rn (x, ∂x ) + FY j DY j (Rn (x, ∂x ) = 2FZ j Rn + FY2j Rn (x, ∂x ). Therefore, the lemma is now proved. Let us set D(x) := (x, ∂x ), that is, D(x) = FX 0 (x, ∂x )X 0∗ +

r  j=1

 FY j (x, ∂x )Y j∗ + FZ j (x, ∂x )Z ∗j ∈ D() ⊗ V ∗ .



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H. Nakashima

By definition, D(x) is a V ∗ -valued differential operator which is invariant under the action of H . For non-commutative variables ai j (1 ≤ i ≤ n, 1 ≤ j ≤ m i ), we set mi n  

mn m1 m2       ai j := a1 j a2 j · · · an j j=1 j=1  j=1   j=1  = a11 · a12 · · · a1m 1 · · · an1 · an2 · · · anm n .

i=1

(2.15)

+1 Theorem 2.4 (Capelli-type identity) Let n = (n 0 , n 1 , . . . , n r ) ∈ Zr≥0 . For j = 0, 1, . . . , r and k = 1, . . . , n j , let S jk be elements in V ∗ defined by

S0k :=

r 2

−k



X 0∗

−

r

(n i + k)Z i∗ ∈ V ∗ , S jk := −k Z ∗j ∈ V ∗ .

i=1

Then, for x ∈ , one has nj r  

  −1 n ∗j D(x) + S j,n j −k = Rn (x, ∂x ) = nσ∗ σ (x)∗ (∂x ).

(2.16)

j=0 k=1

Proof We shall show this theorem by induction on n 0 . We first assume n 0 = 0. Let j, k = 1, . . . , r . Since FZnk (x, ∂x ) = Rnek (x, ∂x ), we have by setting n = nek in Lemma 2.3 (ii) and by (2.11)   D Z j FZnk (x, ∂x ) = nδ jk FZnk (x, ∂x ) + (FZ j FZnk )(x, ∂x )

(2.17)

for any non-negative integer n. If we set j = k, then we have   (x, ∂x ), (D Z j − n) FZn j (x, ∂x ) = FZn+1 j which implies 

   n D Z j − (n j − 1) · · · D Z j − 1 D Z j = FZ jj (x, ∂x ).

The equation (2.17) also shows that for k = j 

 D Z j .FZnkk (x, ∂x ) = 0,

and hence, combining this and (2.18), we obtain by (2.10) nj r   j=1 k=1

  ∗j D(x) + S j,n j −k = R(0,n 1 ,...,nr ) (x, ∂x )

(2.18)

Rings of Invariant Differential Operators on Homogeneous Cones …

213

for any n 1 , . . . , n r ∈ Z≥0 . This shows that the theorem holds for n 0 = 0. Next, let us consider the case n 0 ≥ 1, that is, we assume that (2.16) holds for n 0 = n and show    ∗0 D(x) + S0,n Rn (x, ∂x ) = Rn+e0 (x, ∂x ). By definition of S0,n , we have D(x) + S0,n = D X 0 X 0∗ +

r 

 DY j Y j∗ + D Z j Z ∗j ,

j=1

where we set

D X 0 := D X 0 +

r − n, 2

D Z j := D Z j − n − n j .

Since Lemma 2.3 implies that   r D X 0 Rn (x, ∂x ) = (FX 0 Rn )(x, ∂x ) + Rn (x, ∂x ) 2 and

  D Z j Rn (x, ∂x ) = Rn+e j (x, ∂x ) ( j = 1, . . . , r ),

we obtain D X 0

r 

  D Z j Rn (x, ∂x )

j=1

r = (FX 0 FZ 1 · · · FZ r · Rn )(x, ∂x ) + (FZ 1 · · · FZ r · Rn )(x, ∂x ). 2 Set e[ j,r ] := e j + · · · + er . It follows again by Lemma 2.3 that   ( 21 DY j )2 D Z j+1 · · · D Z r Rn (x, ∂x )   = ( 21 DY j )2 Rn+e j+1 +···+er (x, ∂x ) = 41 ((FY2j + 2FZ j )Rn+e j+1 +···+er )(x, ∂x )   = 41 FY2j Rn+e[ j+1,r ] (x, ∂x ) + 21 Rn+e[ j,r ] (x, ∂x ). Lemma 2.3 (ii) together with (2.10) and (2.14) shows that D Z j−1



    FY2j Rn+e[ j+1,r ] (x, ∂x ) = FZ j−1 FY2j Rn+e[ j+1,r ] (x, ∂x ),

and repeating this we obtain

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H. Nakashima







1  D · · · D Z j−1 FY2j Rn+e[ j+1,r ] (x, ∂x ) 4 Z1   = 41 FZ 1 · · · FZ j−1 FY2j Rn+e[ j+1,r ] (x, ∂x ) j   = FZ 1 · · · ( 21 FˇY j )2 · · · FZ r · Rn (x, ∂x ).

Moreover, we have again by Lemma 2.3 (ii) 1  D 2 Z1

    · · · D Z j−1 Rn+e[ j,r ] (x, ∂x ) = 21 FZ 1 · · · FZ j−1 · Rn+e[ j,r ] (x, ∂x ) 1  = 2 FZ 1 · · · FZ r · Rn (x, ∂x ),

and hence we finally obtain by (2.13)    ∗0 D(x) + S0,n Rn (x, ∂x ) r r j        1 ˇ 2 DZ j − D Z 1 · · · ( 2 DY j ) · · · D Z r Rn (x, ∂x ) = DX0 j=1 j=1   = FX 0 FZ 1 · · · FZ r · Rn + r2 FZ 1 · · · FZ r · Rn (x, ∂x ) r   1  FZ k · Rn + 21 FZ 1 · · · FZ r · Rn (x, ∂x ) ( 2 FY j )2 − k= j   j=1 = Re0 · Rn (x, ∂x ) = Rn+e0 (x, ∂x ). The proof is now completed.



In particular, by putting n = e0 , we get the following corollary. Corollary 2.5 For x ∈ , one has  r  1 (x) · · · r (x) ∗ 0 (∂x ). ∗0 D(x) + X 0∗ = 2 0 (x)r −1 Proof Since e0 σ∗ σ −1 = (1 − r, 1, . . . , 1), where σ , σ∗ are as in (2.12), we obtain the formula by Theorem 2.4. 

References 1. J. Faraut, A. Korányi, Analysis on Symmetric Cones (Clarendon Press, Oxford, 1994) 2. Harish-Chandra, Differential operators on a semisimple lie algebra. Am. J. Math. 79, 87–120 (1957) 3. S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions. Mathematical Surveys and Monographs, vol. 83 (American Mathematical Society, Province, 2000) 4. S. Helgason, Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs, vol. 39, 2nd edn. (American Mathematical Society, Province, 2008) 5. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics, vol. 34 (American Mathematical Society, Province, 2001)

Rings of Invariant Differential Operators on Homogeneous Cones …

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6. R. Howe, T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions. Math. Ann. 290, 565–619 (1991) 7. H. Ishi, Basic relative invariants associated to homogeneous cones and applications. J. Lie Theory 11, 155–171 (2001) 8. H. Nakashima, Basic relative invariants of homogeneous cones. J. Lie Theory 24, 1013–1032 (2014) 9. H. Nakashima, Basic relative invariants of homogeneous cones and their Laplace transforms. J. Math. Soc. Jpn. 70, 323–342 (2018) 10. T. Nomura, Algebraically independent generators of invariant differential operators on a symmetric cone. J. Reine Angew. Math. 400, 122–133 (1989) 11. E.B. Vinberg, The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12, 340– 403 (1963) 12. J.-Q. Zhong, Dimensions of the ring of invariant operators on bounded homogeneous domains. Chinese Ann. Math. 1(2), 261–272 (1980)

An Extension of Pizzetti’s Formula Associated with the Dunkl Operators Nobukazu Shimeno and Naoya Tani

To the memory of Professor Takaaki Nomura

Abstract We give an extension of Pizzetti’s formula associated with the Dunkl operators. It gives an explicit formula for the Dunkl inner product of an arbitrary function and a homogeneous Dunkl harmonic polynomial on the unit sphere. Keywords Pizzetti’s formula · Harmonic polynomials · Dunkl operators 2010 Mathematics Subject Classification. 33C52 · 32C55 · 42C10

1 Introduction Dunkl analysis that was initiated by C. Dunkl is a study of the function theory for the Dunkl operators and the Dunkl Laplacian. It is a deformation of calculus of several variables for partial derivatives and the Euclidean Laplacian, and analogues of classical results have been developed. In this paper, we study analogues in the Dunkl analysis of classical Pizzetti’s formula for spherical mean [14, 18, 19] and its extension given by Bezubik, D¸abrowska and Strasburger [4, Corollary 2.1] and Estrada [12]. The main result is the Extended Pizzetti formula associated with the Dunkl operators (Theorem 3.2):

N. Shimeno (B) School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo 669-1337, Japan e-mail: [email protected] N. Tani Graduate School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo 669-1337, Japan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_13

217

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N. Shimeno and N. Tani 1

ωκ,d

 S d−1

q(y) f (r y)h 2κ (y)dσ (y) =

N  (q(D)nκ f )(0)  r m+2n + o(r m+2N ) (r → 0+ ) n! (λκ + 1)m+n 2

n=0

for a homogeneous Dunkl harmonic polynomial q of degree m and a smooth function f on Rd . Here κ is a parameter associated with a finite reflection group. For κ = 0, the above formula gives the classical result [4, 12] where ω0,d is the surface area of the unit sphere S d−1 , h 0 (y) = 1, D for κ = 0 is the gradient, 0 is the Euclidean Laplacian and λ0 = d/2 − 1. In particular, the case of κ = 0 and q(y) = 1 gives classical Pizzetti’s formula. See the next section for explanation of the notions undefined here. We prove the Extended Pizzetti formula by using the canonical decomposition of a homogeneous polynomial with respect to the Dunkl harmonic polynomials. Pizzetti’s formula follows as a corollary of the Extended Pizzetti formula. Pizzetti’s formula associated with the Dunkl Laplacian was established by Mejjaoli, Trimèche [15, Theorem 4.17] and Ben Salem, Touahri [3, Theorem 2.4]. Our proof is different from theirs. We also deduce the Extended Pizzetti formula from Pizzetti’s formula and Hobson’s formula [22]. As a corollary of the Extended Pizzetti formula, we obtained the Funk–Hecke formula associated with the Dunkl operators, which was originally proved by Xu [24] by a different method.

2 Notation and Preliminaries In this section, we review the Dunkl operators and Dunkl h-harmonics. We refer [6, 9, 20] for details. Let d be a positive integer. Let  ,  be the standard inner product on Rd and put ||x|| = x, x1/2 for x ∈ Rd . Let R ⊂ Rd be a reduced root system, which is not necessarily crystallographic. For α ∈ R, we write rα for the reflection with respect to the hyperplane α ⊥ . Let G denote the finite reflection group generated by {rα : α ∈ R}. We fix a positive system R+ ⊂ R. Let P = P(Rd ) denote the space of polynomials on Rd with real coefficients. For a non-negative integer m, let Pm denote the subspace of P consisting of the homogeneous polynomials of degree m. Let κ : R → R≥0 , α → κα be a G-invariant function on R. We call κ a (nonnegative) multiplicity function. We define λκ =

 d −1+ κα . 2 α∈R

(2.1)

+

For ξ ∈ Rd \ {0}, let ∂ξ denote the directional derivative corresponding to ξ and define the Dunkl operator Dξ by

An Extension of Pizzetti’s Formula Associated with the Dunkl Operators

Dξ f (x) = ∂ξ f (x) +



κα α, ξ 

α∈R+

f (x) − f (rα x) . α, x

219

(2.2)

The Dunkl operators satisfy Dξ Dη =Dη Dξ for all ξ, η ∈ Rd . Let {e1 , . . . , ed } be the standard orthonormal basis of Rd . We write ∂ j = ∂e j , D j = De j . The Dunkl Laplacian κ is defined by d  D 2j . (2.3) κ = j=1

The Dunkl operators are homogeneous of degree −1 and the Dunkl Laplacian κ is homogeneous of degree −2. Put D = (D1 , . . . , Dd ). For p, q ∈ P define  p, qκ = ( p(D)q)(0). Then  · , · κ gives a non-degenerate symmetric bilinear form on P. If p ∈ Pl , q ∈ Pm and l = m, then  p, qκ = 0. Define Hκ := { p ∈ P ; κ p = 0} and Hκ,m = Hκ ∩ Pm . We call an element of Hκ an h-harmonic polynomial or Dunkl harmonic polynomial. We recall the canonical decomposition of a homogeneous polynomial. Define the shifted factorial (a)0 = 1, (a)n = a(a + 1) · · · (a + n − 1), where a is a complex number and n is a positive integer. For P ∈ Pn define [n/2]

projκ,n P(x) =

 j=0

4j

1 ||x||2 j κj P(x). j! (−λκ − n + 1) j

Here [n/2] means the largest integer with [n/2] ≤ n/2. Theorem 2.1 (Canonical decomposition, [8, Theorem 1.11], [9, Theorem 7.1.15], [2, Theorem 5.1]) We have an orthogonal direct sum decomposition Pn = Hκ,n ⊕ ||x||2 Hκ,n−2 ⊕ · · · ⊕ ||x||2[n/2] Hκ,n−2[n/2] . The decomposition of p ∈ Pn is given by [n/2]

p(x) =



||x||2i pn−2i (x) ( pn−2i ∈ Hκ,n−2i )

i=0

with pn−2i (x) =

4i i! (λ

κ

1 projκ,n−2i iκ p(x). + 1 + n − 2i)i

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N. Shimeno and N. Tani

Let h κ (x) denote the weight function defined by h κ (x) =



|α, x|κα .

(2.4)

α∈R+

Let S d−1 denote the unit sphere in Rd and dσ the surface measure on S d−1 . Define  ωκ,d =

S d−1

h 2κ (y)dσ (y).

We recall the orthogonality relation for h-spherical harmonics. Theorem 2.2 ([6, Theorem 3.1.2, Theorem 3.1.9]) Suppose p ∈ Hκ,l and q ∈ Hκ,m . Then 1 ωκ,d

 S d−1

p(y)q(y)h 2κ (y)dσ (y) =

1  p, qκ . 2m (λκ + 1)m

Note that in Theorem 2.2  p, qκ = δlm q(D) p, since  p, qκ = q(D) p is a constant if l = m and  p, qκ = 0 if l = m. If l < m, then by Theorems 2.1 and 2.2, 1 ωκ,d

 S d−1

p(y)q(y)h 2κ (y)dσ (y) = 0 ( p ∈ Pl , q ∈ Hκ,m ).

3 Extended Pizzetti’s Formula In this section, we prove Pizzetti’s formula and its extension associated with the Dunkl operators. We start with the following mean value property for homogeneous polynomials. Theorem 3.1 ([23]) Let p ∈ Pl and q ∈ Hκ,m . Assume l − m is a non-negative even integer and set l − m = 2n. Then 1 ωκ,d

 S d−1

q(y) p(y)h 2κ (y)dσ (y) =

1 q(D)nκ p. 2m+2n n! (λκ + 1)m+n

The integral on the left hand side of the above identity vanishes if l − m is a negative or odd integer. Proof By Theorems 2.1 and 2.2, 1 ωκ,d

 S d−1

q(y) p(y)h 2κ (y)dσ (y) = 0,

An Extension of Pizzetti’s Formula Associated with the Dunkl Operators

221

if l − m is a negative or odd integer. Otherwise, by Theorems 2.1 and 2.2, 

1 ωκ,d

S d−1

q(y) p(y)h 2κ (y)dσ (y) =

1



q(y) pm (y)h 2κ (y)dσ (y) ωκ,d S d−1  1 1 = q(y)nκ p(y)h 2κ (y)dσ (y) n ωκ,d 4 n! (λκ + 1 + m)n S d−1 1 1 = n q(D)nκ p 4 n! (λκ + 1 + m)n 2m (λκ + 1)m 1 = m+2n q(D)nκ p. 2 n! (λκ + 1)m+n

 If p ∈ Hκ,l in Theorem 3.1, we recover Theorem 2.2. If q = 1 in Theorem 3.1, we get Pizzetti’s formula for homogeneous polynomials. Note that such a simple formula as in Theorem 3.1 does not hold for general q ∈ Pm . For κ = 0, Theorem 3.1 was established by Bezubik, D¸abrowska and Strasburger [4, Corollary 2.1]. Our proof closely follows their proof by using the canonical decomposition, the orthogonality relation and the relation of two inner products on polynomials. For κ = 0, Theorem 3.1 was also given by Estrada [10, Proposition 3.3] with a proof similar to that of [4, Corollary 2.1]. An extension of Pizzetti’s formula follows from Theorem 3.1 and Taylor’s theorem. Theorem 3.2 Suppose q ∈ Hκ,m . For a smooth function f on a neighbourhood of 0 ∈ Rd we have 1 ωκ,d

 S d−1

q(y) f (r y)h 2κ (y)dσ (y) =

N  (q(D)nκ f )(0)  r m+2n + o(r m+2N ) n! (λ + 1) 2 κ m+n n=0 (3.1)

as r → 0+ . If f is real analytic in the unit ball around 0 ∈ Rd , then there exists a constant ρ ∈ (0, 1) such that 1 ωκ,d

 S d−1

q(y) f (r y)h 2κ (y)dσ (y) =

∞  (q(D)nκ f )(0)  r m+2n n! (λκ + 1)m+n 2 n=0

(3.2)

for any r with 0 < r < ρ. Proof By Taylor’s theorem f (x) = Theorem 3.1, we have

m+2N l=0

pl (x) + o(||x||m+2N ) with pl ∈ Pl . By

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1 ωκ,d

 q(y) f (r y)h 2κ (y)dσ (y)

S d−1

=

 N 1  m+2n r q(y) pm+2n (y)h 2κ (y)dσ (y) + o(r m+2N ) ωκ,d n=0 S d−1

=

N  q(D)nκ pm+2n  r m+2n + o(r m+2N ). n! (λ + 1) 2 κ m+n n=0

If f ∈ P, then it is clear that q(D)nκ pm+2n = (q(D)nκ f )(0).

(3.3)

(3.3) also holds for any smooth function f by Taylor’s formula associated with the Dunkl operators (cf. [21, Corollary 2.17]). Hence (3.1) follows. d Now assume that f is real analytic in the unit ball of radius ∞1 around 0 ∈ R . Then there exists ρ ∈ (0, 1) such that the multiple Taylor series l=0 pl (x) ( pl ∈ Pl ) of f converges normally to f on {x ∈ Rd ; ||x|| < ρ}. By Theorem 3.1, we have 1 ωκ,d

 S d−1

q(y) f (r y)h 2κ (y)dσ (y)  ∞ 1  m+2n = r q(y) pm+2n (y)h 2κ (y)dσ (y) ωκ,d n=0 S d−1 =

∞  q(D)nκ pm+2n  r m+2n n! (λκ + 1)m+n 2 n=0

for any r ∈ (0, ρ). By Taylor’s formula associated with the Dunkl operators (cf. [21, Corollary 2.17]), (3.3) holds for a real analytic function f around a neighbourhood  of 0 ∈ Rd , hence (3.2) follows. For κ = 0, the above theorem for smooth f is given by Estrada [12, Theorem 5.1]. Putting q = 1 ∈ Hκ,0 in the above theorem, we have an analogue of Pizzetti’s formula as a corollary. Corollary 3.3 For a smooth function f on a neighbourhood of 0 ∈ Rd we have 1 ωκ,d

 S d−1

f (r y)h 2κ (y)dσ (y) =

N  (nκ f )(0)  r 2n + o(r 2N ) n! (λ + 1) 2 κ n n=0

(3.4)

as r → 0+ . If f is real analytic in the unit ball around 0 ∈ Rd , then there exists a constant ρ ∈ (0, 1) such that

An Extension of Pizzetti’s Formula Associated with the Dunkl Operators

1 ωκ,d

 S d−1

f (r y)h 2κ (y)dσ (y) =

∞  (nκ f )(0)  r 2n n! (λκ + 1)n 2 n=0

223

(3.5)

for any r with 0 < r < ρ. Pizzetti’s formula associated with the Dunkl operators was established by [3, 15]. In [22] the first author proved (3.5) for f ∈ P as an application of Hobson’s formula associated with the Dunkl operators. (Note there is an obvious mistake of unnecessary factor (−1)n in [22, Corollary 4.5].) Our proof given here provides an alternative proof of the formula. For κ = 0 (3.5) gives original Pizzetti’s formula [1, 14, 18, 19]. The right hand side of (3.2), which we call the Extended Pizzetti series, is related with the Bessel function. For α ≥ −1/2, let ϕα denote the function defined by ϕα (x) = Γ (α + 1)

 ∞  ||x|| 2n (−1)n Jα (||x||) = Γ (α + 1) , (||x||/2)α n! Γ (α + n + 1) 2 n=0

where Jα denote the Bessel function of the first kind. The Extended Pizzetti series can be written as ∞  √ (q(D)nκ f )(0)  r m+2n 1  r m = (q(D)ϕλκ +m ( −1D r ) f )(0). n! (λκ + 1)m+n 2 (λκ )m 2 n=0

√ Here || −1D||2 in the series expansion of ϕλκ +m (D) is understood to be −(D12 + · · · + Dd2 ) = −κ . In particular, the right hand side of (3.5), which we call the Pizzetti series, can be written as ∞  √ (nκ f )(0)  r 2n = (ϕλκ ( −1D r ) f )(0). n! (λκ + 1)n 2 n=0

We have proved Pizzetti’s formula (Corollary 3.3) as a special case of the Extended Pizzetti formula (Theorem 3.2). Conversely, we will show that Theorems 3.1 and 3.2 follow from Corollary 3.3 as an application of Hobson’s formula. We recall Hobson’s formula associated with the Dunkl operators. Theorem 3.4 (Hobson’s formula [22]) Put ρ = ||x||. If p ∈ Pm , f 0 ∈ C ∞ ((0, ∞)) and f (x) = f 0 (ρ), then 

 1  1 d m−i f 0 (ρ) iκ p(x). p(D) f (x) = i i! 2 ρ dρ i=0 [m/2]

(3.6)

Proposition 3.5 Put ρ = ||x||. Let m and j be non-negative integers and q ∈ Hm,κ . If j < m, then q(D)r 2 j = 0, while if j ≥ m, then

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2m j! ρ 2 j−2m q(x). ( j − m)!

q(D)ρ 2 j =

Proof Since iκ q = 0 for i ≥ 1, it follows from Theorem 3.4 that  q(D)ρ 2 j =

1 d ρ dρ

m

ρ 2 j q(x),

(3.7) 

hence the results follow. 2J

Assume q ∈ Hκ,m and let f (x) = expansion. By Pizzetti’s formula (3.4), 1 ωκ,d

i=0

 q(r y) f (r y)h 2κ (y)dσ (y) =

S d−1

j

pi + o(||x||2J ) ( pi ∈ Pi ) be the Taylor

J j  κ (q f )(0)  r 2 j + o(r 2J ). j! (λ + 1) 2 κ j j=0 j

(3.8)

j

If 2 j < m, then κ (q f )(0) = 0, otherwise, κ (q f )(0) = κ (qp2 j−m ). If 2 j ≥ m, then κj (qp2 j−m ) = ||x||2 j , q(x) p2 j−m (x)κ = q(D)||x||2 j , p2 j−m (x)κ . If j ≥ m, then by Proposition 3.5, j

κ (q p2 j−m ) =

2m j! 2m j! j−m p2 j−m , ||x||2 j−2m q(x), p2 j−m (x)κ = q(D)κ ( j − m)! ( j − m)!

j

otherwise κ (qp2 j−m ) = 0. Thus (3.8) becomes rm ωκ,d

 S d−1

q(y) f (r y)h 2κ (y)dσ (y) =

J  j=m

 r 2 j 2m (q(D)κj−m f )(0) + o(r 2J ). ( j − m)! (λκ + 1) j 2

Hence, we have (3.1) by putting j − m = n, J = m + N . Remark 3.6 We give a remark on the case of κ = 0. In this case, Proposition 3.5 was proved by Estrada [11, Proposition 3.2] by using the canonical decomposition. Classical Hobson’s formula [13, 16, 17] is a special case κ = 0 of Theorem 3.4. Hobson’s formula gives a simpler proof of Proposition 3.5 also for κ = 0. Estrada [12, Proposition 6.1] deduced the Extended Pizzetti formula from Pizzetti’s formula by using Proposition 3.5 for κ = 0.

An Extension of Pizzetti’s Formula Associated with the Dunkl Operators

225

4 Application We can prove an analogue of the Funk–Hecke formula as a corollary of Theorem 3.1. Our proof follows closely to the proof of Funk–Hecke formula in superspace given by De Bie and Sommen [7, §7]. Let Vκ denote the Dunkl intertwining operator (cf. [9, §6.5]). It is a linear operator that is uniquely determined by Vκ Pn ⊂ Pn (n ∈ Z≥0 ), Vκ 1 = 1, Dξ Vκ = Vκ ∂ξ (ξ ∈ Rd \ {0}). If l − m = 2n is an even non-negative integer and q ∈ Hm , then by Theorem 3.1, 1 ωκ,d

 S d−1

Vκ [x, · l ](y)q(y)h 2κ (y)dσ (y) =

1 2m+2n n! (λ

+ 1)m+n

κ

(4.1)

q(D)nκ Vκ [x, · l ]

1 Vκ [q(∂)n x, · l ] 2m+2n n! (λκ + 1)m+n (m + 2n)! = m+2n q(x) 2 n! (λκ + 1)m+n =

for x ∈ S d−1 . Otherwise the integral in the left hand side of (4.1) is zero. On the other hand, for a continuous function ϕ on [−1, 1], define m!   aκ,m (ϕ) = (2λκ )m B λκ + 21 , 21



1

ϕ(t) Cmλκ (t)(1 − t 2 )λκ − 2 dt. 1

−1

(4.2)

Here Cmλ (t) is the Gegenbauer polynomial Cmλ (t) =

m (−1)m (2λ)m 1 2 21 −λ d (1 − t ) (1 − t 2 )λ+m− 2 1 m m dt 2 m! (λ + 2 )m

and B( ·, · ) is the Beta function. Then, by integration by parts, Γ (λκ + 1) aκ,m (t ) = m √ 2 π Γ (λκ + m + 21 )



1

l

−1



dm l 1 t (1 − t 2 )λκ +m− 2 dt. dt m

Here we used basic properties of the Beta function, the Gamma function Γ ( · ) and the shifted factorial. If l − m = 2n is an even non-negative integer, then

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 1 Γ (λκ + 1) 1 l! B n + , λ + m + κ 2 2 2 π Γ (λκ + m + 21 ) (l − m)! (m + 2n)! = m+2n , 2 n! (λκ + 1)m+n

aκ,m (t l ) =

√ m

(4.3)

otherwise aκ,m (t l ) = 0. By (4.1) and (4.3), 1 ωκ,d

 S d−1

Vκ [x, · l ](y)q(y)h 2κ (y)dσ (y) = aκ,m (t l ) q(x)

(4.4)

for l, m ∈ Z≥0 , q ∈ Hκ,m and x ∈ S d−1 . By (4.4) and the Weierstrass approximation theorem, we have the following result. Corollary 4.1 (Funk–Hecke formula [24], [9, Theorem 7.3.4]) Let ϕ be a continuous function on [−1, 1] and q ∈ Hκ,m . Then 1 ωκ,d

 S d−1

Vκ [ϕ(x, · )](y)q(y)h 2κ (y)dσ (y) = aκ,m (ϕ) q(x) (x ∈ S d−1 ).

Original proof of the Funk–Hecke formula [24] (also in the classical case [5]) uses the reproducing kernel. We did not use the reproducing kernel in our proof. By the Funk–Hecke formula, we can prove the reproducing property in the same way as [7, Corollary 5]. Acknowledgements The authors are very grateful to the anonymous referee for valuable comments and suggestions.

References 1. D.H. Armitage and Ü Kuran, The convergence of the Pizzetti series in potential theory, J. Math. Anal. Appl., 171 (1992), 516–531 2. S. Ben Saïd, B. Ørsted, Segal-Bargmann transforms associated with finite Coxeter groups. Math. Ann. 334, 281–323 (2006) 3. N. Ben Salem, K. Touahri, Pizzetti series and polyharmonicity associated with the Dunkl Laplacian. Mediterr. J. Math., 7, 455–470 (2010) 4. A. Bezubik, A. D¸abrowska, A. Strasburger, On spherical expansions of zonal functions on Euclidean spheres. Arch. Math., 90, 70–81 (2008) 5. F. Dai, Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls (Springer, 2013) 6. F. Dai, Y. Xu, Analysis on h-Harmonics and Dunkl Transforms (Birkhäuser, 2014) 7. H. De Bie, F. Sommen, Spherical harmonics and integration in superspace, J. Phys. A: Math. Theor., 40 (2007), 7193–7212 8. C.F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z., 197 (1988), 33–60 9. C..F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, 2nd edn. (Cambridge University Press, Cambridge, 2014)

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10. R. Estrada, Regularization and derivatives of multipole potentials, J. Math. Anal. Appls. 446 (2017), 770–785 11. R. Estrada, Products of harmonic polynomials and delta functions. Advances in Analysis 3, 23–27 (2018) 12. R. Estrada, On Pizzetti’s formula, Asymptot. Anal., 111 (2019), 1–14 13. E.W. Hobson, On a theorem in the differential calculus, Messenger Math., 23 (1894), 115–119 14. G. Łysik, Mean-value properties of real analytic functions, Arch. Math. 98 (2012), 61–70 15. H. Mejjaoli, K. Trimèche, Mean value property associated with the Dunkl Laplacian, Integral Transform. Spec. Funct., 12 (2001), 279–302 16. T. Nomura, A proof of Hobson’s formula with the Euler operator, Kyushu J. Math., 72 (2018), 423–427 17. T. Nomura, Spherical Harmonics and Group Representations (in Japanese) (Nippon Hyoron sha co. Ltd, 2018) 18. P. Pizzetti, Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rend. Reale Accad. Lincei, 18 (1909), 182–185 19. H. Poritsky, Generalizations of the Gauss law of the spherical mean, Trans. Amer. Math. Soc. 43 (1938), 199–225 20. M. Rösler, Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions, Leuven 2002, Springer Lecture Notes in Mathematics, vol. 1817 (2003), pp. 93–135 21. M. Rösler, M. Voit, Dunkl theory, convolution algebras, and related Markov processes, in Harmonic and Stochastic Analysis of Dunkl Processes. ed. by P. Graczyk, M. Rösler, M. Yor (Hermann, 2008), pp. 1–112 22. N. Shimeno, Hobson’s formula for Dunkl operators and its applications, Integr. Transf. Spec. F., 29 (2018), 842–851 23. N. Tani, Harmonic projection operators associated with the Dunkl Laplacian and their applications (in Japanese), Master thesis (Kwansei Gakuin University, 2019) 24. Y. Xu, Funk-Hecke Formula for orthogonal polynomials on spheres and on balls. Bull. London Math. Soc. 32, 447–457 (2000)

On Double Coset Decompositions of Real Reductive Groups for Reductive Absolutely Spherical Subgroups Yuichiro Tanaka

Abstract The aim of this article is to show an induction argument to obtain the double coset decomposition H \G/L of a real reductive Lie group G with respect to reductive absolutely spherical subgroups H and L. As an application, we describe generic double cosets with some exceptions. The exceptions for our approach come from some factorizations of type D4 -groups. Keywords Reductive Lie group · Double coset decomposition · Absolutely spherical subgroup 2020 Mathematics Subject Classification Primary 22E46 · Secondary 22F30 · 53C30

1 Introduction In this article, we show an induction argument to obtain double coset decompositions for a real reductive Lie group with respect to reductive absolutely spherical subgroups from those for Levi subgroups by using T. Matsuki’s results (see Theorem 2.3) on double coset decompositions for symmetric pairs [21]. Here a closed subgroup H of a real reductive Lie group G is absolutely spherical if a Borel subgroup of G C has an open orbit on G C /HC , where the subscript C stands for the complexification. In particular, the fixed points subgroup of an involution is absolutely spherical. The idea of using symmetric pairs to obtain double coset decompositions for non-symmetric pairs comes from T. Kobayashi’s paper [16].

This work was supported by a JSPS Kakenhi (70780063). Y. Tanaka (B) Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Baklouti and H. Ishi (eds.), Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics 366, https://doi.org/10.1007/978-3-030-78346-4_14

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By the induction (Theorem 1.1) combined with computations for some minimal cases (Sect. 4) and reductions of some double coset spaces to more smaller groups (Sect. 5), we describe generic double cosets for reductive absolutely spherical pairs with some exceptions (Theorem 1.3). The exceptional cases to which the argument of this article cannot be applied arise from some factorizations (see Theorem 5.2 for a classification by Onishchik [25]) of type D4 -groups. Akhiezer [1] proved that for any two spherical subgroups H and L of a connected reductive linear algebraic group G over an algebraically closed field of characteristic zero, there exists a torus A of G such that h + a + Ad(g)l = g holds for all g in a Zariski open subset of G. This article takes a direct approach to see the generic structure of double coset spaces with some exceptions. We introduce some notations to state the main results. For a map σ : X → X we denote by X σ the σ-fixed points subset of X . If a Lie group G acts on X , we denote by σg the twisted map g ◦ σ ◦ g −1 for an element g ∈ G. If X is a vector space and σ a linear map, we write X −σ for the (−1)-eigenspace of σ. We write ιg for the inner automorphism by g ∈ G. We denote by g the Lie algebra of G and by exp : g → G the exponential mapping. Let L and H be reductive absolutely spherical subgroups of a connected real reductive Lie group G. We suppose that there are involutions σ and τ of G, which fix L and H pointwise, respectively, and a Cartan involution θ of G commuting with both σ and τ . We take a maximal abelian subspace t of gθ,−σ,−τ = gθ ∩ g−σ ∩ g−τ . Theorem 1.1 Suppose that g−σ,−τ = {0} and that for any t ∈ exp(t) and any nonzero θ-stable abelian subspace a of g−σ,−τt , the centralizer M of a has some finitely M many elements m i and  connected semisimple abelian subgroups Ai of M (1 ≤ i ≤ l) such that the subset 1≤i≤l (M ∩ L)AiM m i (M ∩ ιt H ) contains an open dense subset semisimple abelian of M. Then there exist finitely many elements xi and connected  subgroups Ai of G (1 ≤ i ≤ k) such that the subset 1≤i≤k L Ai xi H contains an open dense subset of G. This theorem easily follows from the fact that gσ = l + (m ∩ gσ ) and gτ = h + (m ∩ gτ ) hold, combined with the double coset decompositions with respect to symmetric pairs. We note that the groups Ai and the elements xi can be obtained explicitly from the elements m i and the standard Cartan subsets (see Theorem 2.3) for σ and τ . We apply an induction argument to the case where H = L and (G, H ) is either complex or compact. Corollary 1.2 1. (cf. [1, Theorem 1.1]) Let H be a reductive complex spherical subgroup of a connected complex reductive Lie group G. Then there exists a complex torus C of G such that the subset H C H contains an open dense subset of G. 2. Let H be an absolutely spherical subgroup of a connected compact Lie group G. Then there exists a torus T of G such that G = H T H . We show the above corollary by using the induction on the dimension of G. From the classification of reductive complex principal spherical subalgebras [3, 17, 23], we find all the cases for which the double coset decompositions cannot be reduced to

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smaller dimensional cases by using the results for symmetric pairs. We then give the double coset decompositions for such cases before starting the induction argument. If σ = τ , then the vector subspace g−τ ,−σ is non-zero in Theorem 1.1. Hence the induction argument can easily be applied to the settings of the corollary. To apply it to more general cases, we need to show that the dimension of G can be reduced by a different argument for the case where g−τ ,−σ is the zero vector space. To state the most general result on the double coset decomposition in this article we introduce the following conditions C1 and C2 on a triple (g, h, l) of reductive Lie algebras, which come from some special geometries of type D4 -groups. C1. g contains 2n-copies of a simple ideal g1 (n ∈ N), a simple or zero-ideal g0 and a simple ideal g2 . The pairs (g, h) and (g, l) have factors (g0 ⊕ g1 , diag(h0 )) ⊕ (g1 ⊕ g1 , diag(g1 ))⊕(n−1) ⊕ (g1 ⊕ g2 , diag(g2 )) and (g1 ⊕ g1 , diag(g1 ))⊕n , respectively, where h0 is a semisimple ideal of h, such that one of the conditions (i)–(iii) below is satisfied up to interchange of h and l and replacement of (g, h, l) with (g, ψ(φ(h)), φ(l)) for some inner automorphism ψ and some automorphism φ of g. C2. g contains (2n + 1)-copies of a simple ideal g1 (n ∈ N), a simple or zeroideal g0 and a simple ideal g2 . The pairs (g, h) and (g, l) have factors (g0 ⊕ g1 , diag(h0 )) ⊕ (g1 ⊕ g1 , diag(g1 ))⊕n and (g1 ⊕ g1 , diag(g1 ))⊕n ⊕ (g1 ⊕ g2 , diag(g1 )), respectively, where h0 is a semisimple ideal of h, such that one of the conditions (i)–(iii) below is satisfied up to interchange of h and l and replacement of (g, h, l) with (g, ψ(φ(h)), φ(l)) for some inner automorphism ψ and some automorphism φ of g. (i) h0 = g0 , and (g0 ⊕ g1 , diag(g0 )) and (g1 ⊕ g2 , diag(g2 )) are of types (B3 ⊕ D4 , B3 ) and (D4 ⊕ B3 , B 3 ), respectively. (ii) h0 = g1 , and (g0 ⊕ g1 , diag(g1 )) and (g1 ⊕ g2 , diag(g2 )) are of types (B4 ⊕ D4 , D4 ) and (D4 ⊕ B3 , B 3 ), respectively. (iii) g0 = {0}, and (g1 , h0 ) and (g1 ⊕ g2 , diag(g2 )) are of types either (D4 , B2 ⊕ A1 ) and (D4 ⊕ B3 , B 3 ), (D4 , B2 ⊕ A1 ) and (D4 ⊕ B4 , D4 ), (D4,C , D4,R,odd ) and (D4,C ⊕ B3,C , B 3,C ) or (D4,C , D 4,R,odd ) and (D4,C ⊕ B4,C , D4,C ), respectively. Here diag(·) stands for the diagonally embedded subalgebra, and the superscript ⊕n for the direct sum of n-copies. A simple non-complex real Lie algebra is of type An , Bn , . . . if its complexification is of type An , Bn , . . ., and the subscript K = R or C is added when we need to indicate explicitly that the corresponding simple Lie algebra is non-complex or complex, respectively. A Lie algebra of type D4,R,odd is either so(1, 7) or so(3, 5). The dashes on B3 , ⊕ and D4,R,odd mean that the corresponding Lie subalgebras are twisted by the outer automorphism of order three. Theorem 1.3 Let L and H be reductive absolutely spherical subgroups of a connected real reductive Lie group G. Suppose that neither the condition C1 nor the condition C2 is satisfied. Then there exist finitely many  semisimple abelian subspaces ji of g and elements xi of G (1 ≤ i ≤ k) such that 1≤i≤k L exp(ji )xi H contains an open dense subset of G (Fig. 1).

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g1

h0 ···

g1

g0

g1

···

g1

g1

g1

g1

g0 h0

···

g1

g1

g1

g1 g1

···

g1

g1 g1

g2 · · ·

(C1)

g2 · · ·

(C2)

g1

g1 ···

g2

g2 ···

g1

g1 g1

Fig. 1 Diagrams of conditions C1 and C2

We note that the subsets exp(ji )xi in Theorem 1.3 can be computed inductively. In the above, we express a triple (g, h, l) as a diagram in the following way. The vertices in the top row consist of the ideals of h (respectively, l), those in the middle row of the ideals of g and those in the bottom row of the ideals of l (respectively, h). There is an edge between a vertex u in the top or the bottom row and a vertex v in the middle row if u projects non-trivially to v. Either if h coincides with l, g is simple or g contains no simple factors of type D4 , then neither the condition C1 nor C2 is satisfied. Especially, we obtain the following as a corollary of the proof of Theorem 1.3. Corollary 1.4 We let L and H be reductive absolutely spherical subgroups of a connected real reductive Lie group G, and suppose either that h coincides with l, that g is simple or that g contains no simple factors of type D4 . 1. (cf. [1, Theorem 1.1]) If G is complex and H and L are its complex Lie subgroups, then there exists a complex torus T of G such that L T H contains an open dense subset of G. 2. If G is compact, then there exists a torus T of G such that G = L T H . Corollary 1.4 contains Corollary 1.2 as a special case, but we discuss these corollaries separately since it would be convenient to firstly introduce the main machinery and give its immediate corollary, and then to deal with the more general case that requires a little complicated argument. Remark 1.5 1. If both (G, H ) and (G, L) are symmetric pairs, then Theorem 1.3 is contained in Matsuki’s results [21]. 2. Akhiezer [1] proved that for a connected reductive linear algebraic group G over an algebraically closed field k of characteristic zero and its spherical subgroups H and L there exists a torus A such that h + a + Ad(g)l = g holds for g in a Zariski open subset. Also in the case k = C he proved that for a real form G R of G there exists a torus A such that gR + a + Ad(g)h = g for g in a Hausdorff open subset. 3. Helminck and Schwarz [9] gave examples of a spherical subgroup H of a connected complex reductive group G such that the isotropy representation of H on

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g/h is not polar. They also exhibited examples of spherical subgroups H and L of G such that the slice representation of H ∩ L on g/(h + l) is not cofree nor coregular. 4. Let G be a classical algebraic group, T a maximal torus and H a connected spherical reductive subgroup over an algebraically closed field of characteristic zero. Anisimov [2] obtained a necessary and sufficient condition for the double coset variety (the categorical quotient) T \\G//H to be an affine space and gave a classification of such pairs (G, H ). 5. Knop, Krötz, Sayag and Schlichtkrull [12] proved that for a real reductive algebraic group G and its maximal compact subgroup K and real spherical subgroup H there exist finite subsets F and F of G such that G = F K AF H holds for a connected semisimple abelian subgroup A. The subsets F and F can be taken as the singleton of the identity element when G/H is a triple space by Danielsen, Krötz and Schlichtkrull [5] and when it is a circle bundle on a symmetric space of non-tube type by Sasaki [29, 30]. Kobayashi [15] had given a conjecture that G = K AH holds in general if H is reductive. His conjecture is true [31]. 6. Complex semisimple spherical pairs were classified by Krämer [17], Brion [3] and Mikityuk [23]. A classification of reductive real spherical pairs has been accomplished recently by Knop, Krötz, Pecher and Schlichtkrull [13, 14]. The double coset decomposition G σ \G/G τ for two involutions τ and σ of G has been studied extensively by Flensted-Jensen [6], Heintze, Palais, Terng and Thorbergsson [7], Helminck and Schwarz [8–10], Hoogenboom [11], Lassalle [18], Matsuki [19– 21], Miebach [22], Oshima and Matsuki [26], Richardson [27] and Rossmann [28] among others.

1.1 Organization of This Article In the former part (Sects. 2 and 3), we show the induction argument and its immediate corollary. The induction argument is shown in Sect. 2. We apply it to the double coset decomposition for complex groups and compact groups in Sect. 3. In the latter part (Sects. 4, 5 and 6), we deal with the double coset decomposition in a more general setting. For this, we firstly discuss the double coset decompositions for the nonsymmetric polar cases in Sect. 4. Secondly, we deal with some special cases for which our induction argument does not work because of the appearance of factorizations of real reductive groups in Sect. 5. Such factorizations are given by Onishchik’s classification results [25]. After these preparations, we give a proof of Theorem 1.3 in Sect. 6. In the following, a pair (g, h) of a real Lie algebra g and its subalgebra h is nontrivial if g = h and effective if h does not contain any non-zero ideal of g. A pair (g, h) of a real reductive Lie algebra g and its subalgebra h is absolutely spherical if gC = hC + b holds for the complexification hC of h and a Borel subalgebra b of the complexification gC of g, and in this case h is an absolutely spherical subalgebra.

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A pair (g, h) is reductive  if h is a reductive subalgebra, and principal if the center n n (gi ∩ zh ), where zg is the center of g and i=1 gi zh of h is given as zg ⊕ i=1 the decomposition of the semisimple part gss of g into the sum of simple ideals gi (1 ≤ i ≤ n). A pair (g, h) is indecomposable if it cannot be represented nontrivially as the direct sum (g1 ⊕ g2 , (h ∩ g1 ) ⊕ (h ∩ g2 )) of pairs. Similarly, a triple (g, h, l) is indecomposable if it cannot be represented non-trivially as the direct sum if (g1 ⊕ g2 , (h ∩ g1 ) ⊕ (h ∩ g2 ), (l ∩ g1 ) ⊕ (l ∩ g2 )) of triples. (g, h, l) is effective  neither h nor l contains any non-zero ideals of g. For a decomposition g = i gi as the sum of ideals gi of g, we denote by pi : g → gi the projection to the ith ideal gi . We write e for the identity element of a group and G o for the identity component of a topological group G. For a subspace V of a real semisimple Lie algebra g we denote by V ⊥ the orthogonal complement with respect to the Killing form. For a pair (g, h) of Lie algebras of a real reductive Lie group G and its real reductive subgroup H , we always assume that there are some connected complex reductive algebraic group G C and its connected complex reductive algebraic subgroup HC whose Lie algebras gC and hC are the complexifications of g and h, respectively.

2 Induction The aim of this section is to show Theorem 1.1. For this, we prepare two lemmas that enable us to reduce our double coset decompositions to those of Levi subgroups. After introducing Matsuki’s results on double coset decompositions, we give a proof of Theorem 1.1. Lemma 2.1 Let G be a connected real reductive Lie group and H a connected reductive absolutely spherical subgroup. Suppose that there is an involution τ of G such that G τ contains H . We take a Cartan subspace a of the (−1)-eigenspace g−τ and write M for the connected centralizer of a in G τ . Then a Borel subgroup of MC has an open orbit on G τC /HC . Proof We take a Borel subalgebra bm + aC + nC of gC , where bm is a Borel subalgebra of mC and nC the nilradical of a parabolic subalgebra of gC , which contains mC + aC as its Levi part. Let AC and NC be the analytic subgroups of G C , which correspond to aC and nC , respectively. Since G C /HC is spherical and G τC AC NC is open dense in G C , there exists an h ∈ G τC such that gC = Ad(h)hC + bm + aC + nC .

(2.1)

By comparing with the Iwasawa decomposition gC = gτC + aC + nC , we obtain gτC = Ad(h)hC + bm .

(2.2) 

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Since MC has an open orbit on G τC /HC and since MC , G τC and HC are reductive, we also obtain the following (see Theorem 5.1). Lemma 2.2 Let G, H , τ and M be as in the previous lemma. We have G τo = M H . To reduce double coset decompositions of a real reductive Lie group to those of its Levi subgroups, we use Matsuki’s results on double coset decompositions with respect to symmetric pairs. Let G be a connected real semisimple Lie group and σ, τ involutions of G. The set G ss of semisimple elements with respect to σ and τ is defined by G ss = {g ∈ G : στg = σ Ad(g)τ Ad(g)−1 is semisimple}. We take a Cartan involution θ of G, which commutes with both σ and τ by taking a conjugate of σ or τ if necessary, and write g = k + p for the decomposition into the (+1)-eigenspace k and the (−1)-eigenspace p with respect to θ.  Theorem 2.3 ([21, Theorem 3]) We have G ss = 1≤i≤k G σo Ci G τo where Ci (1 ≤ i ≤ k) are representatives of standard Cartan subsets. Here a standard Cartan subset is of the form Ci = exp(ai ) exp(ti )ti with a ⊂ ai ⊂ p, ti ⊂ t ⊂ k−σ ∩ k−τ and ti ∈ exp(t) for a maximal abelian subspace t of k−σ ∩ k−τ and an abelian subspace a of p−σ ∩ p−τ such that ai + ti is a maximal abelian subspace of g−σ ∩ g−τti (1 ≤ i ≤ k) with t1 = t, a1 = a and t1 = e. Two standard Cartan subsets C1 = exp(a1 ) exp(t1 )t1 and C2 = exp(a2 ) exp(t2 )t2 are equivalent if there is a pair (h, l) ∈ (G σo ∩ G θ ) × (G τo ∩ G θ ) satisfying h exp(t)l −1 = exp(t) such that h exp(t1 )t1l −1 = exp(t2 )t2 . Theorem 2.4 1. ([21, Corollary of Theorem 3]) Suppose that G is complex and σ and τ are holomorphic involutions. Then we have G ss = G σ exp(t + a)G τ . 2. ([21, Theorem 1]) Suppose that G is compact. Then we have G = G σ exp(t)G τ . Proof of Theorem 1.1 We let L and H be reductive absolutely spherical subgroups of a connected real reductive Lie group G, and σ and τ involutions of G, which fix L and H , respectively. We may assume that G is semisimple by taking the quotient with respect to the center and that both H and Lare connected. Let Ci s (1 ≤ i ≤ k) be standard Cartan subsets such that G ss = 1≤i≤k G σo Ci G τo , where Ci = exp(ai ) exp(ti )ti as in Theorem 2.3 with a ⊂ ai ⊂ p and ti ⊂ t ⊂ k−σ ∩ k−τ for an abelian subspace a of p−σ ∩ p−τ and a maximal abelian subspace t of k−σ ∩ k−τ such that t + a becomes a maximal abelian subspace of g−σ ∩ g−τ . Here ai + ti is contained in g−σ ∩ g−τti with ti ∈ exp(t). We let Mi to be the conτt nected centralizer of ai + ti in G. We note that G σo = (Mi ∩ G σo )L and G o i = τti (Mi ∩ G o )ιti H by Lemma 2.2 and that (Mi , L ∩ Mi ) and (Mi , Mi ∩ ιti H ) are absolutely spherical pairs by Lemma 2.1. Now we assume that there exist finitely Mi i many semisimple abelian subspaces j M j ⊂ mi and elements x j ∈ Mi (1 ≤ j ≤ ki )  Mi i (L ∩ Mi ) exp(j M such that the subset j )x j (Mi ∩ ιti H ) contains an open dense 1≤ j≤ki

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subset of Mi . We note that G ss = τt

 1≤i≤k

G σo ) exp(ai ) exp(ti )(Mi ∩ G o i )(ιti H )ti is contained in we find that the subset  1≤i≤k

⎛ L⎝



⎞ (L ∩

1≤ j≤ki

M M Mi ) exp(j j i )x j i (Mi



G σo exp(ai ) exp(ti )ti G τo =

∩ ιti H )⎠ (ιti H )ti =

 1≤i≤k



L(Mi ∩

1≤i≤k

L Mi (ιti H )ti . Then



M

M

L exp(j j i )x j i ti H

1≤i≤k 1≤ j≤ki

contains an open dense subset of G. Therefore, we obtain Theorem 1.1.



3 Application The aim of this section is to show Corollary 1.2. We make a list of reductive absolutely spherical pairs for which we cannot take appropriate (see the lemma below) symmetric pairs. For this we use the classification results of reductive complex principal spherical subalgebras [3, 17, 23]. Proposition 3.1 Let (g, h) be a non-trivial indecomposable reductive absolutely spherical pair such that g is semisimple and either both of g and h are complex or both are compact. Suppose that there is no symmetric subalgebra h of g such that h is contained in h . Then (g, h) is either (so(7, C), g2 (C)), (g2 (C), sl(3, C)), (so(7), g2 ) or (g2 , su(3)). For the proof of the proposition, we prepare some lemmas. The following is an observation on the above four pairs. Lemma  3.2 Let (g, h) be a reductive absolutely spherical pair with g semisimple. Let g = i gi be the decomposition as the sum of simple ideals gi with the corresponding projection pi : g → gi . Suppose that there is an index k such that either the restriction pk |h is surjective or the pair (gk , pk (h)) is isomorphic to one of the four pairs in Proposition 3.1. Then the image of the center of h under the projection pk is the zero vector space, and there is only one simple ideal hk of h such that pk (h k ) = 0. Proof We can see this directly for each of the four pairs.



The following is an observation on a general absolutely spherical pair. Lemma 3.3 Let (g, h) be an absolutely spherical pair and g = g1 ⊕ g2 a decomposition as the sum of two ideals g1 , g2 of g with pi : g → gi the corresponding projection (i = 1, 2). Then (gi , pi (h)) is absolutely spherical (i = 1, 2). Proof Let b = b1 ⊕ b2 be a Borel subalgebra of gC such that gC = b + hC . Then  we obtain (gi )C = ( pi (g))C = bi + ( pi (h))C for i = 1, 2. The following is an observation on a complex absolutely spherical pair.

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Lemma 3.4 Let (g, h) be an absolutely spherical pair such that both g and h are complex. Then (g, h) is complex spherical. Proof Since (g, h) is an absolutely spherical pair, there exists a Borel subalgebra b of g such that gC g ⊕ g = b ⊕ b + h ⊕ h. Hence g = b + h.  Let us show Proposition 3.1 using the above lemmas.

 Proof of Proposition 3.1 We retain the setting of Proposition 3.1. Let g = i gi be the decomposition as the sum of simple ideals gi with pi : g → gi the projection to the ith ideal gi . Suppose that there is some index k such that the restriction of pk to h is not surjective and the pair (gk , pk (h)) is isomorphic to none of the four pairs. Then by the classifications of reductive complex spherical subalgebras of complex simple Lie algebras [17] (Table 2), we find that there exists a symmetric pair (g k , hk ) such that  hk ⊃ pk (h). Using such an hk , we obtain a symmetric subalgebra i=k gi ⊕ h k of g, which contains h. Suppose that for any index i the restriction of pi to h is surjective. Then (g, h) is of diagonal type, that is, h is simple and gi = h for any i by the indecomposability of (g, h). Let diag(h) be the diagonal  subalgebra of g1 ⊕ g2 . Then we obtain a symmetric subalgebra h = diag(h) ⊕ i=1,2 gi of g = ⊕i gi , which contains h. By the above two arguments, we only need to consider the case where there is some index k such that the pair (gk , pk (h)) is isomorphic to one of the four pairs, and for any other index j either the restriction p j |h is surjective or the pair (g j , p j (h)) is isomorphic to one of the above four pairs. Then by Lemma 3.2 the image of the center of h under the projection pi is the zero vector space, and there is only one simple ideal hi of h such that pi (hi ) = 0 for each index i. By contradiction, let us see that p j (hk ) = 0, that is, h j = hk for any index j not equal to k. Assume that there is some index j not equal to k such that h j = hk . Then (gk ⊕ g j , diag(hk )) is absolutely spherical with  Lemma 3.3 by taking g1 and g2 in the statement of Lemma 3.3 to be gk ⊕ g j and i= j,k gi , respectively. We can see that this is impossible by counting the dimensions of Lie algebras. Therefore h j = hk , that is, p j (hk ) = 0 for any index j not equal to k. Thus (g, h) contains (gk , hk ) as a direct factor. By the indecomposability of (g, h), (g, h) coincides with (gk , hk ), that is, one of the four pairs.  Let us see double coset decompositions for the above four case. Lemma 3.5 1. Let (G, H ) = (G2 (C), SL(3, C)) or (SO(7, C), G2 (C)). Then there exists a semisimple abelian subspace j of g such that H exp(j)H contains an open dense subset of G. 2. Let (G, H ) = (G2 , SU(3)) or (SO(7), G2 ). Then there exists an abelian subspace j of g such that G = H exp(j)H . Proof This result should be known but we give a proof for the convenience. We deal with the complex case only since the same argument can be applied to the compact case. We put (G , H ) = (SO(7, C), SO(6, C)) or (SO(8, C), Spin(7, C))

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accordingly to (G, H ) = (G2 (C), SL(3, C)) or (SO(7, C), G2 (C)). Then we have G/H G /H . By Theorem 2.4, H exp(j)H contains an open dense subset of G for a non-zero semisimple abelian subspace j of the orthogonal complement of h in g with respect to the Killing form. Let M be the connected centralizer of j. Then  H = M H and thus H exp(j)H contains an open dense subset of G. Proof of Corollary 1.2 We deal with the complex case only since the same argument can be applied to the compact case. Let H be a reductive complex spherical subgroup of a connected complex reductive Lie group G. By the induction on the dimension of G, we shall prove that there exists a semisimple abelian subspace j of g such that H exp(j)H contains an open dense subset of G. We may assume G is semisimple and simply connected, H connected and (g, h) indecomposable. If (g, h) is of type (B3 , G2 ) or (G2 , A2 ), then the corollary follows from Lemma 3.5. Hence we suppose that (g, h) is not of type (B3 , G2 ) or (G2 , A2 ). Then by Proposition 3.1, there exists a holomorphic involution τ of G such that G τ contains H . We take a Cartan involution θ commuting with τ , and a maximal abelian subspace j of g−τ . By Theorem 2.4, G τo exp(j)G τo contains an open dense subset of G. Let M be the connected centralizer of j. Then G τo = (M ∩ G τ )H and (M, M ∩ H ) is spherical. By the induction hypothesis, there exists a semisimple abelian subspace j M of m such that (M ∩ H ) exp(j M )(M ∩ H ) contains an open dense subset of M. Thus, we find  that the subset H exp(j M )H contains an open dense subset of G. The rest of this article is devoted to the proof of Theorem 1.3.

4 Polar Spaces In this section, we discuss double coset decompositions for the polar cases. Proposition 4.1 Let L and H be reductive absolutely spherical subgroups of a connected real reductive Lie group G. Suppose that either (g, h) or (g, l) is of type (B3 , G2 ) or (G2 , A2 ). Then there exist finitely many  semisimple abelian subspaces ji of g and elements xi of G (1 ≤ i ≤ k) such that 1≤i≤k L exp(ji )xi H contains an open dense subset of G. If either G is compact or (G, H, L) is complex, then we can take k to be 1 and x1 to be the identity element, and further, G = L exp(j1 )H holds when G is compact. We give matrix realizations of some simple Lie algebras before we prove the proposition (see [32] for more information of realizations of exceptional Lie groups). Let {e1 , ..., e8 } be a canonical basis of R8 . We define linear mappings E i j : R8 → R8 (1 ≤ i, j ≤ 8) by E i j e j = ei and E i j ek = 0 for 1 ≤ k = j ≤ 8. We put G i j = E i j − E ji (1 ≤ i < j ≤ 8), and then {G i j : 1 ≤ i < j ≤ 8} forms a basis of so(8) and {G i j : 2 ≤ i < j ≤ 8} forms that of so(7). We put ([32, Lemma 1.3.1])

On Double Coset Decompositions of Real Reductive Groups …

F12 = F56 = F13 = F57 = F14 = F58 = F15 = F37 = F16 = F38 = F17 = F35 = F18 = F36 =

1 (G 12 + G 34 + G 56 + G 78 ), 2 1 (G 12 − G 34 + G 56 − G 78 ), 2 1 (G 13 − G 24 − G 57 + G 68 ), 2 1 (−G 13 + G 24 + G 57 + G 68 ), 2 1 (G 14 + G 23 + G 58 + G 67 ), 2 1 (G 14 − G 23 + G 58 − G 67 ), 2 1 (G 15 − G 26 + G 37 − G 48 ), 2 1 (G 15 + G 26 + G 37 + G 48 ), 2 1 (G 16 + G 25 − G 38 − G 47 ), 2 1 (−G 16 + G 25 + G 38 − G 47 ), 2 1 (G 17 − G 28 − G 35 + G 46 ), 2 1 (−G 17 − G 28 + G 35 + G 46 ), 2 1 (G 18 + G 27 + G 36 + G 45 ), 2 1 (G 18 − G 27 + G 36 − G 45 ), 2

239

1 F34 = (G 12 + G 34 − G 56 − G 78 ), 2 1 F78 = (G 12 − G 34 − G 56 + G 78 ), 2 1 F24 = (−G 13 + G 24 − G 57 + G 68 ), 2 1 F68 = (G 13 + G 24 + G 57 + G 68 ), 2 1 F23 = (G 14 + G 23 − G 58 − G 67 ), 2 1 F67 = (G 14 − G 23 − G 58 + G 67 ), 2 1 F26 = (−G 15 + G 26 + G 37 − G 48 ), 2 1 F48 = (−G 15 − G 26 + G 37 + G 48 ), 2 1 F25 = (G 16 + G 25 + G 38 + G 47 ), 2 1 F47 = (−G 16 + G 25 − G 38 + G 47 ), 2 1 F28 = (−G 17 + G 28 − G 35 + G 46 ), 2 1 F46 = (G 17 + G 28 + G 35 + G 46 ), 2 1 F27 = (G 18 + G 27 − G 36 − G 45 ), 2 1 F45 = (G 18 − G 27 − G 36 + G 45 ). 2

We can realize spin(7) as the fixed points subset of the map  : so(8) → so(8), G i j → Fi j (1 ≤ i < j ≤ 8) [32, Theorem 1.3.5]. Also, the following linear mappings span the compact simple Lie algebra g2 of type G2 [32, Theorem 1.4.3]. λG 34 + μG 56 + νG 78 ,

−λG 24 − μG 57 + νG 68 ,

λG 23 + μG 58 + νG 67 , λG 25 − μG 38 − νG 47 ,

−λG 26 + μG 37 − νG 48 , −λG 28 − μG 35 + νG 46 ,

λG 27 + μG 36 + νG 45

(λ + μ + ν = 0).

We can realize su(3) ⊂ g2 as {X ∈ g2 : X e2 = 0} [32, Theorem 1.5.1]. Then we obtain matrix realizations of so(8, C), so(7, C), spin(7, C), g2 (C) and sl(3, C) by complexifying the above linear mappings. The map θ(X ) = X¯ defines the Cartan involutions of these Lie algebras, where X¯ denotes the complex conjugate. We also consider a map η j1 , j2 ,..., jk : so(8, C) → so(8, C), X → I j1 , j2 ,..., jk X¯ I j1 , j2 ,..., jk , where

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I j1 , j2 ,..., jk is defined by

2m

2m+1 ei ( l=1 jl + 1 ≤ i ≤ l=1 jl ), ei →

2m+1

2m+2 jl + 1 ≤ i ≤ l=1 jl ), −ei ( l=1

k for 0 ≤ m ≤ [ k−1 ] and l=1 jl = 8. We can realize symmetric subalgebras and 2 real forms of so(8, C), so(7, C), spin(7, C), g2 (C) and sl(3, C) by taking the fixed points subsets with respect to the map θ ◦ η j1 , j2 ,..., jk or the map η j1 , j2 ,..., jk . We put η± = η1,1,1,1,1,1,1,1 for simplicity. The following remark can be used for finding representatives of the standard Cartan subsets in Theorem 2.3. Remark 4.2 Let g, τ , σ, θ, t, a, ti , ai and ti be as in Theorem 2.3. We have στ (X α ) = ti2α X α for an eigenvector X α with eigenvalue α ∈ t∗C with respect to the adjoint action −θ,στt

of tC on gC i if ai  a for a standard Cartan subset exp(ti ) exp(ai )ti . This shows that we can obtain elements ti for representatives of the standard Cartan subsets by comparing ti2α = exp(2α(log ti )) with eigenvalues of στ on gC . After that, we can find abelian subspaces ti + ai ⊂ g−σ,−τti for such ti s. Proof of Proposition 4.1 We divide the proof into eight cases (i)–(viii) according to the types of the pairs (G, H ) and (G, L). We note that the conclusion of Proposition 4.1 does not depend on the coverings of G. In the following, a triple (G, H, L) of Lie groups is a factorization if G = H L holds. (i) The case when both (G, H ) and (G, L) are compact and non-symmetric. Let (G, H, L) be either (G2 , SU(3), SU(3))  or (SO(7), G2 , G2 ). Then for a onedimensional subspace a of h⊥ , we have h⊥ = h∈H Ad(h)a. Therefore we obtain G = H exp(a)L. Here we note that the pairs (G2 , SU(3)) and (SO(7), G2 ) are compact polar [4]. (ii) Either (G, H ) or (G, L) is a compact symmetric pair and (G, H, L) is not a factorization. Let (G, L , H ) be either (G2 , SU(3), SU(2) ×{±1} SU(2)) or (SO(7), G2 , SO(3) × SO(4)). We denote by τ the involution of G, which corresponds to H . Since the pair (G, L) is polar of rank one, we have l⊥ = a + [l, Z ] for any non-zero element Z of (l⊥ )−τ and the one-dimensional subspace a = RZ [4]. We note that [lτ , Z ] ⊂ (l⊥ )−τ and [l−τ , Z ] ⊂ (l⊥ )τ . This implies that a + [l ∩ h, Z ] = (l⊥ )−τ . Because of the facts that the action of L ∩ H on (l⊥ )−τ is isometric with respect to the Killing form, that the decomposition a + [l ∩ h, Z ] = (l⊥ )−τ is a direct sum and that a is of one dimension, L ∩ H has an open orbit on the unit sphere in (l⊥ )−τ . Since L ∩ H is compact, this implies that the action is transitive on the unit sphere. Hence (l⊥ )−τ =  h∈L∩H Ad(h)a. Then we obtain G = H exp(a)L by the same argument as the proof of a Cartan decomposition for compact symmetric pairs, which is given in [11, Chap. 6]. (iii) The case when either (G, H ) or (G, L) is a compact symmetric pair and (G, H, L) is a factorization.

On Double Coset Decompositions of Real Reductive Groups …

241

Let (G, L , H ) be either (SO(7), G2 , SO(6)) or (SO(7), G2 , SO(2) × SO(5)). Then we have G = H L by [25] (see Theorem 5.2 below). (iv) The case when G is non-compact and either H or L is compact We suppose that (g, h, l) be given as either (so(7, C), g2 (C), so(7)), (g2 (C), sl(3, C), g2 ), (so(3, 4), g2(2) , so(3) ⊕ so(4)), (g2(2) , sl(3, R), su(2) ⊕ su(2)) or K of G such (g2(2) , su(1, 2), su(2) ⊕ su(2)). We take a maximal compact subgroup  that K ∩ H is maximally compact in H . We have h⊥ ∩ k⊥ = h∈K ∩H Ad(h)a for a one-dimensional subspace a of h⊥ ∩ k⊥ . Let us see this for the pair (G, H ) = (G2(2) , SL(3, R)). We note that the non-compact part of the homogeneous space G2(2) /SL(3, R) has three dimensions, and hence consider the three-dimensional real representation of SO(3). We can see that a one-dimensional vector subspace serves as a slice for this action. We can find a one-dimensional slice similarly for each of the other four pairs (so(7, C), g2 (C)), (g2 (C), sl(3, C)), (so(3, 4), g2(2) ) and (g2(2) , su(1, 2)) by considering the seven-dimensional real representation of g2 , the sixdimensional real representation of su(3), the four-dimensional real representation of so(4) and the four-dimensional real representation of su(2), respectively. Then for each of the five pairs, we use a decomposition G = K exp(h⊥ ∩ k⊥ ) exp(h ∩ k⊥ ) [24, Theorem 5] and obtain G = K exp(a)H . (v) The case when both H and L are non-compact, (G, H, L) is not a factorization and both (G, H ) and (G, L) are non-symmetric. Let (g, h, l) be either (g2(2) , sl(3, R), sl(3, R)), (g2(2) , sl(3, R), su(1, 2)), (g2(2) , su(1, 2), su(1, 2)), (g2 (C), sl(3, C), sl(3, C)), (so(3, 4), g2(2) , g2(2) ) or (so(7, C), g2 (C), g2 (C)). We only deal with the case where (g, h, l) = (g2(2) , sl(3, R), sl(3, R)) since the same argument can be applied to the other cases. Let (G, L , H ) = (G2(2) , SL(3, R), SL(3, R)). Here we realize SO(7, C), G2 (C) η η and SL(3, C) as above, and G, H = L as G2 (C)o± , SL(3, C)o± , respectively. We η± put G = SO(7, C)o and σ = θ ◦ η2,6 so that G = SOo (4, 3) ⊃ G σ o = SOo (3, 3) ⊃ by the factorization G = G · G σ H = L. Then we have L\G/H L\G /G σ o o (see Theorem 5.2 below). We take a one-dimensional subspace t = RG 24 of (g )θ,−σ . Here we note that G 24 = X + Y for X = G 24 + G 57 + 2G 68 ∈ g and Y = −G 57 − 2G 68 ∈ (g )σ , and that [X, Y ] = 0. By Theorem 2.3, we have G ss =

k 

σ G σ o exp(bi )ti G o

i=1

for some abelian subspaces bi ⊂ g−σ,−σti and some elements ti = exp(s√ i G 24 ) with b1 = t, s1 = 0 and 0 ≤ si < 2π. We note that σσ = id and hence exp(2si −1) = 1 (2 ≤ i ≤ k) (see Remark 4.2). This implies si = 0 or π (2 ≤ i ≤ k), and thus we can = exp(πG 24 ). Then b2 and b3 can be taken as take k and ti s√as k = 3, t2 = e and t3 √ b2 = b3 = R −1G 23 . We note that −1G√23 = U + V with [U, V ] = 0 for U = √ −1(G 23 − 21 (G 58 + G 67 )) ∈ g and V = 2−1 (G 58 + G 67 ) ∈ (g )σ . We define Mi σti to be the connected centralizer of bi in G σ o ∩ (G )o , which is of type B2 . Since σ so(6, C) = so(5, C) + sl(3, C), we have g = l + mi and thus G σ o = L Mi . Hence we obtain

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G ss =

3 

L Mi exp(bi )ti G σ o =

i=1

3 

L exp(bi )ti G σ o .

i=1

We then put j1 = RX , j2 = j3 = RU , x1 = x2 = e and x3 = exp(π X ). We have G ss =

3 

L exp(bi )ti G σ o =

i=1

3 

L exp(ji )xi G σ o ,

i=1

3 and thus i=1 L exp(ji )xi H contains an open dense subset of G. The same argument works for the other cases (g, l) = (g2(2) , su(1, 2)), (g2 (C), sl(3, C)), (so(3, 4), g2(2) ) and (so(7, C), g2 (C)) by taking (g , g σ ) to be (so(3, 4), so(2, 4)), (so(7, C), so(6, C)), (so(4, 4), so(3, 4)) and (so(8, C), so(7, C)), respectively. For the last two cases we use the factorization so(7, C) = so(5, C) + g2 (C) instead of so(6, C) = so(5, C) + sl(3, C). We note that if (G, H, L) is complex, then we can apply Theorem 2.4 instead of Theorem 2.3. (vi) The case when G is complex and L a non-compact real form of G. We discuss the case when G is of type G2 and the case when G of type B3 separately. (vi-1) Let (G, L , H ) be (G2 (C), G2(2) , SL(3, C)). Here we realize G = SO(7, C), η G2 (C) and SL(3, C) as above, and L as G o± . We put τ = θ ◦ η2,6 and σ = τ η± . Then we have L\G/H L\G /G o by the factorization G = G · G τo (see Theorem 5.2 below). We take a one-dimensional subspace t = RG 23 of (g )θ,−σ,−τ . Here we note that G 23 = X + Y for X = G 23 − 2G 58 + G 67 ∈ g and Y = 2G 58 − G 67 ∈ (g )τ , and that [X, Y ] = 0. By Theorem 2.3, we have G ss =

k 

τ G σ o exp(bi )ti G o

i=1

for some abelian subspaces bi ⊂ g−σ,−τti and some elements ti = exp(si G 23 ) with b1√= t, s1 = 0 and 0 ≤ si < 2π. We note that (στ )2 = id and hence exp(2si −1) = ±1 (2 ≤ i ≤ k) (see Remark 4.2). This implies si = 0, π2 , π or 3π (2 ≤ i ≤ k). Thus we can take k and ti s as k = 5, t2 = e, t3 = 2 exp(πG 23 ), t4 = exp( π2 G 23 ) and t5 = exp( 3π G ). Then bi can be taken as √ √ 2 23 √ b2 = b3 = R −1G 24 and b4 = b5 √ = R −1G 35 . We note that −1G 24 = U√ + V with [U, V ] = 0 for U = −1(G √ 24 + G 57 + 2G 68 ) ∈ g and V = + W with [Z , W ] = − −1(G 57√+ 2G 68 ) ∈ (g )τ , and that −1G 35 = Z √ 0 for Z = −1(G 35 + G 28 + 2G 46 ) ∈ g and W = − −1(G 28 + 2G 46 ) ∈ (g )τt4 = (g )τt5 . We define Mi to be the connected centralizer of bi in τt (G )σo ∩ (G )o i , which is of type B2 . Since so(7, C) = so(5, C) + g2 (C), we have g σ = l + mi and thus G σ o = L Mi . Hence we obtain

On Double Coset Decompositions of Real Reductive Groups …

G ss =

5 

L Mi exp(bi )ti G τo =

i=1

5 

243

L exp(bi )ti G τo .

i=1

We then put j1 = RX , j2 = j3 = RU , j4 = j5 = RZ , x1 = x2 = e, x3 = X ). We have exp(π X ), x4 = exp( π2 X ) and x5 = exp( 3π 2 G ss =

5  i=1

L exp(bi )ti G τo =

5 

L exp(ji )xi G τo ,

i=1

5 L exp(ji )xi H contains an open dense subset of G. and thus i=1 (vi-2) Let (G, L , H ) be either (SO(7, C), SOo (1, 6), G2 (C)), (SO(7, C), SOo (2, 5), G2 (C)) or (SO(7, C), SOo (3, 4), G2 (C)). We only deal with the case where (G, L , H ) = (SO(7, C), SOo (1, 6), G2 (C)) since the same argument can be applied to the other cases. Let (G, L , H ) = (SO(7, C), SOo (1, 6), G2 (C)). We realize G = SO(8, C), H = Spin(7, C), G and H as above, and L = SOo (1, 7) and L as (G )η1,1,6 and G η1,1,6 , respectively. By Theorem 5.3 we have G = G · H = L · H and hence L\G/H L\L /(L ∩ H ). Here we note that l ∩ h = g2 ⊂ (l )θ and so (L , L) is symmetric and (L , L ∩ H ) absolutely spherical. Then by the Cartan decomposition for symmetric pairs ([6, Theorem 4.1], [28, Theorem 10]) we have L = L exp(a)(L )θo √ for a one-dimensional subspace a = R −1G 12 √of (l )−θ ∩ l⊥ . We note that √ −1G 12 = X + Y with [X, Y ] = 0 for X = − 3−1 (G 34 + G 56 + G 78 ) ∈ g √

and Y = 3−1 (3G 12 + G 34 + G 56 + G 78 ) ∈ h . We have (L )θo = M · (L ∩ H )o for the connected centralizer M = SO(6) of a in L θ , and hence L = L exp(a)(M(L ∩ H )o ) = L exp(a)(L ∩ H )o .

We put j = RX and then have G = L exp(a)H = L exp(j)H . Thus, we find that G = L exp(j)H . For the other cases (G, L , H ) = (SO(7, C), SOo (2, 5), G2 (C)) and (G, L , H ) = (SO(7, C), SOo (3, 4), G2 (C)) we can take (G , L , H ) to be (SO(8, C), SOo (3, 5), Spin(7, C)). For these two cases we have h ∩ l = g2(2) and use Theorem 2.3 instead of [6, 28]. (vii) The case when both H and L are non-compact, (G, H, L) is not a factorization and either (G, H ) or (G, L) is symmetric. Let (G, L , H ) be either (SOo (3, 4), SOo (1, 2) × SOo (2, 2), G2(2) ), (SOo (3, 4), SO(3) × SOo (1, 3), G2(2) ) or (SO(7, C), SO(3, C) × SO(4, C), G2 (C)). We only deal with the case where (G, L , H ) = (SOo (3, 4), SOo (1, 2) × SOo (2, 2), G2(2) ) since the same argument can be applied to the other cases.

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Let (G, L , H ) be (SOo (3, 4), SOo (1, 2) × SOo (2, 2), G2(2) ). We put σ = θ ◦ η1,7 and τ = θ ◦ η4,1,3 . We realize SO(8, C), SO(7, C), Spin(7, C) and G2 (C) as η η above, and G = SOo (4, 4), G, H = Spino (3, 4) and H as SO(8, C)o± , SO(7, C)o± , η± η± Spin(7, C)o and G2 (C)o , respectively. We also realize L = SOo (1, 2) × SOo (3, 2) θ◦η θ◦η and L as (G )o 5,3 and G o 5,3 , respectively. We have the natural bijections L\G/H L\G /H L\L /(L ∩ H ) by the factorizations G = G · H and G = H · L (see Theorem 5.2 below). We note that l ∩ h is of type B1 ⊕ B1 . We take a onedimensional subspace t = RG 15 of (l )θ,−σ,−τ . Here we note that G 15 = X + Y for X = 13 (G 26 − G 37 + G 48 ) ∈ g and Y = 13 (3G 15 − G 26 + G 37 − G 48 ) ∈ h , and that [X, Y ] = 0. By Theorem 2.3, we have L ss = L exp(t)(L )τo . We define M to be the connected centralizer of t in L, which is of type B1 ⊕ B1 . Then we have (l )τ = l ∩ h + m and thus (L )τo = M(L ∩ H )o . Hence we obtain L ss = L exp(t)M(L ∩ H )o = L exp(t)(L ∩ H )o . We then put j = RX . We find that L exp(t)H = L exp(j)H contains an open dense subset of G , and thus L exp(j)H contains an open dense subset of G. We can take (G , H , L ) to be (SOo (4, 4), Spino (3, 4), SO(3) × SOo (1, 4)) and (SO(8, C), Spin(7, C), SO(3, C) × SO(5, C)) for (G, L , H ) = (SOo (3, 4), SO(3) × SOo (1, 3), G2(2) ) and (SO(7, C), SO(3, C) × SO(4, C), G2 (C)), respectively. (viii) The case when both H and L are non-compact and (G, H, L) is a factorization. Let G = SO(7, C) and (L , H ) = (SO(6, C), G2 (C)) or (SO(2, C) × SO(5, C), G2 (C)), or G = SOo (3, 4) and (L , H ) = (SOo (2, 4), G2(2) ), (SOo (3, 3), G2(2) ), (SO(2) × SOo (1, 4), G2(2) ), (SO(2) × SOo (3, 2), G2(2) ) or (SOo (1, 1) × SOo (2, 3), G2(2) ). Then G = L · H holds (Theorem 5.2). The classifications of non-symmetric principal indecomposable reductive complex spherical subalgebras [3, 17, 23] (Table 2 below) and indecomposable complex semisimple symmetric subalgebras (see Table 8) of a simple Lie algebra of type B3 or G2 show that the above eight cases exhaust all the possibilities of the triples (g, h, l) up to interchange of h and l and replacement of (g, h, l) with (g, ψ(φ(h)), φ(l)) for some inner automorphism ψ and some automorphism φ of g. Hence the proof is finished. 

5 Factorization In this section, we deal with the cases where the assumption g−τ ,−σ = {0} in Theorem 1.1 is not satisfied. For this, we recall Onishchik’s results (see [25] for the details). A triple (g, g , g ) of real Lie algebras g, g and g is a factorization if g = g + g holds.

On Double Coset Decompositions of Real Reductive Groups …

245

Theorem 5.1 ([25, Theorem 3.1]) Let G be a connected reductive Lie group and G , G its connected reductive subgroups. Then (g, g , g ) is a factorization if and only if G = G G holds. A factorization (g, g , g ) is proper if g = g and g = g, and semisimple if g, g and g are real semisimple. Theorem 5.2 ([25, Theorem 4.1]) All proper semisimple factorizations (g, g , g ) of non-compact simple Lie algebras g such that gC is simple are listed in Table 1 up to interchange of g and g and replacement of (g, g , g ) with (g, ψ(φ(h)), φ(l)) for some inner automorphism ψ and some automorphism φ of g.

Table 1 Factorization g g su(2 p, 2q)

sp( p, q)

su(n, n) sl(2n, R) so(3, 4)

sp(n, R)

so(2 p, 2q)

so(2 p, 2q − 1) so(2 p − 1, 2q) so(n − 1, n) so(4 p, 4q − 1) so(4 p − 1, 4q) so(4 p, 4q − 1)

so(n, n) so(4 p, 4q)

g2(2)

g

g ∩ g

No

su(2 p, 2q − 1) su(2 p − 1, 2q) su(n − 1, n) sl(2n − 1, R) so(2, 4) so(3, 3) so(1, 4) so(2, 3) su( p, q)

sp( p, q − 1) sp( p − 1, q) sp(n − 1, R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

sl(n, R) sp( p, q) sp( p, q) ⊕ sp(1)

so(4 p − 1, 4q) so(2n, 2n)

so(2n − 1, 2n)

so(8, 8)

so(7, 8)

so(4, 4)

spin(3, 4)

sp(n, R) sp(n, R) ⊕ sp(1, R) so(1, 8) so(4, 5) so(3, 4) so(2, 4) so(3, 3) so(1, 4) so(2, 3) so(1, 4) ⊕ so(3) so(2, 3) ⊕ so(1, 2)

su(1, 2) sl(3, R) su(2) su(1, 1) su( p, q − 1) su( p − 1, q) sl(n − 1, R) sp( p, q − 1) sp( p − 1, q) sp( p, q − 1) ⊕ sp(1) sp( p − 1, q) ⊕ sp(1) sp(n − 1, R) sp(n − 1, R) ⊕ sp(1, R) so(7) so(3, 4) g2(2) su(1, 2) sl(3, R) so(3) so(1, 2) so(3) ⊕ so(3) so(1, 2) ⊕ so(1, 2)

15 16 17 18 19 20 21 22 23 24 25 26

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For a real reductive Lie algebra g we denote by gsts the sum of all the simple ideals not of type A1 and by g f the sum of all the abelian ideals and simple ideals of type A1 . Theorem 5.3 ([25, Theorem 4.2]) Let g be a simple complex Lie algebra and g = g + g a proper semisimple factorization. Then we have the following two possibilities. (1) g and g sts are simple complex Lie algebras, g = g + g sts and g f is simple or trivial. (2) g = so(8, C), g = spin(7, C), g = so(1, 7) or so(3, 5) and g ∩ g is a compact or non-compact real form of type G2 , respectively, up to interchange of g and g and replacement of (g, g , g ) with (g, ψ(φ(h)), φ(l)) for some inner automorphism ψ and some automorphism φ of g. Theorem 5.4 ([25, Theorem 3.3]) Let g be a real reductive Lie algebra and g and g g its real reductive subalgebras. Let p f : g → g f be the projection. Then g = g + g g g if and only if gsts = g sts + g sts and g f = p f (g f ) + p f (g f ). A factorization (g, g , g ) is effective if g and g contain no non-zero ideals of g, and indecomposable if it cannot be expressed as the sum of two factorizations of nonzero Lie algebras. We shall apply these theorems to our double coset decompositions when the assumption g−σ,−τ = 0 is not satisfied in Theorem 1.1. The following is the main result of this section. Proposition 5.5 We retain the setting of Theorem 1.3. We suppose that (g, h, l) is effective and indecomposable with g semisimple, that neither (g, h) nor (g, l) consists only of indecomposable factors of non-symmetric polar pairs of types (G2 , A2 ) and (B3 , G2 ) and that g = gσ + gτ holds for any pair (σ, τ ) of involutions such that l ⊂ gσ and h ⊂ gτ . Then there exist a proper reductive subalgebra g of g and its reductive absolutely spherical subalgebras h and l such that the assumption of Theorem 1.3 is satisfied for (g , h , l ) and the inclusion G → G induces an open surjective map L \G /H → L\G/H with respect to the quotient topology for the analytic subgroups G , H and L of G with Lie algebras g , h and l , respectively. The following remark will be used for the proof of this proposition. Remark 5.6 1. Let G β , G γ be connected reductive Lie groups, H β , H γ their respective reductive subgroups and L α a reductive subgroup of G α = G β × G γ with p β : G α → G β , p γ : G α → G γ the projections. If p γ (lα ) + hγ = gγ , then the multiplication map L α × G β × H α → G α is open surjective for H α = H β × H γ by Theorem 5.1. 2. Let G a be a topological group and G b , G c its closed subgroups. Let (H a , L a ), (H b , L b ) and (H c , L c ) be pairs of closed subgroups of G a , G b and G c , respectively. Suppose that the inclusions G c → G b and G b → G a induce natural maps f b : L c \G c /H c → L b \G b /H b and f a : L b \G b /H b → L a \G a /H a and that f b and f a ◦ f b are open surjective. Then f a is also open surjective.

On Double Coset Decompositions of Real Reductive Groups …

247

3. Let G a be a connected reductive Lie group and H a , L a and G b be its reductive Lie subgroups such that gb + ha = ga . Then the multiplication map L a × G b × H a → G a is open surjective by Theorem 5.1. To show the proposition, we use classification results of Brion [3], Kramer [17] and Mikityuk [23] for indecomposable principal reductive complex spherical pairs and of Knop, Krötz, Pecher and Schlichtkrull [13, 14] for strictly indecomposable reductive real spherical pairs. Here a pair (g, h) of a real reductive Lie algebra g and its Lie subalgebra h is real spherical if g = h + p holds for a minimal parabolic subalgebra p of g. A pair (g, h) of reductive Lie algebras is strictly indecomposable if (g, hn ) is indecomposable, where hn is the sum of all the non-compact simple ideals of h. As a part of the classification [13, 14] also obtains a classification of strictly indecomposable reductive absolutely spherical pairs. Theorem 5.7 ([3, 17, 23]) Non-symmetric indecomposable principal reductive complex spherical pairs (g, h) are listed in Table 2. For non-symmetric indecomposable principal compact spherical pairs, we only need to take the compact real forms of the above pairs. Theorem 5.8 ([13, 14]) Non-symmetric, non-complex and non-compact strictly indecomposable reductive absolutely spherical pairs (g, h) are listed in Tables 3 and 4. Taking real forms of Table 2, we also find non-strictly indecomposable, noncompact indecomposable principal reductive absolutely spherical pairs in Table 5. We start the proof of Proposition 5.5. The following lemma shows that gsts = {0} in the setting of Proposition 5.5. Lemma 5.9 Let (g, h) and (g, l) be effective reductive absolutely spherical pairs such that g is semisimple and consists of type A1 -factors only. Then there exist symmetric subalgebras hτ and gσ of g, which contain h and l, respectively, such that g  = gτ + g σ . Proof If g contains a type A1 -simple complex factor, then we let P be a longest pass with no cycles, which consists only of type A1 -simple complex factors in the diagram of (g, h, l). If g contains no type A1 -simple complex factors, then we let P be a longest pass with no cycles, which consists of type A1 -simple non-complex factors in the diagram of (g, h, l). We write (g(0) , h(0) , l(0) ) for the triple of ideals of (g, h, l), which serves as the vertices of P. Then (g(0) , h(0) , l(0) ) is given as (g(0) , h(0) , l(0) ) =

(g0 ⊕ g⊕2n ⊕ g0 , diag(g0 )⊕n+1 , diag(g0 )⊕n ) or 0 ⊕2n−1 ⊕ g0 , diag(g0 )⊕n , diag(g0 )⊕n ) (g0 ⊕ g0

up to interchange of h and l, where g0 is of type A1 and n is a natural number. We assign numbers to ideals g0 of g(0) , h(0) and l(0) as gi0 , i = 1, 2, 3, . . . (see Fig. 2),

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Table 2 Complex spherical pair g h g2 (C) so(7, C) sl(m + n, C)

sl(3, C) g2 (C) sl(m, C) ⊕ sl(n, C)

so(4n + 2, C) e6 (C) sl(2n + 1, C)

sl(2n + 1, C) so(10, C) sp(n, C) C ⊕ sp(n, C) spin(7, C) ⊕ C spin(7, C) g2 (C) sl(n, C) ⊕ C sp(n − 1, C) ⊕ C

so(10, C) so(9, C) so(8, C) so(2n + 1, C) sp(n, C) sl(n, C) ⊕ sl(n + 1, C) sl(n + 2, C) ⊕ sp(m, C)

diag(sl(n, C)) ⊕ C sl(n, C) ⊕ diag(sl(2, C)) ⊕sp(m − 1, C)

sl(n, C) ⊕ diag(sl(2, C)) ⊕C ⊕ sp(m − 1, C) sp(n + 1, C) ⊕ sp(l + sp(n, C) ⊕ 1, C) diag(sp(1, C)) ⊕sp(l, C) ⊕ sp(m, C) ⊕sp(m + 1, C) sp(n + 2, C) ⊕ sp(n, C) ⊕ sp(2, C) diag(sp(2, C)) sp(n + 1, C) ⊕ sp(n, C) ⊕ sp(m, C) sp(2, C) ⊕sp(m + 1, C) ⊕diag(sp(1, C))⊕2 so(n, C) ⊕ so(n + diag(so(n, C)) 1, C) so(7, C) ⊕ so(8, C) diag(so(7, C) ) sp(n + 1, C) ⊕ sp(n, C) ⊕ sp(m, C) diag(sp(1, C)) ⊕sp(m − 1, C)

gτ (or gσ )

No.

g2 (C) so(7, C) sl(m, C) ⊕ sl(n, C) ⊕ C sl(2n + 1, C) ⊕ C so(10, C) ⊕ C sl(2n, C) ⊕ C

a b 1

2 3 4 5 so(8, C) ⊕ C 6 so(8, C) 7 so(7, C) or spin(7, C) 8 so(2n, C) 9 sp(n − 1, C) ⊕ 10 sp(1, C) sl(n, C) ⊕ sl(n, C) ⊕ 11 C sl(n, C) ⊕ sl(2, C) ⊕ 12 C ⊕sp(1, C) ⊕ sp(m − 13 1, C)

sp(n, C) ⊕ sp(1, C)⊕3 14 ⊕sp(l, C) ⊕ sp(m, C) sp(n, C) ⊕ sp(2, C)⊕2 15 sp(n, C) ⊕ sp(m, C)

16

⊕sp(1, C)⊕4 so(n, C) ⊕ so(n, C)

17

so(7, C) ⊕ spin(7, C) 18 sp(n, C) ⊕ sp(1, C)⊕2 19 ⊕sp(m − 1, C)

On Double Coset Decompositions of Real Reductive Groups … Table 3 Absolutely spherical pair g h g2(2)

sl(3, R) su(2, 1) so(3, 4) g2(2) sl(m + n, C) (m = n) sl(m, C) ⊕ sl(n, C) ⊕ R sl(m + n, R) (m = n) sl(m, R) ⊕ sl(n, R) su(m, n) (m, n = 0)

su( p, q) ⊕ su(m − p, n − q) (2( p + q) = m + n) sl(m + n, H) (m = n) sl(m, H) ⊕ sl(n, H) so(4n + 2, C) so(4m, 4n + 2) (m = 0) so(2n + 1, 2n + 1) so∗ (4(n + m) + 2) e6 (C) e6(6) e6(2) e6(−14)

e6(−26) sl(2n + 1, R) su(2m + 1, 2n) (n = 0) su(n + 1, n) (n = 0) sl(2n + 1, C) sl(2n + 1, R) su(2m + 1, 2n) (n = 0) su(n + 1, n) (n = 0) so(5, 5)

gτ (or gσ )

249

No.

g2(2)

a-1 a-2 so(3, 4) b sl(m, C) ⊕ sl(n, C) ⊕ 1’ C sl(m, R) ⊕ sl(n, R) ⊕ 1-1 R su( p, q) ⊕ R 1-2

sl(2n + 1, C) ⊕ R su(2m, 2n + 1)

⊕su(m − p, n − q) sl(m, H) ⊕ sl(n, H) ⊕ 1-3 R gl(2n + 1, C) 2’ u(2m, 2n + 1) 2-1

sl(2n + 1, R) su(2m, 2n + 1) so(10, C) ⊕ R so(5, 5) so(6, 4) so∗ (10) so(10) so(8, 2) so∗ (10) so(9, 1) sp(n, R) sp(m, n)

gl(2n + 1, R) u(2m, 2n + 1) so(10, C) ⊕ C so(5, 5) ⊕ R so(6, 4) ⊕ R so∗ (10) ⊕ R so(10) ⊕ R so(8, 2) ⊕ R so∗ (10) ⊕ R so(9, 1) ⊕ R gl(2n, R) u(2m, 2n)

2-2 2-3 3’ 3-1 3-2 3-3 3-4 3-5 3-6 3-7 4-1 4-2

sp(n, R) sp(n, C) ⊕ R sp(n, R) ⊕ R sp(m, n) ⊕ R

u(n, n) gl(2n, C) gl(2n, R) u(2m, 2n)

4-3 5’ 5-1 5-2

sp(n, R) ⊕ R spin(3, 4) ⊕ R

u(n, n) so(4, 4) ⊕ R

5-3 6-1 (continued)

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Y. Tanaka

Table 3 (continued) g so(6, 4) so(2, 8) so(1, 9) so∗ (10)

h

gτ (or gσ )

spin(7) ⊕ R

so(8) ⊕ R

spin(1, 6) ⊕ R spin(2, 5) ⊕ R spin(3, 4) spin(7) g2(2)

so(1, 7) ⊕ R so(3, 5) ⊕ R so(4, 4) so(8) so(3, 4) or spin(3, 4)

No.

so(4, 5) so(1, 8) so(4, 4) so(3, 5) so(1, 7) so(2m + 1, 2n) (n = 0) so(n + 1, n) (n = 0) sp(n, R)

g2 u(m, n)

so(7) so(2m, 2n)

gl(n, R) sp(n − 1, R) ⊕ R

sp(m + 1, n) (n = 0)

sp(m, n) ⊕ R

so(n, n) sp(n − 1, R) ⊕ sp(1, R) sp(m, n) ⊕ sp(1)

6-2 6-3 6-4 6-5 6-6 7-1 7-2 8-1 8-2 8-3 9-1 9-2 10-1 10-2

and denote by p2i−1 : g → g2i−1 the corresponding projection. We write g(0) for the 0 complementary ideal of g with respect to g(0) . j Suppose that there is an ideal g0 of h(0) or l(0) , which projects non-trivially to either g10 with j = 2, g04n+1 with j = 4n, 4n + 2 or g04n+3 with j = 4n + 2. We may j assume g0 is an ideal of l(0) and it projects to g04n+3 non-trivially with j = 4n + 2 since the other cases are similar. Then we obtain symmetric subalgebras ⎞ ⎞ ⎛ j ⎛ j n− 4j n− 4j 2 2     j+2+4i ⎠ j+4i gτ = ⎝ ⊕ g(0) , gσ = ⎝ g2i−1 ⊕ g0 g2i−1 ⊕ g0 ⎠ ⊕ g(0) 0 0 i=1

i=0

i=1

i=0

of g, which contain h and l, respectively, such that g = gτ + gσ . j Suppose that neither h(0) nor l(0) has an ideal g0 that projects non-trivially to either 4n+1 4n+3 1 with j = 4n, 4n + 2 or g0 with j = 4n + 2. We divide the g0 with j = 2, g0 argument into two cases (i) and (ii). (i) If p1 (h) = p1 (l) = g10 , then there exists a simple ideal of h or l, which projects non-trivially to both g10 and g04n+3 . We obtain a pair of symmetric subalgebras (gτ , gσ ) = (diag(g0 )⊕n+1 ⊕ g(0) , diag(g0 )⊕n ⊕ diag(g0 )1,4n+3 ⊕ g(0) ), which contains (h, l) up to interchange and satisfies g = gτ + gσ , where diag(g0 )1,4n+3 is the diagonal subalgebra of g10 ⊕ g04n+3 .

On Double Coset Decompositions of Real Reductive Groups …

251

Table 4 Absolutely spherical pair g

h

gτ (or gσ )

No.

sl(n, R) ⊕ sl(n + 1, R)

diag(sl(n, R)) ⊕ R

sl(n, R) ⊕ sl(n, R) ⊕ R

11-1

su( p + 1, q) ⊕ su( p, q)

diag(su( p, q)) ⊕ R

su( p, q) ⊕ su( p, q) ⊕ R

11-2

sl(n, R) ⊕ diag(sl(2, R))

sl(n, R) ⊕ sl(2, R) ⊕ R

12-1

⊕sp(m − 1, R)

⊕sp(1, R) ⊕ sp(m − 1, R)

su( p + 1, q + 1) ⊕ sp(m, R)

su( p, q) ⊕ diag(su(1, 1)) ⊕sp(m − 1, R)

su( p, q) ⊕ su(1, 1) ⊕ R ⊕sp(1, R) ⊕ sp(m − 1, R)

12-2

sl(n + 2, R) ⊕ sp(m, R)

sl(n, R) ⊕ diag(sl(2, R))

sl(n, R) ⊕ sl(2, R) ⊕ R

13-1

(n  = 0)

⊕R ⊕ sp(m − 1, R)

⊕sp(1, R) ⊕ sp(m − 1, R)

su( p + 1, q + 1) ⊕ sp(m, R)

su( p, q) ⊕ diag(su(1, 1)) ⊕R ⊕ sp(m − 1, R)

su( p, q) ⊕ su(1, 1) ⊕ R ⊕sp(1, R) ⊕ sp(m − 1, R)

13-2

sp(n − 1, C) ⊕ sp(1, R)

sp(n − 1, C) ⊕ sp(1, C)

14-1

⊕sp(m − 1, R)

⊕sp(1, R) ⊕ sp(m − 1, R)

sp(n + 1, R ⊕ sp(l + 1, R)

sp(n, R) ⊕ diag(sp(1, R))

sp(n, R) ⊕ sp(1, R)⊕3

⊕sp(m + 1, R)

⊕sp(l, R) ⊕ sp(m, R)

⊕sp(l, R) ⊕ sp(m, R)

sp(n + 2, R) ⊕ sp(2, R)

sp(n, R) ⊕ diag(sp(2, R))

sp(n, R) ⊕ sp(2, R)⊕2

15-1

sp(n + 2, R) ⊕ sp(1, 1)

sp(n, R) ⊕ diag(sp(1, 1))

sp(n, R) ⊕ sp(1, 1)⊕2

15-2

sp( p + 1, q + 1) ⊕ sp(1, 1)

sp( p, q) ⊕ diag(sp(1, 1))

sp( p, q) ⊕ sp(1, 1)⊕2

15-3

sp(n + 1, R) ⊕ sp(2, R)

sp(n, R) ⊕ sp(m, R)

sp(n, R) ⊕ sp(m, R)

16

( p, q  = 0) sl(n + 2, R) ⊕ sp(m, R)

( p + q  = 0) sp(n, C) ⊕ sp(m, R)

14-2

⊕sp(m + 1, R)

⊕diag(sp(1, R))⊕2

⊕sp(1, R)⊕4

so(m, n) ⊕ so(m, n + 1)

diag(so(m, n))

so(m, n) ⊕ so(m, n)

17-1

so(1, 6) ⊕ so(1, 7)

diag’(so(1, 6))

so(1, 6) ⊕ spin(1, 6)

18-1

so(2, 5) ⊕ so(3, 5)

diag’(so(2, 5))

so(2, 5) ⊕ spin(2, 5)

18-2

so(3, 4) ⊕ so(4, 4)

diag’(so(3, 4))

so(3, 4) ⊕ spin(3, 4)

18-3

sp(n, C)

sp(n − 1, C) ⊕ sp(1, R)

sp(n − 1, C) ⊕ sp(1, C)

19-1

sp(n + 1, R) ⊕ sp(m, R)

sp(n, R) ⊕ diag(sp(1, R))

sp(n, R) ⊕ sp(1, R)⊕2

⊕sp(m − 1, R)

⊕sp(m − 1, R)

(m, n  = 0)

sp(n − 1, C) ⊕ sp(1)

19-2 19-3

(ii) If either p1 (h) or p1 (l) does not coincide with g10 , then we obtain a pair of symmetric subalgebras (gτ , gσ ) = (diag(g0 )⊕n+1 ⊕ g(0) , p1 (l) ⊕ diag(g0 )⊕n ⊕ p4n+3 (l) ⊕ g(0) ) or ( p1 (h) ⊕ diag(g0 )⊕n ⊕ g(0) , diag(g0 )⊕n ⊕ p4n+1 (l) ⊕ g(0) ), which contains (h, l) up to interchange and satisfies g = gτ + gσ .  τ

σ

We consider the case gsts = {0}. Here we note that (gsts , (gsts ) , (gsts ) ) and (gsts , (gτ )sts , (gσ )sts ) are factorizations if (g, gτ , gσ ) is a factorization by Theorem 5.4. We list indecomposable factors of (gsts , (gsts )τ , (gsts )σ ). In the following, K stands for R or C. Lemma 5.10 Let (g, h) and (g, l) be effective reductive absolutely spherical pairs. Suppose that gsts = {0} and that (g, gτ , gσ ) is a factorization for any pair (τ , σ)

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Table 5 Absolutely spherical pair g h su(1, q) ⊕ su(q) diag(su(q)) ⊕ R (q = 0) su( p, q + 2) ⊕ sp(m) su( p, q) ⊕ ( p = 0) diag(su(2)) ⊕sp(m − 1) su( p, q) ⊕ diag(su(2)) ⊕R ⊕ sp(m − 1) sp(n, C) ⊕ sp( p + sp(n − 1, C) ⊕ sp(1) 1, q) ⊕sp( p, q) so(m) ⊕ so(m, 1) diag(so(m)) (m = 0)

g30

g50

···

gi−1 0

g40

su(q) ⊕ su(q) ⊕ R

11-3

su( p, q) ⊕ su(2) ⊕ R 12-3 ⊕sp(1) ⊕ sp(m − 1) 13-3

sp(n − 1, C) ⊕ sp(1, C) ⊕sp(1) ⊕ sp( p, q) so(m) ⊕ so(m)

gi+1 0

14-3

17-2

g04n+2

g30 g20

· · · g4n−1 g4n+1 g4n+3 0 0 0 g4n 0

gi0 gi+2 0

g40 g10

No.

gi+2 0

g20 g10

gτ (or gσ )

g50

···

gi−1 0

gi+1 0 gi0

g4n 0 · · · g4n−3 g04n−1 g4n+1 0 0 g4n−2 0

Fig. 2 Zigzag pass

of involutions of g such that gτ ⊃ h and gσ ⊃ l as in Tables 2, 3, 4 and 5. Then any indecomposable factor of (gsts , (gsts )τ , (gsts )σ ) is given as either (g⊕2n 0 , g1 ⊕ ⊕n ⊕n , g ⊕ diag(g ) , diag(g ) ⊕ g2 ) for diag(g0 )⊕n−1 ⊕ g2 , diag(g0 )⊕n ) or (g⊕2n+1 1 0 0 0 some simple Lie algebra g0 and its reductive subalgebras g1 and g2 , and some natural number n up to interchange of τ and σ (see Fig. 3) and replacement of (τ , σ) with (ψφτ φ−1 ψ −1 , φσφ−1 ) for some inner automorphism ψ and some automorphism φ of g such that one of the following conditions 1 and 2 is satisfied. 1. g0 is of type either G2 , Am (m ≥ 2), Bm (m ≥ 2) or Dm (m ≥ 3), gi = g0 and (g0 , g j ) is symmetric or trivial, where (i, j) = (1, 2) or (2, 1). 2. (g0 , g2 , g1 ) is of type either (Dm,K , Bm−1,K , Am−1,K ⊕K) with m ≥ 3, (A2m−1,K , A2m−2,K ⊕K, Cm,K ) with m ≥ 2, (D4,K B 3,K , D3,K ⊕K), (D4 , B2 ⊕ A1 , B 3 ), (D4 , B 3 , B3 ) or (D4,C , D4,R,odd , B 3,C ).

On Double Coset Decompositions of Real Reductive Groups …

g0 g0

g0 g0

g0

g1

g0

g1

g0

g0

g0 g0

253

···

g0

g0

g0 ···

g0

g0

g0 g0

g2

g0 g0

···

g0

g0

g0

g0

g0 ···

g0

g0 g0

g0 g2

Fig. 3 Factorizations with respect to symmetric subalgebras

Proof A trivial factor appears in the pairs (g, gτ ) of Tables 2, 3, 4 and 5 only for No. a, b, 11, 15, 17 and 18, and the column of g in Table 1 contains no simple Lie algebras of type Cn+2 (n ≥ 1). This implies that if either (g0 , g1 ) or (g0 , g2 ) is trivial, then g0 is of type either G2 , Am (m ≥ 2), Bm (m ≥ 2) or Dm (m ≥ 3). Hence, we obtain the condition 1. Suppose that neither (g0 , g1 ) nor (g0 , g2 ) is trivial. In Theorems 5.2 and 5.3 both g and g are symmetric subalgebras of g by adding abelian ideals or type A1 -factors to g and g if necessary only for No. 1–4, 9–11 and 20–26 in Table 1, for the complexifications of those, for the compact real forms of the complexifications and for (g, g , g ) = (so(8, C), spin(7, C), so(1, 7)) and (so(8, C), spin(7, C), so(3, 5)). This implies that the triple (g0 , g2 , g1 ) is of type either (Dm,K , Bm−1,K , Am−1,K ⊕K) (m ≥ 3), (A2m−1,K , A2m−2,K ⊕K, Cm,K ) (m ≥ 2), (D4,K B 3,K , D3,K ⊕K), (D4 , B2 ⊕ A1 , B 3 ), (D4 , B 3 , B3 ) or (D4,C , D4,R,odd , B 3,C ) up to automorphisms of g0 . Hence we obtain the condition 2.  We consider the case when neither (g0 , g1 ) nor (g0 , g2 ) is trivial for any indecomposable factor of (gsts , (gsts )τ , (gsts )σ ). Lemma 5.11 Let (g, h) and (g, l) be effective reductive absolutely spherical pairs. Suppose that (g, gτ , gσ ) is a factorization and that any indecomposable factor of (gsts , (gsts )τ , (gsts )σ ) is effective for any pair (τ , σ) of involutions of g such that gτ ⊃ h and gσ ⊃ l as in Tables 2, 3, 4 and 5. Then we have (gsts , hsts , lsts ) = (gsts , (gτ )sts , (gσ )sts ). Proof By Lemma 5.10, any indecomposable factor of (gsts , (gτ )sts , (gσ )sts ) is given ⊕n−1 ⊕ g2 , diag(g0 )⊕n ) or (g⊕2n+1 , g1 ⊕ diag(g0 )⊕n , as either (g⊕2n 0 , g1 ⊕ diag(g0 ) 0 diag(g0 )⊕n ⊕ g2 ), where (g0 , g2 , g1 ) is of type either (Dm,K , Bm−1,K , Am−1,K ) (m ≥ 3), (A2m−1,K , A2m−2,K , Cm,K ) (m ≥ 2), (D4,K B 3,K , D3,K ), (D4 , B2 , B 3 ), (D4 , B 3 , B3 ) or (D4,C , D4,R,odd , B 3,C ) up to interchange of τ and σ and replacement of (τ , σ) with (ψφτ φ−1 ψ −1 , φσφ−1 ) for some inner automorphism ψ and some automorphism φ of g. Comparing this list with Tables 2, 3, 4 and 5, we find that (gsts , gsts ∩ h) and

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(gsts , gsts ∩ l) have only symmetric pairs and pairs of No. 1 and 2 of Tables 2, 3, 4 and 5 as their indecomposable factors, and the lemma follows.  The following lemma shows Proposition 5.5 when (gsts , (gsts )τ , (gsts )σ ) contains effective factors only. Lemma 5.12 Let (g, h) and (g, l) be effective reductive absolutely spherical pairs. Suppose that (g, gτ , gσ ) is a factorization and that any indecomposable factor of (gsts , (gsts )τ , (gsts )σ ) is effective for any pair (τ , σ) of involutions of g such that gτ ⊃ h and gσ ⊃ l as in Tables 2, 3, 4 and 5. Then the conclusion of Proposition 5.5 follows. g

g

Proof By Lemma 5.11, we can take g , h and l to be g f , p f (h) and p f (l), respectively.  We consider the case when (gsts , (gsts )τ , (gsts )σ ) contains a non-effective factor. By the choice of symmetric subalgebras in Tables 2, 3, 4 and 5, we can see that either (g, hss ) or (g, lss ) involves a factor of the form No. a, b, 11, 17 or 18, namely, a factor of type either (G2 , A2 ), (B3 , G2 ), (Bm ⊕ Dm , Dm ), (Dm ⊕ Bm−1 , Bm−1 ), (D4 ⊕ B3 , B 3 ) or (Am ⊕ Am−1 , Am−1 ) (Lemma 5.10). The following lemma shows Proposition 5.5 when (g, h) or (g, l) contains a factor of type (G2 , A2 ). Lemma 5.13 Let (g, h) and (g, l) be effective reductive absolutely spherical pairs such that g is semisimple and (g, h, l) is indecomposable. Suppose that (g, gτ , gσ ) is a factorization for any pair (τ , σ) of involutions of g such that gτ ⊃ h and gσ ⊃ l as in Tables 2, 3, 4 and 5. If (g, h) or (g, l) contains an indecomposable pair of type (G2 , A2 ) properly, then the conclusion of Proposition 5.5 holds. Proof We may assume (g, h) contains an indecomposable factor (g0 , h0 ) of type (G2 , A2 ). Then by No. a in Tables 2, 3, 4 and 5, (g, gτ ) contains a trivial factor (g0 , g0 ) of type G2 . By Lemma 5.10, (gsts , (gsts )τ , (gsts )σ ) contains an indecomposable factor (g(0) , gτ(0) , gσ(0) ) of the form ⊕n−1 (g⊕2n ⊕ g1 , diag(g0 )⊕n ) or 0 , g0 ⊕ diag(g0 ) (g(0) , gτ(0) , gσ(0) ) = , g0 ⊕ diag(g0 )⊕n , diag(g0 )⊕n ⊕ g1 ) (g⊕2n+1 0 for some reductive subalgebra g1 of g0 and some natural number n, where (g0 , g1 ) is symmetric or trivial. By inspecting Tables 2, 3, 4 and 5, we find the following: ˜ be an indecomposable reductive absolutely spherical pair with g˜ Let (˜g, h) ˜ is either symmetsemisimple and involving a type G2 -simple factor. Then (˜g, h) ric or of type (G2 , A2 ). This implies that g(0) = g, and hence we can take (g , h , l ) to be either (g0 , h0 , τ p1 (h)) or (g0 , h0 , p1σ (l)) for the projection p1τ : gτ → g1 or p1σ : gσ → g1 , respectively (see Fig. 4). Here, we apply Remark 5.6.1 for the openness and the surjectivity , h0 , diag of the map G → L\G/H by taking (gβ , gγ , hβ , hγ , lα ) to be (g0 , g⊕2n−1 0 ⊕n (g0 )⊕n−1 ⊕ p1τ (h), l) or (g0 , g⊕2n  0 , h0 , diag(g0 ) , l).

On Double Coset Decompositions of Real Reductive Groups …

h0

255

g0

g0

g0

···

g0

g0 g0

g0

g0 g0

···

g0

g0 pσ1 (l)

h0 g0 pσ1 (l) Fig. 4 Contraction of the diagonal component

The following lemma shows Proposition 5.5 when (g, hss ) or (g, lss ) contains a factor of type either (Am ⊕ Am−1 , Am−1 ), (Bm ⊕ Dm , Dm ), (Dm ⊕ Bm−1 , Bm−1 ) or (D4 ⊕ B3 , B 3 ) with some exceptions. Lemma 5.14 Let (g, h) and (g, l) be effective reductive absolutely spherical pairs such that g is semisimple, (g, h, l) is indecomposable and neither of conditions C1 nor C2 is satisfied. Suppose that (g, gτ , gσ ) is a factorization for any pair (τ , σ) of involutions of g such that gτ ⊃ h and gσ ⊃ l as in Tables 2, 3, 4 and 5. Then the conclusion of Proposition 5.5 holds if one of the following conditions 1 and 2 is satisfied. 1. (g, hss ) or (g, lss ) contains a factor of type (Bk ⊕ Dk , Dk ) (k ≥ 4), (Dk ⊕ Bk−1 , Bk−1 ) (k ≥ 4), (D4 ⊕ B3 , B 3 ) or (Ak ⊕ Ak−1 , Ak−1 ) (k ≥ 2). 2. (g, hss ) or (g, lss ) contains a factor of type (B2 ⊕ D2 , D2 ), (B3 ⊕ D3 , D3 ) or (D3 ⊕ B2 , B2 ) but a factor of type (B3 , G2 ) appears in neither (g, hss ) nor (g, lss ). Proof Let m be the largest number among the natural numbers k such that a factor of type (Dk ⊕ Bk−1 , Bk−1 ) or (A2k−1 ⊕ A2k−2 , A2k−2 ) appears in either (g, hss ) or (g, lss ) if such an m exists. Then by the assumption that neither of conditions C1 nor C2 is satisfied there is a triple (g0 , g2 , g1 ) of type (Dm , Bm−1 , Am−1 ⊕K), (A2m−1 , A2m−2 ⊕K, Cm ), (D4 , B 3 , D3 ⊕K) or (D4 , B 3 , B3 ) such that (g0 ⊕ (g2 )ss , (g2 )ss ) appears in (g, hss ) or (g, lss ) as an indecomposable factor under the condition 1 or 2 up to replacement of (g, h, l) with (g, ψ(φ(h)), φ(l)) for some automorphism φ and some inner automorphism ψ of g. Let us see how such a triple arises. a. We consider the case where (g, hss ) or (g, lss ) contains a factor of type (Bk ⊕ Dk , Dk ) or (Dk+1 ⊕ Bk , Bk ) for some k ≥ 4. By Lemma 5.10, we obtain a triple of type (Dm , Bm −1 , Am −1 ⊕K), where m is the largest number among the natural numbers k such that a factor of type (Dk ⊕ Bk −1 , Bk −1 ) appears in either (g, hss ) or (g, lss ).

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b. We suppose that (g, hss ) or (g, lss ) contains a factor of type (D4 ⊕ B3 , B3 ) or (D4 ⊕ B3 , B 3 ) but no factors of types (Dk ⊕ Bk−1 , Bk−1 ) (k ≥ 5) appear in (g, hss ) and (g, lss ). Then by Lemma 5.10 and by the assumption that neither of conditions C1 nor C2 is satisfied, we obtain a triple (g0 , g2 , g1 ) of type (D4 , B3 , A3 ⊕K), (D4 , B 3 , D3 ⊕K) or (D4 , B 3 , B3 ). c. We consider the case where (g, hss ) or (g, lss ) contains a factor of type (B3 ⊕ D3 , D3 ). By Lemma 5.10 and by the assumption that a factor of type (B3 , G2 ) appears in neither (g, hss ) nor (g, lss ), at least one of the two pairs involves a factor of type (D4 ⊕ B3 , B3 ) or (D4 ⊕ B3 , B 3 ) and thus the arguments in the cases a. and b. can be applied. d. Let us suppose that (g, hss ) or (g, lss ) contains a factor of type (Ak ⊕ Ak−1 , Ak−1 ) for some k ≥ 2 but no factors of type (B3 ⊕ D3 , D3 ) appear in (g, hss ) and (g, lss ). By Lemma 5.10 we obtain a triple (g0 , g2 , g1 ) of type (A2m −1 , A2m −2 ⊕K, Cm ), where m is the largest number among the natural numbers k such that a factor of type (A2k −1 ⊕ A2k −2 , A2k −2 ) appears in either (g, hss ) or (g, lss ). e. We suppose that (g, hss ) or (g, lss ) contains a factor of type (B2 ⊕ D2 , D2 ) or (D3 ⊕ B2 , B2 ) but no factors of types (B3 ⊕ D3 , D3 ) and (A4 ⊕ A3 , A3 ) appear in (g, hss ) and (g, lss ). By Lemma 5.10, we obtain a triple of type (D3 , B2 , A2 ⊕K). Let a triple (g0 , g2 , g1 ) be as above. By Lemma 5.10, we have a pair of a non-effective indecomposable factor (g(1) , gτ(1) , gσ(1) ) and an effective indecomposable factor (g(0) , gτ(0) , gσ(0) ) of (gsts , (gsts )τ , (gsts )σ ) such that g(1) contains (g2 )ss as its factor, and (g(0) , gτ(0) , gσ(0) ) is of the form (g(0) , gτ(0) , gσ(0) )

(g0 ⊕ g⊕2n−2 ⊕ g0 , g1 ⊕ diag(g0 )⊕n−1 ⊕ g2 , diag(g0 )⊕n ) or 0 = ⊕ g0 , g1 ⊕ diag(g0 )⊕n , diag(g0 )⊕n ⊕ g2 ) (g0 ⊕ g⊕2n−1 0

up to interchange of σ and τ . Here g(0) and g(1) are connected to each other in the diagram of (g, h, l) in the sense that there is a non-zero ideal isomorphic to (g2 )ss of h or l, which projects non-trivially to both of the last summand g0 of g(0) and the factor (g2 )ss of g(1) . We only consider the case when (g(0) , gτ(0) , gσ(0) ) is of the form , g1 ⊕ diag(g0 )⊕n , diag(g0 )⊕n ⊕ g2 ) since the same argument can be applied (g⊕2n+1 0 ⊕n−1 ⊕ to the other case when (g(0) , gτ(0) , gσ(0) ) takes the form (g⊕2n 0 , g1 ⊕ diag(g0 ) ⊕n g2 , diag(g0 ) ) up to interchange of σ and τ . Then g, h and l are given as ⊕ (g0 ⊕ (g2 )ss ) ⊕ g(1) , g = g0 ⊕ g⊕2n−1 0 h = (g1 )ss ⊕ diag(g0 )⊕n ⊕ h(1) , l = diag(g0 )⊕n ⊕ (g2 )ss ⊕ l(1) for some ideals g(1) , h(1) and l(1) of g, h and l, respectively. Here h(1) is embedded into the sum of the first summand g0 and the summand (g2 )ss ⊕ g(1) of g, l(1) is into the sum of the third summand g0 and the summand g(1) of g, and the ideal (g2 )ss of l is into the summand (g0 ⊕ (g2 )ss ) of g. We divide the argument into four cases (i)–(iv) according to the type of (g0 , g2 , g1 ).

On Double Coset Decompositions of Real Reductive Groups …

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g0

(g1 )ss g0

g0

···

g0 g0

g0

g0

g0

···

g0

g0

h(1) g0

g2 g2

g(1) l(1)

h(1) K

g2

g(1)

K (g1 )ss ∩ g2 l(1) Fig. 5 Contraction of a factorization of type (Dm , Bm−1 , Am−1 )

(i) Suppose that (g0 , g2 , g1 ) is of type (Dm , Bm−1 , Am−1 ⊕K) and that h(1) projects trivially to the direct summand g0 . In this case we can take g , h and l as g = K ⊕ g2 ⊕ g(1) , h = h(1) and l = K ⊕ ((g1 )ss ∩ g2 ) ⊕ l(1) (see Fig. 5). Here, we apply Remark 5.6.2 by taking (ga , ha , la ), (gb , hb , lb ) and (gc , hc , lc ) to be (g, h, l), (g , h , l ) and (g2 ⊕ g(1) , h(1) , ((g1 )ss ∩ g2 ) ⊕ l(1) ), respectively, and considering the corresponding analytic subgroups. For the openness and the surjectivity of the map G c → L a \G a /H a , we apply Remark 5.6.1 by taking ⊕ g0 , h(1) , (g1 )ss ⊕ diag(g0 )⊕n , l). (gβ , gγ , hβ , hγ , lα ) to be (g2 ⊕ g(1) , g⊕2n 0 (ii) Suppose that (g0 , g2 , g1 ) is of type (Dm , Bm−1 , Am−1 ⊕K) or (D4 , B 3 , D3 ) and that the image of the projection of h(1) to the direct summand g0 is K. In this case, we can take g , h and l as g = K ⊕ g2 ⊕ g(1) , h = h(1) and l = K ⊕ ((g1 )ss ∩ g2 ) ⊕ l(1) (see Fig. 6). Here, we apply Remark 5.6.2 by taking (ga , ha , la ), (gb , hb , lb ) and (gc , hc , lc ) to be (g, h, l), (g , h , l ) and (g2 ⊕ g(1) , h(1) , ((g1 )ss ∩ g2 ) ⊕ l(1) ), respectively, and considering the corresponding analytic subgroups. For the openness and the surjectivity of the map G c → L a \G a /H a , we apply Remark 5.6.1 by taking (gβ , gγ , hβ , hγ , lα ) to be ⊕ g0 , h(1) , (g1 )ss ⊕ diag(g0 )⊕n , l). (g2 ⊕ g(1) , g⊕2n 0 (iii) Suppose that (g0 , g2 , g1 ) is of type (A2m−1 , A2m−2 ⊕K, Cm ). In this case we can take g , h and l as g = (g2 )ss ⊕ g(1) , h = h(1) and l = (g1 ∩ (g2 )ss ) ⊕ l(1) 1 , (1) (1) (1) where l1 is the image of the projection of l to g (see Fig. 7). Here we apply Remark 5.6.1 for the openness and the surjectivity of the map G → ⊕ g0 , h(1) , g1 ⊕ L\G/H by taking (gβ , gγ , hβ , hγ , lα ) to be ((g2 )ss ⊕ g(1) , g⊕2n 0 ⊕n diag(g0 ) , l). (iv) Suppose that (g0 , g2 , g1 ) is of type (D4 , B 3 , B3 ). In this case, we can take g , h and l as g = g2 ⊕ g(1) , h = h(1) and l = (g1 ∩ g2 ) ⊕ l(1) (see Fig. 8). Here we apply Remark 5.6.1 for the openness and the surjectivity of the map G → ⊕ g0 , h(1) , g1 ⊕ L\G/H by taking (gβ , gγ , hβ , hγ , lα ) to be (g2 ⊕ g(1) , g⊕2n 0 ⊕n diag(g0 ) , l). 

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g0

(g1 )ss g0

g0

···

g0 g0

g0

g0

g0

···

h(1) g0

g0

g0

g2 g2

g(1) l(1)

h(1) g2

K

g(1)

K (g1 )ss ∩ g2 l(1) Fig. 6 Contraction of a factorization of type (Dm , Bm−1 , Am−1 ⊕K)

g1

g0

g0

g0

···

g0 g0

g0

g0

g0

···

h(1) g0

g0

g0

(g2 )ss g(1) (g2 )ss

l(1)

h(1) (g2 )ss g(1) g1 ∩ (g2 )ss l(1) 1 Fig. 7 Contraction of a factorization of type (A2m−1 , A2m−2 ⊕K, Cm )

g1

g0

g0

g0 g0

···

g0

g0 g0

g0

···

g0

g0

h(1) g0

g2 g2

h(1) g2

g(1)

g1 ∩ g2

l(1)

Fig. 8 Contraction of a factorization of type (D4 , B 3 , B3 )

g(1) l(1)

On Double Coset Decompositions of Real Reductive Groups …

259

Finally, we consider the case when (g, hss ) or (g, lss ) contains a factor of type (B3 , G2 ). Lemma 5.15 Let (g, h) and (g, l) be effective reductive absolutely spherical pairs such that g is semisimple, (g, h, l) is indecomposable and neither of conditions C1 nor C2 is satisfied. Suppose that (g, gτ , gσ ) is a factorization for any pair (τ , σ) of involutions of g such that gτ ⊃ h and gσ ⊃ l as in Tables 2, 3, 4 and 5. If (g, hss ) or (g, lss ) contains a factor of type (B3 , G2 ) properly, then the conclusion of Proposition 5.5 holds. Proof If (g, hss ) or (g, lss ) contains a factor of type (G2 , A2 ), (Am ⊕ Am−1 , Am−1 ) (m ≥ 2), (Bm ⊕ Dm , Dm ) (m ≥ 4), (Dm ⊕ Bm−1 , Bm−1 ) (m ≥ 4) or (D4 ⊕ B3 , B 3 ), then we can apply Lemma 5.13 or 5.14. Hence we may assume neither (g, hss ) nor (g, lss ) contains factors of such types. We may also assume that (g, hss ) contains a factor (g0 , h0 ) of type (B3 , G2 ). Then by No. b in Tables 2, 3, 4 and 5, (g, gτ ) contains a trivial factor (g0 , g0 ) of type B3 . By Lemma 5.10, (gsts , (gsts )τ , (gsts )σ ) contains an indecomposable factor (g(0) , gτ(0) , gσ(0) ) that takes the form (g(0) , gτ(0) , gσ(0) )

⊕n−1 (g⊕2n ⊕ g1 , diag(g0 )⊕n ) or 0 , g0 ⊕ diag(g0 ) = , g0 ⊕ diag(g0 )⊕n , diag(g0 )⊕n ⊕ g1 ) (g⊕2n+1 0

for some reductive subalgebra g1 of g0 and some natural number n, where the first summand g0 of gτ(0) contains h0 and (g0 , g1 ) is symmetric or trivial. We only consider , g0 ⊕ diag(g0 )⊕n , diag(g0 )⊕n ⊕ the case when (g(0) , gτ(0) , gσ(0) ) takes the form (g⊕2n+1 0 g1 ) since the same argument can be applied to the other case. Then (g, h, l) is of the form (g, h, l) = (g0 ⊕ g⊕2n−1 ⊕ g0 ⊕ g(0) , h0 ⊕ diag(g0 )⊕n ⊕ h(0) , diag(g0 )⊕n ⊕ l0 ⊕ l(0) ), 0

where g(0) is the complement of g(0) in g with p (0) : g → g(0) the corresponding projection, h(0) = h ∩ g(0) , l(0) = l ∩ g(0) , and l0 is an ideal of l, which is embedded into the direct sum of the last two summands g0 and g(0) of g such that l0 is isomorphic to the image of the projection of l to the third summand g0 of g. Here we note that the image is contained in g1 and that (g0 , g1 ) is of type either (B3 , D3 ), (B3 , B2 ⊕K) or (B3 , D2 ⊕ B1 ) otherwise (g0 , g1 ) is trivial, g1 contains l0 and l0 is of type G2 as No. b of Tables 2, 3, 4 and 5. We divide the argument into three cases (i)–(iii) according to the type of (g0 , g1 ). (i) Suppose that (g0 , g1 ) is of type (B3 , D3 ) or (B3 , B2 ⊕K) and that (g0 , l0 ) is not of type (B3 , A2 ⊕K). We note that a proper reductive subalgebra g 1 of g1 can be an absolutely spherical subalgebra of g0 only when (g0 , g1 , g 1 ) is of type (B3 , D3 , A2 ⊕K). Hence l0 is isomorphic to g1 since (g0 , l0 ) should be absolutely spherical. Then we can take (g , h , l ) to be (g(0) , h(0) , p (0) (h0 ∩ g1 ) ⊕ l(0) ) (see Fig. 9). We apply Remark 5.6.1 for the openness and the surjectivity of the map G → L\G/H by taking (gβ , gγ , hβ , hγ , lα ) to be (g(0) , g0 ⊕ g⊕2n−1 ⊕ g0 , h(0) , h0 ⊕ diag(g0 )⊕n , l). 0

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h0

g0

g0

g0 g0

···

g0

g0 g0

g0

···

g0

g0

h(0) g0

g(0) g1 l(0)

h(0) g(0) p(0) (h0 ∩ g1 ) l(0) Fig. 9 Contraction of a factorization of type (B3 , G2 , D3 ) or (B3 , G2 , B2 )

The following is a remark on the absolute sphericity of the pair (g(0) , p (0) (h0 ∩ g1 ) ⊕ l(0) ). • We note that if (g0 , g1 ) is of type (B3 , B2 ⊕K), then for reductive Lie algebras g 0 and g 1 , a pair (g0 ⊕ g 0 , g1 ⊕ g 1 ) cannot be absolutely spherical when the type B2 -factor of g1 projects non-trivially to g 0 by Tables 2, 3, 4 and 5. Hence we find that if (g0 , g1 ) is of type (B3 , B2 ⊕K) and g(0) is non-zero, then p (0) (g1 ) = K. Since (so(7, C), so(5, C)) is not spherical, (g(0) , l(0) ) is absolutely spherical. • We also note that if (g0 , g1 ) is of type (B3 , D3 ), then for a semisimple Lie algebra g 0 to which g1 maps non-trivially and for a reductive Lie subalgebra g 1 of Z g 0 (g1 ), a pair (g0 ⊕ g 0 , g1 ⊕ g 1 ) can be absolutely spherical only when the projection of g1 to g 0 is surjective by Tables 2, 3, 4 and 5. Hence we find that if (g0 , g1 ) is of type (B3 , D3 ) and g(0) is non-zero, then g(0) takes (0) (0) (0) (0) the form g(0) 1 ⊕ g2 , where g1 = p (g1 ) and g2 is its complement. Then (0) (0) (0) (0) (g(0) 1 , p (h0 ∩ g1 )) is of type (D3 , A2 ) and thus (g , p (h0 ∩ g1 ) ⊕ l ) = (0) (0) (0) (0) (g1 , p (h0 ∩ g1 )) ⊕ (g2 , l ) is absolutely spherical. (ii) Suppose that either (g0 , g1 ) is trivial or of type (B3 , D2 ⊕ B1 ). We note that a proper reductive subalgebra g 1 of g1 cannot be an absolutely spherical subalgebra of g0 when (g0 , g1 ) is of type (B3 , D2 ⊕ B1 ) by Tables 2, 3, 4 and 5. Hence in the latter case we have l0 = g1 since (g0 , l0 ) is absolutely spherical. We can take (g , h , l ) to be (g0 , h0 , l0 ) (see Fig. 10). Here we apply Remark 5.6.1 for the openness and the surjectivity of the map G → L\G/H by taking ⊕n (gβ , gγ , hβ , hγ , lα ) to be (g0 , g⊕2n 0 , h0 , diag(g0 ) , l). (iii) Suppose that (g0 , g1 ) is of type (B3 , D3 ) and (g0 , l0 ) of type (B3 , A2 ⊕K). Then we can take g , h and l to be g1 ⊕ g(0) , (g1 ∩ h0 ) ⊕ h(0) and l0 ⊕ l(0) (see Fig. 11). We put g = g0 ⊕ g(0) and apply Remark 5.6.1 for the openness and the surjectivity of the map G → L\G/H by taking (gβ , gγ , hβ , hγ , lα ) to be (0) ⊕n (g0 ⊕ g(0) , g⊕2n 0 , l0 ⊕ l , diag(g0 ) , h), where G is the analytic subgroup with Lie algebra g . Then we apply Remark 5.6.3 by taking (ga , gb , ha , la )

On Double Coset Decompositions of Real Reductive Groups …

h0

261

g0

g0

g0

···

g0

g0

g0

g0

g0

···

g0

g0

g0

l0

h0 g0 l0 Fig. 10 Contraction of the diagonal component

g0

h0 g0

g0 g0

···

g0

g0 g0

g0

···

g0

g0

h(0) g0

g(0) l0 l(0)

g1 ∩ h0 h(0) g1

g(0)

l0

l(0)

Fig. 11 Contraction of a factorization of type (B3 , G2 , D3 )

to be (g0 ⊕ g(0) , g1 ⊕ g(0) , h0 ⊕ h(0) , l0 ⊕ l(0) ) and find that the map G → L \G /H is open surjective, where H and L are the analytic subgroups with Lie algebras h0 ⊕ h(0) and l0 ⊕ l(0) , respectively. We find that G → L\G/H is open surjective.  Proof of Proposition 5.5 The proposition follows from Lemmas 5.9, 5.10, 5.12, 5.13, 5.14 and 5.15. 

6 Double Coset Decomposition In this section, we give a proof of Theorem 1.3. We prepare some lemmas. Let L and H be reductive absolutely spherical subgroups of a connected real semisimple Lie group G.

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Lemma 6.1 Suppose that (g, h, l) is indecomposable and effective and that (g, h) consists only of non-symmetric polar factors of types (G2 , A2 ) and (B3 , G2 ). Then there exist finitely many  semisimple abelian subspaces ji of g and elements xi of G (1 ≤ i ≤ k) such that 1≤i≤k L exp(ji )xi H contains an open dense subset of G. Proof Suppose that (g, h) contains an indecomposable factor (g1,1 , h1 ) of type (G2 , A2 ). Let (g2,1 , l1 ) be an indecomposable factor of (g, l) such that g1,1 is a simple ideal of g2,1 . If g2,1 is simple, then (g, h) is of type (G2 , A2 ) and we can apply Proposition 4.1. If g2,1 is not simple, then by the classification (Tables 2, 3, 4 and 5) g2,1 is the direct product g ⊕ g of two copies of a simple Lie algebra g of type G2 , and l1 is its diagonal subalgebra diag(g ). This implies that g = g2,1 = g ⊕ g and h is the direct product of two simple ideals of type A2 . By contracting the diagonal factor (g ⊕ g , diag(g )), we can apply Proposition 4.1. The same argument works  for the case when (g, h) contains an indecomposable factor of type (B2 , G2 ). Let (σ, τ ) be a pair of involutions of g such that h ⊂ gτ , l ⊂ gσ as in Tables 2, 3, 4 and 5. Taking a conjugate of τ (and H accordingly), we may assume there is a Cartan involution θ commuting with both τ and σ. We let g(1) be the centralizer of a θ-stable abelian subspace of g−σ,−τ and put h(1) = g(1) ∩ h and l(1) = g(1) ∩ l. The following lemma can be checked by inspecting Table 2 of reductive spherical subalgebras. Lemma 6.2 From Table 2, we can find the following. (1) (1) 1. If (g(1) , h(1) ) contains a factor (g(1) 1 ⊕ g2 , h1 ) of type (D4 ⊕ B3 , B3 ), then (1) (1) (1) (g1 ⊕ g2 , h1 ) is also a factor of (g, h). (1) (1) (1) 2. If (g(1) , h(1) ) contains a factor (g(1) 1 ⊕ g2 , h1 ) of type (D4 ⊕ B4 , D4 ) and g1 is (1) (1) an ideal of g, then (g(1) 1 ⊕ g2 , h1 ) is also a factor of (g, h). (1) (1) (1) (1) 3. If (g , h ) contains a factor (g(1) 1 ⊕ g2 , h1 ) of type (D4 ⊕ B4 , D4 ) and if (1) (1) (g(1) , l(1) ) contains a factor (g1 , l0 ) of type (D4 , B2 ⊕ A1 ) or (D4,C , D 4,R,odd ), (1) (1) (1) (1) then (g(1) 1 ⊕ g2 , h1 ) is a factor of (g, h) and (g1 , l0 ) that of (g, l). (1) (1) (1) 4. If (g(1) , h(1) ) contains a factor (g1 ⊕ g2 , h1 ) for some type D4 -simple ideals (1) (1) (1) (1) g(1) and some type D4 -simple ideal h(1) and if g(1) 1 and g2 of g 1 of h 1 or g2 (1) (1) (1) is an ideal of g, then (g1 ⊕ g2 , h1 ) is also a factor of (g, h). (1) (1) 5. If (g(1) , h(1) ) contains a factor (g(1) 1 ⊕ g2 , h1 ) of type (D4 ⊕ B3 , B3 ), then either (1) (1) (1) (g(1) 1 ⊕ g2 , h1 ) is also a factor of (g, h) or (g, h) has a factor (g1 ⊕ g2 , h1 ) of (1) (1) type (D4 ⊕ B4 , D4 ) such that g2 ⊂ g2 and h1 ⊂ h1 . (1) (1) Proof Since h(1) 1 is embedded into both of g1 and g2 , there is an indecomposable (1) factor (s, (s ∩ h)ss ) of (g, hss ) such that s contains g1 and g(1) 2 as simple factors of a Levi subalgebra. We divide the argument accordingly to the type of g(1) 2 . (1) (1) a. Let us suppose that g(1) 2 is of type B3 and assume that at most one of g1 and g2 (1) (1) appears as a simple factor of g. Since g1 is of type D4 and g2 of type B3 , the type of (s, sτ , (s ∩ h)ss ) is (D4 ⊕ B4 , D4 ⊕ D4 , diag(D4 )) as No. 17 of Table 2.

On Double Coset Decompositions of Real Reductive Groups … Table 6 Types of subalgebras u (uσ )ss D5 D5 B4 ⊕ D5

A4 D4 B4 ⊕ B4

263

(v ∩ l)ss

vss

No.

None G2 diag(D4 )

None D4 B4 ⊕ D4

2 6 17

(1) (1) This shows Lemma 6.2.5. Here we note that (g(1) 1 ⊕ g2 , h1 ) is non-symmetric and hence (s, s ∩ h) should be non-symmetric and that a proper Levi subalgebra of a semisimple Lie algebra exhibited on the first column does not involve two simple factors of types D4 and B3 in the other cases of the table. The type of the pair (m, m ∩ h) of the centralizer m of a non-zero semisimple abelian subspace of s−τ and its intersection with h is given as (B3 ⊕ D4 , B3 ) and hence not as (B3 ⊕ D4 , B 3 ). This shows Lemma 6.2.1. (1) (1) b. Let us suppose that g(1) 2 is of type B4 and assume that at most one of g1 and g2 (1) (1) appears as a simple factor of g. Since g1 is of type D4 and g2 of type B4 , the type of (s, sτ , s ∩ h) is (D5 ⊕ B4 , B4 ⊕ B4 , diag(B4 )) as No. 17 of Table 2. This shows Lemma 6.2.2 since g(1) 1 of type D4 is contained in s but not an ideal of s, namely, not that of g. (1) Let us further suppose that (g(1) , l(1) ) contains a factor (g(1) 1 , l0 ) of type (D4 , B2 ⊕ A1 ) or (D4,C , D 4,R,odd ). We write s = s1 ⊕ s2 , where s1 is of type D5 and s2 of type B4 and take an indecomposable factor (u, (u ∩ l)ss ) of (g, lss ) such that u contains s1 . If (u, u ∩ l) is non-symmetric, then (u, (u ∩ l)ss ) itself or its complexification is given in No. 2, 6 or 17 of Table 2 since u contains a simple factor s1 of type D5 . By computing the centralizers v of semisimple abelian subspaces of u−σ such that v involve type D4 -factors, we find that a factor of type (D4 , B2 ⊕ A1 ) or (D4,C , D 4,R,odd ) does not appear in (v, (v ∩ l)ss ) (Table 6). Hence (u, u ∩ l) should be symmetric, that is, u ∩ l = uσ and hence (u, u ∩ l) or its complexification is given as either (so(10, C), so( p, C) ⊕ so(10 − p, C)), (so(10, C), gl(5, C)) or (so(10, C) ⊕ so(10, C), diag(so(10, C))) since u contains a simple factor s1 of type D5 . The centralizer v of an abelian subspace of u−σ which contains a type D4 -factor is given in Table 7. We can see that a factor of type (D4 , B2 ⊕ A1 ) or (D4,C , D 4,R,odd ) does not appear in (vss , (v ∩ l)ss ) (Table 7). This shows Lemma 6.2.3. (1) (1) c. Let us suppose that g(1) 2 is of type D4 and that g1 or g2 is an ideal of g, and thus of s. Since a Lie algebra of type D4 ⊕ D4 does not appear as a factor of a Levi subalgebra of a semisimple Lie algebra exhibited on the first column of Table 2, (s, s ∩ h) should be symmetric. If s is simple, then it coincides with g(1) 1 or g(1) 2 but this contradicts to the fact that s contains the both of them. Hence, s is the direct sum of two copies of a simple Lie algebra of type D4 and thus (1) s = g(1) 1 ⊕ g2 . This shows Lemma 6.2.4.



264 Table 7 Lie subalgebras u (uσ )ss so(10, C)

so( p, C) ⊕so(10 − p, C) so(10, C) gl(5, C) so(10, C) ⊕ so(10, C) diag(so(10, C))

Y. Tanaka

(v ∩ l)ss

vss

so( p − 1, C) ⊕so(9 − p, C) None diag(so(8, C))

so(8, C) None so(8, C) ⊕ so(8, C)

These observations show the assumption in Theorem 1.3 is satisfied during the induction argument. Namely, we have the following. Lemma 6.3 If (g, h, l) satisfies neither the condition C1 nor C2, then so does (g(1) , h(1) , l(1) ). Proof If (g(1) , h(1) , l(1) ) satisfies the condition (i) under C1 or C2, then the condition (i) or (ii) holds for (g, h, l) by Lemma 6.2.1, 4 and 5. When the condition (ii) holds for (g(1) , h(1) , l(1) ), so it does for (g, h, l) by Lemma 6.2.1, 2 and 4. If (g(1) , h(1) , l(1) ) satisfies the condition (iii), then so (g, h, l) does by Lemma 6.2.1, 3 and 4.  Table 8 Complex symmetric pair g sl(n, C) sl(2n, C) so(2n + 1, C) sp(n, C) so(2n, C) e6 (C)

e7 (C)

e8 (C) f4 (C) g2 (C)

h so(n, C) s(gl(m, C) ⊕ gl(n − m, C)) sp(n, C) so(m, C) ⊕ so(2n + 1 − m, C) sp(m, C) ⊕ sp(n − m, C) gl(n, C) so(m, C) ⊕ so(2n − m, C) gl(n, C) sp(4, C) sl(6, C) ⊕ sl(2, C) so(10, C) ⊕ C f4 (C) sl(8, C) so(12, C) ⊕ sp(1, C) e6 (C) ⊕ C so(16, C) e7 (C) ⊕ sp(1, C) sp(3, C) ⊕ sp(1, C) so(9, C) sl(2, C) ⊕ sl(2, C)

On Double Coset Decompositions of Real Reductive Groups …

265

Now we can show the double coset decomposition. Proof of Theorem 1.3 Let L and H be reductive absolutely spherical subgroups of a connected real reductive Lie group G. By the induction on the dimension of the commutator subgroup of G, we shall prove that there exist finitely many abelian  subspaces ji of g and elements xi of G (1 ≤ i ≤ k) such that 1≤i≤k L exp(ji )xi H contains an open dense subset of G. We may assume that G is semisimple, both L and H are connected and (g, h, l) is indecomposable and effective. If (g, h) consists only of non-symmetric polar factors of types (G2 , A2 ) and (B3 , G2 ), then we can apply Lemma 6.1. Thus, we may assume that each of (g, h) and (g, l) contains an

Table 9 Real form gC sl(n, C)

sl(2n, C) so(2n + 1, C) sp(n, C)

so(2n, C)

e6 (C)

e7 (C)

e8 (C)

f4 (C)

g2 (C)

g

k

su(n) sl(n, R) su(m, n − m) sl(n, H) so(2n + 1) so(m, 2n + 1 − m) sp(n) sp(n, R) sp(m, n − m) so(2n) so(m, 2n − m) so∗ (2n) e6 e6(6) e6(2) e6(−14) e6(−26) e7 e7(7) e7(−5) e7(−25) e8 e8(8) e8(−24) f4 f4(4) f4(−20) g2 g2(2)

su(n) so(n) s(u(m) ⊕ u(n − m)) sp(n) so(2n + 1) so(m) ⊕ so(2n + 1 − m) sp(n) u(n) sp(m) ⊕ sp(n − m) so(2n) so(m) ⊕ so(2n − m) u(n) e6 sp(4) su(6) ⊕ su(2) so(10) ⊕ R f4 e7 su(8) so(12) ⊕ sp(1) e6 ⊕ R e8 so(16) e7 ⊕ sp(1) f4 sp(3) ⊕ sp(1) so(9) g2 su(2) ⊕ su(2)

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indecomposable factor of type neither (G2 , A2 ) nor (B3 , G2 ). By Proposition 3.1, there are involutions σ and τ of g such that h ⊂ gτ and l ⊂ gσ . If g−τ ,−σ = {0} for any such pair of involutions, then by Proposition 5.5 there exist a connected real reductive subgroup G of G, whose commutator subgroup is of dimension smaller than G and its reductive absolutely spherical subgroups H and L such that the assumption of Theorem 1.3 is satisfied for (G , H , L ) and the inclusion G → G induces an open surjective map L \G /H → L\G/H . Hence, we can apply the induction hypothesis to (G , H , L ) and obtain the double coset decomposition L\G/H generically. Therefore, we may further assume that g−τ ,−σ = {0}. We may assume τ and σ lift to G by taking a covering and may replace σ (and L accordingly) with its conjugate so that there exists a Cartan involution θ of G, which commutes with τ and σ if necessary. We take a maximal abelian subspace t of gθ,−τ ,−σ . By Lemma 6.3 the triple (M, M ∩ L , M ∩ ιt H ) where M is the connected centralizer of a nonzero θ-stable abelian subspace of g−σ,−τt (t ∈ exp(t)) also satisfies the assumption of Theorem 1.3. Then we can apply Theorem 1.1 and the proof is finished.  Proof of Corollary 1.4 The corollary follows from the induction argument as the proof of Theorem 1.3. We note that we can take k to be 1 and x1 to be the identity element in Theorem 1.1 if either G is compact or (G, H, L) is complex by the proof of Theorem 1.1 combined with Theorem 2.4 (Tables 8 and 9).  Acknowledgements The author would like to express his sincere gratitude to Professor Toshiyuki Kobayashi for his generous support and constant encouragement. He is also grateful to an anonymous referee for comments on this article.

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