Unitary Representation Theory of Exponential Lie Groups 9783110874235, 3110139383, 9783110139389

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de Gruyter Expositions in Mathematics 18

Editors

Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, The University of Melbourne, Parkville, R.O.Wells, Jr., Rice University, Houston

de Gruyter Expositions in Mathematics

1

The Analytical and Topological Theory of Semigroups, K. H. Hofmann, J. D. Lawson, J. S. Pym fEdsJ

2

Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues

3

The Stefan Problem, A. M. Meirmanov

4

Finite Soluble Groups, K. Doerk, T. O. Hawkes

5

The Riemann Zeta-Function, A.A. Karatsuba, S. M. Voronin

6

Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin

7

Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, Μ. V. Zaicev

8

Nilpotent Groups and their Automorphisms, Ε. I. Khukhro

9

Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug

10

The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini

11

Global Affine Differential Geometry of Hypersurfaces, A .-M. Li, U. Simon, G. Zhao

12

Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub

13

Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Nazarov, B. A. Plamenevsky

14

Subgroup Lattices of Groups,

R.Schmidt

15

Orthogonal P. H. Tiep

and

16

The Adjunction Theory of Complex Projective Varieties, M. Beltrametti, A. Sommese

17

The Restricted 3-Body Problem: Plane Periodic Orbits,

Decompositions

Integral

Lattices,

A. I. Kostrikin,

A.D.Bruno

Unitary Representation Theory of Exponential Lie Groups by Horst Leptin Jean Ludwig

W DE G Walter de Gruyter · Berlin · New York 1994

Authors Horst Leptin Fakultät für Mathematik Universität Bielefeld Universitätsstraße 25 D-33615 Bielefeld Germany

1991 Mathematics

Subject Classification:

Jean Ludwig Departement de Mathematiques et d'Informatique U . F. R. de Mathematiques, Informatique et Mecanique lie du Saulay F-57045 Metz Cedex Ol France 22-02; 22Exx

Keywords: Lie groups, solvable groups, exponential groups, unitary representations, orbit method, harmonic analysis

© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Leptin, Horst, 1927Unitary representation theory of exponential Lie groups / Horst Leptin, Jean Ludwig. p. cm. — (De Gruyter expositions in mathematics ; 18) Includes bibliographical references. ISBN 3-11-013938-3 1. Lie groups. 2. Representations of groups. I. Ludwig, Jean, 1947. II. Title. III. Series. QA387.L46 1994 94-27983 512'.55—dc20 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication Data Leptin, Horst: Unitary representation theory of exponential Lie groups / by Horst Leptin ; Jean Ludwig. — Berlin ; New York : de Gruyter, 1994 (De Gruyter expositions in mathematics ; 18) ISBN 3-11-013938-3 NE: Ludwig, Jean:; GT

© Copyright 1994 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Disk Conversion: Lewis & Leins, Berlin. Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

Table of Contents

Preface

VII Chapter 1

Solvable Lie Groups, Representations § § § § § §

1 2 3 4 5 6

Bases in solvable Lie algebras, exponential groups Invariant measures, group algebras Induced representations The dual of an exponential group Kernels of restricted and induced representations Smooth functions and kernel operators

1 1 9 15 33 52 65

Chapter 2

Variable Structures § 1 Variable structures § 2 Unitary representations of variable groups and algebras § 3 Variable Lie groups and Lie algebras

99 99 103 114

Chapter 3

The Duals of Exponential Variable Lie Groups

133

§ 1 The continuity of the Kirillov map § 2 The continuity of the inverse Kirillov map, part 1 § 3 The continuity of the inverse Kirillov map, part II

133 140 178

Definitions and Symbols References

197 199

Preface

The reason for and the purpose of this book is to present a full proof of Theorem 1 in chapter III: The Kirillov map Κ from the space g*/G of coadjoint G-orbits onto the unitary dual G of the connected, simply connected exponential Lie group G with Lie algebra g is a homeomorphism. This theorem solves one of the basic problems of the unitary representation theory of locally compact groups for the class of exponential Lie groups, namely the concrete and explicit determination of the unitary dual G of the group G as a topological space. It is, in a sense, the widest possible extension of the KirillovI. Brown theorem for the nilpotent case. Let us briefly sketch its content: Let G be a connected, simply connected solvable Lie group, g its Lie algebra and Ε the exponential mapping from g into G. We say that G is exponential, if £ is a diffeomorphism from g onto G. Let I be a real linear form on g, i.e. an element of the linear dual g* of g. A subalgebra f) of g is called subordinate to I, if /(f)') = 0, f)' = [f), f)] the commutator algebra of f). In this case the formula χ(Ε(χ))

= eil(x),

xsl),

defines a unitary abelian character of the subgroup Η = E{fj) corresponding to f). We set ττ(1, f)) =

indg*

for the unitary representation of G, induced from the character χ of Η. After some basic work of Dixmier, A. A. Kirillov published in 1962 his fundamental paper [17], in which he proved: If G, resp. g is nilpotent, then 7r(/, f)) is irreducible if and only if dim f) is maximal among the dimensions of subalgebras of g subordinate to /. If 77-(/, f)) € G, then 7r(/, 1)) = π(1) is independent of f). G acts on g* via the coadjoint representation. The elements /, I' e g* are in the same G-orbit if and only if π(1) and π(1') are equivalent. Every π e G is of the form π (I) for some / e g * . These results can be expressed in the following more concise form: For / e g * let Ω/ c g* be the orbit of I and set K(il/) = 7T(0 e G

VIII

Preface

Then Κ : Πι K(ili) is a bijection from the orbit space g*/G onto the unitary dual G of G. The mapping Κ (or sometimes its inverse) is called the Kirillov map. Kirillov's result parametrizes set-theoretically in a most satisfactory way the dual G. However, Q*/G and G both bear natural topologies and already Kirillov himself had proven that Κ is continuous. His conjecture, namely that AT is a homeomorphism, was proved only in 1973 by I. D. Brown [6]. It is easy to see that, in the nilpotent case, it suffices to prove the conjecture only for g-step nilpotent algebras g with k free generators (see [6]). The core of Brown's proof consists in the fact, that these algebras, resp. groups have an abundance of symmetries, more precisely: Every linear automorphism of g/g' extends to an algebra automorphism of g. Already in 1957 Takenouchi had shown that also for an exponential group G every Π € G is monomial, i.e. of the form Π(1, f)) for some / e g* and certain /-subordinate algebras f) C g [34]. Later, in 1965, Bernat was able to extend Kirillov's theorem to general exponential groups: There exists a canonical bijection Κ : G*/G G, see [1], While for nilpotent groups the representation π (I, {)) is irreducible if and only if dim f) is maximal among the /-subordinate subalgebras f), already the "ax + b"~ algebra Κα φ Εb with [a, b] = b shows, that this is no longer true for exponential groups. The answer to this irreducibility question was given in 1967 by L. Pukanszky: 7r(/, fj) is irreducible if and only if the linear submanifold / + f) 1 of g* is contained in the orbit fit [28]. One year later Pukanszky proved the continuity of the Kirillov map Κ for general exponential groups [29]. Left open for a long time was the question whether Κ was also open, i.e. a homeomorphism from Q*/G onto G. The first substantial result in this direction came 1984 from H. Fujiwara [10]. He proved that g*/G contains a dense open set, which Κ maps homeomorphically onto a dense open set of G. This book, finally, contains in its chapters II and III the complete proof of the Kirillov conjecture: For all exponential groups, Κ is a homeomorphism from Q*/G onto G. It is worth mentioning that the analogous problem for the primitive ideal-space of the enveloping algebras of exponential Lie algebras recently has been solved by O. Mathieu [22]. Contrary to the nilpotent case there are no "free models" for general exponential groups and in general the inner structure of these groups is on the one hand various, on the other rather rigid, which implies that there is no way to extend Brown's method directly to the exponential case. This led Jean Ludwig to the idea to force the necessary amount of flexibility of the objects by extending the category of groups and algebras to the category of variable objects. This idea and the basic steps of the actual proof in chapter III are due to him. Ludwig reported on it first at the conference on "Harmonische Analyse und Darstellungstheorie topologischer Gruppen" in Oberwolfach, summer 1987. The textbook literature on solvable Lie groups and their representations is very limited, still the 1972-volume [2] by Bernat et al. is the main source in this field

Preface

IX

and apparently no monograph exists, which exposes the theory beyond the type Igroups, in particular the fundamental work of Pukanszky [30], [31]. In this context we point the interested reader to the excellent survey article of C. C. Moore in the Proceedings of the 1972 Williamstown conference [23]. In any case, this situation caused us to include a relatively long first chapter in this book, the content of which will be sketched below. Chapter I starts in § 1 with a general discussion of exponential Lie groups. § 2 contains fundamental facts on homogeneous spaces, quasiinvariant measures and group algebras. In § 3 we define and study induced representations. The central result is Mackey's imprimitivity theorem. We give a complete proof of the theorem in its general form for arbitrary locally compact groups, with a minor, unessential restriction: For the sake of transparency we suppose that on the homogeneous space G/H there exists a relatively invariant measure. Also included in § 3 are results on intertwining operators and an irreducibility criterion for induced representations. In § 4 we come back to exponential Lie groups. After studying polarizations we prove Bernat's results, i.e. the bijectivity of the Kirillov map. § 5 contains two theorems with the precise description of the kernels of the restriction TT\H of an irreducible representation of G onto a closed subgroup H, resp. of the induction ind^7r of TT 6 Η. This problem, resp. the problem of decomposing TT\H and ind^7r of course has been studied extensively in the literature; from the many papers by Corwin, Greenleaf, Grelaud, Lipsman and others we cite only Fujiwara [10], who proves a precise formula for the decomposition of TT\H into irreducible representations of H. § 6 is more or less independent of the rest of this book. Based on Ludwig's paper [21] it treats the following problem: Let π = νηά^χ e G be given. The quotient G/P is diffeomorphic with some R"1 and the representation space of π can naturally be identified with L 2 (R m ). Then for any / e L'(G) the operator 7r(/) is a kernel operator, i.e. there exists a function i / on R m χ Κ™ such that (7τ(/)ξ)(χ) — JRm Κf(x, y)$(y)dy. Problem: Which kernels occur in this fashion? For nilpotent G it is known that for every Schwartz function F e ^(R m χ Μ"1) there exists / e L'(G) with F — KF, see [14]. For exponential groups such a result cannot be expected, for the following reason: The parametrization of G/H via R m depends substantially upon the choice of "coexponential bases" in g for the polarization p. In the nilpotent case a change of the basis induces a bipolynomial diffeomorphism of the parameter space R m , which is compatible with polynomial decay and Fourier transform. In the general exponential case however exponential functions enter the picture and these are incompatible in particular with Fourier transform. Whether our theorem 12 is optimal we do not know, in any case it is the best we could prove for the moment. Certainly it guarantees the existence of finite rank operators in the image TT(L}(G)) for every π e G. For more applications of this theorem see [21]. It is in order to mention here also the remarkable result of D. Poguntke [27]: Let G be a connected locally compact group and Π E G.

χ

Preface

If 7 t ( C * ( G ) ) contains the compact operators, then 7 7 - ( L ' ( G ) ) contains rank-one projections. This is false for non connected groups [26]. Chapter II contains the foundation of the theory of variable structures, in particular of variable groups and algebras and their representation theory. Obviously this theory has intimate connections with other existing theories, e.g. with the modern theory of groupoids, with bundles and above all with contractions. Here we do not discuss these connections and don't even cite the papers of Renault, Fell, Saletan etc. In § 3 we show that there exists a functor 1 from the category of variable Lie groups into (but not onto!) the category of variable Lie algebras. For every variable Lie group G with algebra 1(G) there exists the exponential mapping Ε : 1(G) -> G which also in the variable case is a local homeomorphism. For exponential G the map £ is a global homeomorphism. In chapter III finally we prove the Kirillov conjecture for general variable exponential groups. § 1 contains the continuity part of the proof. Already here the use of variable structures simplifies matters substantially. In order to show that also the inverse A" 1 of the Kirillov map is continuous we of course apply induction on dim G = dim g, where G, resp. 0 is the underlying manifold, resp. vector space of the variable structure. With relatively simple arguments the problem is reduced to the case where all algebras p g, constituting the variable Lie algebra g — Ρ Χ Q — (J Ρ Χ pQ, have the same one dimensional center 3. Then, as usually peP

for real solvable Lie groups, one has to distinguish two cases: Case 1: Infinitely many quotients pQ/% contain a one dimensional minimal ideal; or Case 2: Infinitely often there exist minimal two dimensional ideals in p Q/$. Although the basic ideas of the proof appear already in Case 1, treated in § 2, considerable not only technical complications occur in Case 2, which is treated in § 3. The process of writing this book was for more than one reason complicated and non linear. Certainly this will have left traces, although we have tried to do our best in detecting, correcting and removing misprints, inconsistent notations etc; in this respect we just hope for the lenience of the reader. The same wish we have with respect to colleagues who feel that their work is insufficiently or even not at all regarded. As one can see we have quoted only such facts and results which are indispensable for our purposes, in particular for our proofs. Moreover, to keep the book selfcontained, in particular in chapter I, we gave our own proofs whenever possible. This means under no circumstances that e.g. we claim priority. We hope that this attitude is justified by the fact that this book is not a monograph on an established, well defined mathematical field, but the representation of new methods and results in such a field. Bielefeld and Metz, December 1993

Horst Leptin Jean Ludwig

Chapter 1

Solvable Lie Groups, Representations

§ 1 Bases in solvable Lie algebras, exponential groups We assume the reader to be familiar with the basic facts of the theory of Lie groups and their algebras: If G is a real η dimensional Lie group then its Lie algebra g is the η dimensional vector space of the left translation invariant vector fields on G. For every JC € g there exists a unique one parameter subgroup in G, which we Rx tx denote by e or [e \t e R}. Thus t etx e G is the smooth homomorphism from Μ into G, for which

(*/)(£) = if(ge'x)\r=o • The exponential mapping E.Q-+G is then defined by E(x) = ex. We will also use the notation exp. It is known that Ε is an analytic mapping. If {£>,}" is a linear basis for 9, then

defines a local coordinate system in a neighborhood of the unit element e of G,

the so called canonical coordinates of thefirstkind. More important for many purposes are the canonical coordinates of the second kind. These are defined by the mapping

(ξι, L·, •··, ξn)-> Ε(ξφι)Ε(ξ2^)

• Ε(ξΜ

.

= {b\, bi, · · • , b„] of g is a Malcev

They are particularly usefull, when the basis

basis or a Jordan-Holder basis: Let η Qk = gkm := Y^Rbi i=k

be the subspaces generated linearly by bk, b k +\, • ·· ,b,

2

1. Solvable Lie Groups, Representations

Definition. The basis of g is a Malcev basis or briefly an Μ basis, if all g* are subalgebras of g and g*+i is an ideal in g* for k = 1, 2 , . . . , n, with gn+l = 0 . is a Jordan-Holder basis or JH basis, if 20 is a Malcev basis and the sequence is a refinement of a composition series. So an Μ basis 20 is a JH basis, if for k = 1,2,... ,k either g* is an ideal in g or g*+i is an ideal and g*_i/g* + i is an irreducible g module. Evidently Μ bases, resp. JH bases exist if and only if g is solvable. It is clear that for every normal series in g, i.e. every series g = f)i D fo D · • • D l)m D 0 of ideals Iy in g, there exists a JH basis 20 such that fyj = (20) for suitable kj, j = 1,2,..., m. To see this we may assume that {i)7} is a composition series. This implies that all quotients t)j/l)j+ \ are g irreducible, hence dim (f) 7 /i) 7+ i) < 2. If dim (f) 7 -/f)j + |) = 2, then [ f j j , Ϊ)y] c i)y+i and any subspace ο with dim α = dim ί);+1 + 1 and f)j D α D f)7+i is an ideal in l)j. Adding such subspaces to the chain of the t)j we obtain a refinement g = gi D g2 D · · · 3 g« of the series {f)y}. Then any set 20 = {bk\ C g with g^ = Rbk(Bgk+1 is an JH-basis with f)y = g^(2ö) for suitable kj. Now let 2ft = {bj} be a fixed M-basis of the solvable Lie algebra g. Then the following proposition holds: The mapping Ε®:ξ

= {ξ i,6,···,

ξη)

Ε*{ξ) = Ε{ξφχ)Ε{ξ2^)...

Ε(ξΜ

is bijective and bianalytic from R* onto the connected, simply connected solvable Lie group G, defined by g. The well known proof is an immediate consequence of the following, equally well known fact: Let Ϊ) be an ideal of codimension 1 in g and Η the corresponding analytic closed normal subgroup of G. If g = R a φ f) and R = E(Ra) is the one parameter subgroup defining a, then G = R κ Η is the semidirect product of R and Η and {ξ, χ) —>· Ε {ξ a) χ is a bianalytic map from R χ Η onto G. If a Malcev basis 20 is fixed, we will frequently identify i e l " with its image £"&(£) ε G, so G, as a manifold, "is" R" and the one dimensional subspace j j c e l " , jc,= 0 for ι ^ j} is the one parameter group eRb'. In this sense we can write

(1) where the yj are real analytic functions in ξ. Assume that f) = g* for some k is an ideal, hence Η = {x e R" ; jt,· = 0, i < k} is the corresponding normal subgroup in G. Then the functorial properties of the exponential mapping imply, that the coset E{x)H for χ G g depends only on the coset χ + ί) in g. Consequently the functions yj(£) in (1) depend for j < k only on the i < k. This implies

§ 1 Bases in solvable Lie algebras, exponential groups

3

(2)Lemma. Let be a JH basis and j\ — 1, • • ·, ji the sequence of indices, for which the Qjt are ideals, thus jk+\ < jk + 2. Then Ε is a local diffeomorphism in

= Σ

resp. D

G 0 if and only if all determinants Dk = ^ k

=

(

-

(jc°) (jk+i = jk + 1).

are nonzero.

This is clear, because the product Π Dk is the determinant of the Jacobian f k= ι ^ ' of Ε in JC°. We will now study the conditions, under which the exponential mapping is a global diffeomorphism, i.e. G is an exponential group in the sense of the Definition. An exponential group is a connected, simply connected solvable Lie group G, for which the exponential map £ is a diffeomorphism from the Lie algebra g of G onto G. We start with the indecomposable groups of dimension 2 and 3. η — 2 : There exists exactly one algebra s 2 = R e θ Rb with [a, b] = b with the group S 2 = R κ R with (*i, * 2 ) 0 ί , yi) = (*i + y\,e~ytx2

+ >2) ,

thus a = δι — jc292 , b = d2 . Here (and in the sequel) we use the notation dj — So clearly eRa = Rx

{0}, eRb = [ 0 | x l and Ε{ξχα)Ε{ξ^)

= {ξχ,ξ2)

.

Let k

00

-V Λ

V"

Z

1

7

e

~

1

Then one verifies easily that in S2 the coordinates of the first kind are given by which shows that S2 is exponential. η = 3 : There are two different types: The three dimensional Heisenberg algebra F>, = R f l ® R f c © R c

with [a, b] = c

4

1. Solvable Lie Groups, Representations

and group Hi = R 3 with ( x i , x 2 , x 3 ) ( y i , y 2 , y 3 ) = thus a = 3i , b =

+ y i , x 2 + y2,x3 + y3 + xiy2) .

+ X\d3 , c = 83 , Ε(ξχα

+

+ Μ

= (ξι,ξ2,

& +

,

and Hi is exponential. Now let Λ = Λι + ϊλ 2 6 C x . We set s(A) = RZ>i

®Rb2®Rb3

with [^1, ^2] =

+ A2^3

[b 1, b3] = —A2&2 + λιί>3 .

and group 5(A) = R x C with (*, w)(y, v) = (x + y, e~Xyu + υ) which means that 5(A) = 1 κ C is a semidirect product with minimal normal subgroup C, if A2 φ 0. With x2 + 1x3) = (xi, x2, x3) e 5(A) the bj are given by b\ = 3i -

(A1X2 - A2jc3)32

-

(A2JC2 + AiX3)93

, b2 = d2 , b3 = 9 3

and the exponential mapping in the complex coordinates ξ2 + ϊξ3 by (3)

Ε ( ξ φ ι + i 2 b 2 + 6 * 3 ) = ( ξ ι , Ζ ( - λ ξ Ο ( ξ 2 + ι'6)) ·

We see that 5(A) is exponential if and only if Αι φ 0, hence S(i) is (up isomorphisms) the only nonexponential simply connected solvable Lie group dimensions = 3. In this case the complement of the image of Ε is the set of elements (2irk, 0) e 5(0 with k e Ζ \ {0} and E(2Μλ + χ) = E(2Μλ) (2TTk, 0) f o r all k e Ζ \ {0}, x e R

to in all =

b2®Rb3.

It turns out that the absence of S(i), resp. s(i) is responsible for a simply connected solvable group G to be exponential:

Theorem 1. Let

G be a connected,

sion η and let Q be its Lie algebra.

simply

The following

(i)

No factor

group of G has closed

(ii)

No ad χ, χ e g, has non zero purely

(iii) Ε is

injective.

(iv)

surjective.

Ε is

connected

subgroups

solvable

conditions isomorphic

imaginary

Lie group of are

equivalent:

with

S(i).

eigenvalues.

dimen-

§ 1 Bases in solvable Lie algebras, exponential groups

5

(ν) Ε is a diffeomorphism from 9 onto G. Proof. If (i) does not hold then g contains an ideal f) and elements x, y, ζ £ i) with (4)

[x, y] - — z(mod I)), [x, z] = ;y(mod fj) .

This means ad χ (y + iz) = i(y + /z)(mod f)) , hence i is an eigenvalue of ad x. On the other hand, if i is an eigenvalue of ad x, then there is _y + iz in the complexification g c of 9 with (ad JE — i)(y + iz) = 0, i.e. [x, Y] = —z, z] = y. Let α be a maximal ideal with ζ £ α and b c 9 an ideal with α C b and such that b/α is minimal in 9/0. Then necessarily ζ e b, hence y = [x, z] C b and ( Μ ι φ b)/o = s(i), which implies J/A = S(i) for the subgroups J and A of G corresponding to b and a. Thus we proved (i)(ii). For the rest of the proof we use induction. The theorem is true for dimensions 1,2 and 3, so let us assume that it is true for all connected, simply connected solvable groups of dimension less than n. Let m be a nonzero minimal ideal in 9 and Μ its corresponding connected normal subgroup in G. We set 9 = g/m, resp. G — G/M for the quotients, similarly ä = a + m e g, χ = χ Μ e G for a € g, χ e G. Now assume that (i) and (ii) hold. Let a,b € G with E{a) = E(b), hence E{a) = E(ä) = E(b), which implies ä = b, because dim G < η — 1 and clearly (ii) holds also for g and G. But ä = b implies that R ö + m = R H m = i) is a subalgebra and dim f) < 3. Certainly (i) is true for f), hence Ε is injective and so E{a) = E{b) implies a = b, which means that (iii) is true for g and G. Now let χ G G. As G satisfies (i) and (ii) there exist ä = a + m € g with χ — xM = E(a), which means χ — E(a)y with y 6 M. Again f) = ]RiZ + m i s a subalgebra of dimension < 3, hence £ is a bijection from I) onto the corresponding subgroup H. As χ € Η there exists be f) with E(b) = x. This shows that Ε is surjective, i.e. (iv) holds for g and G. To prove (iv) =>· (ii) let a be an element in g and A = ad a. With the same notation as above we see that E(Tt) = E(u) for u e g implies that Ε is surjective from 9 onto G, hence (i) through (v) hold for g and G, in particular A has no purely imaginary eigenvalues on g/m. Again consider the subalgebra ϊ) := Μα + m with corresponding subgroup H. For χ € Η there exists « e g with χ = E(u), hence χ — E(u) in G. But χ — E(tä) for some / e M , and because Ε is one to one on g it follows that ü = tä, i.e. u e f). This proves that Ε is surjective from i) to Η and consequently f) is not isomorphic to s(i), in particular A has no purely imaginary eigenvalues on m. Next we show (iii)=>-(i): We assume that Ε is injective, but that there exist an ideal t in g and a subalgebra f) with \)jt = 5(1), which means that there are elements a, b, c in g, independent modulo 6, with [a, b] + c, [a, c] — b e t. We know that we can choose a and b such that for the cosets a = a + t, we have

6

1. Solvable Lie Groups, Representations

Ε (a) = E(a + b) in the quotient H/K c G/K. This means that there exists q € Κ with E{a)q = Ε (a + b) in G. But j := R a φ 6 is a proper subalgebra of 0 and Ε is injective on j, hence Ε is bijective and consequently there exists d e j with E(d) = E{a)q — E(a + b). Since Ε is injective this implies d = a + b, which is a contradiction, because a + b j. We have proved the equivalence of conditions (i)-(iv). Evidently (v) implies (iii) and (iv), so finally we have to show that a bijective Ε is a diffeomorphism, i.e. that the Jacobian of Ε is regular in every point χ € q. For this we will use lemma (2). Let m be a minimal non zero ideal in g. If m is central we may assume that m = R b n . Then, with the notation of (1) and ξ = , £ > , · · · , ξη~ι, 0), we have yn(&

= yn(,£)

+ ξη ,

hence = 1. We take m as last ideal of a JH-series and 20 as the corresponding JH-basis. Then we can apply induction and lemma (2): From dim g/m = η — 1 we see that Dk φ 0 for k < I, while D/ = J g = 1. If m is not central, then the centralizer c is an ideal in g. If dim g/c = 2, then also dim m = 2 and we could find some χ e 9 such that ad x| m would have eigenvalues ±1, which is excluded by (ii). Thus g = R a ® c . Let be a composition series passing through c and m, i.e. with f)2 — c, f)/ = m, and let S8 be a corresponding JH-basis. Then f)2 =

η ^2Rbt, 1=2

m =

Rbn

or m = Ri>„_i Θ

and ad

bu

restricted to

m, has an eigenvalue Λ = Λ] + ιλ 2 with λι φ 0. We will only treat the case dim m = 2. Because m is central in f)2 we see that for any χ e i)2 the restrictions of ad and ad {£\b\ + x) onto m coincide, hence for ξι φ 0 the subalgebra + φ m is isomorphic with s(A) and formula (3), after applying the isomorphism ξη-ι

·

1+

ξn • i

ξη-Φη-ι

+

ξηΚ

from C onto m, gives for χ =

n-2 Σ ξ

Λ the

1=2

relation

E

(

έ

^

)

=

E

+

where

βχ{ξχ),

Ε

β2(ξχ)

iih-J

~

2(t0tn)b„-l

+

{β2{ξ\)ξη-\+βλ{ξΧ)ξη)Κ),

are defined by ? ( - λ £ ι ) = ^(£1) +

=

e

O l (& , y 2 ( i ) , · · · , yn-2(i),

ϊβ2{ξ\)

u>, {ξ'),

. From (1) we get

wz{g))

§ 1 Bases in solvable Lie algebras, exponential groups with ξ

=

Ε ^ Σ Μ ^

{ξ\, ξ2, ·•· , i n - i )

e

7

Κ" 2 . Hence the last two coordinates of

are given by >>„_,(£) = u ; , ( f ) +

This yields for the Jacobian of

" e2(ii)tn

ξ„)

(dy\ V3

_(βλ{ξλ),

(y n -i, yn)'· -e2{£x)\ J

and D, = εχ{ξχ)2 + β2(ξι)2

= |?(-λ£,)|2

>0

for all ξ\ € R, because λ] φ 0. This ends the proof of Theorem 1. It is clear from theorem 1 that closed connected subgroups and simply connected quotients of exponential groups are also exponential; moreover we have Proposition 1. Let G be as in theorem 1. If Η is a closed connected subgroup of G with corresponding subalgebra I), then Η is exponential if and only if Η — E(\)). This follows immediately from theorem 1 and the functorial properties of the exponential mapping. Let Η be a closed subgroup of the Lie group G. The coset space G/H of all left cosets jc = xH, χ e G, is also a manifold. If G is solvable, connected and simply connected of dimension η and Η is connected of dimension /, then G/H is diffeomorphic with We want to describe this diffeomorphism explicitly by means of the exponential map and suitable bases in the Lie algebra g of G. Let f) be the Lie algebra of the closed subgroup H. Then g/f) is an f) module, hence there exists f) composition series of g/f), i.e. f) invariant subspaces g = gi D . . . D g r D g r +i = f), such that the quotients gj/gj+i are f) irreducible, so dim (gj/gj+ι) is 1 or 2. Slightly incorrectly we will call such a series {gy};=i>...,r+i an f) composition series of g/f). Definition. A coexponential basis for f) in g is a set Söf, = {b\, b2,..., elements bj e g, independent modulo f), such that (i) {bj Θ f)}j=i

d> with bj — Σ i=j

of g/f) and

ba) of

is a refinement of an f) composition series

1. Solvable Lie Groups, Representations

8 (ii) the mapping

d :

(Σ^Α^)

Ε(ξφχ)Ε{ξφ2)···Ε{ξ^ά^

7=1

maps bi χ Η diffeomorphically onto G. d It follows that for a coexponential basis Söf, the mapping χ = Σ ξ j b j —» ι e)H € G/H is a diffeomorphism of bi onto G/H.

Proposition 2. If G is connected, simply connected solvable, then for every subalgebra f) of 0 there exist coexponential bases.

Proof. Let t be a subalgebra with f) c t c g. If {b\,..., bd) is a coexponential basis for Ε in g and {c\,... ,Ck} a coexponential basis for \) in t, then obviously {b\,... ,bd , ci,... ,Ch) is coexponential for Ϊ) in g. So we only need to consider the case that {j is a maximal proper subalgebra of g. If g' = [g, g] c i), then i) is an ideal and any b φ ί) can be taken as a coexponential basis. Otherwise g = g' + I). As dim g' < dim g, induction allows us to assume that g' contains a coexponential basis {b}, resp. {b\, b2} for g'n(). We claim, that this basis is also coexponential for ij in g. We consider only the case [bx, b2}. With Β = E(Rbi)E(Rb2) C G' we have G = B(G' η Η), hence G = G'H = BH. From jc>> = x'y', x, x! e B, y, y' G H, we get x'~lx = y'y~l e G'DH, hence χ = χ', y = because {b\, b2} is coexponential for g'nf) in g'. Thus E^ is bijective from (Rb] ®Rb2) χ Η =: bxH onto G. Clearly £•(, is a local diffeomorphism from a neighborhood of {0} χ {e} onto a neighborhood η of the unit element e in G. Let {b^,..., bn) be a basis of f), so that χ = Σ $jbj —> ι F{x) = Ε{ξφ\)Ε(ξφ2)... E(£nbn) is a diffeomorphism F of a neighborhood of O e M " onto a neighborhood of e. If q : χ —»· q{x) e g is a polynomial mapping of g into g without constant and linear terms, then also χ —> E(q(x))F(x) is a local diffeomorphism. Now fix an element (χ 0 , ho) = {£\b\ +£%b 2 , ho), in b x H. Then, since the ad y for y e g' are uniformly nilpotent, it is easy to see that there exists a g' valued polynomial 0 for all χ e G, hence A g (jc) = det Ad X - 1 . If G is solvable and simply connected, then the Haar measure is explicitly given by dx = άξ]άξ2 • · · άξη, where the ξ j are the canonical coordinates of the second kind defined by a given Μ basis 95 of the Lie algebra g of G. This follows immediately by induction on dimg and the following trivial fact: If the locally compact group Γ is the semidirect product of the closed subgroups Λ and B, i.e. Γ = Α χ Β with normal B, then the (left) Haar measure dy of Γ is the product measure of the Haar measures dx of A and dy of B, i.e. d y — dxdy. If Η is a connected closed subgroup of G and SSf, — {b[, b2, . . . , b^} a coexpod

©

—

nential basis for f), then b = ^ parametrizes the coset space G = G/H. It is _ ι known that G carries an essentially unique relatively G-invariant positive Radon measure dx, but this measure is in general not equivalent to the Lebesgue measure on b. However, for particular choices of S^t, one can describe dx explicitly and simply by use of the Lebesgue measure on b. Again let G, Η, g and I) be as above. As we had mentioned, the Haar moduli Δc and Δ« of G and Η are given by Δ 0 ( λ ) = det Adgjc -1 , Δ//(>) = det Ad^y" 1 . We set

=

d e t A
A p ( / ) is a bounded representation of L'(G) in the Banach algebra X(LP(G)) of bounded operators of LP(G). _ More generally let us now take a quotient space G = G/H of G and a χ relatively invariant measure dx on G. For g e LP(G), 1 < ρ < oo, and a e G we set (\p(a)g)(x)

= x(a)rg(a-lx)

.

Then obviously A p (a) : g — A p ( a ) g defines a linear isometry on LP(G), i.e. Ihp(a)g\p = \g\p for all a and g, and λρ : a —> λρ(α) is a continuous isometric representation of G in LP(G), i.e. a homomorphism of G into the isometries of LP(G) for which all mappings a —> λ p ( a ) g , g e LP(G), are continuous from G

14 into

1. Solvable Lie Groups, Representations LP(G).

So for any / g

LX(G)

the vector valued integrals

A p (/)(g) := [ f(x)\p(x)gdx JG

e L"(G)

exist and |A p (/)(g)| p < |/|i|g| Pl _i.e. Ap : / ->• A p (/) is a representation of in the bounded operators !£(LP(G)). As usually we write

L\G)

(6)

Of particular interest is the case ρ = 2. Then the L 2 (G) are Hilbert spaces and the representation Λ2 of G is unitary, i.e. we have A 2 (* _ l ) = A2O)*, the adjoint operator of λ2(χ), for all χ e G. The representation A2 of l ) (G) is a *-representation, which means that A 2 (/*) = A 2 (/)* holds for all / g Ll(G). This is an immediate consequence of (6).

The left regular representation A of L1 (G), i.e. the representation A2 of L}(G) in L2(G), is faithful, so if we define l/l* = |A(/)| for / e L1 (G), where |Γ| is the usual norm of the bounded operator Τ of L 2 (G), then / ->• | / U is a norm on Ll(G), satisfying the identity | / * /*|* = | / | 2 . The C* algebra of G, denoted by C*(G), is the completion of Ü (G) with respect to the norm |/|*. Thus C*(G) is canonically isomorphic to the norm closed hull of the image λ 2 ( ΐ ' (G)) in the C* algebra X(L2(G)) of all bounded operators of L2(G). We remind the reader of the general definition of the group C* algebra: C*(G) is the completion of L1 (G) with respect to the maximal C* norm and is characterized by the fact that L1 (G) is dense in C*(G) and that every continuous ^-representation of L'(G) in a Hilbert space extends uniquely to such a representation of C*(G) in the same space. C*(G) coincides with the closure of A2(L' (G)) in if (L 2 (G)) if and only if G is amenable, so in particular, as in the present case, when G is solvable. Later we will use the adjoint algebra sAb of an involutive Banach algebra -ύ (see [18] or [15]). This is the subset s&b of all bounded operators Τ of the Banach space si, for which there exists another bounded operator T* of si, such that

cTa)*b = a*(T*b) holds for all a, b e si. Obviously sib is a subalgebra of the algebra l£(si) of bounded operators of si. For χ e si let l x be the left multiplication operator lx(a) = χ a of si. Then clearly lx e sib with (lx)* — Ix*. For our purposes it is sufficient to assume that \lx\ = |jc| for all χ € si (where |T| denotes the operator norm of Τ e !£(si). This is condition (L*), ρ 266, in [18]). Under this assumption the following facts are easy to see:

§ 3 Induced representations -

15

Τ* is uniquely defined for Τ e si" and Τ —> Τ* is an isometric involution of äT. sib is closed in i£(si), hence sib is an involutive Banach algebra. The mapping χ —• lx is an isometric *-isomorphism of si into sib, hence can be considered as a Banach algebra extension of si. In this sense si is a closed two sided ideal in sdb. If SÖ is another essential Banach algebra extension of si, i.e. si is a two sided ideal in si and xs& = 0, χ e SÖ, implies χ = 0, then there exists a unique homomorphism h : SÖ —>· s&b with h(a) = a for a Ε si. •-representations of si in Hilbert spaces extend in a functorial way to * representations of si b . For details and more information see [18].

§ 3 Induced representations One of the most powerful tools in representation theory is the notion of induced representations. Given a group G and a subgroup H, the induction process attaches to a given representation σ of Η in a functorial way a representation ind σ of G. In this paragraph we assume G to be locally compact, i.e. not necessarily Lie, and Η closed in G. Furthermore, "representation" always means "continuous unitary representation". For the matter of simplicity we also assume that there exists a relatively invariant measure dx on the coset space G = G/H, hence there exists a positive real character χ on G which extends χ^ on Η and with dax — χ(α)~ιdx for a € G. This allows us to define the left regular representation 2 2 A of G in the Hilbert space L (G): For a G G and / 6 L (G) we set ( X ( a ) f )(x) = χ(α)ϊ f (a~lx). Then it is clear, that A : a —> λ(α) is a representation of G in L2(G). Let σ be a representation of Η in a Hilbert space Sj. The space SF of all measurable functions from G into ή is a G space: For a e G, f e It we define A(a)f by (A(a)f)(x) = f(a~lx). Now σ defines a G invariant subspace of 8F: We set =

{/

e 9f;

f(xy)

=

a(y)*f(x)

for all

y

e

Η

and almost all

χ e G}

.

Then 9 v is stable under all Λ(α) and for / , g € 9 v we see that the function χ —» (/(jc)lg(x)) depends only on x, hence we write if(x)\8(x))

and |/(*)| = ( / l / ) 0 c ) 5 =

\f\(x)·

=

(f\8)(x)

The subspace of 3 v

ind ή : = { / e

; J_ \f\(x)2dx

< oo}

1. Solvable Lie Groups, Representations

16

is a Λ invariant Hilbert space with inner product

(.f\8)

:=

[ j f \ g ) ( x ) d x . JG

For a € G, f € ind 5} we find |Λ(α)/| 2 = / | / | ( α - ' J ) 2 ^ = ./G

χ{ά)~λ

|/|

2

hence, the mapping (7)

(indgo-)(fl) : /

x(a)*A(a)f

defines a unitary operator (ind^o-)(a) of the Hilbert space ind fj. Now obviously the following definition makes sense. Definition. The mapping ind^o-: χ -» (ind^ir)(j:) , where ( ind^ Ωξ maps isometrically into ind Sj. Let / be a continuous function in ind f j . Then ξ := /\r, is continuous on R and |f(r)| - \ f ( r , e ) \ = |/|((r, e)~) implies \ξ\ = | / | , consequently ξ € X . Clearly / = from which it follows easily that Ω is onto. Let y -> / be the automoφhism of H, defined by t e R, so that the product in G is given by (r, z) = (rt, y'z). For ξ € ££ and a = (t,u) e G we get ( i n d % σ ( α ) ) Ω ξ ( κ , y) = Ωξ(Γικ,

(«'"'TV)

=

§ 3 Induced representations

17

Hence, if we set (a(t,u)t)(r):=a(urir)£(rlr)

(8) we see that

σ(α) = n*ind£o-(a)il for all a € G. This means that σ : a —• σ{α) is a representation of G in L2(R, 5}), equivalent to ind^o - : Proposition 3. The induced representation ind^er of G is equivalent to the representation σ of G in L2(R, SJ), defined by (8). We return to the general situation of an arbitrary locally compact group G, a closed subgroup Η and a relatively invariant measure dx von G — G/H. We leave the proof of the following useful fact as an easy exercise to the reader: (9)

For u e ^ ο ( ^ ) and ξ € f) define the function u#£ by u#£(x) = / u(xy)σ(y)ξdy JΗ

.

The mapping u ξ —> u#£ defines a linear injection from the tensor product ^o(G) ® f ) onto a dense subspace of ind fj.

c

One of the basic problems in the representation theory of G is to decide whether a given representation π of G is induced from a proper subgroup. There is an easy necessary condition. Assume that π = ind^er for some representation σ of H. Then we can canonically identify the Banach space L°°(G) with a von Neumann subalgebra of r)), Sj(tt) = ind 5)(σ): The bounded measurable function q e L°°(G) defines an operator Q e i£(iö(7r)) by virtue of the definition (Ö/)W for / e

f)(7r).

= q{x)fix),

x e G ,

The mapping δ :q

8{q) = Qe

is a *-algebra isomorphism of L°°(G) onto a commutative von Neumann subalgebra A of 5 £ ( I 3 0 R ) ) . N O W L°°(G) is a G space: For a E G, q E L°°(G) we define a qa A o o ( a ) ~ l q , i.e. q (x) — q(ax). With this notation we can write =

(10)

8{qa) = ττ(α)*δ(oo(G) in a Hilbert space $), i.e. a * isomorphism of ^ (G) into 5£(Sj), (ii) π is a continuous unitary representation of G in ft, (iii) the relation (10) holds for all q e ^ ( G ) and all a e G . Two systems ( δ ι , π Ο and (82, π2) are unitarily equivalent if there exists a unitary operator U from the space of (δ;, π\) onto the space f j 2 of (δ2, ττ2) such that U8\ = ö2U and UTT\ = ττ2U. The system (δ, ind^er), constructed above from the representation σ of H, will be called the imprimitivity system of (G, H) induced by σ and we say that an imprimitivity system is induced, if it is in this sense induced by some representation σ of H. The most fundamental result in this context is Theorem 4 (the imprimitivity theorem). Every system of imprimitivity (δ, π) for (G, H) is unitarily equivalent to an induced system. The proof follows essentially Blattner's ideas [4], with the simplification by Müller-Römer [24], %oo(G) is in a natural way a G algebra, i.e. there is a canonical homomorphism of G into the automorphisms of the C* algebra ^00 (G), defined by f fa, fa(x) — f(ax) for a e G. The corresponding twisted Lx algebra L'(G, ^00(G)) is called the imprimitivity algebra of (G, H) and it is easy to see that the *-representations of L'(G, ^ooiG)) are in a canonical 1 : 1 correspondence with the imprimitivity systems for (G, Η): The representation A, defined by (δ, π), is given by

It is certainly sufficient to prove the theorem for cyclic representations. So let us assume that £ is a cyclic vector in the Hilbert space f) of (δ, 7τ), resp. of A, i.e. that the Λ ( / ) ε , / e Ll(G, ^ooiG)), span a dense subspace of fy. Clearly the space %j(G χ G) of compactly supported continuous functions on the product space G χ G can be identified with a dense subspace of L1 (G, c€oo(G)). As usually / e ^oiG x G) defines the function χ —» f(x) e ^oo(G) with f(x)Qy) = f(x,y)· The explicit expression for the convolution of two functions

19

§ 3 Induced representations /, j 6

x

G) is then given by / JG

f*g(x,y)=

f(xt,r[y)g(r\y)dt

and the involution by = AG(x-l)f(x-\xy)

f*(x,y)

.

It follows immediately that %o(G χ G) is a dense *-subalgebra of L'(G, ^(»(G)). We will realize S) as a function space on G x G . For this we define a Radon measure d\ on G χ G by _/ A(f)e is unitary from ft := Voq(G χ G)/N, { / ; A ( / ) ε = 0 }, into S). If we define (δ', π') on ft by (S'(u)f)(x,

y) = u(xy)f(x, l

(n'(a)f)(x,y)

= f(a~ x,y)

with X

=

y) for u 6 ^ « , ( 0 ) for α e G ,

we see that (δ', π') and (δ, π) are unitarily equivalent, hence we will assume that Ϋ) is the completion of ft and δ = δ', ττ = π'. Our next step is to move back from G x G to the product space G x G : By Τf{x,

y) =

f JH

fix,yz)dz

we define the usual positive linear mapping Τ from %oiG χ G) onto %oiG χ G). The mapping Τ has a right inverse: Let β be a. Bruhat function for Η (see [32]), i.e. a nonnegative continuous function on G, such that for any compact C c G the restriction ß\ ch has compact support and Τ β = 1. Then clearly ß f , defined by i ß f ) i x , y) = ßiy)fix,

y), is in

χ G) for all / € o(G χ G), for which g* • / ( x , y) = Jc Twt(x, y)dt. It follows from (12) that (13) f _Twt(x,y)d\(x,y)= JGxG

β(ίγ~ι)/(ίγ~ιχί7:ϊ)§(ί^)Α0(γ-[)άμ(χ,γ).

ί JGxG

The integrand on the right hand side is a compactly supported continuous function in (f, JC, y) e G χ G χ G = G 3 , hence it is integrable with respect to the product measure άμ{χ, y)dt on G 3 . We apply Fubini, integrate first over t, substitute t —> ry and again change the order of integration. The result is [ ß(t) ( [ JG

f(tx, ~x~^)g(ty, ~Ρ)άμ{χ,

\JGxG

y)) dt =: [ ß(t){f J

{t)\g\t))

μάί

JG

where we have set f ( t ) ( x ) = f(tx, χ-') hence / # ( 0 e ^o(G) and f#:t—>· ^oiG). Furthermore, (14)

(α\ν)μ

for / g %(G χ G) ,

/ # ( 0 is a continuous mapping from G into

:=

α{χ)ν{γ)άμ{χ, JGxG

defines an inner product on %)(G) .

y)

21

§ 3 Induced representations

Similar arguments show that we can also integrate the left hand side of (13) over t. The result, using Fubini and (11), is the identity (15)

(f\g)A

= Jg

β(ί)(/*(ί)\8*(ί))μώ.

We claim now that (14) is positive semidefinite on ^ ( G ) , hence it defines a Hilbert space £ , and (a(z)u)(x)=x(z)-Liu(z~lx),

(16)

zeH,xeG

defines a unitary representation σ of Η in £ . Furthermore, we will show that (δ, π) is induced from σ. This will finish the proof of theorem 4. By definition of άμ and θ we see that for ζ e Η (ισ(ζ)ιι|σ(ζ)υ) μ = =

[

AG(xyhr]Ac(zrlAH(xM(yhzrl)v((xyhzrl)dhdÄ(x,y).

_ [

JGXG

JH

Substituting hz~{ for h in the inner integral gives (σ(ζ)ιι\σ(ζ)ν)μ = («|ι>)Μ. Now we take / e ^>o(G χ G) and q € %o(G). Let q χ / be the function (jc, y) q(xy)f(x, 30. hence q x f e %(G χ G) and (q χ f)*(x)(y) = q{x)f{x){y) . With q χ / both for / and g in (15) we get (17)

/ ß(x)q{x)(f(x)\f(x))fXdx

> 0

JG

for all q e %o(G). For ζ e Η we have f(xz)(y) hence f#(xz) (18)

= fixzy,

= x(z)^a(z~l)(f*(x))

r 1 7 ) = Axzy,

(.zy)~]) = / # ( x ) ( z y ) ,

and

(f(xz)\f(xz))p =

X(z)-](f\x)\/«(χ))μ.

As (17) implies JH ß{xz){f*(xz)\f*{xz))ßdy > 0 for all jc e G (observe that ( f i x ) \f*(x)) is continuous in x!), (17) and (18) yield (fix)\/*(χ))μ

> 0 for all / e %)(G χ G) and all jc g G .

For u € ^ o i G ) with supp u = K, we can find ν e ^ ( G ) , such that ι>(>>) = 1 for all j - 1 e K. We set f{x, y) = u(x)v(y). Then f ( e ) ( y ) = f ( y , ^ = uiy)v(^) = u(y), i.e. f ( e ) — u. It follows that (u\u) > 0, hence the inner product (14) is positive semidefinite and (16) defines a continuous unitary representation of Η in

1. Solvable Lie Groups, Representations

22

Let ττσ = ind^o". The space Srf of ττσ consists of measurable functions φ : G —> £ with φ(χζ) = σ(ζ)*φ(χ) for ζ € Η and almost all JC eG. The representation of £, defined by

(Ω/)(χ) = * ( * ) * / ( * ) is continuous from G into £ and satisfies the identity (Ω/)(χζ) = σ(ζ)*(Ω/)(JC) for ζ e H. We remark that this is the first time where we used the assumption that χ — extends to a real character of g. It is easy to see how this has to be modified in the general case. Now recall that, given the Bruhat function β for H, a χ relatively invariant measure dx on G is given by f^F(x)dx = JG F(x)ß(x)x(x)~ldx, see [32], ch. 8, § 1. Hence we derive from (15) for f,ge%)(GxG): [ (Ω/(χ)\Ω§(χ))μβ(χ)χ(χΓιάχ JG

fjilf \ίϊ8)μ(χ)άχ= JG =

f ß(x)(f*(x)\g*{x))dx JG

=

=

(/IgV

This means that Ω : / —• Ω / defines a unitary operator from S) into f f . To show that Ω is onto, we apply (9): the elements with u and ψ in ^ ( G ) span a dense subspace in ff. On the other hand u and φ define a function / € ^ ( G χ G) by f(x,y)=

/ u(xyz)χγ,^)=χ(α)12(ΩΩ(α-ιχ)(γ)

thus, after (7): Ωττ{ά) = ( i n d £ o - ) ( ά ) Ω .

^H)

= ,

23

§ 3 Induced representations Similarly we see for q € ^oo(G): (nScr(q)f)(x)(y) which means Ü,Sa(q)

= q(xy)(Üf)(x)(y)

,

= a(h(x))*g(x) is measurable on G and the identity f ( x y ) = < r ( y ) * f ( x ) holds for all χ e G, y e H. Evidently Uf = g and consequently U is a unitary operator from ind onto L2(G,

Let

ft). a

e

G,

χ e G

and

g e L2(G,

We set

η (α, χ) = 5(χ) _ 1 α5(α _ 1 χ) e Η and define

7r(a)g

(19)

e

L2(G,

$))

by

( τ f ( a ) g )( x ) =

1 and qa(a) = 0. As χ σ(α) is continuous, \χσ(α)\ > 1, resp. | Λ σ(ίϊ)| < \ describe neighborhoods

28

1. Solvable Lie Groups, Representations

of p, resp. q, hence we can assume that |V(a)|>l

for all χ € A

|V(a)|< ί

for all χ e ΖΪ .

Let si be the C* subalgebra of C*(H), /s

generated by a. Then si =

( Z ) with

Ζ = si the spectrum of si. For χ e G let Zj be the hull in Ζ of the kernel of the restriction of χσ onto si. If a € ^οο(Ζ) is the Gelfand transform of a, then all Zj, χ e B, are contained in the closed set Υ = {ζ e Ζ; α (ζ)


1}. We choose

h e

( Z ) with 0 < h < 1, h(C) = 1 for ζ e X , h(C) = 0 for ζ e Y. Let b be Λ the element in si with b = h and c = ab. Then obviously xa(c) = χσ(α) ^ 0 for J € A, V ( c ) = 0 for χ e B, hence V(c)

cet:=f]h,

y^OforxeA.

xeB

From (21) it follows that (n(S)g)(x) = Ja(S)g(x) for all S e C*(H),_which implies 7τ{1)χ Β 9) π = 0, i-e. XBir(t) = = 0. On the other hand χσ acts irreducibly on f ) and is nontrivial on t for all χ e A. Hence χσ(ΐ)$) = S) for χ € A and consequently ΧΑ^Φ^π = ^ Φ χ α ^ τ τ = Xa$)tt- If T, hence also T* would commute with π(Η), we would have ΤΤ*π{ΐ) = Τπ{1)Τ* = ΤχΒπ(ϊ)Τ* - 0. But TT* = TT*χα implies ΤΤ*$)π = ΤΤ*χΑ^π = ΤΤ*ττ(ΐ)χΑ?)π = 0, which contradicts Τ φ 0. It follows that Τ ( χ , ξ ) . For χ e Α, and define ζξ e A by ( χ , ζξ)

ζ ε G we write as usually xz

z~lxz

= ( x z , ξ ) . Thus Λ is a locally compact G space with

action (ζ, ξ ) —• ζ ξ . Let ξ denote the closure of the orbit the set of all ξ , ξ e A.

=

in A and 53 = 55(A)

We provide 53 with the finest topology for which the

surjective mapping ξ —• ξ is continuous and claim: The mapping ξ —»• ξ is also open: Let U C A be open. A s ζξ = ξ we may assume that U is G-invariant. Then the closed complement U' = W

is also G invariant, hence ϋ £ U implies •§ c U'

and consequently ξ Φ ϋ for every ξ eJ7. This means that U is the inverse image of its image U in 03 and implies that U is open. Now we will also assume that A is second countable, i.e. A,

hence A has a

countable basis of open sets. Then also 53 is second countable, moreover 53 is a r 0 -space: $ e 53 is in the closed hull of ξ if and only if ϋ e ξ , thus ϋ and ξ have the same hull in 53 if and only if ϋ — ξ . Let π be a unitary representation of G and let π a be its restriction to A. Then π a defines a representation of C*(A) we get a representation ρ of ^ooiA)

and as C * ( A ) is isomorphic to ^^

in ή with p ( f ) =

(A)

7 t a ( / ) for the Gelfand

30

1. Solvable Lie Groups, Representations

transform / of the element / 6 C*(A). G acts on L ' ( A ) c C*(A) by / fz with fz(y) = Δ Λ (ζ) _ 1 /(ζ>>ζ _ 1 ) where Δ Α denotes the modulus of the automorphism y -» of A. It follows immediately that irA{fz) = π(ζ)*πΑ(/)π{ζ) for all X / e L {A) and consequently for all / € C*(A). As the transpose of / fz on ^oo(A) is given by g —> g z with g z (£) = g( z £), we find that

(23)

p(gz) =

π(ζ)*ρ(§)π(ζ)

for all g g ζ eG. Let ωπ be the hull of the kernel of ρ in A; hence ωπ is G invariant and the smallest closed subset of A such that ρ factors over the quotient algebra < ^>οο(Α)β{ω )^·% {ω ), with 1ί{ω ) the ideal of all / e ^οο(Α) vanishing on π 00 π π ω π . The following fact is basic:

(24) If π is irreducible then ωπ € 03, i.e. there exists some χ € A, such that the orbit G χ is dense in ωπ. Remark. This is a special case of a result of Guichardet, see [13]. Proof. By Stone's theorem there exists a projection valued Borel measure Ρ on A such that p{g) = fagfödPfö for g e ^ » ( A ) . For a Borel set X c A the formula (23) implies π{ζ)*Ρ(Χ)π{ζ) = P(z~lX) for all ζ e G, with = {ζ~[ξ·,ξ e X}. It follows that for a G invariant Borel set X the projection P(X) commutes with 7r, hence in this case P(X) = 0 or 1. Now the mapping ξ —• ξ is continuous and open from A onto 93, so Ρ defines also a projectionjyajued measure on denoted by P: If X is the inverse image of X C 93, then P(X) = P(X). Evidently these inverse images are all G invariant, hence P(Z) = 0 or 1 for all Borel sets Ζ c 93. This fact together with the identity P(U Π V) + P(U U V) = P(U) + P(V) implies that P(U)P(V) = P(U η V) for all Borel sets U, V c Now let {Uj) be a countable basis of the open sets in 93. We set

Sj = Uj if P(Uj) = 1, Sj = CUj

if P(Uj) = 0 .

~ k _ It follows that P(Sι Π S2 Π · · · Π Sk) = Π p(sj) = 1 for ι oo for S = Π Ξ^ and consequently 5 ^ 0 . Let χ e S.

a11

keN,

„ hence P(S) = 1

k=ι

If ξ e 93 and ξ Φ χ, then, since 93 is To, there exists Uj either with χ e Uj, ξ £ Uj or with ξ e Uj, χ hg on L 2 (C), defined by functions h € L°°(C), which are constant on all spirals {e^'z \ t e R}, ζ e Cx. The algebra of these functions is isomorphic to L°°(T), hence A// and XH are reducible. For dim α = 1 the representation acts on L 2 (R) and is given by (A„(/, *)£)00

= e±'eie'x>g(e'y)

,

hence the subspace of all g € L 2 (R) with g(;y) = 0 for y < 0 is invariant, so again \H is reducible. Let us remark that we really used the fact that G is exponential. For G — C κ C with action ζ - » euz for u e Η = C tx {0} the representation \ H is irreducible.

§ 4 The dual of an exponential group We return in this paragraph to exponential groups, so G will always denote a connected, simply connected solvable exponential Lie group and g its Lie algebra. Our notation will be the same as in § 1.

34

1. Solvable Lie Groups, Representations

Recall that a representation π of a locally compact group Γ is called monomial, if it is induced from a one dimensional representation χ, i.e. from an abelian character of some closed subgroup Η of Γ : π = indf^. Our first basic result is an easy consequence of (26): (29) Every irreducible unitary representation of the exponential group G is monomial. Proof. We proceed by induction on η = dim G. The theorem is true for η = 1, so we suppose η > 1 and that the theorem holds for all exponential groups of lower dimensions. As a first consequence of this assumption we can assume that the kernel of π is discrete, hence the center Ζ of G either is trivial or Z = R and then the restriction π\ζ is not trivial. We fix a minimal noncentral ideal α in g. Let 3 denote the center of g, thus dim 3 < 1. There are four possibilities for 0: (a,) 3 Π α = ( 0 ) , dim α = 1 , (a 2 ) 3 Π α = ( 0 ) , dim α = 2 , (bi) jca, dim α = 2 , (b2> 3C0, dim α = 3 . In all cases α is abelian. This is clear for (ai), (a 2 ) and (bj). In case (b 2 ) the quotient a/3 is a minimal, hence abelian ideal in g/3. Because a/3 is minimal and dim a/3 = 2, the algebra g acts nontrivially on a/3, thus the annihilator f) of a/3 is an ideal of codimension < 2. But dimg/f) = 2 is excluded, because G is exponential. So there exists a basis {u, v, z} for α with 3 = Mz and a e g with [a, u] = u — λν, [α, υ] = Am 4- ν, Λ φ 0, [μ, υ] Ε 3· Then 0 = [α, [u, υ]] = [|[α, μ], υ] + [μ, [α, υ]] = 2[μ, ν], hence [μ, υ] = 0. The normal subgroup A — expa is isomorphic with R 7 , j = 1, 2 or 3, and actually we may consider the exponential mapping exp as an identification of α with A. Then the restriction of the inner automorphism, defined by an element g e G, on A is the same as the restriction of Ad g on a, i.e. exp(Ad g ( x ) ) = g - e x p ; t - g _ 1 or briefly Ad g(jt) = gxg~:- In this picture the dual group A is the linear dual a* of a and the action of G is the coadjoint action on a*. Let c be the centralizer of α in g and C = expc the corresponding normal subgroup of G. Then the dimension of the quotient F := G/C is 1 in the cases (ai), (a2), 1 or 2 in case (bi) and 3 in case (b 2 ). We will study the action of G, resp. F on A = a* in these cases. (aj): Let α = Em. Then g = Ra 0 c where we can choose a e g so that [a, u] = u, hence F = e x p ] R a and Ad exp ta(u) = e'u. Let a* = M

s for some s Φ 0. Again our claim follows as above. Now let dim F = 2. Then F acts nontrivially on a/3, hence there exists fceg with [b, u] = u( mod 3). It is an easy exercise to show that one can find elements a, b g g and a basis {u, z] of 0 such that g = Εα φ E6 φ c and [a, u\ = ζ, [b,u\ = u, [a, b] - — &(mod c). This implies Ad exp ta{u) = u + tz, Ad exp sb(u) — esu, hence if we think of the elements of ο as column vectors, in particular if we set u = ,ζ = , then F acts as the group of all matrices ^

^ ^. As in this model o* is the space of vectors

(,ξ, η), we see that we have the following orbits in a*: 0;

ω± = { ( ± / , 0 ) ; r > 0 } ;

ω5 = {(t, s); t eR},

sφ 0 .

The o)s are closed and as before we have ωπ = ω5 for some s φ 0. The matrices stabilizing the element (0, s) e cos = ωπ are those of the form

So the

stabilizer of the character χο of A, given by (0, s) G ω4·, is the closed subgroup Η = (exp Εα) κ C of G. Again 7τ is monomial.

36

1. Solvable Lie Groups, Representations

(b2>: In this case α defines a two dimensional ideal ac = Cw φ Cz in the complexified algebra g c . If w = u + iv with u, ν £ g, then α = Ew φ Ευ 0 Ez. Let φ and φ be the C-valued linear forms on g with [x, w] = φ{χ)\ν + ψ{χ)ζ . As g is exponential and φ{χ) an eigenvalue of ad χ it follows that ψ{χ) — q'7] = Sjjz, i.e. I is the Heisenberg algebra t) Puk(/, g).

§ 4 The dual of an exponential group

43

The next proposition characterizes the Pukanszky polarizations: Proposition 4. For a polarization p for f the following properties are (i) ρ € Puk ( / , 0), i.e. / + p 1 = A d * P ( f ) (ii) f + p± C A d * G ( / ) (iii) p€P(f + h, fl) for all h e p±

equivalent:

Proof. The implication (ί)=φ·(ϋ) is trivial. If / + h = A d * a ( / ) for some a e G, then clearly i(f + h, 0) = / ( / , 0), moreover, hep1 implies that p is subordinate to / + h. But d i m 0 / p = / ( / + h, 0), hence ρ e P(f + h, 0). This proves (ii) ==>• (iii). To show (iii) (i) we observe that for any I € 0* with Z(p') = 0 we have 1 Ad*P(/) - / c p i.e. Ad*P(l) C / + p 1 , in particular A d * P ( f + h) C / + p x if (iii) holds. Furthermore (iii) implies d i m A d * P ( / + h) = d i m ( / + p x ) , hence A d * P ( f + h) is open in / + ρ χ . As the flat subspace f + p± is the disjoint union of the Ad*/ > -orbits there can be only one orbit: / + p-1- = A d * P ( f ) . It is trivial that there exist maximal isotropic subspaces for Bf for any / e g * , however, it is not clear that they can be choosen to be subalgebras of 0 and everything but trivial that Puk ( / , 0) for a given / e 0* is non empty. Fortunately there is a very elegant and useful construction of Pukanszky polarizations in exponential Lie algebras. In § 1 we had defined Jordan-Holder bases by means of certain composition series of 0. These series were sequences {0*}* of subalgebras 0* of 0 such that 0*+i is an ideal in 0* for k — 1, 2 , . . . , η = dim0, 0„+i = 0, and which contain a composition series, thus either 0* is an ideal in 0 or and 0* + i are ideals in 0 and 1 /0/t+i is a two dimensional irreducible 0 module. We will call these series Jordan-Holder sequences or JH sequences. (36) Let f) be a subalgebra in the exponential Lie algebra 0. If if = {0y }y is a JH sequence of q, then tfo = {öj Π \)}j is a JH sequence of I). Proof. Let t)j = Qj Π f). Of course i/o is to be understood as the chain of different t)jk with dimf) 7 t , = dimf)^ + 1. If t)j is no ideal in f), then Qj is no ideal in 0, hence Qj-\ and 0 ; + i are ideals in 0 and f)7_i and t)j + \ are ideals in f), moreover t)j-i/i)j+i

=

1 +Qj+\)/$j+i

=Hj-i/9j+i

.

hence is an irreducible 0 module. Since i)j is no ideal in f), it is also a non trivial \) module. Then (30) implies that f)y_i/f) 7+ i is i) irreducible, which proves our claim. We use this result for Proposition 5. Let if — be a Jordan-Holder sequence of the exponential algebra 0. For / € 0* let f k = f\Bk be the restriction of f to 0*. thus f \ €

Lie If

44

1. Solvable Lie Groups, Representations

we write t* = g{' for the radical of the symplectic form B/k in Qk, then η o:=o(/,

is a Pukanszky polarization

Sf)

:=J2tk k=\

for f in g.

Proof.

We use induction on d i m g = n. Obviously ^ = {0/th=2,...,n is a JH η sequence in g 2 , hence 02 '•= Σ *k = ^2) € Puk ( / 2 , 02), in particular k=2 [02,02] C Ü2- We show first that 0 is a subalgebra, i.e. that a e t\, u e r „ i >

1, implies [a, u] € D. This is clear if g, is an ideal in g: If in this case

χ e g „ then [[α, m]x] = [[a,

u] + [a, [u, *]]. But [a, χ] € g, and u e r „ hence

/ , ( [ [ a , JC], «]) = 0, similarly, /,·([α, [u, Λ:]]) = 0, because a e t i . It follows that fi([[a,

M], JC]) = 0, which means [a, u] e r, C 0.

Let us now assume that g, is not an ideal. Then g,_i and g,+i are ideals and 0i— 1 /0i+1 is 0 irreducible. We set ν = [α, μ] and claim that (37)

/([(/)• This finishes the proof of (39). To apply (39) to the proof of proposition 5 we set 1} = g2 and p = 0. Since χι = + x>2 and D2 € Puk (/1^, f)) by induction, we see that ρ = Qf + p n f ) and ρ π — po = t)2- It follows that Ό e Puk (g, /), and proposition 5 is proved. As an important corollary we note: (40) a) Let a Pukanszky b) Let a be polarization

[aj}j be a decreasing sequence of ideals in g and f e g*. There exists polarization p for f in g with p Π α,· e Puk (/|Qj, a ; ) for all j. a commutative ideal in g. For every / e g * there exists a Pukanszky p for f containing a.

Proof, a) Refine {Oy} to a JH sequence Sf of g and set p = o(/, if). Then aj = g, η for some i, hence p n a p D, = Σ t * = t'(//. {&k}k>d € Puk (/,, g,). Since ρ Π aj k=i is subordinate to /, we have ρ Π α ; = D, e Puk (/|Qy, aj). b) follows from a) by taking {a,} = {o}. Polarizations of the kind described in proposition 5 we will call Vergne polarizations. Theorem 7. Let f e g*, p e P ( f , g) and π = ir(f, p, G). Then IT is irreducible if and only if p satisfies Pukanszky's condition, i.e. p e Puk (/, g). Moreover, in this case π is independent of the choice of p in Puk (/, g). Proof by induction on η = dim G. If / is zero on a nontrivial ideal a, then a is contained in all ρ € P ( f , g) and π is trivial on A = E(a), so the situation is essentially reduced to the quotient group G/A and we can apply induction. We leave the details to the reader and assume for the rest of the proof that f(a) φ 0

46

1. Solvable Lie Groups, Representations

for every ideal α φ 0 in g; in particular the dimension of the center 3 of g is 0 or 1. Let π = 7τ(/, p, G) be irreducible and let us fix a minimal noncentral ideal 0 in g. We distinguish two cases with respect to / e g*, p e P(f, g) and o: A. p contains o. Then αf is a proper subalgebra of g and p c αΛ In the cases (a(/ + ( a f ) x ) C Ad*PAd*A(/) = A d * P ( f ) , i.e. f + p± = Ad*/>(/). Thus ρ e Puk(/, g). Β. p does not contain a. Then t :— ρ + ο is a proper superalgebra of p. As we have seen the irreducibility of 77 excludes now the cases (a(«*') = 0, = 1. Then φ*ϋ(Η) c k e r L , w ττ\, but τ τ \ ( φ * ϋ ( Η ) ) = τ T \ ( L 1 ( H ) ) φ 0. From theorem 6 we derive that π = ind π\ is irreducible. Now let us assume that 3 = Rz is the center of g and the only minimal nonzero ideal in g. Then we are in case (bi) or Φ2) and ro = 0/3 is a minimal g module. Again let e the centralizer of α in g and ö the annihilator of ro. If ö = g, then ιυ is

48

1. Solvable Lie Groups, Representations

a trivial g module, thus dim to = 1, α = My Θ Rz, f(y) = 0 and there exists χ e g with g = Μ* φ c, [λγ, y] = z. Then as before, A = Ε (a) is central in Η = E(c) and tf is contained in e. This implies the existence of a Vergne polarization p with α C p C e. For a = exp(£iy + ξιζ) e A we have now χ ι (a) = ehence 2 77,(a) = e'^ - 1 and - notations analogous to our first case - ττ\(α) = • 1. Theorem 6 shows that π = i n d ^ i e G in the same way as before. Now suppose that 0 ^ 0 . Then 0 = R a φ 0 and ad a acts irreducibly on ro. We take a Vergne polarization p for a JH sequence through a, e and ö and assume again that c D, hence also ρ C D. We claim that even p e e : Let f\ = / | D and 1 χ € ö^ . On ρ 40 we had seen that the bilinear form β defined on (D/e) χ (α/3) by [jc, y] = ß{x, y)z is nondegenerate. Since χ e implies ß(x, y) = f([x, y]) = 0 for all y € α we get χ € c. The same argument works (in case (b2)) for any algebra j properly between e and 9. The subalgebra f) R a φ e is not an ideal in 0, however ρ is also a Pukanszky polarization in I) for f\ = / s o ττ\ : = τ τ ( / ι , ρ , Η), Η — £(ϊ)), is irreducible. We will now apply theorem 5 to show that then also π = ind^7Ti is irreducible. To this end it is sufficient to prove that the subalgebra ä c ind -fK^i)) of multiplication operators with bounded measurable functions on GJΗ is contained in the von Neumann algebra TT{G)". It is easy to see that we can choose the element a in t) so that Qo = j i e α; /(jc) = 0} is invariant under ad a. This implies that we can identify A with Ao x R, where Ao = £(ao) = R in case (b|) and Ao = C in case Φ2) and such that (42) holds for χ € Ao- Thus if φ is a function in the representation space f)(7r 1), χ = exp ta · c e Η with c e E(t), and (y, s) e Ao χ R, then (iri(y, s) Ω(ρ*) of the Kirillov map. Now we fix IT e G. Clearly the restriction Ω(7γ)|(, of Ω(ττ) onto Ϊ) is Ad*H invariant, hence we can define Ω„(π) := {Θ e Ω(ί)*); Θ c Ω(π)|(,} i.e. Ω(,(77•) is the subset in Ω(ί)*) of all Ad*H orbits in f)*, contained in the restriction Ω(7γ)|(,. We claim Theorem 9. The set {7γ(Θ) ; Θ e of the restriction TT\H in C*(H), i.e.: ker*7r|// -

is dense in the hull of the kernel ker*7r|w

f~j ker*7r(©) ΘεΩ^ιτ)

55

§ 5 Kernels of restricted and induced representations

Proof by induction on d i m G + dim G/H =: d{G,H). We need the following observation: Let si be a C*-algebra and a C*-subalgebra of the adjoint algebra slb, see [18]. If π is a representation of si, then π extends uniquely to siWe denote the restriction of this extension to by π\ β(ξ) defines an isometry from f)i onto ?)ι· A direct calculation shows that β commutes with the left translations A(y) for y e B, hence β intertwines the restriction of i n d p ^ i = ττ\ onto Β with i n d p ^ = ττ2. But p2 is

§ 5 Kernels of restricted and induced representations

59

a Pukanszky polarization for / | b in b, consequently 7Γ2 = v { f |b) is irreducible. It follows that then also ττ\β and 7t\h are irreducible. To finish case (α) we have to show that π\π = ττ' corresponds to the orbit of f = f ^ in f)*. From ( a ) and ρ (JL α we derive ρ + b = g, hence ρ Π f) + b = f) and consequently ρ Π f) £ α. So we can take a € ρ Π f), clearly with f ( a ) = 0 , s o that 0 = Ma 0 a, [) = Ι α 0 b, ρ = R a ® pi and q : = ρ Π = Μα ® p 2 . For the corresponding groups we get G = R κ A , Η = R κ Β , Ρ = R κ Ρ ι and Q : = £ ( ρ (Ί ϊ)) = R κ Ρ 2 , with R = £ ( Ε α ) . In proposition 6, we have shown how π can be realized on the space 53(77-)) and the same arguments show that S j ( t t 2 ) can be taken as the space of ρ := rndgxif, q). Furthermore, we had defined the isomorphism β from S)(tt\) onto ^(77-2), which in-tertwined t t \ \ b and 7r2, hence 7r|ß and 7r2 are intertwined. Evidently also p \ B is transfered into π 2 . On the other hand, we leave it to the reader to verify that also π\κ and p\r are transfered to the same representation τ of R on 9 ) { τ γ 2 ) , given by ( r ( r ) ^ ) ( x ) = c { r ) 2 £ ( r - 1 j c r ) for r 6 R , χ € Β and with a modular function c as described in proposition 6. It follows that ρ and 7r' are equivalent, hence q 6 Puk ( / ' , ( ) ) and π' = 7r(fl/ λΗ c ker* ττ. This follows easily from the following two facts: Since G is amenable, the constant 1 on G is a uniform limit on compacta of a sequence {i/^}* of positive definite functions ψ* on G, associated to see [8] p. 44. If φ is a positive definite function on G, associated to ττ, then the products φψκ are associated to ττ A« and converge uniformly on compacta to φ, hence ker* ττ D ker* ττ λΗ, see [7], § 18.

§ 6 Smooth functions and kernel operators An important consequence of (43) is the fact that the orbit G£ of ξ in the vector space V under the exponential group G is locally closed, i.e. that every element in Gif has a neighborhood in Gξ which is closed in V. This is equivalent to the fact that G£ is open in its closure Gif in V and also with the local compactness of G$ in the relative topology. To prove our assertion it is sufficient to show that there exists a neighborhood of ξ in V, which is diffeomorphic to the product of a neighborhood of ξ in G£ and an open set in K m , m = dimV — dimG£. For this purpose and with the same notation as in (43) we take a coexponential basis {ci, C2, • • •, Ck} for g^ in g, hence k — dim(g/g^) = dim and the af. = are linearly independent in V. We choose elements . . . , a n with η = dim V, such k that {a,·}" is a basis in V. For jc e R let e(jc) = E(x\C\)E{x2C2) · -. E(xkCk). Then

F ·. (x, y)

n—k F(x, y) = ε(χ)ξ + ^yjak+j 7=1

is a smooth mapping of Rk χ Κ" - * = Ε" into V. It follows from (43) that χ {0}) = G£, more precisely, χ —» F(x, 0) e V is a diffeomorphism from k R onto Since J£(0, 0) = a, for i resp. ι n, used above.

(JCJ, . . . , xn) e IR", resp. y e Mm is identified m J^/fc; € n, with the bases {α,·} of g and {£,} of ι

70

1. Solvable Lie Groups, Representations

Proof.

We fix

χ

=

Σχ'αί

0· Then the one parameter subgroup expRjc together ι with the nilpotent normal subgroup Ν = exp η generate a closed subgroup exp E x · N, which is isomorphic with the semidirect product Ε κ Ν. As before let A,· be the restriction of ad a, onto n, hence A = Σ xtAt = ad x\n. In our identification of exp Ejc · Ν with Ε κ Ν and of Ν and η, the element exp tx exp y, >' e ri, corresponds to ( / , y ) e l « n and the product is given by (s, u)(t, v) = (s + t, (e~tAu)()v). We wish to compute the one parameter group exp R(x + y) in the coordinates ρ

of Ε κ η. For this we use the decomposition η = {bi}

used above and set

q

η Σ χ ί

=

with corresponding basis ι

^

α

hence exp tq

=

(t,

0) e Ε χ η. The vector

ι fields

Bi

q

that

of the

is given by

m 4- — Σ (Ax)kj~.

*=i

operator

linear in χ and y, with ξ =

ρ

suc

h that

1

(61)

C of the form f(x,y)= \μ\J is a complete topological vector space, in fact even a Frechet space, which contains all compactly supported smooth functions. The functions in Sfr>J decay uniformly faster than any inverse polynomial in the R* directions and faster then any e~ c ^ in the R r directions, in particular c L'(R r χ Μ5). For r = 0, Sf0,s is the ordinary Schwartz space on R*. From (64) we derive (65)

is invariant under

Tr i

, i.e. if f € yr s, Τ e %r

c gi, hence

+ pnicgi + pni

which proves ρ = q + ρ Π i with q = ρ Π gj. Since q' C ρ' Π g' c g0 we see that q/g^ is commutative. If g' C t C gi and r' = [t, r] c g(,, i.e. r/q' 0 is commutative, then [r + ρ η i, r + ρ η i] = r' + [r, ρ η i] + (ρ η i)' c g£ + [gi, i] + ρ' , hence r + ρ Π i is isotropic. This implies r = q if r D q, hence q is maximal. We remark that, using the arguments in the proof of proposition 5, one can show that [ρ Π i, g^] c ρ Π i, so that gi + ρ (Ί i is always a subalgebra of g. Later the following remark will be useful: (74) If ρ^ = t>(/, then

k), k = 1,2, are Vergne polarizations through i, i.e. if i e dim(pi + i) = dim(p2 + i)

in particular pi + i = p2 + i if p2 C pi + i.

82

1. Solvable Lie Groups, Representations

This follows from the formula dim(p^ + i) + dim p* Π i = dim p* + dim i and the fact that the p* Π i are polarizations for /|j in i, hence dim pi Π i = dimp2 Π i. Finally, let ^h be an i special coexponential basis for 0 in g thus in particular a JH basis for g and let κ be the diffeomorphism of Rr° χ R'5° onto G, defined by with so - dimi, ro = dim(g/i). Then we write &K,M(G, Ϊ)

for the space of all C valued functions / on Rk χ R w χ G, for which the function -*/(£

ϋ,σ(χ,γ))

from Rk+r