Automorphic Forms Beyond GL2 : Lectures from the 2022 Arizona Winter School 9781470474928, 9781470476717

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Mathematical Surveys and Monographs Volume 279

Automorphic Forms Beyond GL 2 Lectures from the 2022 Arizona Winter School

Ellen Elizabeth Eischen Wee Teck Gan Aaron Pollack Zhiwei Yun Hang Xue, Editor

10.1090/surv/279

Automorphic Forms Beyond GL 2 Lectures from the 2022 Arizona Winter School

Mathematical Surveys and Monographs Volume 279

Automorphic Forms Beyond GL 2 Lectures from the 2022 Arizona Winter School

Ellen Elizabeth Eischen Wee Teck Gan Aaron Pollack Zhiwei Yun Hang Xue, Editor

EDITORIAL COMMITTEE Alexander H. Barnett Michael A. Hill Bryna Kra (chair)

David Savitt Natasa Sesum Jared Wunsch

2020 Mathematics Subject Classification. Primary 11F70, 11F27, 11F67, 14D24, 20G41, 14F08.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-279

Library of Congress Cataloging-in-Publication Data Names: Eischen, Ellen E., contributor. | Gan, Wee Teck, contributor. | Pollack, Aaron, 1986contributor. | Yun, Zhiwei, 1982- contributor. | Xue, Hang, 1985- editor. | Arizona Winter School (25th : 2022 : Tucson, Ariz.) Title: Automorphic forms beyond GL2 : lectures from the 2022 Arizona Winter School / Ellen Eischen, Wee Teck Gan, Aaron Pollack, Zhiwei Yun ; Hang Xue, editor. Description: First edition. | Series: Mathematical surveys and monographs, 0076-5376 ; volume 279 | Includes bibliographical references. Identifiers: LCCN 2023053456 | ISBN 9781470474928 (paperback) | 9781470476717 (ebook) Subjects: LCSH: Automorphic forms–Congresses. | AMS: Number theory – Discontinuous groups and automorphic forms – Representation-theoretic methods; automorphic representations over local and global fields. | Number theory – Discontinuous groups and automorphic forms – Theta series; Weil representation; theta correspondences. | Number theory – Discontinuous groups and automorphic forms – Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols. | Algebraic geometry – Families, fibrations – Geometric Langlands program: algebro-geometric aspects. | Group theory and generalizations – Linear algebraic groups and related topics – Exceptional groups. Classification: LCC QA353.A9 A94 2024 | DDC 512.7–dc23 LC record available at https://lccn.loc.gov/2023053456

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

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Contents Preface Hang Xue, editor Chapter 1. Automorphic forms on unitary groups E. E. Eischen Bibliography Chapter 2. Automorphic Forms and the Theta Correspondence Wee Teck Gan Bibliography Chapter 3. Modular forms on exceptional groups Aaron Pollack Bibliography Chapter 4. Rigidity method for automorphic forms over function fields Zhiwei Yun Bibliography

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Preface Hang Xue, editor

The Southwest Center for Arithmetic Geometry (SWC) was founded in 1997 by a group of seven mathematicians working in the southwest United States, and has been continuously supported by the National Science Foundation since that time. In the beginning, the SWC served as a true center for Arithmetic Geometry in the Southwest, but survives today in name only, having been subsumed by its principal activity, the Arizona Winter School (AWS). The AWS was started with the ambitious goal of creating an intense and immersive workshop in which graduate students – especially those who may not be studying at traditional centers of number theory – would work under the guidance of leading experts to solve research-level problems at the forefront of the field. The very first school was held in the Spring of 1998 under the title “Workshop on Diophantine Geometry Related to the ABC Conjecture.” In the twenty-one years that have followed, the AWS has been held annually each March on a different topic in arithmetic geometry and related areas, and has become a pillar of graduate education and training in these subjects throughout the country and abroad. Over the years, the Arizona Winter School model has been adjusted and refined to meet the needs of an ever-growing and increasingly diverse audience: the five-day meeting, organized around a different central topic each year, now features a set of four lecture series by leading and emerging experts. A month before the school begins, each speaker proposes one or more research projects related to their lectures, and is assigned 10-15 graduate students who will work on these projects. At that time, speakers also provide detailed lecture notes for their courses. During the school, students attend lectures daily from 9am to 5pm, and work each evening (often into the early hours of the morning!) with speakers and designated assistants on these research projects. Students not assigned to these research project groups have the option to join one of two problem sessions devoted to solving advanced exercises related to the lecture series, or one of four study groups which focus on understanding the course lecture notes in detail; these additional activities are supervised by young researchers and allow students not assigned to one of the research projects to meaningfully engage with the workshop material on many levels. On the last day, the students from each research project group present their work to the whole school. The result is an extremely focused and immersive five days of mathematical activity for all. In 2016, we began using specialized educational software expressly made for recording lectures together with dual HD web-cameras (one for the speaker and one for the primary document camera). The result has been a marked improvement vii

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in the quality and availability of the lecture videos. The lectures are live-streamed with a rich viewing experience where the user can toggle between the speaker feed and the document camera, or view both simultaneously in several different formats. This enhancement allows remote participation in the workshop which lowers the barrier for entry to the AWS for students from a broad variety of circumstances. At AWS 2022, we offered problem sessions and study groups both in the virtual format thereby allowing virtual participation in the evening sessions. This volume is comprised of the lecture notes which were prepared for the twenty-fifth Arizona Winter School on “Automorphic Forms beyond GL(2),” held March 5–9, 2022 at the University of Arizona in Tucson. The speakers were Ellen Eischen, Wee Teck Gan, Aaron Pollack, and Zhiwei Yun. We are greatly indebted to these authors for their hard work in making both the twenty-fifth AWS and this proceedings volume a reality. Akshay Venkatesh gave opening and closing lectures at the Winter School, which discussed the analogy between automorphic forms and topological field theories. We thank the National Science Foundation (NSF) for their longstanding and continued support of the Arizona Winter School, the Clay Mathematics Institute for their partnership in organizing the 2022 AWS, and the University of Arizona Department of Mathematics for their support. The editor is partially supported by the National Science Fundation (grant DMS #1901862) while editing this volume. Finally, we owe a great deal to the other members of the Southwest Center, both past and present, for their effort, perseverance, and vision in running the Arizona Winter School for more than twenty years, and for helping it to become the one-of-a-kind workshop that it is. Hang Xue

10.1090/surv/279/01

Automorphic forms on unitary groups E. E. Eischen

Abstract. This manuscript provides a more detailed treatment of the material from my lecture series at the 2022 Arizona Winter School on Automorphic Forms Beyond GL2 . The main focus of this manuscript is automorphic forms on unitary groups, with a view toward algebraicity results. I also discuss related aspects of automorphic L-functions in the setting of unitary groups.

1. Introduction This manuscript has two main purposes: (1) Introduce automorphic forms on unitary groups from several perspectives. (2) Illustrate some strategies in number theory (especially concerning algebraicity of values of automorphic L-functions), exploiting the fact that structures associated to unitary groups provide a convenient setting in which to work. We start with some familiar examples. Since it can be easy to get bogged down in sophisticated machinery, it is prudent to keep some familiar, yet relevant, examples in mind from the beginning. In the mid-1700s, Leonhard Euler proved that the values of the Riemann zeta series ζ(s) = ∑n≥1 n−s at positive even integers are algebraic (rational, in fact) up to a well-defined transcendental factor. More precisely, he proved that for each positive integer k, B2k , ζ(2k) = (−1)k+1 (2π)2k 2(2k)! where B2k ∈ Q is the 2kth Bernoulli number (the 2kth coefficient in the Taylor t ∞ tn series expansion ete t −1 = ∑n=0 Bn n! ). A century later, Ernst Kummer proved that far beyond merely being rational, the numbers − B2k2k arising in the values of ζ(2k), which turn out to be the values at 1−2k of the analytic continuation of ζ(s), satisfy striking congruences mod powers of a prime number p [95]. Rather than viewing these properties as a cute curiosity, Kummer was interested in information they encoded about cyclotomic fields. Indeed, he showed that p does not divide the class number of the cyclotomic field Q (e2πi/p ) if and only if p does not divide the numerators of the Bernoulli numbers B2 , B4 , . . . , Bp−3 , in which case he could prove special cases of Fermat’s Last Theorem [93, 94]. The author is grateful for support from NSF Grant DMS-1751281. ©2024 American Mathematical Society

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The Riemann zeta function is the Dirichlet L-function attached to the trivial character, but one could also ask what happens if we replace the trivial character by, say, some nontrivial algebraic character. In this case, once again, we obtain algebraic values at certain integer inputs, and the values have algebraic meaning. Furthermore, in the twentieth century, these L-functions began to play a significant role in Galois-theoretic statements. For example, picking up where Kummer had left off a century earlier, Kenkichi Iwasawa linked the behavior of Galois modules over towers of cyclotomic fields to p-adic zeta functions (p-adic analytic functions encoding the congruences first observed by Kummer). It turned out that the congruences observed by Kummer encoded not only information about the sizes of class groups but also about structures of collections of class groups, viewed as Galois modules [79, 80]. So far, our discussion has only concerned L-functions attached to characters, but automorphic forms on unitary groups are already lurking nearby. From class field theory, we have a correspondence between Hecke characters and representations of abelian Galois groups. In fact, a Hecke character of A×L , the ideles of a number field L, is an automorphic form on GL1 (AL ) or, equivalently when L is a CM field, on idelic points of the general unitary group GU (1) of rank one. In other words, we have a correspondence between Galois representations of abelian extensions and automorphic forms (and of their L-functions), at least in this simple case. The values of such L-functions can be shown to be algebraic, up to a well-determined transcendental factor. One of the most powerful techniques we have for proving such algebraicity results (as well as for proving analogues of Kummer’s congruences) is to express the values of these L-functions in terms of modular forms, i.e. automorphic forms on GL2 . These are closely related to automorphic forms on the general unitary group GU (1, 1) of signature (1, 1), thanks to Isomorphism 4 below. It turns out that structural aspects of modular forms are useful for proving results about these Lfunctions, including about the rationality and congruences exhibited by their values at certain points. In the simplest example of this phenomenon, the rationality of ζ(1 − 2k) follows from the rationality of the Fourier coefficients in the nonconstant terms in the Fourier expansion of the Eisenstein series of weight 2k and level 1, (1)

G2k (q) = ζ(1 − 2k) + 2 ∑ σ2k−1 (n)q n , n≥1

where q = e2πiz , z is a point in the upper half plane, and σ2k−1 (n) = ∑d∣n d2k−1 . This is the simplest implementation of an idea of Erich Hecke (that was later fleshed out by Helmut Klingen and Carl Siegel, as explained in [3, Section 1.3] and [78]) to study algebraicity of values of zeta functions by exploiting properties of Fourier coefficients of modular forms [86, 133, 134]. Moving a step further, we could investigate properties of the values of a Rankin– Selberg convolution of a weight k holomorphic cusp form f (q) = ∑n≥1 an q n that we assume to be primitive (i.e. a1 = 1 and f is a common eigenfunction of the Hecke operators of level N ) and a weight  < k holomorphic modular form g(q) = ∑n≥0 bn q n with algebraic Fourier coefficients an and bn for all n. That is, we consider the zeta series ∞

an bn . s n=1 n

D(s, f, g) = ∑

AUTOMORPHIC FORMS ON UNITARY GROUPS

Goro Shimura proved that for all integers k, , m satisfying  < k and (2)

3 k+−2 2

< m < k,

D(m, f, g) ¯ ∈ π k Q, ⟨f, f ⟩

¯ denotes an algebraic closure of Q and ⟨, ⟩ denotes the Petersson inner where Q product [125, Theorem 3]. A key step in proving Expression (2) is to express the values of D(s, f, g) as a Petersson inner product involving an Eisenstein series. More precisely, Shimura and Robert Rankin proved in [115, 125] that (3)

(r) D(k − 1 − r, f, g) = cπ k ⟨f˜, gδλ E⟩,

(−1) 4 Γ(k−−2r) N where c = Γ(k−1−r)Γ(k−−r) ∏p∣N (1 + p−1 ) (with N the level of the mod3 ular forms), E denotes a particular holomorphic Eisenstein series of weight λ ∶= (r) k −  − 2r and level N , f˜(z) ∶= f (−¯ z ), and δλ is a Maass–Shimura differential (r) operator that raises the weight of a modular form of weight λ by 2r. In fact, δλ E is also an Eisenstein series but, unlike the Eisenstein series we have encountered so far in this manuscript, is not holomorphic (although it is nearly holomorphic). Expression (2) then follows from the algebraicity of the Fourier coefficients of the Eisenstein series E, the fact that the Maass–Shimura operator preserves certain ¯ of the space of level N properties of algebraicity, and the decomposition over Q modular forms of weight k into an orthogonal basis of cusp forms (so the pairing in Equation (3) becomes a scalar multiple of ⟨f, f ⟩). In Section 4, we will see a vast extension of this idea of expressing L-functions in terms of Eisenstein series (and other automorphic forms) to glean information about rationality properties of the values of L-functions. Much more broadly, one might ask about analogous behavior for L-functions associated to other arithmetic data. In the 1970s, Pierre Deligne formulated vast conjectures about certain values of L-functions at integer points [26]. His conjectures concern L-functions attached to motives M , which include L-functions attached to algebraic Hecke characters (i.e. the example mentioned above) and to holomorphic modular forms (the next natural example to consider, given their connection with 2-dimensional Galois representations). Roughly speaking, he conjectured that if an integer m is critical for the motive M , then L(m, M ) is a rational multiple of a period associated to M . The details of how to formulate our aforementioned results about Artin L-functions in this setting are the subject of [26, Section 6]. More generally, though, this conjecture extends well beyond the low-dimensional examples discussed so far. In parallel with the case mentioned above, Ralph Greenberg also extended Iwasawa’s conjectures (concerning p-adic aspects) to this more general setting [56–58]. Meanwhile, the Bloch–Kato conjectures predict the meaning of the order of vanishing at s = 0 of L(s, M ) [7]. Given that we just named three significant sets of conjectures about L-functions, it is natural to seek tools to prove them. At present, our main route is through automorphic forms and automorphic representations (generated by automorphic forms). That is, rather than dive in and prove such conjectures directly for the data to which the L-functions are attached, the most fruitful approach to date has been to work with automorphic forms associated to that data. In our familiar examples above, we saw that automorphic forms on GL1 and GL2 played key roles. We also noted, though, that we could instead view these automorphic forms as being defined on unitary groups. In fact, it turns out that the relative simplicity r k−1

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of the above cases (relative to aspects of the higher-dimensional situation, which is absolutely not a suggestion that the already sophisticated 1- and 2-dimensional cases should be considered simple!) obscures the fact that unitary groups can be more effective than GLn (the group with which most people would probably be inclined to start) for extending certain techniques from dimensions 1 and 2 to higher-dimensional settings. 1.1. Why work with unitary groups? Unitary groups form a particularly convenient class of groups with which to work, due to certain algebraic and geometric properties. In particular, in analogue with the case of modular curves in the setting of modular forms, unitary groups have associated moduli spaces with integral models. This enables us to study algebraic aspects of automorphic forms (which arise as sections of vector bundles over Shimura varieties, in analogue with modular forms that arise as sections of a line bundle over modular curves). While the locally symmetric spaces for GLn for n ≥ 3 lack the structure of Shimura varieties, systems of Hecke eigenvalues for GLn can be realized in the cohomology of unitary Shimura varieties [22, 75, 119, 132]. Related to this, we also have substantial additional results about Galois representations in this setting (e.g. [18–20, 70, 135]), which enable us to study L-functions of Galois representations by instead studying L-functions of certain cuspidal automorphic representations. In addition, thanks to representation-theoretic properties of unitary groups, we have convenient models for the L-functions associated to certain automorphic representations of unitary groups. These models are useful both for proving analytic properties and for extracting algebraic information (and even p-adic properties, as seen in [31]). In fact, additional automorphic forms (Eisenstein series, of which the function G2k from Equation (1) is the simplest case) come into play in the study of these L-functions, so that we can eventually turn questions about L-functions into questions about properties of automorphic forms. Working with unitary groups has enabled major developments (which go beyond the scope of this manuscript but several of which are mentioned here as motivation for studying automorphic forms on unitary groups), including a proof of the main conjecture of Iwasawa Theory for GL2 [139] and the rationality of certain values of automorphic L-functions (including [54, 61, 63, 65, 69, 131]), as well as progress toward cases of the Bloch–Kato conjecture (including [87, 88, 138, 141]), and the Gan–Gross–Prasad conjecture (many recent developments, including [9, 10, 72, 83, 103, 144–147]). Because of their well-established power but also because many challenges remain, automorphic forms on unitary groups continue to play a significant role in research in number theory. Current research spans a variety of topics. In addition to applications to sophisticated problems like those mentioned above, there are also fundamental challenges associated with studying unitary Shimura varieties themselves (which arise independently of the automorphic theory but substantially impact the automorphic theory and associated L-functions). While unitary groups are related to symplectic groups and there is overlap in approaches to the two classes of groups, unitary groups generally present more challenges (including in terms of geometry, e.g. as documented in [117] and [39]) than one sees in the symplectic setting. At a more basic-sounding level, there is still much progress to be made even for computing examples on low-rank unitary groups (although some progress has been made under certain conditions, e.g. in [109, 143]).

AUTOMORPHIC FORMS ON UNITARY GROUPS

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Given all this, this manuscript focuses on the following topics: ● Introduction to automorphic forms on unitary groups from several perspectives (including analytic, algebraic, and geometric), as well as connections between those perspectives ● Aspects of L-functions associated to certain automorphic representations of unitary groups, with a view toward algebraicity results ● Constructions of examples of automorphic forms on unitary groups 1.1.1. Why present this particular set of topics (and not others) here? One could fill hundreds of pages discussing automorphic forms and their roles (even if one restricts to unitary groups) and still have a lot of ground left to cover. For this manuscript, the author has aimed to select a cohesive and relevant collection of topics that give readers a taste of some fundamental parts of this area while also meeting three key criteria: (1) These topics have arisen repeatedly in recent research developments, including those carried out by this manuscript’s author. (2) These topics are closely tied to questions that the author gets asked. (3) These topics are appropriate for graduate students and others getting started in this area. To assist readers who hope to explore further, this manuscript cites many papers that go into further details about specific topics. There are also a number of excellent, more extensive resources (such as books) that cover a lot of related material on automorphic forms, and readers are also encouraged to delve into those resources, for example [2, 4, 14, 21, 52, 131]. nGiven the three key criteria here, those familiar with the author’s research might wonder why the manuscript barely mentions p-adic automorphic forms and p-adic methods. While p-adic methods meet (at least) the first two key criteria, it is imperative to understand the basics of analytic and algebraic aspects of automorphic forms on unitary groups before moving on to the p-adic setting. An additional prerequisite for successfully working with p-adic automorphic forms is a strong understanding of p-adic modular forms (the GL2 setting), which is at odds with the focus of this workshop (namely, the beyond GL2 setting). In fact, as discussed in [36, Section 1], all known constructions of p-adic L-functions appear to be adaptations of the specific techniques employed in the proof of the algebraicity of the values of the corresponding L-function. (For example, Serre’s approach to proving Kummer’s congruences and constructing p-adic zeta functions in [121] builds directly on the ideas of Hecke, Klingen, and Siegel employing the constant terms of Eisenstein series summarized above. For additional examples, see Remark 4.2.3.) So this manuscript covers a subset of the prerequisites necessary for embarking on a study of p-adic methods for automorphic forms on unitary groups. One advantage of several of the topics presented here is that the broad strategies or ideas here might be transferrable to other groups, even if the technical details of how to carry out those strategies are specific to the group at hand. For example, the recipe for proving algebraicity of critical values of L-functions employed by Michael Harris in the setting of unitary groups (including in [65, 69]) has also been successfully adapted to certain other cases. Indeed, his strategy is an extension of the one introduced by Shimura for proving algebraicity of critical values of Rankin–Selberg convolutions of modular forms (on GL2 ) mentioned above. Technical ingredients (like Shimura varieties) that play a crucial role in Harris’s work are not necessarily

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available for other groups, but nevertheless, the recipe can be inspiring for how to proceed in other settings. To give a specific example, note that the overall strategy for proving algebraicity of critical values of Spin L-functions for GSp6 in [40] is an (admittedly substantial, when it comes to certain technical issues) adaptation of Harris’s (and Shimura’s) approach, even though this setting lacks what appear at first to be crucial ingredients from the unitary setting. Thus, while a primary goal is to prepare readers to work with automorphic forms on unitary groups, a significant secondary objective is to help readers gain some intuition that might apply more broadly. 1.2. Who is this manuscript for? This manuscript is for graduate students and others who are getting starting started doing research involving automorphic forms on unitary groups. It introduces aspects of some core topics that can serve as a launchpad for mathematicians hoping to explore further. The manuscript is especially aimed at those looking to understand connections between algebraic, geometric, and analytic aspects of automorphic forms on unitary groups. Deservedly, there has been much recent attention on the setting of unitary groups. Questions I have received suggest, though, that there is substantial demand for an accessible entry point, as it can be challenging to enter such a developed and dynamic field. A key aim of this manuscript is to provide a welcoming entrance to some of the extensive work in this area. Exercises related to the material presented here can be found in the problem sets prepared by Lynnelle Ye. 1.2.1. Recommended material to learn first. This manuscript is written for readers who are already familiar with automorphic forms on GL2 , i.e. modular (and Hilbert modular) forms. You will get the most out of this manuscript if you already know about modular forms and some of their uses in various settings. This includes the classical analytic perspective (e.g. as in [120, Chapter VII] and [90, Chapter III]), the algebraic geometric perspective (e.g. as in [84, Section 1] and [55, Chapters 1 and 2]), and the automorphic or representation-theoretic perspective (e.g. as in [49] and [14]). Not knowing some of this background material will not be problematic, but not knowing most of it will make it difficult to have the intuition necessary to follow portions of the material presented here. If you find that you need to develop your understanding of the GL2 setting further before proceeding, there are also other excellent resources for learning the fundamentals of modular forms, including [15, 27, 28, 111, 116, 136] and the online resources from the Arizona Winter Semester 2021: Virtual School in Number Theory.1 If you know the GL2 case well, then you will have the intuition necessary to pause periodically and think about how the material presented here specializes to the n = 1 case. Especially when you are stuck, it can be useful to specialize to the setting of modular forms to see what insight that more familiar setting offers. Sometimes this will be particularly helpful (for instance, when studying algebraicity of values of L-functions, which follows from an extension of Shimura’s approach in the setting of modular forms). Even in cases where the n = 1 case is too simple to offer much insight into the case of higher rank groups (e.g. singular forms are just constant functions in the n = 1 case), it can still offer a helpful reality check along the way. What might at first look like abstract formalities or a mess of heavy 1

https://www.math.arizona.edu/~swc/aws/2021/index.html

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7

notation will often instead become a natural extension of what you already know from the setting of modular forms for GL2 . 1.3. Acknowledgments. This manuscript would not be possible without the expertise and perspective I gained while working with my collaborators on research projects concerning automorphic forms on unitary groups. In particular, I would like to thank Ana Caraiani, Jessica Fintzen, Maria Fox, Alex Ghitza, Michael Harris, Jian-Shu Li, Zheng Liu, Elena Mantovan, Angus McAndrew, Chris Skinner, Ila Varma, and Xin Wan. Conversations I had with each of these collaborators substantially shaped my understanding of automorphic forms on unitary groups. I am especially grateful to Mantovan for her insightful answers to my questions about geometry, which have clarified my understanding of some of the geometric aspects of the material presented here. I am also especially grateful to Skinner and Harris, as well as my postdoctoral mentor Matthew Emerton, for helping me get started in this area and patiently answering my questions. I hope that this manuscript serves as a resource for some of their (and others’) future students (and perhaps saves them from some of the tedious sort of questioning to which I subjected them when I did not have access to such a written resource). I am grateful to the people who provided feedback on portions of earlier versions of this manuscript, including Utkarsh Agrawal, Francis Dunn, Bence Forr´ as, Maria Fox (who also was the discussion leader), Sean Haight, Andy Huchala, Angus McAndrew, Phil Moore, Yogesh More, Sam Mundy (who also was the project assistant), Samantha Platt, Wojtek Wawr´ ow, Pan Yan, Lynnelle Ye (who also wrote accompanying problem sets), and the anonymous referees. I am also grateful to the organizers of the Arizona Winter School (Alina Bucur, Bryden Cais, Brandon Levin, Mirela Ciperiani, Hang Xue, and David Zureick-Brown) for organizing the excellent workshop and inviting me to be a lecturer. 2. Unitary groups and PEL data Before introducing automorphic forms on unitary groups, it is prudent to establish some basic information about unitary groups and PEL data. This section includes properties of associated moduli spaces and certain representations that will occur in our definitions of automorphic forms in Section 3. 2.1. A first glance at unitary groups. Readers who are already familiar with unitary groups are encouraged to skip this short introduction to unitary groups and start with Section 2.2, where we associate a unitary group to PEL data. This section briefly introduces unitary groups to make sure all readers, including beginners, start off having seen at least basic definitions of the groups with which we work. Let K be a quadratic imaginary extension of a totally real field K + . Let V be an n-dimensional vector space over K, and let ⟨, ⟩ be a nondegenerate K-valued Hermitian pairing on V , i.e. ⟨, ⟩ ∶ V × V → K is linear in the first variable and conjugate-linear in the second variable, and ⟨v, w⟩ = ⟨w, v⟩ for all v and w in V . Note that we can linearly extend any such Hermitian pairing to a K + -algebra S and the S-module V ⊗K + S.

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Definition 2.1.1. The unitary group associated to (V, ⟨, ⟩) is the algebraic group U ∶= U (V, ⟨, ⟩) whose S-points, for each K + -algebra S, are given by U (S) = {g ∈ GLK⊗K + S (V ⊗K + S) ∣ ⟨gv, gw⟩ = ⟨v, w⟩} . This is a group scheme defined over K + . Given g ∈ U (S), we define g † to be the unique element of U (S) such that ⟨gv, w⟩ = ⟨v, g † w⟩ for all v, w ∈ V. Note that since ⟨, ⟩ is nondegenerate, we also have that gg † is the identity element in U , and g †† = g. (We have ⟨v, g †† w⟩ = ⟨g † v, w⟩ = ⟨w, g † v⟩ = ⟨gw, v⟩ = ⟨v, gw⟩. The fact that ⟨, ⟩ is nondegenerate then shows g = g †† .) Definition 2.1.2. The subgroup SU ∶= SU (V, ⟨, ⟩) of the unitary group U with determinant 1 is called a special unitary group. (It is a subgroup of the special linear group.) The unitary group U is a subgroup the group of unitary similitudes defined as follows. Definition 2.1.3. The general unitary group or unitary similitude group or group of unitary similitudes associated to (V, ⟨, ⟩) is the algebraic group GU ∶= GU (V, ⟨, ⟩) whose S-points, for each K + -algebra S, are given by GU (S) = {g ∈ GLK⊗K + S (V ⊗K + S) ∣ ⟨gv, gw⟩ = ν(g)⟨v, w⟩, ν(g) ∈ GL1 (S)} . The homomorphism ν ∶ GU → GL1 is called a similitude factor. We have an exact sequence g↦ν(g)

1 → U → GU → GL1 → 1. Sometimes (e.g. when realizing GU as an algebraic group), it is convenient to identify GU with the group of tuples {(g, ν(g)) ∈ GLK⊗K + S (V ⊗K + S) × GL1 (S) ∣ ⟨gv, gw⟩ = ν(g)⟨v, w⟩} . If we choose an ordered basis v1 , . . . , vn for V , then we may identify ⟨, ⟩ with an n × n-matrix A with coefficients in S and V with S n (via vi ↦ ei with ei the i-th standard basis vector, viewed as a column vector for the moment), via ⟨vi , vj ⟩ = t ei Aej . Note that our convention in this manuscript is always to write the superscript t on the left side of the matrix of which we are taking the transpose. Remark 2.1.4. If S = R, then an ordered choice of basis identifies V ⊗K + R with Cn , viewed for the moment as row vectors. If ⟨, ⟩ is a Hermitian pairing on V , then ¯ where the lefthand superscript t there is a Hermitian matrix A (i.e. A = A∗ ∶= t A, denotes the transpose and¯denotes the complex conjugate) such that ⟨v, w⟩ = vAw∗

AUTOMORPHIC FORMS ON UNITARY GROUPS

9

for all v, w ∈ V . After a change of basis, the Hermitian matrix corresponding to the nondegenerate pairing ⟨, ⟩ can be written in the form 0 ), −1b

1 Ia,b ∶= ( a 0

with a + b = n. The tuple (a, b) is called the signature of ⟨, ⟩. If ab = 0, we say the unitary group preserving ⟨, ⟩ is definite. When a = b, we can also choose a basis with respect to which the matrix corresponding to ⟨, ⟩ is iη, where 0 η ∶= ηa ∶= ( 1a

−1a ). 0

When the group under consideration has signature (a, b), it is conventional to write SU (a, b), U (a, b), or GU (a, b). Sometimes, this notation is reserved for the group of matrices preserving Ia,b if a ≠ b and ηn if a = b, and this is the convention we will employ going forward. It is also conventional to write U (A) for the matrix group preserving a Hermitian matrix A (and SU (A) for the subgroup of matrices of determinant 1 and GU (A) for the corresponding similitude group). In Section 2.2, we will introduce groups G and G1 that are defined over Q and are closely related to ResK + /Q GU and ResK + /Q U , respectively. (Given a group H defined over K + , ResK + /Q denotes the restriction of scalars functor, i.e. (ResK + /Q H) (S) ∶= H(S ⊗Q K + ) for each Q-algebra S.) Note that (ResK + /Q GU ) (R) ≅ (ResK + /Q U ) (R) ≅

∏

GU (aτ , bτ )

∏

U (aτ , bτ ) .

τ ∶K + ↪C τ ∶K + ↪C

In this case, the signature is the tuple (aτ , bτ )τ . Given such a tuple (a, b), we write Ua,b (R) for ∏τ U (aτ , bτ ) (and similarly for SUa,b and GUa,b ). If the unitary group is definite of signature (n, 0) or (0, n), we often write n in place of the pair (n, 0) or (0, n). 2.1.1. A close relationship between GL2 and GU (1, 1). In Section 1, we noted that automorphic forms on GU (1, 1) and GL2 are closely related. Later, we will see that the symmetric space for GU (1, 1) is a finite set of copies of the upper half plane (the symmetric space for GL2 ). There is also a close relationship between automorphic forms on GU (1, 1) and modular forms. The relationship stems from the isomorphism (4)

GU (1, 1) ≅ (GL2 × ResK/K + Gm ) /Gm ,

where Gm is embedded as α ↦ (diag(α, α), α−1 ) and, following the usual convention, Gm denotes the multiplicative group. 2.1.2. Unitary groups over local fields. Above, we defined the signature of a unitary group associated to a Hermitian pairing ⟨, ⟩ on a C-vector space. More generally, we will encounter Vv ∶= V ⊗K + Kv+ , where v is a finite place of K + . If v splits as ww ¯ in K, then the decomposition K ⊗K + Kv+ ≅ Kw ⊕ Kw¯ . induces a decomposition Vv = Vw ⊕ Vw¯ . Note that the nontrivial element of Gal(K/K + ) swaps the two summands in the direct sum here, and U (Kv+ ) fixes

10

E. E. EISCHEN

each summand. Furthermore, still assuming that v splits as ww ¯ in K, we get isomorphisms (5)

U (Kv+ ) ≅ GLn (Kw ) ≅ GLn (Kv+ ).

On the other hand, in the case where v is inert, Kw /Kv+ is a quadratic extension of p-adic fields (for v a prime over (p)), in which case the structure is described in, e.g. [68, Section 1]. 2.2. PEL data. In this section, we introduce data of PEL type, along with corresponding moduli spaces. In analogue with the setting of modular forms, which can be viewed as sections of a line bundle over a modular curve (a moduli space of elliptic curves with additional structure), we will later define automorphic forms as sections of a vector bundle over a higher-dimensional generalization of the modular curve (namely Shimura varieties, which serve as moduli spaces for certain abelian varieties with additional structure). As we shall see in Section 2.2.3, the data of PEL type introduced below correspond to a moduli problem. In analogue with the moduli problem of classifying pairs consisting of an elliptic curve and a level structure, our moduli problem concerns tuples of abelian varieties with not only a level structure but also additional structures (polarization and endomorphism). Similarly to the case of modular forms, realization of our automorphic forms in terms of algebraic geometry will enable us to work over base rings (and schemes) beyond just C. This is essential for considering questions about algebraicity or rationality. For an excellent introduction to Shimura varieties, the reader is encouraged to consult [101]. Note that because we are concerned with unitary groups in this manuscript, we have specialized from the beginning to PEL data that will correspond to unitary Shimura varieties. For detailed references specific to the case of unitary groups, the reader is encouraged to consult [91, Section 5] and [101, Section 5.1]. We consider tuples D ∶= (D, ∗, OD , V, ⟨, ⟩, L, h) consisting of: ● A finite-dimensional semisimple Q-algebra D, each of whose simple factors has center CM field K ● A positive involution ∗ on D over Q, by which we mean an anti-automorphism (i.e. reverses the order of multiplication in D) of order 2 such that traceD⊗Q R (xx∗ ) > 0 for all nonzero x ∈ D ⊗Q R ● A ∗-stable Z-order OD in D ● A nonzero finitely generated left D-module V ● A nondegenerate Q-valued alternating form ⟨, ⟩ on V such that ⟨bv, w⟩ = ⟨v, b∗ w⟩ for all b ∈ D and v, w ∈ V ● A lattice L ⊆ V preserved by OD that is self-dual with respect to ⟨, ⟩. ● A ∗-homomorphism h ∶ C → EndD⊗Q R (V ⊗Q R) (and by ∗-homomorphism, z ) for all z ∈ we mean an R-algebra homomorphism such that h(z)∗ = h(¯ C), such that the symmetric R-linear bilinear form ⟨⋅, h(i)⋅⟩ on VR ∶= V ⊗Q R is positive definite Remark 2.2.1. To see that the pairing (⋅, ⋅) ∶= ⟨⋅, h(i)⋅⟩ is symmetric, observe that for all v, w ∈ V , (v, w) ∶= ⟨v, h(i)w⟩ = ⟨h(−i)v, w⟩ = −⟨h(i)v, w⟩ = ⟨w, h(i)v⟩ = (w, v). Remark 2.2.2. We view K as embedded in D via the diagonal embedding of K into each of the simple factors of D.

AUTOMORPHIC FORMS ON UNITARY GROUPS

11

Such a tuple D is called a PEL type datum or a datum of PEL type. This is the setup given in [91, Section 5]. (N.B. Because we have specialized to the unitary case and so are, for example, excluding the Siegel case, we have required that K be a CM field and not a totally real field.) One can also consider an integral PEL datum, as in, for example, [101, Section 5.1.1]. To distinguish the previous case from the integral case, we sometimes call the previous case a rational PEL datum. In the integral case, we consider tuples (O, ∗, L, ⟨, ⟩, h) consisting of: ● An order O in a finite-dimensional semisimple Q-algebra D ● A positive involution ∗ on O ● A O-module L that is finitely generated as a Z-module ● A nondegenerate alternating pairing ⟨, ⟩ ∶ L × L → 2πiZ such that ⟨bv, w⟩ = ⟨v, b∗ w⟩ for all b ∈ O and v, w ∈ L ● A ∗-homomorphism h ∶ C → EndO⊗Z R (L ⊗Z R) such that the symmetric R-linear bilinear form (2πi)−1 ⟨⋅, h(i)⋅⟩ on L ⊗Z R is positive definite. To our integral PEL datum, following [101, Section 5.1.3], we attach a group scheme G over Z whose R-points, for each ring R, are given by (6) G(R) ∶= {(g, r) ∈ EndO⊗Z R (L ⊗Z R) × R× ∣ ⟨gv, gw⟩ = r⟨v, w⟩ for all v, w ∈ L ⊗Z R} . Remark 2.2.3. Note that from an integral PEL type datum, one obtains a (rational) PEL datum as above by tensoring the integral data with Q (and renormalizing the bilinear forms). As explained in [101, Section 5.1.1], in the moduli problem that we will discuss below, rational PEL data are most naturally suited to working with isogeny classes of abelian varieties, while integral PEL data are most readily suited to the language of isomorphism classes of abelian varieties. Let K + be the fixed field of ∗. Then K/K + is a quadratic imaginary extension. Let n = dimK V. Note that VC ∶= VR ⊗R C decomposes as VC = V1 ⊕ V2 , where V1 is the submodule on which h(z) (more precisely, h(z) × 1) acts by z and V2 is the submodule on which h(z) (more precisely, h(z) × 1) acts by z. The reflex field E of D is defined to be the field of definition of the isomorphism class of the C-representation V1 of D. For more details about how to view the reflex field and its geometric significance, see [101, Section 5.1.1] or [99, Section 1.2.5]. Remark 2.2.4. Let p be a prime number. For defining PEL type moduli problems over OE,(p) , it is also useful to replace our PEL datum with a p-integral PEL datum, i.e. (as in [101, Equation (5.1.2.2)]) (O ⊗Z Z(p) , ∗, L ⊗Z Z(p) , ⟨, ⟩, h) with L ⊗Z Z(p) required to be self-dual under the resulting Hermitian pairing on L ⊗Z Qp and p ∤ Disc(O). Note that the involution ∗ on D induces an involution on C ∶= EndD (V ). To the PEL datum D, we associate an algebraic group G over Q whose R-points are given by G(R) = {x ∈ C ⊗Q R ∣ xx∗ ∈ R× }

12

E. E. EISCHEN

for any Q-algebra R. Note that this agrees with the definition of G coming from the integral PEL data in Equation (6). The similitude factor of G is the homomorphism ν ∶ G → Gm , defined by g ↦ gg ∗ . We let G1 be the group whose R-points are given by G1 (R) ∶= ker(ν) = {x ∈ C ⊗Q R ∣ xx∗ = 1} . 2.2.1. Decompositions and signatures associated to PEL data. Following the conventions and perspective of [38, Section 2.1.5], we briefly summarize some key decompositions and define the signature of a PEL datum. While these decompositions are basic and the definition of signature occurs in each of the author’s papers in this area, it apparently took many iterations (for this author, at least) to arrive at what feels like an “optimally” concise and useful setup for them, hence the citation of a relatively recent paper for this background material. ¯ Given For any number field L, we let TL denote the set of embeddings L ↪ Q. ∗ τ ∈ TL , we denote its composition with complex conjugation by τ . Going forward, we fix a CM type ΣK for K (i.e. ΣK ⊆ TK contains a choice of exactly one representative from each pair of complex conjugate embeddings τ, τ ∗ ∈ TK ). The decomposition K ⊗Q C = ⊕τ ∈TK C (identifying a ⊗ b with (τ (a)b)τ ∈TK ) induces decompositions Vi = ⊕τ ∈TK Vi,τ , for i = 1, 2 VC = ⊕τ ∈TK Vτ , with Vτ = V1,τ ⊕ V2,τ for all τ ∈ TK . Here, the subscript τ denotes the submodule on which each a ∈ K acts as scalar multiplication via τ (a). Definition 2.2.5. The signature of the unitary PEL datum D is (aτ )τ ∈TK , where aτ ∶= dimC V1,τ for all τ ∈ TK . We also sometimes speak of the signature at τ ∈ ΣK or at σ ∈ TK + , by which we mean (aτ , aτ ∗ ), given the unique τ ∈ ΣK such that τ ∣K + = σ. For such τ ∈ ΣK , we also sometimes set a+σ ∶= a+τ ∶= aτ and a−σ ∶= a−τ ∶= aτ ∗ . It is also common to write (aτ , bτ ) in place of (aτ , aτ ∗ ). Note that for each τ ∈ TK , we have aτ + aτ ∗ = n. More generally, we record some basic facts about decompositions of modules that will be useful to us later. Given a number field L, we denote by LGal the ¯ and we denote by OL the ring of integers in L. If R is an Galois closure of L in Q, OK +Gal -algebra and the discriminant of K + /Q is invertible in R, then we have an isomorphism ∼

OK + ⊗Z R → ⊕τ ∈TK + R a ⊗ r ↦ (τ (a)r)τ ∈TK + . Given an OK + ⊗Z R-module M and τ ∈ TK + , we denote by Mτ the submodule on which each a ∈ OK + acts as multiplication by τ (a), and we have an OK + ⊗Z R-module isomorphism ∼

M → ⊕τ ∈TK + Mτ

AUTOMORPHIC FORMS ON UNITARY GROUPS

13

If R is, furthermore, an OK Gal -algebra, then we can further decompose M as M = ⊕τ ∈TK Mτ = ⊕τ ∈ΣK Mτ ⊕ Mτ ∗ = ⊕σ∈TK + Mσ+ ⊕ Mσ− , where for each τ ∈ TK , Mτ denotes the submodule of M on which each element a ∈ OK acts via scalar multiplication by τ (a), and for each σ ∈ TK + , Mσ+ (resp. Mσ− ) is the submodule of Mσ on which each element a ∈ OK acts as multiplication by τ (a) (resp. τ ∗ (a)) for τ ∈ ΣK the unique element of ΣK such that τ ∣K + = σ. For such τ ∈ ΣK , we also sometimes write Mτ± in place of Mσ± . We also set M ± = ⊕σ∈TK + Mσ± .

(7)

2.2.2. PEL data arising from unitary groups. Following the conventions of [31, Section 2.2], we say that a PEL datum D like above is of unitary type if the following three conditions hold: ● D = K × ⋯ × K, i.e. D is a direct product of finitely many copies of K ● ∗ acts as complex conjugation on each factor K in D = K × ⋯ × K ● OD ∩ K is the ring of integers OK in K, where K is identified with its diagonal embedding in D = K × ⋯ × K Fix a totally imaginary element α of OK . As explained in [31, Section 2.3], given a collection of m unitary groups preserving Hermitian pairings ⟨, ⟩W1 , . . . , ⟨, ⟩Wm on K-vector spaces W1 , . . . , Wm , respectively, we obtain a PEL datum D = (D, ∗, OD , V, ⟨, ⟩, L, h) of unitary type as follows: ● Let D = K m . ● Let ∗ be the involution on D that acts as complex conjugation on each factor K. m ⊆ K m. ● Let OD = OK ● Let V = ⊕i Wi . ● Let ⟨(v1 , . . . , vm ), (w1 , . . . wm )⟩ = ∑i ⟨vi , wi ⟩i , where ⟨, ⟩i ∶= traceK/Q (α⟨, ⟩Vi ) . ● Let L = ⊕Li , where Li ⊆ Vi is an OK -lattice such that ⟨Li , Li ⟩ ⊆ Z. ● Let h = ∏i hi ∶ C → EndK + ⊗Q R (V ⊗Q R) = ∏i EndK + ⊗Q R (Wi ⊗Q R), where hi ∶ C → EndK + ⊗Q R (Wi ⊗Q R) is defined by hi = ∏ hi,τ ∶ C → EndK + ⊗Q R (Vi ⊗Q R) = ∏ EndR (Vi ⊗K,τ C) τ ∈ΣK

τ ∈ΣK

and hi,τ ∶ C → EndR (Vi ⊗K,τ C) is defined as follows. Choose an ordered basis B

(8)

for Wi ⊗K,τ C with respect to which the matrix for ⟨, ⟩ is of the form diag(1r , −1s ) (with r and s dependent on i and τ ), and identify Wi ⊗K,τ C with Cr+s , as well as EndR (Wi ⊗K,τ C) with M(r+s)×(r+s) (C), via this choice of basis. We then define hi,τ (z) = diag(z1r , z¯1s ) for each z ∈ C.

14

E. E. EISCHEN

2.2.3. Moduli problem. In this section, we will describe a moduli problem that classifies abelian varieties together with the structure of a polarization, endomorphism, and level structure. The letters PEL above are an abbreviation for Polarization, Endomorphism, Level structure, part of the data in this moduli problem. The moduli problem here is analogous to the one encountered in the context of modular forms, where we classify elliptic curves with level structure. To define a moduli space in our setting, we also need to include a polarization and endomorphism, although these extra structures will generally not come into play much later in this manuscript. Our formulation here most closely follows the conventions in [16, Section 2.2], [31, Section 2.1], and [101, Section 5.1]. Let G be as in Equation (6), and let K be an open compact subgroup of G(Af ), where Af denotes the finite adeles in the adeles A over Q. We first consider the moduli problem that associates to each pair (S, s) consisting of a connected, locally Noetherian scheme S over E and a geometric point s of S the set of equivalence classes of tuples (A, λ, ι, η) consisting of: ● An abelian variety A over S of dimension g ∶= n[K + ∶ Q] (where n is the dimension of the K-vector space V in the PEL datum D) ● A polarization λ ∶ A → A∨ (where A∨ denotes the dual abelian variety) ● An embedding ι ∶ OD ⊗Z Q ↪ End(A) ⊗Z Q of Q-algebras that satisfies the Rosati condition, i.e. λ ○ ι(b∗ ) = (ι(b))∨ ○ λ for all b ∈ OD ● A K-level structure η, i.e. a π1 (S, s)-fixed orbit of OD -linear isomorphisms ∼

L ⊗Z Af → H1 (A, Af ) that maps ⟨, ⟩ to a A×f -multiple of the pairing on H1 (A, Af ) defined by the λ-Weil pairing In addition, we require that the above tuples satisfy Kottwitz’s determinant condition (as explained in, for example, [91, Section 5], [99, Definition 1.3.4.1], and [16, Section 2.2]). Tuples (A, λ, ι, η) and (A′ , λ′ , ι′ , η ′ ) are considered equivalent if there is an isogeny φ ∶ A → A′ such that λ is a nonzero rational multiple of φ∨ ○λ′ ○φ, ι′ (b) ○ φ = φ ○ ι(b) for all b ∈ OD , and η ′ = φ ○ η. If K is sufficiently small (or more precisely, if K is neat, in the sense of [99, Definition 1.4.1.8]), then this moduli problem is representable by a smooth, quasiprojective scheme MK over E. This is a result of [99, Corollary 7.2.3.10] and [91, Section 5], which also explain that a p-integral version of this moduli problem is representable by a smooth, quasi-projective scheme over OE ⊗Z(p) . (N.B. Given a scheme S = Spec(R), we sometimes write R in place of Spec(R) when the meaning is clear from context.) More precisely, assume that in addition to the conditions above, K = Kp Kp with Kp ⊆ G(Qp ) hyperspecial. (i.e. We require that there exists a smooth group scheme G that is a model of G over Zp , such that the special fiber G is reductive, and such that Kp = G(Zp ), as discussed in much more detail in [52, Section 2.4]. For geometric motivation for the condition of being hyperspecial, see also [101, 110]. For the origins, see Tits’s original article in [140].) Then there is a smooth, quasi-projective scheme MK over OE ⊗ Z(p) that to each pair (S, s) consisting of a connected, locally Noetherian scheme S over OE,(p) and a geometric point s of S associates the set of equivalence classes of tuples (A, λ, ι, η) (where

AUTOMORPHIC FORMS ON UNITARY GROUPS

15

tuples (A, λ, ι, η) and (A′ , λ′ , ι′ , η ′ ) are considered equivalent if there is a primeto-p isogeny φ meeting the conditions from above and furthermore λ is a nonzero prime-to-p multiple of φ∨ ○ λ′ ○ φ) consisting of: ● An abelian variety A over S of dimension g ● A prime-to-p polarization λ ∶ A → A∨ (where A∨ denotes the dual abelian variety) ● An embedding ι ∶ OD ⊗Z Z(p) ↪ End(A) ⊗Z Z(p) of Z(p) -algebras that satisfies the Rosati condition ● A Kp -level structure η, i.e. a π1 (S, s)-fixed orbit of OD -linear isomorphisms ∼

L ⊗Z Ap,∞ → H1 (A, Ap,∞ ) that maps ⟨, ⟩ to a (A∞,p )× -multiple of the pairing on H1 (A, Af ) defined by λ-Weil pairing (The notation Ap,∞ means the adeles away from p and ∞.) Once again, these tuples are also required to satisfy Kottwitz’s determinant condition (as explained in, for example, [91, Section 5], [99, Definition 1.3.4.1], and [16, Section 2.2]). Note that MK ×SpecOE,(p) SpecE = MK . Remark 2.2.6. The condition that K is neat guarantees the representability of our moduli problem by a smooth moduli space. For the purposes of this manuscript, the details of what it means to be neat are unimportant. For the sake of completeness, though, we briefly recall from [99, Definition 1.4.1.8] that K is defined to be neat if each of its elements g = (gp ) is neat. That is, the group ∩p p , where p denotes the group of algebraic eigenvalues of gp (viewed as an element of GL(L ⊗ Qp )), is torsion free. 2.2.4. Compactifications. The moduli spaces MK have toroidal compactifications (as proved in [99]), over which one can define modular forms. There are also minimal compactifications of the spaces MK , as constructed in [99] and summarized in [101]. See [101, Section 5.1.4] for a summary of key developments for compactifications leading up to Lan’s work on toroidal and minimal compactifications in [99], in particular connections with the earlier work of Gerd Faltings and Ching-Li Chai in the setting of Siegel moduli problems in [42, Chapters III–V]. 2.2.5. Complex points and connection with Shimura varieties. We briefly summarize the connection between complex points of our moduli space MK and complex points of unitary Shimura varieties. Let H be the orbit of h under conjugation by G(R), and let K∞ ⊆ G(R) be the centralizer of h. We give H the structure of a real manifold via the identification G(R)/K∞ with H via g ↦ ghg −1 . Furthermore, h induces a complex structure on H. When (G, H) is a Shimura datum (in the sense of [101, Section 2.3]), we say that (G, H) is a PEL-type Shimura datum, and the Shimura variety ShK associated to (G, H) is called a PEL-type Shimura variety. (See [101, Section 2.4] for an excellent introduction to Shimura varieties.) Remark 2.2.7. As noted in [101, Section 5.1.3], it is not necessarily the case that (G, H) is a Shimura datum. When O is an order in an imaginary quadratic field and a ≥ b, though, (G, H) is a Shimura datum, G(R) ≅ GU (a, b), and H ≅ Ha,b , where (9)

Ha,b ∶= {z ∈ Mata×b (C) ∣ 1 − t z¯z > 0} ,

16

E. E. EISCHEN

with > 0 meaning positive definite and Mata×b meaning a × b matrices. For details, see [101, Example 5.1.3.5]. Note that the complex points of ShK can be identified with the double coset space XK ∶= G(Q)/ (H × G (Af ) /K) , with G(Q) acting diagonally on H and G (Af ) on the left and K acting on G (Af ) on the right, as detailed in [91, Section 8] and [101, Section 2]. Even if (G, H) is not a Shimura datum, there is an open and closed immersion XK ↪ MK (C), and there is an open and closed subscheme of MK that is an integral model of ShK . In our setting (i.e. the unitary setting), MK (C) is a disjoint union of finitely many copies of XK , as explained in [16, Section 2.3.2] and [101, Section 5.1.3]. Following the conventions for terminology introduced in [31, Section 2.3], we call MK the moduli space (of PEL-type) and ShK the Shimura variety (of PEL type). Note that due to our insistence that the center of D be a CM field, we have actually narrowed the set of cases under consideration in this manuscript to the unitary setting (what is often called Type A), rather than the broader PEL setting (which includes symplectic and orthogonal groups as well). Remark 2.2.8. Observe that each element h ∈ G(Af ) acts on the right on H × G(Af ) via (z, g) ↦ (z, gh), which induces a map (10)

[h] ∶ XhKh−1 → XK .

In turn, this provides a right action of G(Af ) on the collection {XK }K , which is useful for relating these geometric spaces to automorphic representations. (This provides some motivation for this adelic formulation, which might otherwise seem unnecessary at first glance.) We briefly review the structure of the double coset XK . For more details, see, e.g. [101, Section 2.2]. It turns out that we can express G(Af ) as a finite disjoint union G(Af ) = ⊔i∈I G(Q)+ gi K, with gi ∈ G(Af ) indexed by a finite set I and G(Q)+ ∶= G(Q) ∩ G(R)+ (with G(R)+ the connected component of the identity), and let H+ be a connected component of H on which G(R)+ acts transitively. Then (as in [101, Equations (2.2.21)]), we have XK = G(Q)+ / (H+ × G(Af )/K) = ⊔i∈I G(Q)+ / (H+ × G(Q)+ gi K/K) = ⊔i∈I Γi /H+ , where Γi ∶= (gi Kgi−1 ) ∩ G(Q)+ . As explained in [101, Section 2.2], each Γi is a congruence subgroup of G(Z), i.e. Γi contains the principal congruence subgroup (i.e. the kernel of G(Z) → G(Z/N Z)) for some positive integer N .

AUTOMORPHIC FORMS ON UNITARY GROUPS

17

Remark 2.2.9. In the case of the usual upper half plane h and elliptic curves, we associate to each point z ∈ h, an elliptic curve whose complex points are identified with the torus C/(Zz + Z). The analogue in our setting is as follows (and discussed in detail in [131, Section 4]). Given a unitary PEL datum like above and z ∈ H, Shimura defines a map pz (see [131, Section 4.7] or [32, Section 2.3.2]) so that Cg /pz (L) is an abelian variety, and he explains (in [131, Theorem 4.8]) how to assign a polarization, endomorphism, and level structure so that ΓK /H classifies PEL tuples with level structure corresponding to K. (Here, ΓK ∶= K ∩ GU + (Q).) For additional details on the moduli problem over C, see also [32, Section 2.3.2] or [101, Sections 3.1.5 and 3.2]. We give a more precise description of the lattice pz (L) in Remark 2.2.11, for signature (n, n). 2.2.6. Hermitian symmetric space associated to a unitary group. The symmetric domain H for G as above (with signature (aτ , bτ ) for each τ ∈ ΣK ) is ∏ Haτ ,bτ ,

τ ∈ΣK

where Ha,b is defined as in Equation (9). The elements g = (gτ )τ ∈ΣK ∈ ∏ GU + (aτ , bτ ) τ ∈ΣK

(where the superscript + denotes positive determinant) act on z = (zτ )τ ∈ΣK ∈ ∏τ ∈ΣK Haτ ,bτ by gz = (gτ zτ )τ ∈ΣK , where for a b ) gτ = ( c d with a ∈ GLaτ (C) and d ∈ GLbτ (C), we have gτ zτ ∶= (azτ + b)(czτ + d)−1 . The stabilizer of 0 ∈ Haτ ,bτ is the product of definite unitary groups U (aτ ) × U (bτ ) embedded diagonally in U (aτ , bτ ). So we can identify Haτ ,bτ with U (aτ , bτ )/(U (aτ ) × U (bτ )). In the case of definite unitary groups (i.e. aτ bτ = 0 for all τ ), H = ∏τ ∈ΣK Haτ ,bτ consists of a single point, which parametrizes an isomorphism class of abelian varieties isogenous to aτ + bτ (which is the same for all τ ) copies of a CM abelian variety, as explained in [131, Section 4.8]. In the special case where aτ = bτ = n (i.e. the case considered by Hel Braun when she introduced Hermitian modular forms in [11–13]), it is often convenient to work with an unbounded realization of the space H, namely Hermitian upper half space: Hn ∶= {Z ∈ Matn×n (C) ∣ i(t Z¯ − Z) > 0} , (11)

= {Z ∈ Matn×n (C) = Hermn (C) ⊗R C ∣ Im(Z) > 0}

where Hermn denotes n × n Hermitian matrices and Im(Z) denotes the Hermitian imaginary part.

18

E. E. EISCHEN

Remark 2.2.10. The equality in Equation (11) follows from the identification Matn×n (C) ≅ Hermn (C) ⊗R C Z ↦ Re(Z) + iIm(Z) with Re(Z) and Im(Z) the Hermitian real and imaginary parts, respectively, i.e. 1 ¯ Re(Z) ∶= (Z + t Z) 2 1 ¯ Im(Z) ∶= (Z − t Z). 2i The action here of GU (ηn ) on Hn is similar to the action given above on Ha,b , A B i.e. g = ( ) acts on z ∈ Hn by C D gz = (Az + B)(Cz + D)−1 . The point 0 in Hn,n corresponds to the point i1n ∈ Hn . Note that the stabilizer of i1n ∈ Hn is the product of definite unitary groups U (n)×U (n) embedded diagonally in U (n, n). So we can identify Hn,n with U (n, n)/(U (n) × U (n)). Remark 2.2.11. If we choose a basis B as in (8) and the lattice L is as in Sec+ tion 2.2.2, then if the signature at each place is (n, n), we can define pz ∶ L[K ∶Q] → g C from Remark 2.2.9 by pz (x) = ([zτ 1n ]t x ¯, [t zτ 1n ]t x)τ ∈T

K+

for each x ∈ L and z = (zτ )τ ∈ H (with [zτ 1n ] and [t zτ 1n ] denoting n × 2n matrices). 2.3. Weights and representations. We briefly summarize key information about weights and representations, following [38, Section 2.2] and [39, Sections 2.3– 2.5]. We will use the conventions established here in our definitions of automorphic forms on unitary groups in Section 3. The setup here is also similar to [29, Sections 2.1–2.2], and [30, Sections 2.3– 2.4]. For a more detailed treatment, the reader might consult [82, Chapter II.2] or [43, Sections 4.1 and 15.3]. Let H be the subgroup of G1 (C) that preserves the decomposition VC = V1 ⊕ V2 . Let B be a Borel subgroup of H, and let T ⊂ B be a maximal torus. A choice of basis for VC that preserves the decomposition VC = V1 ⊕ V2 identifies H with ∏τ ∈ΣK (GLaτ × GLaτ ∗ ) = ∏τ ∈TK GLaτ . We write H = ∏τ Hτ , B = ∏τ Bτ , and T = ∏τ Tτ (with each of these products over τ ∈ TK ). We choose such a basis so that, furthermore, Bτ is identified with the group of upper triangular matrices in GLaτ for each τ ∈ TK and Tτ = Gamτ is identified with the group of diagonal matrices in GLaτ for each τ ∈ TK . Let NB denote the unipotent radical of B. We denote by X ∗ ∶= X ∗ (T ) the group of characters of T . Via the isomorphism B/NB ≅ T , we also view X ∗ as characters of B. We define ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ X + ∶= X + (T ) ∶= ⎨(κ1,τ , . . . , κaτ ,τ )τ ∈TK ∈ ∏ Zaτ ,τ ∣ κi,τ ≥ κi+1,τ for all i⎬ . ⎪ ⎪ ⎪ ⎪ τ ∈TK ⎩ ⎭

AUTOMORPHIC FORMS ON UNITARY GROUPS

19

We identify each tuple κ = (κ1,τ , . . . , κaτ ,τ )τ ∈TK ∈ X + with the dominant weight in X ∗ defined by aτ

κ

i,τ ∏ diag(t1,τ , . . . , taτ ,τ ) ↦ ∏ ∏ ti,τ .

τ ∈TK

τ ∈TK i=1

If κi,τ ≥ 0 for all i, τ and, furthermore κi,τ > 0 for some i, τ , then we call κ positive. Suppose R is a Zp -algebra or a field of characteristic 0. To each dominant weight κ, there is an associated representation ρκ = ρκ,R of H(R), which is obtained by application of a κ-Schur functor Sκ (as explained in, e.g. [43, Section 15.3] and summarized in [30, Section 2.4.2]). We call ρκ the representation of (highest) weight κ. If R is of characteristic 0 or of sufficiently large characteristic, the representation ρκ is irreducible (as explained in, e.g. [82, Chapter II.2]). Given a locally free sheaf of modules F over a scheme T , we denote by Sκ (F) the locally free sheaf of modules defined by Sκ (F)(SpecR) ∶= Sκ (F(SpecR)) for each affine open SpecR of T . Remark 2.3.1. In practice, the representations ρκ with which we work are built from compositions of tensor products, symmetric products, and alternating products. For an explicit description over C of the highest weight representations and highest weight eigenvectors, see [131, Section 12.6]. More generally, see also [16, Remark 3.5], which concerns any field of characteristic 0. In that setting, ρκ is isomorphic to − 1 −1 IndH B − (−κ) = {f ∶ N /H → A ∣ f (th) = κ(t) f (h) for all t ∈ T } ,

where B − is the opposite Borel, identified with the group of lower triangular matrices, and N − is its unipotent radical. Here, ρ(g)f (h) ∶= f (hg) for all g, h ∈ H. 3. Automorphic forms on unitary groups In analogue with the case of modular forms on GL2 , there are various ways to define automorphic forms on unitary groups. Each formulation has its own merits, depending on what one is trying to do. We introduce the following viewpoints: analytic functions on the symmetric spaces H (Section 3.1), functions on abelian varieties with PEL structure (Section 3.2), global sections of a sheaf on MK (Section 3.3), and functions on a unitary group (over R in Section 3.4.2 and adelically in Section 3.4.3). We conclude the section with brief discussions of q-expansions (Section 3.6) and automorphic representations (Section 3.5). 3.1. As analytic functions on Hermitian symmetric spaces. We now introduce automorphic forms as analytic functions on Hermitian symmetric spaces, a generalization of the formulation of modular forms as functions on the upper half plane. This perspective was first introduced by Braun in [11–13] and was widely employed by Shimura, e.g. in [131, Section 5]. In this section, we set H = ∏ Hmτ ,nτ τ ∈ΣK

+ GUm,n (R) ∶= ∏ GU + (mτ , nτ ). τ ∈ΣK

20

E. E. EISCHEN

Definition 3.1.1. Suppose mn ≠ 0. The factors of automorphy or automora bτ + phy factors associated to g = (gτ )τ ∈ΣK ∈ GUm,n (R), with gτ = ( τ ) and cτ dτ z = (zτ )τ ∈ΣK ∈ H at a place τ ∈ ΣK are: ● λgτ (zτ ) ∶= λτ (g, z) ∶= λ(gτ , zτ ) ∶= (bτ t zτ + aτ ) and μgτ (zτ ) = μτ (g, z) = μ(gτ , zτ ) ∶= (cτ zτ + dτ ) if we work with the bounded domain H = Hm,n ● λgτ (zτ ) ∶= λτ (g, z) ∶= λ(gτ , zτ ) ∶= (cτ t zτ + dτ ) and μgτ (zτ ) = μτ (g, z) = μ(gτ , zτ ) ∶= (cτ zτ + dτ ) if we work with the unbounded domain H = Hn (and require m = n at all places τ ) Definition 3.1.2. Suppose now that mn = 0. The factors of automorphy or a b + automorphy factors associated to g = ( ) ∈ GUm,n (R) and z ∈ H at a place c d τ ∈ ΣK are: ● λgτ (zτ ) ∶= λτ (g, z) ∶= λ(gτ , zτ ) ∶= gτ and μgτ (zτ ) = μτ (g, z) = μ(gτ , zτ ) ∶= 1 if n = 0 ● λgτ (zτ ) ∶= λτ (g, z) ∶= λ(gτ , zτ ) ∶= 1 and μgτ (zτ ) = μτ (g, z) = μ(gτ , zτ ) ∶= g if m = 0 Definition 3.1.3. The scalar factor of automorphy or scalar automorphy factor a b + associated to g = ( ) ∈ GUm,n (R) and z ∈ H at a place τ ∈ ΣK is: c d jgτ (zτ ) ∶= jτ (g, z) ∶= j(gτ , zτ ) ∶= det(μτ (g, z)). Remark 3.1.4. Note that det(λτ (g, z)) = det(gτ )ν(gτ )−nτ jτ (g, z) + for all g ∈ GUm,n (R) and z ∈ H. We also note that

λτ (gh, z) = λτ (g, hz)λτ (g, z) μτ (gh, z) = μτ (g, hz)μτ (g, z) + for all g, h ∈ GUm,n (R) and z ∈ H.

For each g = (gτ )τ ∈ΣK and z = (zτ ) ∈ ∏τ ∈ΣK Hτ , we define Mg (z) ∶= M (g, z) ∶= (λτ (g, z), μτ (g, z))τ ∈ΣK . Remark 3.1.5. We note that t Mγ (z) maps the lattice pγz (L) from Remark 2.2.9 to pz (L) and defines an isomorphism from Aγz to Az for each γ ∈ ΓK , . Note that Mg (z) ∈ ∏τ ∈ΣK GLaτ (C) × GLbτ (C). Let H(C) = ∏τ ∈ΣK GLaτ (C) × GLbτ (C), and let ρ ∶ H(C) → GL(X) be an algebraic representation with X a finite-dimensional C-vector space. Following the conventions of [131, Equations (5.6a) and (5.6b)], we define (f ∣∣ρ g)(z) ∶= ρ(Mg (z))−1 f (gz) f ∣ρ g ∶= f ∣∣ρ (ν(g)−1/2 g) , where ν(g)−1/2 g ∶= (ν(gτ )−1/2 gτ )τ ∈ΣK . Let Γ be a congruence subgroup of GU + (Q).

AUTOMORPHIC FORMS ON UNITARY GROUPS

21

Definition 3.1.6. With the conventions and notation above, we define an automorphic form of weight ρ and level Γ to be a function f ∶H→X such that all of the following conditions hold: (1) f is holomorphic. (2) f ∣∣ρ γ = f for all γ ∈ Γ. (3) If K + = Q and the signature is (1, 1), then f is holomorphic at every cusp. Remark 3.1.7. By Koecher’s principle ([100, Theorem 2.3 and Remark 10.2]), if we are not in the situation of Condition (3) (i.e. K + = Q with signature (1, 1)), then holomorphy at the boundary is automatic. We will use the terminology automorphic function to refer to a function that meets Condition (2) but not necessarily Conditions (1) and (3). Remark 3.1.8. The definition of automorphic form in [131, Section 5.2] uses ∣ρ in place of ∣∣ρ . In the text following that definition, though, Shimura explains that he works almost exclusively with ∣∣ρ in [131]. Similarly, we will work with ∣∣ρ here. Our choice in this manuscript is motivated by the fact that ∣∣ρ is what arises naturally from algebraic geometry. So this definition will be consistent with our later algebraic geometric definition of automorphic form. Remark 3.1.9. In the case of signature (n, n), automorphic forms on unitary groups were first introduced by Hel Braun in [11–13], where she considered them as functions on Hermitian symmetric spaces, in analogue with the formulation of Siegel modular forms as functions on Siegel upper half space. She called them Hermitian modular forms, in analogue with Siegel modular forms. Example 3.1.10. It is instructive to compare the case of GL2 (i.e. the familiar case of classical modular forms) with the case of GSp2 and GU (1, 1). What do you notice about the symmetric spaces in each of these cases? What do the automorphic forms on each of these groups have to do with each other? Example 3.1.11. When K is a quadratic imaginary field and ⟨, ⟩ is of signature (2, 1) on a K-vector space V , the group U (V, ⟨, ⟩) is also called a Picard modular group, and the automorphic forms on it are called Picard modular forms. 3.2. As functions on a space of abelian varieties with PEL structure. In analogue with the case of modular forms, where we move from functions on the upper half plane to functions on a space of elliptic curves with additional structure, we now reformulate our definition of automorphic forms in terms of functions on a space of abelian varieties with additional structure. Given an abelian variety A = Az (i.e. an abelian variety A together with a polarization, endomorphism, and level structure) parametrized by z ∈ ΓK /H (like in Remark 2.2.9) for some neat open compact subgroup K, let Ω ∶= ΩA/C = H 1 (A, Z) ⊗ C. Note that the action of O on Ω coming from h induces a decomposition of modules Ω = Ω+ ⊕ Ω− = ⊕τ ∈TK Ωτ , where Ω± = ⊕τ ∈ΣK Ω±τ and the rank of Ω±τ is a±τ , where (a+τ , a−τ ) is the signature of the PEL data at τ . (This is an instance of the decomposition described in

22

E. E. EISCHEN

Equation (7).) For each τ ∈ TK , we write Ωτ for Ω+τ if τ ∈ ΣK and for Ω−τ if τ ∉ ΣK . Note that O acts on Ωτ via τ . For each τ ∈ TK , we define EA = ⊕τ ∈TK IsomC (Ωτ , Caτ ), where (aτ , aτ ∗ ) is the signature at τ for each τ ∈ ΣK and IsomC denotes the C-module of isomorphisms as C-modules. We have an action of H(C) = ∏τ ∈TK GLaτ (C) on EA given by ((gτ )τ ∈TK (τ )τ ∈TK ) ((xτ )τ ∈TK ) = (τ (t gτ xτ ))τ ∈TK for each (gτ )τ ∈TK ∈ H(C) and each (τ )τ ∈TK ∈ EA . Remark 3.2.1. Note that the map pz from Remark 2.2.9 induces a choice of basis for H1 (Az , Z) (and hence an element z ∈ EAz ), via the identification of H1 (Az , Z) with pz (L). The following lemma is similar to [16, Lemma 3.7]. Lemma 3.2.2. Let K be a neat open compact subgroup of GU (A∞ ), and let Γ = ΓK ∶= K ∩ GU + (Q). Let ρ ∶ H(C) → GL(X) be an algebraic representation, with X a finite-dimensional C-vector space. There is a one-to-one correspondence between the following two sets: ● the set of X-valued automorphic functions f on H of weight ρ and level Γ ● the set of X-valued functions F on the set of pairs (A, ), with A an abelian variety parametrized by Γ/H and  ∈ EA , that satisfy (12)

F (A, g) = ρ(t g)−1 F (A, )

for all g ∈ H(C). The bijection identifies such a function F of (A, ) with the automorphic function fF defined by fF (z) = F (Az , z ). Proof. Given a function F of (A, ) as in the statement of the lemma, note that by Remark 3.1.5 (which says that the action of Γ preserves the isomorphism class of (A, )), we have F (Aγz , γz ) = F (Az , t Mγ (z)−1 z ) = ρ(Mγ (z))F (Az , z ) for all γ ∈ Γ. So the function fF is well-defined (i.e. independent of the isomorphism class of A). Now, we define a map f ↦ Ff that is inverse to the map F ↦ fF . Given A parametrized by ΓK /H, let z be such that A is isomorphic to Az , and let g ∈ H(C) be such that  = gz . It is straightforward now to check that given an automorphic function f as above, the function Ff (A, ) ∶= ρ(t g)−1 f (z) satisfies Condition (12) and provides the desired inverse map.  Thus, we may view automorphic functions as certain functions that assign an element of X to each pair (A, ). (N.B. If instead we want to restrict the discussion to automorphic forms, then we also need to check the holomorphy conditions from Definition 3.1.6.) Note that as an intermediate step in reformulating our modular functions, we could also have defined them as functions on lattices. For details of that perspective, see [32, Theorem 2.4]. We can also define automorphic forms as certain rules that take complex abelian varieties as A as input, which is the content of Lemma 3.2.3 and follows immediately from Lemma 3.2.2. First, though, we need to introduce the contracted product EA,ρ ∶= EA ×H X ∶= EA × X/ ∼,

AUTOMORPHIC FORMS ON UNITARY GROUPS

23

where the equivalence ∼ is given by (, v) ∼ (g, ρ(t g −1 )v) for all g ∈ H(C). Lemma 3.2.3. [Lemma 3.9 of [16]] Let ρ, X, K, and Γ be as in Lemma 3.2.2. There is a one-to-one correspondence between the following two sets: ● the set of X-valued automorphic functions f on H of weight ρ and level Γ ● the set of rules that assign to each point A parametrized by Γ/H an element F˜ (A) ∈ EA,ρ . The correspondence between F and F˜ is given by (, F (A, )) = F˜ (A). Remark 3.2.4. Note that we could reformulate the elements F˜ from Lemma 3.2.3 as the global sections of a vector bundle over Γ/H. When we formulate automorphic forms algebraic geometrically below, this is a perspective we will introduce. For the algebraic theory, it will be useful to develop a definition of automorphic forms that also works over other base rings. Our formulation in terms of abelian varieties provides some inspiration for how we might extend our discussion to other base rings. Remark 3.2.5. In this section, we will exclude the possibility of K + = Q with signature (1, 1) so that we do not need to worry about holomorphy at cusps (which is essentially the case of classical modular forms). In all other cases, holomorphy at cusps is automatic, as explained in Remark 3.1.7. Let E ′ be a finite extension of E that contains τ (K) for all τ ∈ TK . For each scheme S over Spec (OE ′ ,(p) ) , we define MK,S ∶= MK ×Spec(OE,(p) ) S. Following the conventions for decompositions introduced in Section 2.2.1, we can decompose the sheaf of relative differentials ΩA/S of an S-point A of MK as ΩA/S = ⊕τ ∈TK ΩA/S,τ = ⊕σ∈TK + (Ω+A/S,σ ⊕ Ω−A/S,σ ) . We define sheaves EA/S ∶= ⊕τ ∈TK IsomOS (ΩA/S,τ , OSaτ ) a±

± ∶= ⊕τ ∈ΣK IsomOS (Ω±A/S,τ , OSτ ) EA/S

For the remainder of this section, let R be an OE ′ ,(p) -algebra, let Mρ be a finite free R-module, and let ρ ∶ H(R) → GLR (Mρ ) be an algebraic representation of H(R). For any R-algebra R′ , we extend the action of H(R) linearly to an action of H(R′ ) on (Mρ )R′ ∶= Mρ ⊗R R′ . We define a contracted product EA/R′ ,ρ ∶= EA/R ×H (Mρ )R′ ∶= (EA/R′ × (Mρ )R′ ) / ∼, where the equivalence ∼ is defined by (, m) ∼ (g, ρ(t g −1 )m) for all g ∈ H. Definitions 3.2.6, 3.2.7, 3.3.1 are the analogues in our setting of Nick Katz’s definitions of modular forms in [84, Sections 1.1 – 1.5]. Notice the parallels between

24

E. E. EISCHEN

the functions introduced in Lemma 3.2.2, which are defined over C, and those defined by Definition 3.2.6, which allows us to work over other rings as well. Definition 3.2.6. An automorphic form of weight ρ and level K defined over R is a function f that, for each R-algebra R′ , assigns an element of (Mρ )R′ to each pair (A, ) consisting of an R′ -point A of MK (R′ ) and  ∈ EA/R′ such that both of the following conditions hold: (1) f (A, g) = ρ(t g −1 )f (A, ) for all g ∈ H(R′ ) and all  ∈ EA/R′ (2) If R′ → R′′ is a homomorphism of R-algebras, then f (A ×R′ R′′ ,  ⊗R′ 1) = f (A, ) ⊗R′ 1R′′ ∈ (Mρ )R′′ , i.e. the definition of f (A, ) commutes with extension of scalars for all R-algebras. Definition 3.2.6 is the generalization to our setting of the second definition of modular forms Katz gives in [84, Section 1.1]. Definition 3.2.7 is the generalization to our setting of the first definition of modular forms Katz gives in [84, Section 1.1]. Similarly to Katz’s situation, our two definitions here are equivalent and are simply two ways to formulate an automorphic form. Definition 3.2.7. An automorphic form of weight ρ and level K defined over R is a rule A ↦ f˜(A) ∈ EA/R′ ,ρ , for each R-algebra R′ and R′ -point A in MK (R′ ), that commutes with extension of scalars for all R-algebras, i.e. f˜(A ×R′ R′′ ) = f˜(A) ⊗R′ 1R′′ , for all R-algebra homomorphisms R′ → R′′ . The equivalence between Definitions 3.2.6 and 3.2.7 is given by f˜(A) = (, f (A, )). Remark 3.2.8. For readers seeing these definitions for the first time, it is a useful exercise to see what they say in the case of modular forms and then compare them with Katz’s definitions of modular forms in [84, Section 1.1]. At first glance, the appearance of EA/R might look surprising to readers who are only familiar with the modular forms setting and have not yet considered the setting of higher rank groups. Note, though, that giving an element of EA/R is equivalent to choosing an ordered basis for ΩA/R . In the setting of modular forms, A is replaced by an elliptic curve, in which case an ordered basis is simply a choice of a nonvanishing differential. In other words, in the setting of modular forms, the setup here says to consider functions on pairs (E, ω) consisting of an elliptic curve (of some specified level) and a nonvanishing differential, which coincides precisely with the conventional algebraic geometric formulation of modular forms presented in [84, Section 1.1]. 3.3. As global sections of a sheaf. Those familiar with the algebraic geometric definition of modular forms are likely accustomed to defining a modular form as a global section of a certain sheaf over a moduli space, like in [84, Section 1.5]. This is the formulation we introduce now. Let π ∶ A → MK,OE′ ,(p) denote the universal object over MK,OE′ ,(p) , and define ω ∶= π∗ ΩA/M .

AUTOMORPHIC FORMS ON UNITARY GROUPS

25

Following the conventions for decompositions introduced in Section 2.2.1, we have a decomposition ω = ⊕τ ∈ΣK (ω +τ ⊕ ω −τ ) = ⊕τ ∈TK ω τ . We define a sheaf E = EK on MK,OE′ ,(p) by E ∶= EK ∶= ⊕τ ∈TK IsomOM (ω τ , (OM )aτ ) . For motivation for introducing the sheaf E, see Remark 3.2.8. Following the conventions introduced in Section 2.2.1, we also define sheaves E ± = ⊕τ ∈ΣK Eτ± . For any representation (ρ, Mρ ) of H, we define ω ρ ∶= E ρ ∶= E ×H Mρ to be the sheaf on MK for which, for each OE ′ ,(p) -algebra R, E ρ (R) = (E(R) × Mρ ⊗ R) / ∼, with the equivalence ∼ given by (, m) ∼ (g, ρ(t g −1 )m) for all g ∈ H. Definition 3.3.1. An automorphic form of weight ρ and level K defined over R is a global section of the sheaf ω ρ = EK,ρ = E ρ on MK,R . It is a straightforward exercise to check that Definition 3.3.1 is equivalent to Definition 3.2.7. Remark 3.3.2. When ρ is an irreducible representation of positive dominant weight κ, we often write ω κ or E κ in place of E ρ . Also, note that if κ = (κτ )τ ∈TK , then ω κ = ⊠τ ∈TK ω κτ τ Remark 3.3.3. As noted in [39, Section 2.5], ω κ can be canonically identified with Sκ (ω). Remark 3.3.4. If f is an automorphic form of weight ρκ , we often say that f is of weight κ. Remark 3.3.5. By Koecher’s principle (see [100, Theorem 2.3 and Remark 10.2]), automorphic forms defined over MK extend uniquely to automorphic forms over a toroidal compactification of MK , so long as we exclude the solitary case of [K + ∶ Q] = 1 with signature (1, 1) (the familiar case of modular forms for GL2 ). Remark 3.3.6. The algebraic automorphic forms defined above over R = C are in bijection with finite sets of holomorphic automorphic forms as defined earlier. More precisely, given an automorphic form on MK (C), we get global sections over each of the connected components of MK (C). Recall from Section 2.2.5 each of these connected components is isomorphic to a quotient Γ/H. By GAGA and Lemma 3.2.3 (together with Remark 3.2.4), we then see that each algebraic automorphic form on MK (C) gives rise to a set of holomorphic automorphic forms on H, one for each component of MK (C).

26

E. E. EISCHEN

Remark 3.3.7. The moduli space MK acts as a sort of bridge that allows us to move between different base rings. This, in turn, is useful for studying algebraic aspects of automorphic forms, including certain values of automorphic forms a priori arising over C, a crucial ingredient in our study of L-functions below. In another direction, this space also serves as a starting point for defining p-adic automorphic forms. 3.4. As functions on a unitary group. From the perspective of representation theory, it is useful to define our automorphic forms as functions on a unitary group. Within this perspective, there are several equivalent ways of viewing automorphic forms, namely as C-valued functions on G(R), as C-valued functions on G(A), and as vector-valued functions on G(R) or G(A). The relationship between the C- and vector-valued approaches for functions on G(R) is also the subject of [6, Remark 1.5(2)]. The formulation of automorphic forms as functions on groups is a direct generalization of the formulation for GL2 (i.e. classical modular forms). We begin by reviewing the (likely more familiar) case of GL2 . The setup from Sections 2.2.5 and 2.2.6 enables us to extend that approach to the setting of unitary groups. 3.4.1. Review of the case of GL2 . Before proceeding with unitary groups, we first briefly review how to translate the definition of modular forms as functions on the upper half plane h into the definition in terms of functions on a group, first as functions of SL2 (R), and then as functions of GL2 (A). Although the main focus of this manuscript is automorphic forms on unitary groups, readers might find it helpful first to recall the situation for modular forms. In particular, the definitions of automorphic forms as functions on unitary groups in Sections 3.4.2 and 3.4.3 arise similarly to how they arise for modular forms and GL2 . cos θ sin θ Since SL2 (R) acts transitively on h and SO2 (R) = {αθ ∶= ( )} is − sin θ cos θ the stabilizer of i ∈ h, we can identify SL2 (R)/SO2 (R) with h. Given a modular form f of weight k and level Γ on h, we define φf ∶ Γ/SL2 (R) → C by φf (g) ∶= j(g, i)−k f (gi), where j(g, z) is the canonical automorphy factor, i.e. j(g, z) = cz + d for z ∈ h and a b g=( ) ∈ SL2 (R). Then φf is an example of an automorphic form of weight c d k and level Γ on SL2 (R), i.e. φf is a smooth function that is left Γ-invariant (i.e. φf (γg) = φf (g) for all γ ∈ Γ and g ∈ SL2 (R)) and such that each αθ ∈ SO2 (R) acts on the right by multiplication by ekiθ , i.e. φf (gαθ ) = ekiθ φf (g). Taking this a step further, we can replace SL2 (R) by GL+2 (R)/R>0 , where + denotes positive determinant (or by GL2 (R)/R× , noting that in each case we are taking the quotient of the group by its center). In this case, we define φf ∶ (Γ ⋅ Z(G)) /G(R) → C, with G denoting GL2 or GL+2 (or even SL2 , like above) and Z(G) denoting the center of G, by φf (g) ∶= det(g)k/2 j(g, i)−k f (gi), and φf (g) satisfies the same conditions as in the previous paragraph, but with SL2 replaced by G. It turns out, though, to be convenient to define automorphic forms as functions of GL2 (A). First, we note that if for each prime number p, Kp ⊂ GL2 (Zp ) is a

AUTOMORPHIC FORMS ON UNITARY GROUPS

27

compact open subgroup such that det(Kp ) = Z×p and Kp = GL2 (Zp ) for all but finitely many p, then ⎞ ⎛ GL2 (A) = GL2 (Q) ⋅ GL+2 (R) × ∏ Kp . ⎠ ⎝ p

(13) If K = ∏p Kp and

Γ = GL2 (Q) ∩ (GL+2 (R) × K),

(14)

then Equation (13) induces bijections Γ/h ↔ Z (GL2 (A)) GL2 (Q)/GL2 (A)/ (SO2 (R) × K) Γ/GL+2 (R)

↔ GL2 (Q)/GL2 (A)/K.

Note Z (GL2 (A)) GL2 (Q)/GL2 (A)/ (SO2 (R) × K) = R× GL2 (Q)/GL2 (A)/ (SO2 (R) × K) = R>0 GL2 (Q)/GL2 (A)/ (SO2 (R) × K) . These identifications enable us to reformulate f (and φf ) from above as a function ϕf ∶ GL2 (Q)/GL2 (A) → C defined by ϕf (γg∞ (gp )p ) ∶= φf (g∞ ), for all γ ∈ GL2 (Q), g∞ ∈ GL+2 (R), and gp ∈ Kp . So ϕf is a function of GL2 (A) that is left-invariant under GL2 (Q), right invariant under K, and satisfies ϕf (gαθ ) = ekiθ φf (g) for all αθ ∈ SO2 (R) (and is smooth as a function of GL2 (R)). One can also extend this treatment to include a nebentypus character. We also note that the familiar congruence subgroups Γ0 (N ), Γ1 (N ), and Γ(N ) arise in Equation ˆ respectively, (14) when K is the subgroup K(N ), K0 (N ), and K1 (N ) of GL2 (Z), consisting of matrices congruent mod N to elements of Γ0 (N ), matrices in K(N ) ∗ ∗ ∗ 0 of the form ( ), and matrices in K(N ) of the form ( ), respectively. 0 1 0 1 3.4.2. As functions of unitary groups over R. Given the expression of the symmetric space H in terms of a quotient of G(R), it is natural also to define automorphic forms as certain functions on G(R). The group K∞ is the stabilizer of a fixed point i ∈ H, and similarly to [131, Section A8.2], we identify K∞ with its image in H(C) under the embedding K∞ ↪ H(C) that maps k ∈ K∞ to Mk (i) = M (k, i). Remark 3.4.1. We use the notation i for the fixed point to emphasize the connection with the SL2 case, where H is the upper half plane, K is the group SO2 (R), and i is the number i. More generally, when working with Hn , i can be realized as the diagonal matrix i1n , and K∞ as U (n) × U (n). As discussed in [131, Section A8.2], an automorphic form f viewed as a function on H gives a corresponding function f ρ on G(R) as in Definition 3.4.2 below via f ρ (g) ∶= (f ∣∣ρ g)(i) = ρ(Mg (i))−1 f (gi) for each g ∈ G(R). In the other direction, given f ρ on G(R) as in Definition 3.4.2, we define an automorphic form on H of weight ρ by f (z) = f (gz i) ∶= ρ(Mg (i))f ρ (gz ),

28

E. E. EISCHEN

for each z ∈ H, where gz ∈ G(R) is such that gz i = z. Let ρ ∶ H(C) → GL(V ) be a finite-dimensional C-representation of H(C). We arrive at the following definition, which is similar to the definition of an automorphic form over any reductive group. Definition 3.4.2. An automorphic form of weight ρ and level Γ is a holomorphic function f ∶ G(R) → V that is of moderate growth (in the sense made explicit in Definition 3.4.6 below) and such that f (γgk) = ρ(k)−1 f (g) for each g ∈ G(R) and each k ∈ K∞ . Remark 3.4.3. In the formulation of Definition 3.4.2, the holomorphy condition is equivalent to being killed by certain differential operators (as detailed in [131, Section A8.2]). 3.4.3. As functions of unitary groups over the adeles. Given our expression in Section 2.2.5 of MK (C) as a quotient XK of the adelic points of a unitary group (and also given our expression of XK in terms of quotients of H), it is natural to reformulate our definition of automorphic form in terms of functions on adelic points of a unitary group. More generally, an automorphic form on a reductive linear algebraic group G (unitary group or not) can be formulated adelically as a function on G(A) meeting certain conditions. Indeed, this is the context in which we will be able to work with automorphic representations. The adelic formulation is particularly convenient in the context of L-functions. For example, each Euler factor at a place v in the Euler product for an automorphic L-function corresponds with information from the automorphic form at the place v. This is seen in Tate’s thesis, as well as in the discussion of the doubling method in Section 4 below. (The usefulness of the adelic formulation is also seen in Shimura’s computation of Fourier coefficients in [130], as well as the related computations of Fourier coefficients in [33, 34], where the global Fourier transform factors - under certain conditions - as a product of local Fourier transforms.) The adelic formulation also provides a convenient setting for viewing automorphic forms of different levels at once, for example in a collection of Shimura varieties ShK . Automorphic representations, discussed briefly in Section 3.5, are realized in terms of the right regular action of G(A) on certain functions φ of G(A), i.e. given g ∈ G(A), (gφ)(h) ∶= φ(hg). Definition 3.4.4. Given a subgroup U of G(A), we say that a function φ on G(A) is right U-finite, if the right translates of φ by elements of U span a finite-dimensional vector space. Similarly, we say that φ is Z(g)-finite, where Z(g) denotes the center of the complexified Lie algebra g of G(R), if the translates of φ by elements of Z(g) span a finite dimensional vector space. Definition 3.4.5. We say that a function φ on G(A) is smooth if it is C ∞ as a function of G(R) and locally constant as a function of G(Af ). Definition 3.4.6. We say that a C-valued function φ on G(A) is of moderate growth if there exist numbers n, C ≥ 0, such that ∣φ(g)∣ ≤ C∥g∥n

AUTOMORPHIC FORMS ON UNITARY GROUPS

29

−1 for all g ∈ G(A). Here, ∥g∥ ∶= ∏v max1≤i,j≤n (max (∣gij ∣v , ∣gij ∣v )), where we have identified G with its image in GLn under an embedding G ↪ GLn and gij denotes the ijth entry of the associated matrix in GLn . More generally, a tuple of Cvalued functions (φ1 , . . . , φd ) is of moderate growth if each function φi is of moderate growth.

Definition 3.4.7. For any reductive group G (unitary group or not), an automorphic form on G(A) is defined to be a function φ on G(A) such that φ(g) = φ(αg) for all α ∈ G(Q) (so we may view φ as a function on G(Q)/G(A)) and such that φ additionally is smooth, right (K × K∞ )-finite, of moderate growth, and Z(g)-finite. We say that φ is of level K if φ is fixed by K. Example 3.4.8. Observe that an adelic automorphic form on a definite unitary group of signature (1, 0) is a Gr¨ ossencharacter on A×K . (Given a number field L, we denote by AL the ring of adeles for L.) We can also consider automorphic forms of weight κ as follows. Given an irreducible C-representation (Vκ , ρκ ) of K∞ of highest weight κ, an automorphic form of weight κ is an automorphic form φ ∶ G(A) → Vκ such that φ(gu) = ρκ (u)−1 φ(g) for all g ∈ G(A) and u ∈ K∞ . We also can consider automorphic forms with nebenˆ → Q ¯ × factoring through T (Z/mZ) for some integer m typus character ψ ∶ T (Z) and T a maximal torus like in Section 2.3, i.e. automorphic forms φ for which ˆ φ(gt) = ψ(t)φ(g) for all t ∈ T (Z). Definition 3.4.9. An automorphic form ϕ on G(A) is called a cusp form (or cuspidal automorphic form) if ∫

N (Q)/N (A)

ϕ(ng)dn = 0

for each g ∈ G(A) and each unipotent radical N (Q) of each proper parabolic subgroup of G(Q). Remark 3.4.10. We make a brief note about conventions. Up to this point and sometimes going forward, we work with a group G defined over Q. At times, though, it will convenient to work with groups defined over other fields (e.g. G1 defined over K + ). This will be the case, for example, in our initial discussion of the doubling method in Section 4.3, when we introduce the original approach from [113]. If we replace G by a group H defined over a number field L, we consider the AL -points in place of the A-points, and we write Hv for the component of H at v. 3.5. Representations of G(A) and automorphic representations. In Section 4, we consider L-functions attached to certain automorphic representations. Here, we briefly establish some fundamental information about automorphic representations. The Langlands conjectures (introduced by Robert Langlands) predict a precise correspondence between Galois representations and automorphic representations. In particular, the predictions include that the L-function associated to a Galois representation is the same as a particular L-function associated to the corresponding automorphic representation. The information summarized here is not specific to unitary groups but will be helpful to have available as we move forward. For more detailed introductions to

30

E. E. EISCHEN

automorphic representations (and their role in the Langlands program), see, for example, [1, 14, 50, 51, 89, 92]. We denote by A(G) the space of automorphic forms on G(A), and we denote by A0 (G) the space of cusp forms on G(A). Let Z denote the center of G, and let χ be a character of Z(Q)/Z(A). We denote by A(G)χ the submodule of automorphic forms φ such that φ(zg) = χ(z)φ(g), and we denote by A0 (G)χ the submodule of cuspidal automorphic forms in A(G)χ . Note that G(A) acts on A(G) and A0 (G) by right translation, i.e. g ∈ G(A) acts on an automorphic form φ via (gφ)(h) ∶= φ(hg) for all h ∈ G(A). Moreover, we have that A(G) and A0 (G) are (g, K∞ )-modules (in the sense of, e.g. [92, Definition 2.3]), so are (g, K∞ ) × G(Af )-modules. Definition 3.5.1. A (g, K∞ ) × G(Af )-module (π, W ) is admissible if each irreducible representation of K × K∞ occurs with finite multiplicity in W . Definition 3.5.2. An irreducible representation π is called an automorphic representation if it is an admissible representation such that π is a subquotient of A(G). Definition 3.5.3. An automorphic representation is cuspidal (with central character χ) if it occurs as a submodule of A0 (G)χ for some character χ. We have A0 (G) = ⊕π m(π)π with the sum over all (isomorphism classes of) irreducible admissible cuspidal representations π and m(π) the multiplicity of π, which is always finite. Given an irreducible representation π, we can write π = π∞ ⊗ πf , with π∞ an irreducible representation at archimedean components and πf an irreducible representation of G(Af ). Definition 3.5.4. Let K be a compact open subgroup of G(Af ). An automorphic representation π is of level K if πfK ≠ 0 (where the superscript K denotes the subspace of K-fixed vectors). Every irreducible admissible representation π can be decomposed as a restricted tensor product π = ⊗′v πv of irreducible admissible representations πv , as made precise in Proposition 3.5.5 below. First, we briefly recall the notion of a representation (⊗′v πv , ⊗′v Wv ) of a group H = ∏′ Hv that is the restricted direct product with respect to subgroups Hv′ ⊆ Hv . Consider an infinite set of vector spaces Wv indexed by a set Σ, and suppose that for all but finitely many v ∈ Σ outside a finite subset of S ⊆ Σ, we have chosen a vector ξv○ ∈ Wv . Let ⊗′ Wv be the restricted tensor product of the vector spaces Wv , i.e. the space ⊗′ Wv is spanned by vectors of the form ⊗v ξv with ξv = ξv○ for all but finitely many v. Let (πv , Wv ) be a representation of a group Hv , and let Hv′ be a subgroup of Hv . Let H be the restricted direct product of the groups Hv with respect to the subgroups Hv′ . Suppose that for all but finitely many v, there exists a vector ξv0 ∈ Wv such that Hv′ fixes the vector ξv0 . Then we define a representation (⊗′v πv , ⊗′v Wv ) of H by (⊗′v πv )((gv )v )(⊗ξv ) = ⊗v πv (gv )(ξv ).

AUTOMORPHIC FORMS ON UNITARY GROUPS

31

Proposition 3.5.5 (Tensor Product Theorem [44]). If π is an irreducible, admissible representation of a connected reductive algebraic group H, then π is isomorphic to a restricted tensor product π ≅ ⊗′v πv , where for all but finitely many nonarchimedean v, πv is an irreducible, admissible representation of H(Kv+ ) that contains a nonzero Kv -fixed vector ξv0 , and for each archimedean v, πv is an irreducible, admissible (g, Kv )-module. In our work with the doubling method in Section 4.3, we will rely on the factorization guaranteed by Proposition 3.5.5. In practice, our restricted tensor product will be taken with respect to K. 3.6. q-expansions. For the moment, we consider the case of automorphic forms on unitary groups of signature (n, n) at infinity, and furthermore, we assume that the unitary group under consideration is quasi-split over K + . We immediately see that for each α ∈ Hermn (C), where Hermn denotes n × n Hermitian matrices, 1 ( n 0

α ) 1n

1 is an element of U (n, n)(R) and can be identified with the tuple ( n 0 ∏σ∈TK + U (n, n)(R) via

ασ ) ∈ 1n

α ↦ (ασ )σ∈TK + ,

(15)

where ασ is the matrix whose ij-th entry is τ (αij ) with αij the ij-th entry of α and τ the unique element of ΣK extending σ. There is a Z-lattice M ⊆ Hermn (C) α 1 such that ( n ) is in Γ for all α ∈ M . So if f is an X-valued automorphic form 0 1n of level Γ, then for all α ∈ M , we have f (z + α) = f (z) for all z ∈ H = ∏σ∈TK + Hn . Let M ∨ ∶= {h ∈ Hermn (C) ∣ traceK + /Q (trace(hM )) ⊆ Z} . Then as explained in [130, Lemma A1.4] (see also [131, Section 5.6]), f has a Fourier expansion f (z) = ∑ c(h)e(hz),

(16)

h∈M ∨

trace(h z )

σ σ with c(h) ∈ X, e(hz) = e ∑σ∈TK + for each h ∈ M ∨ (with hσ defined as in (15)), and z = (zσ )σ∈TK + ∈ ∏σ∈TK + Hσ . Sometimes we set

2πi

(17)

q h ∶= e(hz),

in analogue with how we write q = e2πiz in the Fourier expansion of a modular form (e.g. as in Equation (1)). By [131, Proposition 5.7], if K + ≠ Q or the signature is not (1, 1), then the Fourier coefficients c(h) of any (holomorphic) automorphic form are 0 unless h = (hσ )σ∈TK + has the property that hσ is nonnegative at each σ ∈ TK + . An automorphic form is a cusp form if the Fourier coefficients are nonzero only at positive definite matrices.

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E. E. EISCHEN

Remark 3.6.1. In this section, we have focused on Fourier expansions of automorphic forms on unitary groups that are of signature (n, n) at infinity and that, furthermore, are quasi-split over K + . More generally, though, for unitary groups not necessarily meeting these conditions, automorphic forms have Fourier–Jacobi expansions. Fourier–Jacobi expansions play a similar role to Fourier expansions, except that the coefficients are not necessarily numbers and instead are certain functions called theta functions. Like in the case of the Fourier expansions discussed above, these expansions are exponential sums. Unlike the expansions above, though, the exponential function is only applied to a subset of the coordinates parametrizing the symmetric space associated to the unitary group. (The theta function is then evaluated on other coordinates parametrizing the symmetric space.) While this is beyond the scope of this manuscript, a clear and concise summary of Fourier–Jacobi expansions, including their precise form, is provided on [46, p. 1104]. Remark 3.6.2. In analogue with the algebraic q-expansion principle for modular forms (which says that algebraic modular forms are determined by their qexpansions), Kai-Wen Lan has proved an algebraic Fourier–Jacobi principle for unitary groups [99, Proposition 7.1.2.14]. Lan has also proved that the algebraically defined Fourier–Jacobi coefficients agree with the analytically defined Fourier–Jacobi coefficients [98]. These facts will be important in our discussion of algebraicity for Eisenstein series in Section 4.5 4. Automorphic L-functions for unitary groups One reason we care about automorphic forms and automorphic representations is that they can be convenient tools for proving results about L-functions attached to particular arithmetic data (such as Galois representations), as mentioned in Section 1. Like in familiar cases, e.g. the Riemann zeta function from Section 1, the L-functions with which we work have Euler products (analogous to ζ(s) = ∏p prime (1 − 1/ps )−1 for Re(s) > 1), functional equations (analogous to Z(s) = s Z(1 − s), where Z(s) = π − 2 Γ ( 2s ) ζ(s)), and meromorphic continuations to C. 4.1. L-functions. Given a number field L, for each unramified finite place v, let Frobv denote a(n arithmetic) Frobenius conjugacy class in Gal(L/L). Let S denote a finite set of places of K containing the set S∞ of archimedean places of K and the set Sram of ramified places of K. By the Chebotarev density theory, each continuous Galois representation ρ ∶ Gal(L/L) → GLn (C) is completely determined by the set of elements ρ(Frobv ) with v ∉ S. The Euler product for the Artin L-function associated to ρ is LS (s, ρ) ∶= ∏ Lv (s, ρ), v∉S

where −1

Lv (s, ρ) ∶= det (1 − ρ(Frobv )qv−s )

,

AUTOMORPHIC FORMS ON UNITARY GROUPS

33

with qv the number of elements in the residue field at v. Since ρ(Frobv ) is a semisimple conjugacy class and since any semi-simple conjugacy class A is completely determined by its characteristic polynomial det(t − A), we see that Lv (s, ρ) is is independent of our choice of representative for ρ(Frobv ). Under certain conditions, the Langlands conjectures assert that suitable cuspidal automorphic representations π have associated Galois representations ρπ . (In the one-dimensional case, class field theory achieves this. For an introduction to the Langlands conjectures, beyond the scope of this manuscript, see e.g. [1, 97].) More precisely, the prediction is that under suitable conditions, a certain semisimple conjugacy class σv (π) associated to π = ⊗v πv (in the Langlands dual group L G, as explained in, e.g. [8, 23]) is the same as the conjugacy class of ρπ (Frobv ), in which case we obtain as an immediate consequence that LS (s, ρπ ) = LS (s, π) ∶= ∏ Lv (s, πv ) v

Lv (s, πv ) ∶= det(1 − σv (π)q −s )−1 where σv (π) is the semi-simple conjugacy class associated to πv . Remark 4.1.1. In general, the L-function and associated Euler factors also take as input a representation r of the Langlands dual group L G, i.e. one considers LS (s, π, r) ∶= ∏v det(1 − r(σv (π))q −s )−1 . Going forward, though, we will work with the doubling method, which only concerns the standard representation. Hence, in this manuscript, we always take r to be the standard representation, and we omit it from the notation in the input to our L-function. If you are familiar with L-functions associated to modular forms, then you probably expect information about an associated Hecke algebra to show up in the definition of the local Euler factors Lv (s, πv ). Indeed, the semi-simple conjugacy class associated to πv is associated to information from a Hecke algebra, which is generated by double coset operators, in analogue with the situation from modular forms. That is, for g ∈ G(Af ) and open compact subgroups K1 and K2 , we define the Hecke operator [K1 gK2 ] from automorphic forms of level K1 to level K2 to be the action of the double coset K2 gK1 = ⊔gi K1 given by [K2 gK1 ]f (h) = ∑ f (hgi ). i

As noted in [31, Equation (22)], we have ∗

[K2 gK1 ]f = ∑ [gi ] f, i

with [gi ]∗ the pullback of the map (10). When the level K is clear from context, we set T (g) ∶= [KgK]. For additional details of Hecke operators on automorphic forms on unitary groups, see, e.g. [31, Sections 2.6.8 and 2.6.9]. There are also some standard choices of Hecke operators, generalizing the familiar Hecke operators T from modular forms to the setting of unitary groups. In analogue with the case of modular forms (on GL2 ), the Hecke operators generate a Hecke algebra. We briefly summarize some key aspects of Hecke algebras

34

E. E. EISCHEN

here. (See, e.g. [4, Section 1.4] for a more detailed summary of Hecke algebras in our setting.) Given a field L of characteristic 0 and a place v of K + , we denote by H(G(Kv+ ), Kv , L) the algebra of compactly supported L-valued functions that are both left and right Kv -invariant, together with the operation ∗ defined for f1 , f2 ∈ H(G(Kv+ ), Kv , L) by (f ∗ h)(g) = ∫G(K + ) f (x)h(x−1 g) dx. This is the Hecke algebra v of G with respect to Kv over L. (For a more detailed introduction, see, e.g. [52, Sections 3 and 4] or [4, Section 2.1].) Given a smooth representation of V of G(Kv+ ) over L (i.e. V is an L-vector space on which G(Kv+ ) acts continuously and such that each vector of V is invariant under some compact open subgroup of G(Kv+ )), V Kv has a natural structure of a H(G(Kv+ ), Kv , L)-module. If v does not ramify in K and Kv is a maximal compact hyperspecial subgroup of G(Kv+ ), then H(G(Kv+ ), Kv , L) is a commutative algebra. If, furthermore, V is an unramified representation of G(Kv+ ) (i.e. dim V Kv = 1), then the Hecke algebra H(G(Kv+ ), Kv , L) acts on V Kv through a character H(G(Kv+ ), Kv , L) → L. Thanks to a structure theorem proved by Satake, we have that H(G(Kv+ ), Kv , L) is finitely generated as an L-algebra, and, furthermore, when we specialize to the case of a representation π like above, the collection of the values of these characters on a set of generators for H(G(Kv+ ), Kv , L) is an element of the semisimple conjugacy class σv (π) associated to π above [118]. Remark 4.1.2. For context, we remark briefly on how this description relates to the familiar cases of GL1 (Hecke characters) and GL2 (modular forms). In the case of GL1 , then π = ⊗v πv is a Hecke character A× → C× (where A× denotes the ideles over Q). At each v where πv is unramified, πv is completely determined by its value on a uniformizer v , and the set of values σv (π) ∶= πv (v ) completely determines π. In the case of GL2 , we consider an irreducible cuspidal automorphic representation π = ⊗p πp generated by a holomorphic cuspform f (q) = q + ∑n≥2 an q n of weight k and level 1. (You can generalize to newforms of level N , if you would like.) In this case, we have Tp f = ap f for all prime numbers p, and σp (πp ) is in the conjugacy class of diag(αp , βp ) ∈ GL2 (C), with αp βp = pk−1 and αp + βp = ap , i.e. the Euler factor at p is the familiar Euler factor at p for the L-function attached to the cusp form f . 4.2. Strategy for proving algebraicity of certain values of L-functions. In Section 1, we observed a connection between rationality of certain values of particular L-functions (e.g. the Riemann zeta function) and certain Eisenstein series. For example, in Equation (1), we observed that the Riemann zeta function arises as the constant term of the Eisenstein series G2k , and we noted that rationality of ζ(1 − 2k) then followed from properties of G2k . From Section 1, it seems like if we want to prove rationality of L-functions more generally, an approach might be: Try to relate the L-function in question to an Eisenstein series. That is an awfully vague strategy, though. Supposing some version of this approach even works more generally, which Eisenstein series should we use, and in what sense should we “relate” our L-function to this Eisenstein series?

AUTOMORPHIC FORMS ON UNITARY GROUPS

35

Motivated by the example of Shimura’s proof in [124] of algebraicity of certain values of the Rankin–Selberg convolution that we recounted in Section 1, we propose the following recipe for investigating rationality properties: (1) Find a Petersson-style pairing of automorphic forms (integrated against an Eisenstein series) that factors into an Euler product, has a functional equation, and can be meromorphically continued to all of C. (2) Prove the rationality of Eisenstein series occurring in that pairing. (3) Express a familiar automorphic L-function in terms of that pairing, similarly to Equation (3), to obtain an expression analogous to Expression (2). In addition to the case of Rankin–Selberg convolutions of modular forms from the introduction, this approach has been carried out in a number of cases, including by Shimura for Rankin–Selberg convolutions of Hilbert modular forms [126] and by Shimura’s PhD student Jacob Sturm for a higher-rank generalization of the Rankin–Selberg convolution [137]. We refer to that method as the Rankin–Selberg method. Although it turns out that the precise pairing in the Rankin–Selberg method does not work in certain settings (including for unitary groups, the main focus of this manuscript), we have substitutes in certain situations. For unitary groups, we have the doubling method, which is discussed in detail in Section 4.3. In addition to Shimura’s work in the setting of unitary groups (e.g. in as compiled in [130, 131]), Harris has proved extensive results employing the doubling method to investigate the algebraicity of values of L-functions associated to automorphic representations of unitary groups (in particular, [65, Theorem 3.5.13]), as well as associated results about rationality of Eisenstein series, including [63, 66, 67, 69, 71]. One of the keys that enables proofs of algebraicity of certain values of L-functions (and associated Eisenstein series) is the realization of automorphic forms as sections over an integral model for a PEL-type moduli space, as in Section 3.3. This recipe has also been carried out with certain other pairings. For example, Harris proved algebraicity of critical values of L-functions attached to Siegel modular forms in [62], again relying heavily on the geometry of an associated Shimura variety. Recently, using a pairing introduced by Aaron Pollack in [112], the author, Giovanni Rosso, and Shrenik Shah have proved algebraicity of critical values of Spin L-functions for GSp6 . Perhaps surprisingly, that pairing involves working with an Eisenstein series defined on a group that has no known Shimura variety or moduli problem. As these examples show, the above recipe is powerful, even though specific details of how it is carried out in different situations can vary significantly. Remark 4.2.1. Finding a pairing in terms of which one can express an Lfunction, as required for the above three-step recipe, is highly non-trivial and is a serious research problem on its own. So even though we just provided several examples where the recipe has been successfully carried out, one is not guaranteed to be in a position to carry out even the first step. Furthermore, even if one has such a pairing, it is not guaranteed that the pairing will be suitable for obtaining results about algebraicity. (Such pairings are often useful for studying analytic aspects of L-functions, hence the interest in them even when they seem not to be well-suited to proving algebraicity results.) For example, the pairing introduced in [5] for Spin L-functions for GSp2n , for n = 3, 4, 5, appears not to be amenable to proving algebraicity results. That is also the case for the approach to “twisted

36

E. E. EISCHEN

doubling” in [17]. In general, proofs of algebraicity rely on the pairing and its input having an algebraic or geometric interpretation. Remark 4.2.2. Here, we are focusing on the use of Eisenstein series as a tool for proving algebraicity of certain values of L-functions. Eisenstein series also play a key role in governing analytic behavior, like the functional equation and meromorphic continuation, of L-functions. This is the case, for example, with the Langlands– Shahidi method, which realizes the reciprocals of certain L-functions in the constant terms of Fourier expansions of Eisenstein series [122, 123]. There are also other approaches to investigating algebraicity, e.g. [76]. Remark 4.2.3. As noted in Section 1.1.1, all known methods for constructing p-adic L-functions are adaptations of the specific techniques used to prove algebraicity results for the corresponding C-valued L-functions. In fact, Haruzo Hida’s approach to constructing p-adic Rankin–Selberg L-functions in [73] builds directly on Shimura’s proof of algebraicity of Rankin–Selberg convolutions summarized above. Similarly, the construction of p-adic L-functions for unitary groups in [31,41] builds on the work with the doubling method. As noted in Remark 4.3.3, the doubling method specializes in its simplest case to a variant of Damerell’s formula, a key ingredient in the proof of algebraicity for L-functions associated to Hecke characters of CM fields, which in turn led to Katz’s construction of p-adic L-functions associated to Hecke characters of CM fields [85]. Remark 4.2.4. Our focus is on algebraicity of the values of L-functions. In analogue with Expression (2), we are especially interested in deriving results of the form L(π, m) ¯ ∈ π c Q, ⟨ϕ, ϕ⟩ where L(π, m) is the value at some integer m of an L-function associated to an irreducible cuspidal automorphic representation π that contains a cusp form ϕ and π c denotes a power of the transcendental number π. (Results of this form arise from applications of integral representations of L-functions, which express certain automorphic L-functions as integrals of automorphic forms, like in the doubling method in Section 4.3 below and the Rankin–Selberg method. This approach is seen, for example, in the beginnings of the introductions to [45, 114].) In Section 1, we also mentioned Deligne’s conjectures about the meanings of the values of L-functions at certain points. Although our focus in the next sections will be on proving algebraicity results like those above, we briefly bring up this still more challenging aspect now. Deligne defined an integer m to be critical for an L-function if neither m nor the point symmetric to it with respect to the central point of the L-function are poles of the Γ-factors of the L-function (analogues of the factors that show up in the functional equation for the Riemann zeta function like at the beginning of Section 4). Deligne predicts that the critical values (i.e. values at the critical points) of an L-function associated to a motive M are rational multiples of the period of M (an algebraic invariant of M coming from cohomology). In general, the values of automorphic L-functions are not readily seen to satisfy Deligne’s conjecture, even when we have algebraicity results. By exploiting geometry in the unitary setting, though, Harris explained connections between his algebraicity results and motivic periods in the case of K + = Q in [65], and this work was later

AUTOMORPHIC FORMS ON UNITARY GROUPS

37

extended by his PhD student Lucio Guerberoff to the case of K + of degree > 1 in [61, Theorem 4.5.1]. These results and related ones are surveyed in [74]. Remark 4.2.5. In the above three-step recipe, we skipped over the fact that the Eisenstein series with which we need to work are not necessarily holomorphic. At the beginning of this manuscript, in Equation (1), we recalled the holomorphic Eisenstein series G2k . Shortly after that, though, we encountered a nonholomorphic Eisenstein series in Equation (3). The prototypical example from the setting of modular forms of weight λ, level N , and character χ is χ(n) , Eλ,N (z, s, χ) = ∑ λ 2s (0,0)≠(m,n)∈Z×Z (mN z + n) ∣mN z + n∣ which converges for Re(2s) > 2 − λ. In fact, the Eisenstein series denoted by E ∗ in Equation (3) is the holomorphic modular form Eλ,N (z, χ) defined to be spe(z,s,χ)

λ,N ∗ cialization to s = 0 of Eλ,N (z, s, χ) ∶= 2L(2s+λ,χ) , but we also need to work with ∗ Eisenstein series at points s where the Eisenstein series Eλ,N (z, s, χ) is not holomorphic. Such Eisenstein series can be obtained via application of the Maass–Shimura (r) (r) operator δλ from Section 1, which is defined by δλ ∶= δλ+2r−2 ○ ⋯ ○ δλ+2 ○ δλ with 1 λ ∂ δλ ∶= 2πi ( 2iy + ∂z ). We have

E

∗ Eλ+2r,N (z, −r, χ) =

Γ(λ) (r) ∗ (−4πy)r δλ Eλ,N (z, χ), Γ(λ + r) (r)

∗ i.e. Eλ+2r,N (z, −r, χ) is a scalar multiple of the term δλ E in Expression (3) from Section 1. ∗ (z, s, χ) are clearly not holomorphic, alAt s ≠ 0, the Eisenstein series Eλ,N ∞ though they are C . This might seem potentially problematic for carrying out our recipe concerning algebraicity. Fortunately, though, the operators δλ can be reformulated geometrically over a modular curve MN (C), and that geometric formulation provides enough structure to preserve algebraic aspects of modular forms. In brief, the map of sheaves

Eλ → Eλ+2 corresponding to δλ is induced by the composition of maps (18)

∇

∼

1 ⊗k 1 ⊗k 1 ⊗k ω ⊗k ↪ (HdR )  → (HdR ) ⊗ ΩMN → (HdR ) ⊗ ω ⊗2 ↠ ω k+2 ,

1 ∶= R1π∗ (Ω●A/MN ), ∇ is the Gauss–Manin connection, the isomorphism is where HdR the identity map tensored with the Kodaira–Spencer morphism, and the surjection is the projection onto the first factor of the Hodge decomposition ω ⊕ H 0,1 . Each of these maps is algebraic and can be defined over any OE,(p) -algebra, except for the final one. The final map has enough structure, though, that it preserves essential algebraic structure. In particular, when applied to a holomorphic modular form, δλ preserves algebraicity at CM points, which is essential for proving algebraicity via the aforementioned Damerell’s formula. In addition, the holomorphic projection (r) (r) H(gδλ E) is a holomorphic modular form with the property ⟨f, H(gδλ E)⟩ = (r) (r) ⟨f, gδλ E⟩, so ⟨f, H(gδλ E)⟩ still becomes a scalar multiple of ⟨f, f ⟩ and we still obtain Expression (2). The composition of maps (18) can naturally be formulated over our PEL moduli space MK to construct differential operators on unitary groups, and these operators

38

E. E. EISCHEN

preserve similar algebraic properties of automorphic forms in this setting. We do not elaborate on them here. Note, though, that more book-keeping is involved in this setting, the resulting forms in this case can be vector-valued, and the Kodaira– Spencer map in this case is ∼

ΩMK → ⊗τ ∈TK + ω +τ ⊗ ω −τ . The Maass–Shimura differential operators and their algebraicity properties have been studied in detail for automorphic forms on unitary groups and related cases in, e.g. [30, 32, 38, 62, 64, 85, 105, 128, 129, 131]. Remark 4.2.6. In the remainder of this section, we highlight the ingredients needed to carry out our three-step recipe for proving algebraicity for automorphic L-functions associated to cuspidal automorphic representations of unitary groups. We emphasize aspects that are unlikely to be viewed as “straightforward” extensions of the tools occurring in the setting of Shimura’s work with Rankin–Selberg convolutions of modular forms, and consequently, a detailed introduction to the doubling method forms a large portion of the remainder of this section. 4.3. The doubling method, in the setting of unitary groups. We now introduce the aforementioned doubling method, which provides an integral representation of our L-functions, i.e. a global integral that unfolds to a product of local integrals in terms of which we can realize an Euler product for our L-functions. The approach we present here is due to [47, 113]. There are also helpful notes in [24]. The doubling method is an instance of a pullback method or Rankin–Selberg style integral, i.e. we will be integrating a pullback of an Eisenstein series against a pair of cusp forms. As noted in [113, Section 1], the doubling method is based on the Rankin–Selberg method (a generalization to GLn of the Rankin–Selberg convolution from Equation (3)), but unlike the Rankin–Selberg method, it does not require uniqueness of Whittaker models (and hence allows us to work with unitary groups). The doubling method can also be formulated algebraically geometrically so that we can study algebraic aspects of values of L-functions. 4.3.1. Setup for the doubling method. To begin, let U ∶= U (V, ⟨, ⟩) be as in Definition 2.1.1 (with V an n-dimensional vector space over the CM field K). Let W = V ⊕ V . We define a Hermitian pairing ⟨, ⟩W on W by ⟨(u, v), (u′ , v ′ )⟩W ∶= ⟨u, u′ ⟩V − ⟨v, v ′ ⟩V for all u, v, u′ , v ′ ∈ V . Following the conventions of [130, Equation (1.1.7)], we sometimes denote the pairing ⟨, ⟩W by ⟨, ⟩ ⊕ −⟨, ⟩.

(19) Let

UW ∶= U (W, ⟨, ⟩W ). Then UW is a unitary group of signature (n, n) at each place, and we have an embedding U × U = U (V, ⟨, ⟩) × U (V, −⟨, ⟩) ↪ UW ,

(20) ′

via (g, g )(u, v) ∶= (gu, g ′ v) for all g, g ′ ∈ G and u, v ∈ V . This doubling of our group is what leads to the name doubling method. (This is the formulation of unitary

AUTOMORPHIC FORMS ON UNITARY GROUPS

39

groups in the original setup for the doubling method in [47, 113]. In Section 4.4, we will also address the relationship with the unitary groups associated to PEL data, by giving a set of PEL data that induces an analogous embedding.) Remark 4.3.1. The doubling method is formulated in terms of integrals over the groups U (AK + ) and U (Kv+ ) and subquotients of these groups. For these integrals, we work with a Haar measure on these groups. For the purposes of this manuscript, we do not need to be more precise. The reader seeking more information about appropriate choices of normalizations could consult [31, Section 1.4.3]. Let P be the Siegel parabolic subgroup of UW preserving the maximal isotropic subspace V Δ ∶= {(v, v) ∣ v ∈ V } ⊂ W. We also define VΔ ∶= {(v, −v) ∣ v ∈ V } ⊂ W. With respect to the decomposition W = V Δ ⊕ VΔ , we have that for each K + algebra R, (21) P (R) A = {( 0

t

1 0 )( n A¯−1 0

X ) ∣ A ∈ GLK⊗K + R (V ⊗K + R) and X ∈ Hermn (K ⊗K + R)} . 1n

Given a K + -algebra R and a character ψ of (K ⊗K + R)× , we view ψ as a character of P (R) via A ( 0

t

B ) ↦ ψ(det A). A¯−1

4.3.2. Siegel Eisenstein series. Let χ ∶ K × /A×K → C× be a unitary Hecke character. Given s ∈ C, let fs,χ ∶ UW (AK + ) → C be an element of U

(AK + ) (χ∣ ⋅ ∣−s ) K+ )

I(s, χ) ∶= IndP W (A

∶= {f ∶ UW (AK + ) → C∣f (ph) = χ(p)∣p∣−s+n/2 f (h)} ,

with the absolute value here denoting the adelic norm. (In practice, we usually restrict ourselves to the smooth, K-finite functions in I(s, χ).) The elements α ∈ UW (AK + ) act on elements U

(AK + ) (χ∣ ⋅ ∣−s ) K+ )

f ∈ IndP W (A via

(αf )(h) = f (hα) for all h ∈ UW (AK + ). The induced representation I(s, χ) factors as a restricted U (Kv+ ) (χv ∣ ⋅ ∣−s ), with the induced representation for the local tensor product ⊗v IndP W (Kv+ ) groups defined analogously.

40

E. E. EISCHEN

We define a Siegel Eisenstein series on UW by Efs,χ (h) ∶=

(22)

fs,χ (γh).

∑

γ∈P (K + )/UW (K + )

This series converges absolutely and uniformly for Re(s) > n2 . In addition, Efs,χ is a meromorphic function of s and is an automorphic form in h. Define a Hecke character χ ˇ by ∶= χ(¯ χ(x) ˇ x)−1 , let NP denote the unipotent radical of P , and let w be the Weyl element interchanging V Δ and VΔ . The Eisenstein series Efs,χ (h) satisfies the functional equation Efs ,χ (h) = EM (s,χ)fs,χ (h), where M (s, χ) ∶ I(s, χ) → I(1−s, χ) ˇ is the intertwining operator defined for Re(s) > n by 2 [M (s, χ)fs,χ ](h) = ∫ =∫

NP (AK + )

fs,χ (wnh) dn

Hermn (AK )

1 X fs,χ (w ( ) h) dX. 0 1

4.3.3. The doubling integral. Let π be an irreducible cuspidal automorphic representation of U , and let π ′ denote the contragredient of π. Given ϕ ∈ π and ϕ′ ∈ π ′ , we define the doubling integral by (23) Z (ϕ, ϕ′ , fs,χ ) ∶= ∫

[U×U](K + )/[U×U](AK + )

Efs,χ (g, h)ϕ(g)ϕ′ (h)χ−1 (det h) dg dh.

The notation [U × U ] here denotes the image of U × U inside UW . Remark 4.3.2. The character χ is absent from the original treatment in [113], but it is a straightforward exercise to show that if we define the Eisenstein series as above to include χ, then the doubling integral must take this form. Indeed, we include χ in our analysis below. The integral Z (ϕ, ϕ′ , fs,χ ) inherits the analytic properties of Eisenstein series. In particular, Z (ϕ, ϕ′ , fs,χ ) extends to a meromorphic function of s and satisfies the functional equation Z (ϕ, ϕ′ , fs,χ ) = Z (ϕ, ϕ′ , M (s, χ)fs,χ ) . The doubling integral Z (ϕ, ϕ′ , fs,χ ) plays a role in our setting analogous to the role played by the Rankin–Selberg integral for GLn . Remark 4.3.3. Note that in the case of definite groups (i.e. signature(n, 0)), we can choose our input data so that the integral unfolds into a finite sum of values of automorphic forms and can be reinterpreted as values of those automorphic forms at CM points of the corresponding Shimura variety on which the Eisenstein series is defined. In the special case where n = 1, this allows us to recover a variant of Damerell’s formula, an expression of values L(0, χ), with χ a Hecke character of type A0 on a CM field, as a finite sum of values of a Hilbert modular form at CM Hilbert–Blumenthal abelian varieties. Damerell initiated the study of the algebraicity properties of these values, and this study was later completed by Goldstein–Schappacher [59, 60], Shimura [124], and Weil [142]. For those seeking

AUTOMORPHIC FORMS ON UNITARY GROUPS

41

more details, note that a nice history of the development of Damerell’s formula is provided in [77, §5]. 4.3.4. Unfolding the doubling integral into an Euler product. Theorem 4.3.4. We have the equality Z (ϕ, ϕ′ , fs,χ ) = ∫

U(AK + )

fs,χ (g, 1)⟨π(g)ϕ, ϕ′ ⟩ dg,

where ⟨ϕ, ϕ′ ⟩ ∶= ∫

(24)

U(K + )/U(AK + )

ϕ(g)ϕ′ (g) dg.

Remark 4.3.5. The pairing ⟨, ⟩ is the unique U -invariant pairing of π with π ′ , up to a constant multiple. We likewise also write ⟨, ⟩ for the corresponding integral over U (Kv+ ). Remark 4.3.6. It is natural to wonder what we might get if we replace π ′ by some other irreducible cuspidal automorphic representation π ˜ ≇ π ′ and ϕ′ ∈ π ˜ . In ′ that case, we would have ⟨π(g)ϕ, ϕ ⟩ = 0, and as a consequence of Theorem 4.3.4, we would then have that Z (ϕ, ϕ′ , fs,χ ) is identically zero. Remark 4.3.7. Going forward, we suppose we have factorizations π = ⊗′v πv and π ′ = ⊗′v πv′ (where these restricted tensor products are over the places of K + ), and we suppose ϕ = ⊗′v ϕv with ϕv ∈ πv and ϕ′ = ⊗′v ϕ′v with ϕ′v ∈ πv′ . Let S be the set of all finite places of K + that ramify in K or where π, π ′ , and χ are ramified. For all finite places v ∉ S, let ϕv ∈ πv and ϕ′v ∈ πv′ be normalized Kfixed vectors such that ⟨ϕv , ϕ′v ⟩ = 1. We also suppose that fs,χ = ⊗′v fs,χv , with H(K + ) fs,χv ∈ Iv (s, χ) ∶= IndP (K +v) (χv ∣ ⋅ ∣−s ). v

Corollary 4.3.8. The integral Z (ϕ, ϕ′ , fs,χ ) factors as an Euler product (over all places v of K + ): Z (ϕ, ϕ′ , fs,χ ) = ⟨ϕ, ϕ′ ⟩ ∏ Zv (ϕv , ϕ′v , fs,χv ), v

where ′

∫U(Kv+ ) fs,χv (gv , 1)⟨πv (gv )ϕ, ϕ ⟩ dgv , ⟨ϕv , ϕ′v ⟩ with the denominator equal to 1 for all v ∉ S. Zv (ϕv , ϕ′v , fs,χv ) =

Proof of Corollary 4.3.8. By the uniqueness of the U -invariant pairing between π and π ′ , we have ⟨π(g)ϕ, ϕ′ ⟩ = c ∏⟨πv (gv )ϕv , ϕ′v ⟩, v

with c =

⟨ϕ,ϕ′ ⟩ . ∏v ⟨ϕv ,ϕ′v ⟩

The corollary now follows immediately from Theorem 4.3.4. 

Proof of Theorem 4.3.4. The theorem will follow from an analysis of the orbits of U × U acting on X ∶= P /UW , which we now explain. We write [U × U ]γ for the stabilizer of a point γ ∈ X. We re-express the Eisenstein series Efs,χ (h) as Efs,χ (h) =

⎞ ⎛ fs,χ (γγ0 h) , ∑ ⎠ [γ]∈P (K + )/UW (K + )/[U×U](K + ) ⎝[γ0 ]∈[U×U]γ (K + )/[U×U](K + ) ∑

42

E. E. EISCHEN

where, for each γ ∈ UW (K + ), [U × U ]γ (K + ) is the stabilizer of P (K + )γ ∈ P (K + )/UW (K + ) under the right action of [U × U ](K + ) and [γ] is the orbit of P (K + )γ in P (K + )/UW (K + ) under the right action of [U × U ](K + ). Inserting this expression for the Eisenstein series into the doubling integral, we have Z (ϕ, ϕ′ , fs,χ ) =

∑

∑

[γ]∈P (K + )/UW (K + )/[U×U](K + ) [γ0 ]∈[U×U]γ (K + )/[U×U](K + )

(∫

[U×U](K + )/[U×U](AK + )

=

fs,χ (γγ0 (g, h)) ϕ(g)ϕ′ (h)χ−1 (det h) dg dh)

∑

I(γ),

[γ]∈P (K + )/UW (K + )/[U×U](K + )

where for each γ ∈ UW (K + ), I(γ) ∶= ∫

[U×U]γ (K + )/[U×U](AK + )

fs,χ (γ(g, h)) ϕ(g)ϕ′ (h)χ−1 (det h) dg dh.

Note that for each γ ∈ UW (K + ), [U × U ]γ (K + ) = {(g, h) ∈ [U × U ](K + ) ∣ P (K + )γ(g, h) = P (K + )γ} = {(g, h) ∈ [U × U ](K + ) ∣ γ(g, h)γ −1 ∈ P (K + )} . First we consider the case of the identity γ = γ0 = 1 ∈ UW (K + ). The stabilizer of the identity γ0 = 1 ∈ UW (K + ) is [U × U ]γ0 (K + ) = P (K + ) ∩ [U × U ](K + ) = {(g, g) ∣ g ∈ U (K + )} =∶ U Δ (K + ). In this case, we have fs,χ (γ0 (g, h)) = fs,χ ((g, h)) = fs,χ ((h, h)(h−1 g, 1)) = χ(det(h))fs,χ (h−1 g, 1). So I(γ0 ) = ∫

U Δ (K + )/[U×U](AK + )

fs,χ (h−1 g, 1) ϕ(g)ϕ′ (h) dg dh.

Noting that U × U ≅U Δ × (U × 1) ≅ U × U (g, h) ↔(h, h)(h−1 g, 1) ↔ (h, h−1 g) and writing g1 = h−1 g, we have I(γ0 ) = ∫ =∫

U(AK + ) U(AK + )

∫

U(K + )/U(AK + )

fs,χ (g1 , 1) π(g1 )ϕ(h)ϕ′ (h) dh dg1

fs,χ (g1 , 1) ⟨π(g1 )ϕ, ϕ′ ⟩ dg1 .

The other orbits are negligible, i.e. the stabilizer of a point in the orbit contains the unipotent radical of a proper parabolic subgroup of [U × U ](K + ) as a normal subgroup. (This definition of negligible comes from [113, §1, p.2 ].) We will now show that for each negligible orbit [γ], I(γ) = 0, thus completing this proof (and also justifying the name negligible). For the remainder of this proof, suppose that γ belongs to a negligible orbit, i.e. there is a proper

AUTOMORPHIC FORMS ON UNITARY GROUPS

43

parabolic subgroup of U × U whose unipotent radical N γ is a normal subgroup of [U × U ]γ . We write I(γ) = ∫

[U×U]γ (AK + )/[U×U](AK + )

I(γ, h1 , h2 ) dh1 dh2 ,

where I(γ, h1 , h2 ) =∫

fs,χ (γ(g1 , g2 )(h1 , h2 )) ϕ(g1 h1 )ϕ [U×U]γ (K + )/[U×U]γ (AK + ) × χ−1 (det(g2 h2 )) dg1 dg2 .

′

(g2 h2 )

We will now prove that I(γ, h1 , h2 ) = 0, thus completing the proof. Let M ∶= N γ /[U × U ]γ . We further decompose that integral as I(γ, h1 , h2 ) = ∫

M (K + )/M (AK + )

I(γ, h1 , h2 , m1 , m2 ) dm1 dm2 ,

where I(γ, h1 , h2 , m1 , m2 ) = ∫

N γ (K + )/N γ (AK + )

Fs,χ,γ,h1 ,h2 ,m1 ,m2 (n1 , n2 ) dn1 dn2 ,

Fs,χ,γ,h1 ,h2 ,m1 ,m2 (n1 , n2 ) ∶= fs,χ (γ(n1 , n2 )(m1 ,m2 )(h1 , h2 ))ϕ(n1 m1 h1 )ϕ′ (n2 m2 h2 )χ−1 (det(n2 m2 h2 )). Since N γ is the unipotent radical of a proper parabolic subgroup of U × U , we can write N γ = N1 × N2 , with each subgroup Ni the unipotent radical of a parabolic subgroup of U . So noting that det ni = 1 and writing γ(n1 , n2 ) = pγ for some p ∈ P , we have I(γ, h1 , h2 ) =∫

M (K + )/M (AK + )

fs,χ (pγ(m1 , m2 )(h1 , h2 ))

× I1 (m1 , h1 )I1 (m2 , h2 )χ−1 (det(m2 h2 )) dm1 dm2 , where, for i = 1, 2, Ii (mi , hi ) = ∫

Ni (K + )/Ni (AK + )

ϕi (ni mi hi ) dni ,

with ϕ1 = ϕ and ϕ2 = ϕ′ . Since N is nontrivial, at least one of the subgroups Ni is nontrivial. So since ϕ and ϕ′ are cusp forms, Ii (mi , hi ) = 0 for at least one of i = 1, 2. So I(γ) = 0 for each γ in a negligible orbit. Finally, to conclude the proof of Theorem 4.3.4, we need to show that if γ ∉ [1], then γ ∈ X = P /UW is in a negligible orbit. First, note that since UW acts transitively on the space of maximal isotropic subspaces of W and P stabilizes the maximal isotropic subspace V Δ , we identify X = P /UW with the variety that parametrizes the maximal isotropic subspaces of W . Under this identification, the U × U -orbit of P 1 ∈ P /UW corresponds to the U × U -orbit of the maximal isotropic subspace V Δ . We need to show that the other orbits are negligible, i.e. the stabilizer contains the unipotent radical of a proper parabolic subgroup of U × U as a normal subgroup. Let V + = V × ⟨0⟩ ⊆ W , and V − = ⟨0⟩ × V ⊆ W . By [113, Lemma 2.1], for

44

E. E. EISCHEN

any maximal isotropic subspace L ⊆ W , we have dimK L ∩ V + = dimK L ∩ V − , and furthermore, the U × U -orbits of maximal isotropic subspaces parametrized by X are the spaces Xd ⊆ X defined by Xd ∶= {L ⊆ W ∣ L is a maximal isotropic subspace such that dimK (L ∩ V + ) = d} Observe that the orbit of V Δ is then X0 . For the remainder of the proof, fix d > 0, and let L ∈ Xd . Let R ⊂ U × U be the stabilizer of L. To complete the proof, we will find a proper parabolic subgroup of U × U whose unipotent radical is normal in R. Let P ± denote the parabolic subgroup of U preserving the flag V ⊃ π ± (L) ⊃ L± , where π ± is the orthogonal projection of W onto V ± and L± ∶= L ∩ V ± . Note that since d > 0, P ± is a proper parabolic subgroup of U . So P + × P − is a proper parabolic subgroup of U × U . Let Nd = N + × N − , with N ± the unipotent radical of P ± . To conclude the proof, it suffices to show that Nd is a normal subgroup of R. Note that R ⊂ P + × P − . So since Nd is normal in P + × P − , it suffices to show that N ⊂ R. Note that N ± induces the identity on π ± (L)/L± . Now consider the projections π± ∶ π ± (L) → π ± (L)/L± . Then π± (nv) = π± (v) for all v ∈ L± and n ∈ Nd . Finally, note that for the maximal isotropic subspace, there is an isometry ι ∶ π ± (L)/L± → π ∓ (L)/L∓ determined by ι(π+ v+ ) = π− v− if and only if (v+ , v− ) ∈ L. So for all n = (n+ , n− ) ∈ Nd , ι(π+ (n+ v+ )) =  ι(π+ v+ ) = π− (v− ) = π− (n− v− ). So Nd ⊂ R, as desired. Remark 4.3.9. The doubling method actually applies not only to unitary groups but to classical groups more generally. Indeed, our approach above often did not use features unique to unitary groups but relied primarily on an embedding U × U ↪ UW of a group into its “doubled group” and the following two properties, which can also be extended to other classical groups: (1) The stabilizer of the identity γ0 = 1 is U Δ . (2) All the orbits other than the orbit of γ0 = 1 are negligible. For example, explicit constructions for symplectic and orthogonal groups are also discussed in [113, Section 2]. In practice, [113] treats the input and construction axiomatically and then shows how this axiomatic treatment can be applied for each of the classical groups. 4.3.5. From local doubling integrals to Euler factors for standard Langlands Lfunctions. We already saw that Z(fs,χ , ϕ, ϕ′ ) satisfies a functional equation, but we want to relate it to a familiar L-function. In Sections 4.3.6 through 4.3.9, we give a brief overview of the realization of the Euler factors L(s, π, χ) for the standard Langlands L-function in terms of the local integrals Zv (ϕv , ϕ′v , fs,χv ). For the remainder of the manuscript, we set fv ∶= fs,χv . As noted in [35, Section 4.1], at all finite places v ∉ S, the strategy for computing the local integrals arising in the doubling method for unitary groups (as well as for symplectic groups) is to reduce them to integrals computed by Godement and Jacquet for GLn [81]. For certain other groups, the computations reduce to other integrals computed by Godement and Jacquet in [53]. 4.3.6. Factors at split nonarchimedean places. At split finite places v, U (Kv+ ) ≅ GLn (Kv+ ), as in Isomorphism (5). In [113, Section 3] (see also [24, Section 2.3] and [31, Section 4.2.1]), Ilya Piatetski-Shapiro and Stephen Rallis explain how to choose

AUTOMORPHIC FORMS ON UNITARY GROUPS

45

a local Siegel section fv so that the computation reduces to integrals computed by Godement and Jacquet for GLn in [53]. At these places, we have (25)

1 dn,v (s, χv ) Zv (ϕv , ϕ′v , fv , s) = Lv (s + , πv , χv ) , 2

where n−1

dn,v (s, χv ) = dn,v (s) = ∏ Lv (2s + n − r, χv ∣Kv+ ηvr ) , r=0

Kv+

attached by local class field theory to the extension ηv is the character on Kw /Kv+ (with w a prime of K lying over v). (N.B. In spite of how it might look, the shift by 1/2 in Equation (25) above and in Equation (26) below are not typos, but rather are consistent with the corresponding statements in [102, Theorem 3.1 and end of Section 3].) Here Lv (s + 12 , πv , χv ) ∶= L(s, BCK/K + (πv ) ⊗ (χv ○ det)), where BCK/K + (πv ) denotes the base change of πv from U /K + to GLn /K (as developed in [96]). 4.3.7. Factors at inert nonarchimedean places. At inert nonarchimedean primes, the unitary group is not isomorphic to a general linear group. So we will need a different approach from the one we just employed for split finite primes. Note that there is a close relationship between the symplectic group Sp2n and the unitary group U (n, n) of signature (n, n). In [113, Section 6], Piatetski-Shapiro and Rallis computed the local integrals for Sp2n . (They did this by choosing a local section fv that allows them to reduce their calculation to a calculation of the aforementioned integrals for GLn computed Godement and Jacquet.) In [102, Section 3], Jian-Shu Li completed the computation of the local integrals for unitary groups by reduction to these calculations for Sp2n completed by Piatetski-Shapiro–Rallis, and similarly to the result at split finite places above, we obtain 1 dn,v (s, χv ) Iv (ϕv , ϕ′v , fv , s) = Lv (s + , πv , χv ) . (26) 2 Remark 4.3.10. For those interested in constructing p-adic L-functions, note that one must modify the input at p and compute the corresponding integrals at p. The approach in [31] assumes the prime p splits, so the local integrals in that paper reduce immediately to (elaborate!) calculations for general linear groups. In the case of p inert, at least for unitary groups of signature (a, a), in the spirit of Li’s calculation mentioned above, a starting point would be to adapt the calculations carried out by Zheng Liu for symplectic groups in [106], as discussed in [35, Section 4.1]. 4.3.8. Factors at ramified nonarchimedean places. At ramified places, it is posU(K + ) sible to construct a section fv ∈ IndP (Kv+ ) (χv ∣ ⋅ ∣−s ) such that v

∫

U(Kv+ )

fv (g, 1)⟨πv (g)ϕv , ϕ′v ⟩dg

is constant (see, e.g. [113, p. 47] or [31, Section 4.2.2]). 4.3.9. Factors at archimedean places. The author and Liu computed the archimedean zeta integrals in terms of a product of Γ-factors, without restriction on the weights or signature [37]. For over 30 years, though, the question of how to compute them remained open, except in special cases. The first progress in this direction was due to Paul Garrett, who showed that the archimedean zeta integrals

46

E. E. EISCHEN

are algebraic up to a particular power of the number π [48]. In addition, for certain choices of Siegel sections fv , he computed the archimedean integrals when a certain piece of the weight of the cuspidal automorphic representation π is one-dimensional. (Prior to Garrett’s contribution, Shimura had addressed the case of scalar weight [130, 131].) The work in [37] builds on Liu’s work for the symplectic case in [107], as well as work for unitary groups of signature (n, 1) in [104, 108]. 4.4. The doubling method, revisited from the perspective of algebraic geometry. Section 4.3 introduced the doubling method, as it was originally presented in [47, 113]. This formulation is not only important for obtaining an Euler product, but also for establishing key analytic properties, such as the functional equation and meromorphic continuation of the L-function. On the other hand, if we want to study algebraic aspects of the values of the L-function, it is not necessarily immediately apparent how to translate this analytic formulation (in terms of integrals) into algebraic information. The key is to reformulate the doubling pairing in terms of PEL data and the corresponding moduli spaces. This is the approach employed by Harris in his study of critical values of L-functions associated to automorphic representations of unitary groups in [65], and it is also the approach used in the constructions of p-adic Lfunctions for unitary groups in [31]. This approach is also the main topic of [71]. Now, the groups arising in the full PEL moduli problem are general unitary groups, i.e. have similitude factors. As seen in [65, Sections 3.1 and 3.2], though, it is straightforward to adapt the doubling method to the case of similitude groups. The inclusion U × U ↪ UW from (20) extends to an inclusion G(U × U ) ↪ GUW ∶= GU (W, ⟨, ⟩W ),

(27) where

G(U × U ) ∶= {(g, h) ∈ GU (V, ⟨, ⟩) × GU (V, −⟨, ⟩) ∣ ν(g) = ν(h)} . Then the doubling integral defined in Equation (23) is replaced by a similar integral (with the input automorphic forms now defined on similitude groups) but over (Z (AK + ) (G (U × U )) (K + )) / (G (U × U )) (AK + ) , where Z denotes the identity component of the center and is identified with the center of GU (V, ⟨, ⟩) = GU (V, −⟨, ⟩). Likewise, the invariant pairing from Equation (24) is replaced by an integral over Z(A)GUV (K + )/GUV (AK + ), where GUV ∶= GU (V, ⟨, ⟩). We also note that the definition of the Eisenstein series from Equation (22) can be extended to GUW . Similarly to the unitary setting above, we denote by GP the parabolic subgroup of GUW preserving V Δ in VΔ ⊕ V Δ . In place of Equation (21), we have (28) GP (R) λA = {( 0

t

1 0 )( n A¯−1 0

X ) ∣ λ∈R× , A∈GLK⊗K + R (V ⊗+K R), X ∈Hermn (K ⊗K + R)} . 1n

AUTOMORPHIC FORMS ON UNITARY GROUPS

47

Given a K + -algebra R and a character ψ of (K ⊗K + R)× , we obtain a character of GP (R) via λA p=( 0

t

B ) ↦ ψ(det A(p))∣λ(p)∣−ns K+ , A¯−1

where A(p) = A and λ(p) = λ. Correspondingly, as detailed in, e.g. [33, Section 3.1], we replace I(s, χ) from above by the induction from GP (AK + ) to GUW (AK + ) of −s+n/2 , p ↦ χ(det A(p))∣λ(p)∣−ns K + ∣ det A(p)∣

and we sum over GP (K + )/GUW (K + ) instead of P (K + )/UW (K + ) in the definition of the Eisenstein series. Remark 4.4.1. From the discussion in Section 2.2.2, it is straightforward to write the doubling integral and subsequent discussion in terms of algebraic groups defined over Q. 4.4.1. PEL data for the doubling method. Ultimately, we are interested in studying algebraicity of values of the above L-functions. For this, it will be useful to relate the above approach to the PEL data and moduli spaces introduced in Section 2.2. Similarly to [31, Section 3.1], we choose PEL data of unitary type that will induce an embedding of moduli spaces corresponding to the inclusion (27). In particular, we fix the following PEL data of unitary type, following the conventions of Section 2.2.2: D+ ∶= (K, ∗, OK , V, ⟨, ⟩Q , L, h) is a PEL datum of unitary type associated with (V, ⟨, ⟩) D− ∶= (K, ∗, OK , V, −⟨, ⟩Q , L, z ↦ h(¯ z )) D+,− ∶= (K × K, ∗ × ∗, OK × OK , V ⊕ V, ⟨, ⟩Q ⊕ −⟨, ⟩Q , L ⊕ L, z ↦ h(z) ⊕ h(¯ z )) , where ⟨, ⟩Q ⊕ −⟨, ⟩Q is defined as in (19) D ∶= (K, ∗, OK , V ⊕ V, ⟨, ⟩Q ⊕ −⟨, ⟩Q , L ⊕ L, z ↦ h(z) ⊕ h(¯ z ))

We write G+ , G− , G+,− , and G for each of the algebraic groups corresponding, respectively, to this PEL data, as in Equation (6). The subscripts have been chosen to emphasize the connection with the unitary similitude groups immediately above, in particular: ● G+ is defined in terms of a pairing ⟨, ⟩ ● G− is defined in terms of −⟨, ⟩ and is canonically isomorphic to G+ ● G+,− is canonically embedded in G+ × G− ● G is defined in terms of ⟨, ⟩ ⊕ −⟨, ⟩, and G+,− is canonically embedded inside G These four groups play analogous roles to the groups GU (V, ⟨, ⟩), GU (V, −⟨, ⟩), G(U × U ), and GU (W, ⟨, ⟩W ), respectively, from above. We choose compatible compact open subgroups K± , K+,− , K inside the groups G± , G+,− , G. We continue to use the subscripts here to refer to objects corresponding to these PEL data. Like in [31, Equation (38)], for each OE,(p) -scheme S (or each E-scheme S, depending on which moduli problem we are working with), this induces natural S-morphisms (29)

M+,−,K+,− → MK

(30)

M+,−,K+,− → M+,K+ ×S M−,K− .

Via the map (29), we can also pullback (i.e. restrict) automorphic forms, viewed as global sections of the vector bundle E ρ , on MK to obtain automorphic forms

48

E. E. EISCHEN

on M+,−,K+,− . Via the map (30), we can evaluate those automorphic forms on products of pairs of abelian varieties with PEL structure. As an exercise to check their understanding, the reader might write down the corresponding maps of abelian varieties with PEL structure induced by these two maps of moduli spaces. This setup, together with the algebraicity of the Eisenstein series discussed in Section 4.5, enables us to view the doubling integral as a pairing between algebraic automorphic forms over the moduli space M+,−,K+,− . 4.5. Algebraicity of Eisenstein series. The strategy outlined above for studying rationality (or algebraicity) of values of automorphic L-functions relies on the rationality (or algebraicity) of the Eisenstein series that are input to the doubling method. Thanks to the algebraicity properties of the Maass–Shimura operators mentioned earlier that can be used to study Eisenstein series at points s where they might not be holomorphic, we focus our discussion here on holomorphic Eisenstein series. In the case of modular forms, the algebraic q-expansion principle tells you that modular forms f are determined by their q-expansions (i.e. value of f at the Tate curve with the canonical differential, as in [84, Sections 1.1 and 1.2] or [25]). More precisely, we have the following proposition. Proposition 4.5.1 (Corollary 1.6.2 of [84]). Let f be a holomorphic modular form of level n, defined over a Z[1/n]-algebra S. Suppose the q-expansion coefficients of f lie in a Z[1/n]-subalgebra R ⊆ S. Then f is defined over R. Because the algebraic q-expansions of a modular form f agree with the analytic q-expansions (i.e. Fourier expansions) of f (as noted in [85, Equation (1.7.6)]), we can use the Fourier coefficients of a modular form to determine that it is defined over, say, Q or some localization of a ring of integers. In other words, to determine that f is defined over some Z[1/n]-algebra R, we can complete the following two steps: (1) Determine the Fourier expansion of f . (2) Observe that the coefficients of f are contained in R (and apply Proposition 4.5.1). Fortunately, as noted in Remark 3.6.2, Lan has proved an analogous algebraic Fourier–Jacobi expansion principle for automorphic forms on unitary groups [99, Proposition 7.1.2.14], and he has also shown that their analytically defined Fourier– Jacobi expansions also agree with algebraically defined Fourier–Jacobi expansions [98]. (For unitary groups of signature (n, n) at each archimedean place, like in Section 3.6, we can write this in a variable q that makes it reasonable to call this a q-expansion principle again.) This takes care of Step (2) in our setting. Still, we are left with a potentially challenging problem: choosing an Eisenstein series and determining its Fourier cofficients. The choice of Eisenstein series is influenced by our use of the doubling method introduced in Section 4.3. To assist with computing the Fourier coefficients, we have a convenient result of Shimura. Like above, we will work with Siegel Eisenstein series Ef ∶= Efs,χ associated to f ∶= fs,χ ∈ I(s, χ) factoring as f = ⊗v fv , with fv ∈ Iv (s, χ). Recall that Ef is an automorphic form on a group that is of signature (n, n) at each real place, which can be identified with the matrices preserving η = ηn at each real place (as in Remark 2.1.4) and that is assumed to be quasi-split. In this case, Ef has

AUTOMORPHIC FORMS ON UNITARY GROUPS

49

an adelic Fourier expansion, similar to the one in Equation (16), whose Fourier coefficients each factor as a product of local Fourier coefficients. More precisely, we have the following result for Ef∗ (x) ∶= Ef (xηf−1 ), where ηf ∶= ∏v∤∞ η. Theorem 4.5.2 (Section 18.10 of [130]). Ef∗ (q) =

∑

cβ q β ,

β∈Hermn (K)

with cβ complex numbers that factor as a precisely determined constant multiple of a product (over all places v of K + ) of local Fourier coefficients cβ,v defined to be the Fourier transform of fv at β. Choosing a section f that factors over places of K + not only played a role in producing the Euler product for our L-function, but also plays a crucial role in breaking our global Fourier coefficients into a product of local Fourier coefficients. Remark 4.5.3. We stated Theorem 4.5.2 in such a way as to highlight the fact that each Fourier coefficient decomposes as a product local factors, at the cost of omitting the setup necessary to give more precise statements about adelic Fourier expansions. Since we do not need those details anywhere else in this manuscript, we simply refer the reader to [34, Section 2.2.4] or [33, Section 3.1] for the details specific to our situation here or to [130, Section 18] for a more general and more detailed discussion of the adelic Fourier coefficients of Eisenstein series. In loc. cit. and [127], Shimura also computed local Fourier coefficients for these Eisenstein series (for specific choices of fv ), which one can use to determine the ring of definition of the Eisenstein series. For other purposes, such as constructing p-adic families of Eisenstein series and p-adic L-functions (like in [31, 33, 34]), one needs different choices at primes dividing p from those chosen by Shimura, and determining them constitutes a significant step in the p-adic situation. We also note that Shimura’s rationality results for Eisenstein series were further extended by Harris in [63, 69]. Remark 4.5.4. At the risk of opening a can of worms, we briefly elaborate on details for the statement of Theorem 4.5.2. The reader wishing to delve still further into this subject after reading this remark, though, should heed Remark 4.5.3. While the equation in Theorem 4.5.2 is formatted to highlight connections with earlier material, the longer-form version of the equation is ¯ −1 1 m th )( Ef (( 0 0 1

0 )) = ∑ h β∈Herm

n (K)

cβ (h)eAK + (trace(βm)),

for each h = (hv )v ∈ GLn (AK ) and m ∈ Hermn (AK ), with cβ (h) a complex number that depends only on f , β, and h. Here, eAK + ((x)) = ∏v ev (xv ) for each adele x = (xv )v , ev (xv ) = e2πixv if v ∣ ∞, and ev (xv ) = e−2πiy with y ∈ Q such that traceKv+ /Qq (xv ) − y ∈ Zp at each place v over a rational prime q. The product cβ (h) then is given as a rational multiple of the product ∏v cβ,v (h), with cβ,v the Fourier transform of fv : cβ,v (h) ∶= ∫

¯ −1 0 −1 1 mv t h )( v fv (( )( 0 1 1 0 0 Hermn (K⊗Kv+ )

0 )) ev (−traceβmv ) dmv . hv

50

E. E. EISCHEN

It turns out that the numbers cβ,v are completely determined by their values at 1. In the case of modular forms of weight 2k and level 1, the sections fv can be ord (n) chosen so that for n ≥ 1, cn,∞ (1) = n2k−1 and cn,p (1) = ∑j=0p p−j(2k−1) , so cn (1) = ord (n)

n2k−1 ∏p ∑j=0p p−j(2k−1) = ∑d∣n d2k−1 , the usual divisor function occurring in the Fourier expansion of the weight 2k, level 1 Eisenstein series. For further details, the reader is again urged to consult the aforementioned references.

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10.1090/surv/279/02

Automorphic Forms and the Theta Correspondence Wee Teck Gan

1. Lecture 1: The Ramanujan Conjecture In the first lecture, we shall recall the Ramanujan conjecture for classical modular forms and its reformulation in the language of cuspidal automorphic representations of PGL2 . For more details on this transition and reformulation, please take a look at [6, §4.27 and §4.28]. This reformulation allows one to readily generalize the conjecture to the setting of cuspidal automorphic representations of general connected reductive group G over a number field k. We will then discuss the unitary group analog of a construction of Roger Howe and Piatetski-Shapiro [15] which gives a definitive counterexample to the extended Ramanujan conjecture for the unitary group U3 in three variables. Their original construction gave a counterexample on the group Sp4 , but we will use the same idea to produce a counterexample on U3 via the method of theta correspondence. The unitary case we discussed here was actually considered by Gelbart-Rogawski in [13]. 1.1. The Ramanujan conjecture. About a century ago, Ramanujan considered the following power series of q Δ(q) = q ⋅ ∏ (1 − q n )24 . n≥1

Expanding this out formally, we have: Δ(q) = ∑ τ (n)q n = q − 24q 2 + . . . n>0

Ramanujan conjectured that for all primes p, ∣τ (p)∣ ≤ 2 ⋅ p11/2 . This is the Ramanujan conjecture in question. More generally, for any holomorphic cuspidal Hecke eigenform φ of weight k (and level 1) on the upper half plane h, with Fourier coefficients {an (φ)}n≥1 , the Ramanujan-Petersson conjecture asserts that ∣ap (φ)∣ ≤ 2p(k−1)/2 for all primes p. For Hecke eigenforms, the Fourier coefficients ap (φ) are also the eigenvalues of the Hecke operator Tp . Hence, the Ramanujan conjecture concerns bounds on cuspidal Hecke eigenvalues. It was eventually proved by Deligne [5] as a consequence of his proof of the Weil’s conjectures. ©2024 American Mathematical Society

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WEE TECK GAN

1.2. Cuspidal automorphic representations. The classical theory of modular forms can now be subsumed in a representation theoretic setting. The details of this transition can be found in [6, §3 and §4]. Let us briefly recall this. Let ● k be a number field with ring of adeles A = ∏′v kv , which is a restricted direct product of the local completions kv for all places v of k. ● G be a connected reductive linear algebraic group over k; for simplicity we may take G to be semisimple; ● ρ ∶ G ↪ GLn be a faithful algebraic representation over k, giving rise to a system of open compact subgroups Kv = ρ−1 (GLn (Ov )) ⊂ G(kv ) for almost all v, where Ov is the ring of integers of kv ; moreover, for almost all v, Kv is hyperspecial. ● G(A) = ∏′v G(kv ) be the adelic group, which is a restricted direct product of G(kv ) relative to the family {Kv } of open compact subgroups for almost all v; ● [G] = G(k)/G(A) be the automorphic quotient; the locally compact group G(A) acts on [G] by right translation and there is a G(A)-invariant measure (unique up to scaling). For the theory of classical modular forms, one is taking k = Q and G = PGL2 . Using the natural identification SL2 (Z)/h ≅ PGL2 (Z)/ PGL2 (R)/ O2 (R) ≅ PGL2 (Q)/ PGL2 (A)/ O2 (R) ⋅ ∏ PGL2 (Zp ), p

a classical modular form φ on h corresponds to a function f ∶ PGL2 (Q)/ PGL2 (A) → C defined by

√ f (g) = (φ∣k g)( −1).

Replacing φ by f allows one to extend the notion of modular forms to the setting of general reductive groups G. More precisely, an automoprhic form on G is a function f ∶ [G] → C satisfying some regularity and finiteness properties: ● ● ● ●

f f f f

is is is is

smooth right Kf -finite (where Kf = ∏v 0, i.e. if its matrix coefficients decay sufficiently quickly [6, §4.27]. For unramified representations, we can make do with the following ad-hoc definition. Definition: A Kv -unramified representation π(χv ) as above is said to be tempered if χv is a unitary character, i.e. ∣χv ∣ = 1. As an example, the trivial representation of Gv is certainly Kv -unramified and 1/2 1/2 is contained in the principal series I(δBv ). Since δBv is not a unitary character, the trivial representation of Gv is not tempered (unless Gv is compact). From the point of view of matrix coefficients, the trivial representation has constant matrix coefficients which certainly do not decay at all. 1.5. Representation theoretic formulation of the Ramanujan conjecture. We can now formulate the Ramanujan conjecture for cuspidal representations of a quasi-split group G. Conjecture 1.2 (Naive Ramanujan Conjecture). Let π = ⊗′v πv be a cuspidal representation of a quasi-split G. Then for almost all v, πv is tempered. The transition from Ramanujan’s original conjecture to this representation theoretic formulation is not clear at all and was first realized by Satake. This transition is described in [6, §4.28]. The condition that G be quasi-split is there because the conjecture may be easily shown to be false without it. For example, if G is an anisotropic group, so that [G] is compact, then the constant functions, which afford the trivial representation, are certainly cusp forms but the trivial representation is not tempered (as we have remarked above).

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1.6. Counterexample of Howe-PS. The naive Ramanujan conjecture is expected to be true when G = GLn , where it has in fact been shown in many cases. However, in the 1977 Corvallis proceedings, Howe and Piatetski-Shapiro [15] constructed an example of a cuspidal automorphic representation of the split group Sp4 which violates the naive Ramanujan conjecture above. This has led to the following tweak: Conjecture 1.3 (Revised Ramanujan Conjecture). Let π = ⊗′v πv be a globally generic cuspidal representation of a quasi-split G. Then for almost all v, πv is tempered. Note that for G = GLn , all cuspidal representations are known to be globally generic (see [6, §6]). In this series of lectures, we will follow the same basic idea of Howe-PS and construct a similar counterexample for a quasi-split unitary group U3 . 1.7. -Hermitian spaces and unitary groups. Let us first recall some basics about unitary groups and the underlying Hermitian spaces. We first begin with an arbitrary field F of characteristic 0, and let E be an ´etale quadratic F -algebra (so E is either a quadratic field extension or E = F × F ), with Aut(E/F ) = ⟨c⟩ acting on E by x ↦ xc . With  = ±, let V be a finite-dimensional -Hermitian space over E. This means that V is equipped with a nondegenerate E-sesquilinear form ⟨−, −⟩, so that ⟨v1 , v2 ⟩c =  ⋅ ⟨v2 , v1 ⟩ and

⟨λv1 , v2 ⟩ = λ ⋅ ⟨v1 , v2 ⟩

for v1 , v2 ∈ V and λ ∈ E. If  = +1, one gets a Hermitian form; if  = −1, one gets a skew-Hermitian form. Observe that if δ ∈ E × is a trace 0 element (to F ), then multiplication-by-δ takes an -Hermitian form to a −-Hermitian form. If (V, ⟨−, −⟩) is an -Hermitian space, let U(V ) be its associated isometry group: U(V ) = {g ∈ GL(V ) ∶ ⟨gv1 , gv2 ⟩ = ⟨v1 , v2 ⟩ for all v1 , v2 ∈ V }. Because U(V, ⟨−, −⟩) = U(V, δ ⋅ ⟨−, −⟩) for trace 0 elements δ ∈ E × , the class of isometry groups one obtains for Hermitian and skew-Hermitian spaces is the same. These isometry groups are called the unitary groups: each of them is a connected reductive group with a 1-dimensional anisotropic center Z ≅ E 1 (the torus defined by the norm 1 elements of E × ). If n = dimE V , then we say that V is maximally split (or simply split) if V contains a maximal isotropic subspace of dimension [n/2]. In that case, U(V ) is quasi-split and thus possesses a Borel subgroup B defined over F . Such a Borel subgroup is obtained as the stabilizer of a maximal flag of isotropic subspaces: 0 ⊂ X1 ⊂ ⋅ ⋅ ⋅ ⊂ X[n/2] with dim Xj = j. Let us highlight the special case when E = F ×F is the split quadratic F -algebra. Then V is an F × F -module and hence has the form V0 × V0∨ for an F -vector space

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V0 . Up to isomorphism, any Hermitian E-space is isomorphic to the one defined by ⟨(v1 , l1 ), (v2 , l2 )⟩ = (l2 (v1 ), l1 (v2 )) ∈ E. Then we note that U(V ) ≅ GL(V0 ), via the natural action of GL(V0 ) on V0 × V0∨ . We will largely ignore such split cases in the following, as they can be easily handled. 1.8. Invariants of spaces. Assume now that F is a local field and E/F a quadratic field extension. A Hermitian space V of dimension n has a natural invariant known as the discriminant: disc(V ) ∈ F × /NE/F (E × ). More precisely, if {v1 , . . . , vn } is an E-basis and A = (⟨vi , vj ⟩) is the matrix of inner products of basis elements, then disc(V ) = (−1)n(n−1)/2 ⋅ det(A) ∈ F × /N E × . Using the nontrivial quadratic character ωE/F of F × /NE/F (E × ), it is convenient to encode disc(V ) as a sign: (V ) = ωE/F (disc(V )) = ±. When F is nonarchimedean, Hermitian spaces are classified by the two invariants dim(V ) and (V ), so that for each given dimension, there are 2 Hermitian spaces V + and V − . When F = R, however, Hermitian spaces are classified by their signatures (p, q), with p + q = dim V . Likewise, if W is a skew-Hermitian space, then disc(W ) ∈ δ dim W ⋅ F × /NE/F (E × ) and one sets (W ) = ωE/F (δ − dim W ⋅ disc(W )) = ±. Note however that (W ) depends on the choice of δ. Assume now that F = k is a number field and E/k is a quadratic field extension. Then a Hermitian space V over E is determined by its localizations {V ⊗k kv } as v runs over all places of k; in other words, the Hasse principle holds. Note that half the places v will split in E and for these, the local situation is the split case (so the split case cannot be ignored for global considerations). A family of local Hermitian space {Vv } (relative to Ev /kv ) arises as the family of localizations of a global Hermitian space relative to E/k if and only if: ● for almost all v, (Vv ) = +; ● ∏v (Vv ) = 1. There is an analogous statement for skew-Hermitian spaces which can be formulated, using the observation that multiplication by a nonzero trace 0 element of E switches Hermitian spaces and skew-Hermitian spaces.

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1.9. Examples. Let us consider some examples in small dimension relative to a quadratic field extension of local fields E/F . ● When dim V = 1, one may identify V with E (by the choice of a basis element), and a Hermitian form is given by (x, y) ↦ axy c , with a ∈ F × ; we denote this rank 1 Hermitian space by ⟨a⟩. Then V + = ⟨1⟩ and ⟨a⟩ ≅ ⟨b⟩ if and only if a/b ∈ N E × . In any case, U(⟨a⟩) = E 1 ⊂ E × for any a ∈ F × . We take this occasion to take note of a canonical isomorphism (given by Hilbert’s Theorem 90): E × /F × ≅ E 1 defined by x ↦ x/xc . We will frequently use this isomorphism to identify E 1 as E × /F × . ● When dim V = 2, V + is the split Hermitian space V + = Ee1 ⊕ Ee2 , such that ⟨e1 , e1 ⟩ = ⟨e2 , e2 ⟩ = 0 and ⟨e1 , e2 ⟩ = 1. This 2-dimensional split Hermitian space is also called a hyperbolic plane and is sometimes denoted by H. The stabilizer of the isotropic line Ee1 is a Borel subgroup, containing a maximal torus T = {t(a) = (

a

(ac )−1

) ∶ a ∈ E ×}

and with unipotent radical U = {u(z) = (

1 z ) ∶ z ∈ E, T rE/F (z) = 0}. 1

The other Hermitian space V − is anisotropic (it has no nonzero isotropic vector). One can describe it in terms of the unique quaternion division F -algebra D. For this, one fixes an F -algebra embedding E ↪ D (unique up to conjugacy by D× by the Skolem-Noether theorem) and regard D as a 2-dimensional E-vector space by left multiplication. One can find d ∈ D such that d normalizes E and dxd−1 = xc for x ∈ E, and write D = E ⋅ 1 ⊕ E ⋅ d. Then one defines a Hermitian form on D by ⟨x, y⟩ = projection of x ⋅ y onto E ⋅ 1. In terms of this model, one can describe the unitary group U(V − ) as : U (V − ) ≅ (E × × D× )1 /∇F × = {(e, d) ∶ NE (e) ⋅ NB (b) = 1}/∇F × , where ∇(F × ) = {(t, t−1 ) ∶ t ∈ F × }. The element (e, d) ∈ E × × B × gives rise to the operator x ↦ e ⋅ x ⋅ b−1 on D. If one replaces D by the matrix algebra M2 (F ), the above description of V − and its isometry group U(V − ) gives rise to a description of V + = H and U(V + ). This shows that U(V + ) is intimately connected with GL2 and U(V − ) with D× .

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● Consider now the case when dim V = 3. Let H denote the hyperbolic plane introduced above and recall the 1-dimensional Hermitian space ⟨a⟩. The sum ⟨a⟩ ⊕ H is then a 3-dimensional Hermitian space with (⟨a⟩ + H) = ωE/F (a). As a runs over F × /N E × , one obtains the two equivalence classes of 3-dimensional Hermitian spaces (in the nonarchimedean case). Thus, both these spaces are split and since V + ≅ a ⋅ V − for a ∈ F × ∖ N E × , we see that U(V + ) ≅ U(V − ) is a quasisplit group. Since this unitary group will play a big role in the Howe-PS construction, let us set up some more notation about it. Let ⟨a⟩ = E ⋅ v0 and H = Ee ⊕ Ee∗ with e and e∗ isotropic vectors. Then with respect to the basis {e, v0 , e∗ } of ⟨a⟩ ⊕ H, the inner product matrix takes the form ⎛ a ⎜ ⎝ 1

1 ⎞ ⎟. ⎠

The Borel subgroup B = T U stabilizing the isotropic line E ⋅ e is then upper triangular, with elements of T and U taking the form ⎛ a b t(a, b) = ⎜ ⎝ and

c −1

(a )

⎞ ⎟ ⎠

with a ∈ E × and b ∈ E 1

c c ⎛ 1 0 z ⎞ ⎛ 1 −a x 1 1 0 ⎟⋅⎜ u(x, z) = ⎜ ⎝ 1 ⎠ ⎝

∗ ⎞ x ⎟, 1 ⎠

with x, z ∈ E and T rE/F (z) = 0. 1.10. Basic idea of Howe-PS construction. We can now give a brief summary of the Howe-PS construction of cuspidal representations of U3 which violate the naive Ramanujan conjecture. Let E/k be a quadratic field extension of number fields. Consider a skewHermitian space (W, ⟨−, −⟩) of dimension 3 over E. We would like to produce some cusp forms on U(W ) which violates the naive Ramanujan conjecture. These functions on U(W ) will be obtained by restriction (or pullback) of a simpler class of automorphic forms on a larger group containing U(W ). What is this larger group? By restriction of scalars, we have a 6-dimensional space ResE/k (W ) over k and the k-valued form TrE/k (⟨−, −⟩) defines a symplectic form on ResE/k (W ) with associated symplectic group Sp(ResE/k (W )). This defines an embedding ι ∶ U(W ) ↪ Sp(ResE/k (W )). It turns out that the simple automorphic forms we need are not really living on Sp(ResE/k (W )). Rather, the symplectic group Sp2n (A) has a topological S 1 -cover Mp2n (A) known as the metaplectic group (where S 1 is the unit circle in C× ): 1 → S 1 → Mp2n (A) → Sp2n (A) → 1 and the simpler automorphic forms in question actually live on Mp2n (A). The need to work with this nonlinear cover accounts for much of the technicality of this subject, but one cannot argue with nature.

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These simpler automorphic forms on the metaplectic groups are the theta functions and the automorphic representations they span are called the Weil representations. In order to pull back these theta functions from Mp(ResE/k (W ))(A) to Sp(ResE/k (W )), one needs to construct a lifting of ι to ˜ι ∶ U(W )(A) ↪ Mp(ResE/k (W ))(A). This is highly technical but it can be done and Howe-PS then restricted these theta functions to U(W )(A). Representation theoretically, if Ω ⊂ A(Mp(ResE/k (W ))) denotes an automorphic Weil representation, and ˜ι∗ ∶ A(Mp(ResE/k (W ))) → A(U(W )) denotes the restriction of functions, then one obtains a U(W )(A)-submodule ˜ι∗ (Ω) ⊂ A(U(W )). Now recall that the center of U(W ) is isomorphic to E 1 as an algebraic group. One can spectrally decompose ˜ι∗ (Ω) according to central characters. ˜ι∗ (Ω) = ⊕ Ωχ χ

as χ runs over the automorphic characters of E 1 , or equivalently of E × /k× , where Ωχ = {f ∈ Ω ∶ f (zg) = χ(z) ⋅ f (g) for all z ∈ Z(U(W )) = A1E and g ∈ U(W )(A)}. What we would like to show is that: ● for each χ, Ωχ is an irreducible automorphic representation of U(W ) and is cuspidal for many χ. ● for any χ, Ωχ violates the naive Ramanujan conjecture. One can view the map χ ↦ Ωχ as a lifting of automorphic representations from U1 to U3 . This lifting is an instance of the theta correspondence, which we will discuss in the next two lectures. 2. Lecture 2: Local Theta Correspondence The next two lectures will be devoted to a discussion of the theory of theta correspondence, so as to understand the construction of Howe-PS in its proper context. Two possible references for this are the survey papers of D. Prasad [22] and S. Gelbart [12]. In particular, the latter is concerned with theta correspondence for unitary groups. In this second lecture, we shall focus on the local theta correspondence, for which a basic reference is the book [19] of Moeglin-VignerasWaldspurger. Hence, we will be working over a local field F (of characteristic 0), and for simplicity, we shall assume F is nonarchimedean. 2.1. Basic idea. From the last lecture, we saw that the Howe-PS construction basically gives a map {Automorphic characters of U1 = E 1 } → {Automorphic representations of U3 } sending χ to Ωχ .

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Now given any two groups G and H, one may ask more generally: what are some ways of constructing such a lifting from Irr(G) to Irr(H)? Here, Irr(G) denotes the set of equivalence classes of irreducible representations of G. A standard procedure is as follows. Suppose for simplicity that G and H are finite groups and Ω is a (finite-dim) representation of G × H. Then one may decompose Ω into irreducible G × H-summands: Ω=

⊕

m(π, σ) ⋅ π ⊗ σ.

⊕

π∈Irr(G) σ∈Irr(H)

One can rewrite this as: Ω=

π ⊗ V (π)

⊕ π∈Irr(G)

where V (π) =

⊕

m(π, σ)σ.

σ∈Irr(H)

This gives a map Irr(G) → R(H) (Grothendieck group of Irr(H)) sending π to V (π). Since we are interested in getting irreducible representations of H as outputs, we ask: for what Ω is V (π) is irreducible or zero for any π? If we can find such an Ω, then the map π ↦ V (π) would be a map Irr(G) → Irr(H) ∪ {0}. An example of an Ω that has this property is certainly the trivial representation. However the map so obtained is not very interesting. On the other hand, if dim Ω is too big, then dim V (π) will have to be big for many π’s as well, so that V (π) tends to be reducible. Thus, a simple heuristic is that Ω cannot be too big nor too small. In practice, one can try an Ω arising in the following way. Suppose there is an (almost injective) group homomorphism ι ∶ G × H → E

for some group E.

One can take Ω to be an irreducible representation of E of smallest possible dimension > 1 and then pull it back to G × H. The theory of theta correspondence, which was systematically developed by Howe, arises in this way. 2.2. Reductive dual pairs. Let F be a field of characteristic 0, and let E be an ´etale quadratic F -algebra, with Aut(E/F ) = ⟨c⟩. Let V be a finitedimensional Hermitian space over E and W a skew-Hermitian space. Then V ⊗E W is naturally a skew-Hermitian space over E. By restriction of scalars, we may regard V ⊗E W as an F -vector space which is equipped with a natural symplectic form Tr(⟨−, −⟩V ⊗ ⟨−, −⟩W ). Then one has a natural map of isometry groups ι ∶ U(V ) × U(W ) → Sp(V ⊗E W ) which is injective on each factor U(V ) and U(W ). The images of U(V ) and U(W ) are in fact mutual commutants of each other in the symplectic group, and such a pair of groups is called a reductive dual pair.

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Howe has given a complete classification of all such dual pairs in the symplectic group. A further example (perhaps easier than the one above) is obtained as follows. If V is a symmetric bilinear space (or a quadratic space) and W a symplectic space over F , then V ⊗F W inherits a natural symplectic form (by tensor product) and one has O(V ) × Sp(W ) → Sp(V ⊗ W ). 2.3. Metaplectic groups and Heisenberg-Weil representations. Assume now that F is a local field. The symplectic group Sp(V ⊗E W ) has a nonlinear S 1 -cover Mp(V ⊗E W ) known as the metaplectic group: 1 → S 1 → Mp(V ⊗E W ) → Sp(V ⊗E W ) → 1 The construction of this central extension is a lecture course in itself, but since its construction is such a classic result, we feel obliged to give a sketch. Let us work in the context of an arbitrary symplectic vector space W over F (in place of the cumbersome notation V ⊗E W ). Let H(W ) = W ⊕ F and equip it with the group law 1 (w1 , t1 ) ⋅ (w2 , t2 ) = (w1 + w2 , t1 + t2 + ⟨w1 , w2 ⟩). 2 This is the so-called Heisenberg group. It is a 2-step nilpotent group with center Z = F and commutator [H(W ), H(W )] = Z. The definition of this group law is motivated by the Heisenberg commutator relations from quantum mechanics, hence the name. The irreducible (smooth) representations of H(W ) can be classified and come in 2 types: ● the 1-dimensional representations: these factor through H(W )/[H(W ), H(W )] = H(W )/Z ≅ W and hence are given by characters of W . Observe that these are precisely the representations trivial on the center Z. ● the other irreducible representations have nontrivial central character. For a fixed ψ ∶ Z = F → C× , the Stone-von Neumann theorem asserts that H(W ) has a unique irreducible representation ωψ with central character ψ. Moreover, the representation ωψ is unitary. One can give an explicit construction of ωψ . Let W =X ⊕Y be a Witt decomposition of W so that X and Y are maximal isotropic subspaces. Then H(X) = X + F is an abelian subgroup of H(W ) and one can extend ψ to a character of H(X) by setting ψ(x, 0) = 1 (i.e. extending trivially to X). Then H(W )

ωψ ≅ indH(X) ψ

(compact induction).

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This induced representation is realized on S(Y ) ∶= Cc∞ (Y ), and the action of h ∈ H(W ) can be easily written down: ⎧ (ωψ (0, z)f )(y ′ ) = ψ(z) ⋅ f (y ′ ), for z ∈ F ; ⎪ ⎪ ⎪ ⎪ ⎨(ωψ (y, 0)f )(y ′ ) = f (y + y ′ ), for y ∈ Y ⎪ ⎪ ′ ′ ′ ⎪ ⎪ ⎩(ωψ (x, 0)f )(y ) = ψ(⟨y , x⟩) ⋅ f (y ), for x ∈ X. This action preserves the natural inner product on S(Y ). The symplectic group Sp(W ) acts on H(W ) as group automorphisms via: g ⋅ (w, t) = (g ⋅ w, t). Observe that the action on Z is trivial. Hence the representation g ωψ = ωψ ○ g −1 is irreducible and has the same nontrivial central character as ωψ . By the Stone-von Neumann theorem, these two representations are isomorphic, i..e there exists an invertible operator Aψ (g) on the underlying vector space S of ωψ such that Aψ (g) ○ g ωψ (h) = ωψ ○ Aψ (g) for all h ∈ H(W ). By Schur’s lemma, the operator Aψ (g) is well-defined up to C× . By the unitarity of ωψ , we can insist that Aψ (g) is unitary and hence it is well-defined up to the unit circle S 1 ⊂ C× . Hence we have a map Aψ ∶ Sp(W ) → GL(S)/S 1 . When one pulls back the short exact sequence 1 → S 1 → GL(S) → GL(S)/S 1 → 1 by the map Aψ , one obtains the desired metaplectic group Mpψ (W ): 1 → S 1 → Mpψ (W ) → Sp(W ) → 1 ` ` `Aψ `A˜ψ ∥ ` ` a a 1 → S 1 → GL(S) → GL(S)/S 1 → 1 Hence, this construction produces not just the group Mpψ (W ) but also a natural representation A˜ψ ∶ Mpψ (W ) → GL(S). Thus, we have a representation ωψ of Mpψ (W ) ⋉ H(W ). In other words, the irreducible representation ωψ of H(W ) extends to the semidirect product H(W ) ⋊ Mpψ (W ) (with Mpψ (W ) acting on H(W ) via its projection to Sp(W )). We call this a Heisenberg-Weil representation; its restriction to Mpψ (W ) is simply called a Weil representation. It turns out that the isomorphism class of the extension defining Mpψ (W ) is independent of ψ; so we shall write Mp(W ) henceforth, suppressing ψ. The Weil representation ωψ of Mp(W ) is, in some sense, the smallest infinite-dimensional representation of the metaplectic group. We only make two further remarks here: ● ωψ is a genuine representation of Mp(W ), in the sense that ωψ (z) = z for all z ∈ S 1 , so that ωψ does not factor to a smaller cover of Sp(W ).

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● ωψ is not an irreducible representation of Mp(W ), but rather a direct sum of two irreducible representations (the even and odd Weil representaitons): ωψ = ωψ+ ⊕ ωψ− . Indeed, with ωψ realized on S(Y ) as above, ωψ+ is realized on the subspace of even functions (i.e. f (−x) = f (x)) whereas ωψ− is realized on the subspace of odd functions. 2.4. Schrodinger model. One can ask if it is possible to write down some formulas for the action of elements of Mp(W ), for example on the model S(Y ) of ωψ . Let P (X) be the maximal parabolic subgroup of Sp(W ) stabilizing the maximal isotropic subspace X; this is the so-called Siegel parabolic subgroup. Its Levi decomposition has the form P (X) = M (X) ⋅ N (X), with M (X) ≅ GL(X) and N (X) = {n(B) ∶ B ∈ Sym2 X ⊂ Hom(Y, X)}, where n(B) = (

1 B ) 0 1

(relative to W = X ⊕ Y ).

The metaplectic S 1 -cover splits canonically over N (X) and noncanonically over M (X). Then one can write down formulas for the action of elements lying over g ∈ GL(X) and n(B) ∈ N (X): ⎧ ⎪ ⎪(ωψ (g)f )(y) = ∣ detX (g)∣1/2 ⋅ f (g −1 ⋅ y) ⎨ 1 ⎪ ⎪ ⎩(ωψ (n(B))f )(y) = ψ( 2 ⋅ ⟨By, y⟩) ⋅ f (y). To describe the action of Mp(W ) (at least projectively), one needs to give the action of an extra Weyl group element w=(

0 1 ) −1 0

which together with P (X) generates Sp(W ). The action of this w is given by Fourier transform (up to scalars). The above gives the Schrodinger model of the Weil representation (which is related to the Schrodinger description of quantum mechanics). This concludes our brief sketch of the construction of the metaplectic group and its Weil representations. 2.5. Weil representations of unitary groups. Let us return to the setting of our unitary dual pair ι ∶ U(V ) × U(W ) → Sp(V ⊗E W ). By the above, the metaplectic group Mp(V ⊗E W ) has a distinguished representation ωψ depending on a nontrivial additive character ψ of F . If the embedding ι can be lifted to a homomorphism ˜ι ∶ U(V ) × U(W ) → Mp(V ⊗E W ), then we obtain a representation ωψ ○ ˜ι of U(V ) × U(W ).

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Such splittings have been constructed and classified by S. Kudla [17]. They are not unique but can be specified by picking two characters χV and χW of E × such that dim V dim W and χW ∣F × = ωE/F . χV ∣F × = ωE/F One way of doing this is, for example, fixing a character γ of E × such that γ∣F × = ωE/F (i.e. a conjugate symplectic character) and then taking χV = γ dim V

and χW = γ dim W .

In any case, for a fixed pair (χV , χW ) of splitting characters, Kudla provides a splitting ˜ιχV ,χW ,ψ ∶ U(V ) × U(W ) → Mp(V ⊗E W ) of ι. In fact, the choice of χV gives rise to a splitting ιV,W,χV ,ψ ∶ U(W ) → Mp(V ⊗E W ) over U(W ), whereas the choice of χW gives a splitting ιV,W,χW ,ψ ∶ U(V ) → Mp(V ⊗E W ). Hence, the splitting over the two members of the dual pair can be constructed somewhat independently of each other. This is a manifestation of a basic property of the metaplectic cover: if two elements of Sp(V ⊗E W ) commute, then any lifts of them in Mp(V ⊗E W ) also commute with each other. With a splitting fixed, we may pull back the Weil representation ωψ and obtain a representation ΩV,W,χV ,χW ,ψ ∶= ˜ι∗χV ,χW ,ψ (ωψ )

of U(V ) × U(W ).

We call this ΩχV ,χW ,ψ a Weil representation of U(V )×U(W ); we will often suppress the subscript for ease of notation. While we have not described the splitting ˜ιχV ,χW ,ψ explicitly, we highlight a few basic properties: ● (Change of (χV , χW )) One can easily describe the effect of changing (χV , χW ) on ΩχV ,χW ,ψ . More precisely, if (χ′v , χ′W ) is another pair of splitting characters, then ΩV,W,χ′V ,χ′W ,ψ ≅ ΩV,W,χV , χW ,ψ ⊗ (χ′V /χV ○ i ○ detW ) ⊗ (χ′W /χW ○ i ○ detV ), where i ∶ E 1 → E × /F × , taking note that χ′V /χV and χ′W /χW are characters of E × /F × . ● (Scaling) Given a ∈ F × , one can scale the additive character ψ to obtain ψa (x) = ψ(ax). One can also scale the Hermitian or skew-Hermitian forms on V and W , otaining V a and W a . If we identify U(V ) and U(V a ) as the same subgroup of GL(V ), then one has: ΩV,W,χV ,χW ,ψa ≅ ΩV a ,W,χV ,χW ,ψ ≅ ΩV,W a ,χV ,χW ,ψ . ● (Duality) Recalling that the Weil representation is unitarizable, so that its dual is isomorphic to its complex conjugate, we note: ΩV,W,χV ,χW ,ψ ≅ ΩV,W,χ−1 ,χ−1 ,ψ . V

W

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● (Center) If we take χV = γ dim V

and χW = γ dim W

for a conjugate symplectic character γ, and identify ZU(V ) and ZU(W ) with E 1 (and hence with each other), ˜ιχV ,χW ,ψ agrees on ZU(V ) and ZU(W ) . 2.6. Local theta lifts. In the theory of local theta correspondence, one studies how the representation ΩV,W,χV ,χW ,ψ decomposes into irreducible pieces. For this, one would need to know more about the representation ΩV,W,χV ,χW ,ψ than what has been described above. For example, one may demand if there are formulas for the group action. Ultimately, such formulas would have been derived from those in the Schrodinger model we described earlier and an explicit knowledge of the splitting ˜ιχV ,χW ,ψ . We will come to this in the particular case of interest later on in this lecture. At the moment, let us just formulate some questions one may ask and describe the answers to some of these questions. We will write Irr(U(W )) for the set of equivalence classes of irreducible smooth representations of U(W ). Unlike the case of finite (or compact) groups, the representation Ω is infinite-dimensional and is not necessarily semisimple as a U(V ) × U(W )-module. So when one talks about the decomposition of Ω into irreducible constituents, one can understand this in potentially two ways: ● for which π ⊗ σ ∈ Irr(U(V )) × Irr(U(W )) is π ⊗ σ a subrepresentation of Ω? ● for which π ⊗ σ ∈ Irr(U(V )) × Irr(U(W )) is π ⊗ σ a quotient of Ω? It turns out that it is more fruitful to consider the second question above. Hence, Ω determines a subset of Irr(U(V ) × Irr(U(W )): ΣΩ = {(π, σ) ∶ π ⊗ σ is a quotient of Ω} and thus a correspondence between Irr(U(V )) and Irr(U(W )). This is the correspondence in “theta correspondence”. If (π, σ) ∈ ΣΩ , we say that σ is a local theta lift of π and vice versa. One can reformulate the above definition in a slightly different way, which is more convenient for the question of determining all possible theta lifts of a given π. For π ∈ Irr(U(V )), one considers the maximal π-isotypic quotient of Ω: Ω/

⋂

f ∈HomU(V ) (Ω,π)

Ker(f ),

which is a U(V ) × U(W )-quotient of Ω expressible in the form π ⊗ Θ(π), for some smooth representation Θ(π) of U(W ) (possibly zero, and possibly infinite length a priori). We call Θ(π) the big theta lift of π. An alternative way to define Θ(π) is: Θ(π) = (Ω ⊗ π ∨ )U(V ) , the maximal U(V )-invariant quotient of Ω ⊗ π ∨ . To see the equivalence of these two definitions of Θ(π):

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● we have a natural equivariant projection q ∶ Ω → Ω/

⋂

f ∈HomU(V ) (Ω,π)

Ker(f ) = π ⊗ Θ(π).

In addition, the natural map Ω ⊗ π ∨ → (Ω ⊗ π ∨ )U(V ) induces another equivariant map p ∶ Ω → π ⊠ (Ω ⊗ π ∨ )U(V ) . ● note that HomU(V ) (Ω, π) ≅ Hom((Ω ⊗ π ∨ )U(V ) , C). For f ∈ HomU(V ) (Ω, π), with corresponding f ∈ (Ω ⊗ π ∨ )∗ , one has a commutative diagram p

Ω → π ⊠ (Ω ⊗ π ∨ )U(V ) ` ` `f f` ` ` a a π π This implies that Ker(p) ⊂

⋂

f ∈HomU(V ) (Ω,π)

Ker(f ) = Ker(q).

● conversely, if φ ∈ Ker(q), then p(φ) ∈ π ⊠ (Ω ⊗ π ∨ )U(V ) is killed by every element of (Ω ⊗ π ∨ )∗U(V ) , so that p(φ) = 0. In any case, it follows from definition that there is a natural U(V ) × U(W )equivariant map Ω ↠ π ⊗ Θ(π), which satisfies the “universal property” that for any smooth representation σ of U(W ), HomU(V )×U(W ) (Ω, π ⊗ σ) ≅ HomU(W ) (Θ(π), σ) (functorially). The local theta lifts of π are then the irreducible quotients of Θ(π). 2.7. Howe duality conjecture. The goal of local theta correspondence is to determine the representation Θ(π) or rather its irreducible quotients. Recall that our hope is that Θ(π) is close to being irreducible or at least not too big. To this end, we first note the following basic result of Howe and Kudla: Proposition 2.1 (Finiteness). For any π ∈ Irr(U(V )), Θ(π) is of finite length. In particular, if Θ(π) is nonzero, it has (finitely many) irreducible quotients and we may consider its maximal semisimple quotient (its cosocle) θ(π). Moreover, for any π ∈ Irr(U(V )) and σ ∈ Irr(U(W )), dim HomU(V )×U(W ) (Ω, π ⊗ σ) < ∞. We call θ(π) the small theta lift of π. The local theta lifts of π are precisely the irreducible summands of θ(π).

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We can now formulate a fundamental theorem [11, 29]: Theorem 2.2 (Howe duality theorem). (i) If Θ(π) is nonzero, then it has a unique irreducible quotient. In other words, θ(π) is irreducible or 0. (ii) If θ(π) ≅ θ(π ′ ) ≠ 0, then π ≅ π ′ . Hence, one has an map θχV ,χW ,ψ ∶ Irr(U(V )) → Irr(U(W )) ∪ {0} which is injective when restricted to the subset of Irr(U(V )) which is not sent to 0. Moreover, dim HomU(V )×U(W ) (Ω, π ⊗ σ) ≤ 1 for any π ∈ Irr(U(V )) and σ ∈ Irr(U(W )). Another way to formulate the theorem is to note that the subset/correspondence ΣΩ is the graph of a bijective function between prV (ΣΩ ) ⊂ Irr(U(V )) and prW (ΣΩ ) ⊂ Irr(U(W )), where prV refers to the projection to Irr(U(V )). Thus, we see that local theta correspondence is an instance of the Basic Idea highlighted at the beginning of the lecture. 2.8. Questions. After the Howe duality theorem above, we can ask the following questions: (a) (Nonvanishing) For a given π ∈ Irr(U(V )), decide if θχV ,χW ,ψ (π) is nonzero. (b) (Identity) Describe the map θχV ,χW ,ψ explicitly. In other words, if θχV ,χW ,ψ (π) is nonzero, can one describe it in terms of π in another way? Nowadays, both these questions have rather complete answers, but it is too much to describe these answers for this course. Instead, we will highlight some relevant answers for our application. 2.9. Rallis’ tower. For the nonvanishing question, Rallis observed that it is fruitful to consider theta correspondence in a family. Let W0 be an anisotropic skew-Hermitian space over E (for example a 1-dimensional one), and for r ≥ 0, let Wr = W0 ⊕ Hr where H is the hyperbolic plane. The collection {Wr ∣ r ≥ 0} is called a Witt tower of spaces. Observe that: ● dim Wr mod 2 is independent of r; ● disc(Wr ) or equivalently the sign (Wr ) is independent of r. Hence, in the nonarchimedean case, there are precisely two Witt towers of skewHermitian spaces with a fixed dimension modulo 2, and any given skew-Hermitian space W is a member of a unique Witt tower. One can then consider a family of theta correspondences associated to the tower of reductive dual pairs (U(V ), U(Wr )) with respect to a fixed pair of splitting characters (χV , χW ) (note that we can fix χWr independently of r, since the parity of dim Wr is independent of r). Kudla showed [16]:

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Proposition 2.3. (i) For π ∈ Irr(U(V )), there is a smallest r0 = r0 (π) such that ΘV,Wr0 ,ψ (π) ≠ 0. Moreover, r0 ≤ dim V . (ii) For any r > r0 , ΘV,Wr ,ψ (π) ≠ 0 (tower property). (iii) Suppose that π is supercuspidal. Then ΘV,Wr0 ,ψ (π) is irreducible supercuspidal. For r > r0 , ΘV,Wr ,ψ (π) is irreducible but not supercuspidal. Some remarks: ● We call this smallest r0 (π) from the proposition above the first occurrence index of π in the Witt tower (Wr ) (relative to our fixed (χV , χW , ψ)). ● The fact that r0 ≤ dim V means that when W is sufficiently large, more precisely when W has an isotropic subspace of dimension ≥ dim V , the map θV,W,ψ is nonzero on the whole of Irr(U(V )). When r ≥ dim V , we say that the theta lifting is in the stable range. ● The nonvanishing problem (a) highlighted above is reduced to the question of determining the first occurrence indices for the two Witt towers. In fact, our job is halved because the two first occurrence indices (for a given dim W● mod 2) are not independent of each other. Rather, we have the following theorem of Sun-Zhu [26]: Theorem 2.4 (Conservation relation). Consider the two towers (Wv ) and (Wr′ ) of skew-Hermitian spaces with fixed dim W● mod 2, and let r0 and r0′ be the respective first occurrence indices of a fixed π ∈ Irr(U(V )) (relative to a fixed data (χV , χW , ψ)). Then dim Wr0 + dim Wr′0′ = 2 dim V + 2. In particular r0 and r0′ determine each other. The conservation relation above implies the following dichotomy theorem (which you should try to prove): Corollary 2.5. Suppose that W and W ′ belong to the two different Witt towers of skew-Hermitian spaces (with dim W mod 2 fixed), and dim W + dim W ′ = 2 dim V . Then for any π ∈ Irr(U(V )), exactly one of the two theta lifts ΘV,W,ψ (π) and ΘV,W ′ ,ψ (π) is nonzero. 2.10. U1 × U1 . Let us illustrate the dichotomy theorem in the base case where dim V = dim W0 = 1. Let W0 and W0′ be the two skew-Hermitian spaces of dimension 1. For any χ ∈ Irr(U(V )) = Irr(E 1 ), the dichotomy theorem implies that exactly one of the theta lifts θV,W0 ,ψ (χ) or θV,W0′ ,ψ (χ) is nonzero. Now here is an interesting question: which of these lifts is nonzero? This turns out to be a highly nontrivial and beautiful result of Moen and Rogawski [25, Prop. 3.4] and Harris-KudlaSweet [14]: Theorem 2.6. The theta lift θV,W0 ,ψ (χ) (with respect to splitting characters (χV , χW )) is nonzero if and only if (V ) ⋅ (W0 ) = E (1/2, χE ⋅ χ−1 W , ψ(T rE/F (δ−))). Here, recall that δ is a nonzero trace zero element of E and the definition of the sign (W0 ) depends on δ. Moreover, χE is the character of E × /F × defined by

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χE (x) = χ(x/xc ) and the local epsilon factor on the right is that defined in Tate’s thesis. This theorem shows that the question of nonvanishing of theta lifting has deep arithmetic connections. One way to prove this theorem is via the doubling seesaw argument, which relates the theta lifting to the doubling zeta integral of PiatetskiShapiro and Rallis, a theory that produces the standard L-function and epsilon factor. This is the approach of [14]. We have no time to go into this here, but would like to mention that this doubling zeta integral plays an important role in Ellen Eischen’s lectures. After addressing nonvanishing, the next issue is that of identity: what is θV,W0 ,ψ (χ) if it is nonzero? Suppose we pick χV = χW (as is allowed here). Then the splitting over the two U1 ’s agree (on identifying them with E 1 ) and so the theta lifting is the identity map Θ(χ) = χ on its domain (i.e. outside its zero locus). With our knowledge of how the Weil representation changes when we change (χV , χW ), this allows one to figure out the general case: ΘχV ,χW ,ψ (χ) = χ ⋅ (χ−1 W χV ○ i) on its domain. 2.11. Application to Howe-PS setting. We shall now specialize to the particular case we are interested in. Set V = ⟨1⟩ to be the 1-dimensional Hermitian space with form (x, y) ↦ xy c so that U(V ) = E 1 ≅ E × /F × . We consider the theta correspondence for U(V ) with the two odd-dimensional Witt towers (Wr ) and (Wr′ ), with dim Wr = dim Wr′ = 2r + 1. Because U(V ) is compact, one in fact has a direct sum decomposition: Ω = ⊕ χ ⊗ Θ(χ) χ

as χ runs over the characters of E 1 . Now let us record some consequences of the general results discussed above: ● For any χ, ΘV,Wr ,ψ (χ) and ΘV,W ′ ,ψ (χ) are irreducible or 0. This is because, with U(V ) being compact, any χ is supercuspidal. ● For any χ and r > 0, θV,Wr ,ψ (χ) and θV,Wr′ ,ψ (χ) are both nonzero; this is because we are already in the stable range when r > 0. ● What if r = 0 (i.e. the dual pair U1 × U1 )? This is the situation addressed by the dichotomy theorem (Corollary 2.5): exactly one of θV,W0 ,ψ (χ) and θV,W0′ ,ψ (χ) is nonzero. Exactly which one is nonzero is highly non-obvious but is given by Theorem 2.6. ● Suppose without loss of generality that θV,W0′ ,ψ (χ) = 0. Then ΘV,W1′ ,ψ (χ) is supercuspidal. I will leave it as an exercise for the reader to deduce the above assertions from the results discussed above. Instead, I will describe the proof of some of those results in the special case of U1 × U3 . This is where we do the “dirty work”, which will be formulated as a series of guided exercises below.

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2.12. Guided exercise. The Weil representation Ω for U(V ) × U(W1 ) = U1 × U3 can be given a realization as follows. Let ω be the Heisenberg-Weil representation for (U(V ) × U(W0 )) ⋉ H(V ⊗E W0 ) = (U1 × U1 ) ⋉ H(V ⊗E W0 ) where we recall that H(V ⊗E W0 ) is the Heisenberg group associated to the 2dimensional symplectic space V ⊗E W0 over F . It is in fact not easy to give a concrete model for this representation (even starting with a Schrodinger model for Mp2 ). Then the Weil representation Ω for U(V ) × U(W1 ) can be realized on S(Ee∗ ⊗ V ) ⊗ ω which can be thought of as the space of Schwartz functions on the 1-dimensional E-vector space Ee∗ ⊗ V valued in ω. One can give explicit formulas for the action of U(V ) × B, where B = T U is the Borel subgroup stabilizing the isotropic line E ⋅ e as follows. (a) for h ∈ U(V ) = E 1 , (h ⋅ f )(x) = ω(h) (f (h−1 ⋅ x)) ,

with x ∈ E and f ∈ Ω.

(b) for an element ⎛ a t(a, b) = ⎜ ⎝

b

c −1

(a )

⎞ ⎟∈T ⎠

with a ∈ E × and b ∈ E 1 ,

one has: (t(a, b)f )(x) = χV (a) ⋅ ∣a∣1/2 ⋅ ω(b) (f (ac ⋅ x)) , where we regard b ∈ E 1 as an element of U(W0 ). (c) for an element ⎛ 1 0 z ⎞ 1 0 ⎟ ∈ U, u(0, z) = ⎜ ⎝ 1 ⎠

with z ∈ F ,

one has: (u(0, z)f )(x) = ψ(zN (x)) ⋅ f (x). (d) for an element ⎛ 1 ∗ 1 u(y, 0) = ⎜ ⎝

∗ ⎞ y ⎟, 1 ⎠

one has: (u(y, 0)f )(x) = ω(h(xy, 0))(f (x)), where h(xy) = (xy, 0) ∈ H(V ⊗E W0 ) is regarded as an element in the Heisenberg group. (The two asterisks in u(y, 0) are determined by y; work out what they should be). Now recall that we have: Ω=

⊕

χ∈Irr(E 1 )

χ ⊗ Θ(χ),

and our goal is now to understand Θ(χ) as much as possible.

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Given the above information, here is the guided exercise: Exercise: (i) Let Z = {u(0, z) ∶ z ∈ F } ⊂ U . This is the center of U . Compute the coinvariant space ΩZ = Ω/⟨z ⋅ f − f ∶ z ∈ Z, f ∈ Ω⟩, as a representation of U(V ) × B/Z = E 1 × T ⋅ U /Z. Indeed, show that the natural projection Ω → ΩZ is given by the evaluation-at-0 map ev0 ∶ S(E) ⊗ ω → ω. (ii) From the answer in (i), deduce the following: (a) For any χ ∈ Irr(U(V )), Θ(χ)Z = Θ(χ)U . (b) Suppose that χ has nonzero theta lift θ0 (χ) to U(W0 ) with respect to ω. Then Θ(χ) is nonzero and non-supercuspidal. Indeed, one has a nonzero T = E × × U(W0 )-equivariant map Θ(χ)N → χV ∣ − ∣1/2 ⊗ θ0 (χ), so that by Frobenius reciprocity, there is a nonzero equivariant map U(W )

Θ(χ) → IndB

(χV ∣ − ∣−1/2 ⊗ θ0 (χ))

(normalized induction)

1/2 δB (t(a, b))

taking note that = ∣a∣E . Hence we see that Θ(χ) contains a constituent of the latter principal series representation, which is nontempered (since ∣ − ∣1/2 is not unitary). (c) Suppose that χ has zero theta lift to U(W0 ). From (i), deduce that Θ(χ) is supercuspidal (i.e. Θ(χ)U = 0). (iii) Now compute the twisted coinvariant space ΩZ,ψ = Ω/⟨z ⋅ f − ψ(z) ⋅ f ∶ z ∈ Z, f ∈ Ω⟩ as a representation of U(V ) × Tψ , where Tψ = {t(a, b) ∶ a, b ∈ E 1 } ⊂ T is the stabilizer of (Z, ψ) in T . (iv) Deduce from the answer in (iii) that for any χ ∈ Irr(U(V )), Θ(χ) ≠ 0. What one sees from this guided exercise is that to understand the theta lifts Θ(π) (for example to detect its nonvanishing or supercuspidality), it is useful to consider various twisted coinvairant spaces ΩN,ψ where N ⊂ U(W ) is an abelian subgroup and ψ is a (possibly trivial) character of N . Such twisted coinvariant spaces (or twisted Jacquet modules) are local analogs of the Fourier coefficients of modular forms. They are readily computable from the concrete model of the Weil representation analogous to the one above. It also shows that the Weil representations have an inductive structure with respect to the Rallis tower. In this guided exercise, we see that we are basically reduced to the following two problems: ● the irreducibility of Θ(χ) (as given by the Howe duality theorem); ● the understanding of the U1 × U1 theta correspondence (which we discussed earlier).

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2.13. Split case. Note that the case when E = F × F is also necessary for global applications. In this case, the dual pair is GLm × GLn . The Weil representation is, up to twists by 1-dimensional characters, the natural action of GLm (F ) × GLn (F ) on the space S(Mm×n (F )) of Schwartz functions on the space of m × n matrices. The study of this local theta correspondence is essentially completed in the paper [18] of A. Minguez. Hence we shall say no more about this case in this paper. 2.14. Remarks. We have given a discussion of the theory of classical theta correspondence which is based on reductive dual pairs in the symplectic group. But this idea is clearly more robust. One may ask: ● Can one classify all reductive dual pairs G × H in any simple Lie group E, as opposed to just for E = Sp2n ? ● Is there an understanding of the smallest infinite-dimensional representation Ω of any such E? ● If so, when one pulls back Ω to G × H, does one obtain a transfer or lifting of representations analogous to those described in this lecture? These questions started to be explored in the mid-1980’s. Reductive dual pairs have been classified on the level of Lie algebras by Rubenthaler. The construction and classification of the so-called minimal representations of a simple Lie group E was begun by Kostant, Vogan, Kazhdan, Savin, Torasso and others; see [10]. Finally the study of the resulting theta correspondence began in the 1990’s but the theory is not as systematic as the classical case. It is only recently that one has somewhat complete results in several families of examples. In the project for this course, you will work with a particular instance of this exceptional theta correspondence. 3. Lecture 3: Global Theta Correspondence In this third lecture, we will discuss the global theta correspondence. For the case of unitary groups, a nice reference is the article [12] of Gelbart. We will see that almost all of the considerations and constructions of the previous lecture make sense in the global setting, once they are appropriately construed. We will work over a number field k with associated local field kv for each place v of k and with adele ring A = ∏′v kv . We fix a quadratic field extension E/k and consider a pair of a Hermitian space V and a skew-Hermitian space W relative to E/k. 3.1. Basic idea. Let us return to the basic idea of Lecture 2: in the local setting, the Weil representation allows one to define the local theta lifting θ ∶ Irr(U(V )) → Irr(U(W )) ∪ {0}. If U(V ) × U(W ) is in the stable range (with V smaller), one even has θ ∶ Irr(U(V )) → Irr(U(W )). In the global setting, one might imagine that by taking (restricted) tensor product taken over all places v of a number field k, one gets θ ∶ Irr(U(V )(A)) → Irr(U(W )(A)).

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This is the case, but we are interested not in the lifting of abstract irreducible representations, but rather of cuspidal automorphic representations. Cuspidal automorphic representations are representations which are realized in the space of cuspidal automoprhic forms (i.e. functions on [G]). So what we need is a map {Cusp forms on U(V )} → {Automorphic forms on U(W )}. We are thus interested in procedures which allow one to lift functions on a (measure) space X to functions on another space Y . A standard such procedure is via a kernel function K, i.e. a function K ∶ X × Y → C. Given such a function K, one gets a linear map TK ∶ C(X) → C(Y ) defined by TK (f )(y) = ∫ K(x, y) ⋅ f (x) dx, X

assuming convergence is not an issue. Let’s apply this simple idea to our setting. Recall from Lecture 2 that we have ˜ι ∶ U(V ) × U(W ) → Mp(V ⊗E W ). We shall see that the (global) Weil representation Ω is automorphic on Mp(V ⊗E W ), i.e. there is an equivariant map θ ∶ Ω → A(Mp(V ⊗E W )). So for any φ ∈ Ω, we have an automorphic form θ(φ) on Mp(V ⊗E W ): these are the theta functions. Pulling back θ(φ) by ˜ι, we may regard θ(φ) as a function on [U(V )] × [U(W )]. We can thus use θ(φ) as a kernel function to transfer functions on [U(V )] to functions on [U(W )]. In other words, each θ(φ) gives a linear map θφ ∶ { Cusp forms on U(V )} → {Automoprhic forms on U(W )}. As we consider all these θφ together, we have a map {Cuspidal automorphic representations of U(V )} ` ` ` a {Automorphic representations of U(W )}. 3.2. Adelic metaplectic groups. We shall now give more precise formulation of the above basic idea. Fix a non-trivial additive character ψ = ∏v ψv of F /A. Suppose that W is a symplectic vector space over k. Then for each v, we have seen the metaplectic group 1 → S 1 → Mp(Wv ) → Sp(Wv ) → 1 For almost all v, it is known that the covering splits uniquely over the hyperspecial maximal compact subgroup Kv , so that we may regard Kv as an open compact subgroup of Mp(Wv ). Then one can form the restricted direct product: ′

∏v Mp(Wv )

(with respect to the family {Kv }).

This contains as a central subgroup ⊕v S 1 . If we quotient out the restricted direct product above by the central subgroup Z = {(zv ) ∈ ⊕ S 1 ∶ ∏ zv = 1} v

v

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we get the adelic metatplectic group 1 → S 1 → Mp(W )(A) → Sp(W ) → 1. Note that though we use the notation Mp(W )(A), Mp(W ) is not an algebraic group and we are not taking the group of adelic points of an algebraic group. An important property of this adelic metaplectic cover is that it splits (canonically) over the group Sp(W )(k) of k-rational points, so that one can canonically regard Sp(W )(k) as a subgroup of Mp(W )(A). As a result one can consider the automorphic quotient [Mp(W )] = Sp(W )(k)/ Mp(W )(A) and introduce the space of genuine automorphic forms on Mp(W )(A): these are the automorphic functons f ∶ [Mp(W )] → C such that for all z ∈ S 1 , f (zg) = z ⋅ f (g). 3.3. Global Weil representations. We may consider the global Weil representation ′ ′ ωψ ∶= ⊗v ωψv of ∏v Mp(Wv ). This factors to a representation ωψ of Mp(W )(A) (for clearly Z acts trivially). If W = X ⊕ Y is a Witt decomposition, we have seen that for each v, ωψv can be realized on S(Yv ). Hence, ωψ can be realized on ′

⊗v S(Yv ) = S(YA ). In other words, ωψ is realized on a very concrete space of functions. 3.4. Theta functions. It turns out that one has an Mp(W )(A)-equivariant map θ ∶ S(YA ) → A(Mp(W )) defined by averaging over the k-rational points of Y : θ(f )(g) = ∑ (ωψ (g) ⋅ f )(y). y∈Yk

The fact that θ(f ) is left-invariant under Sp(W )(k) is a consequence of the Poisson summation formula. The functions θ(f ) are called theta functions. The map θ is non-injective. More precisely, since ωψv = ωψ+ v ⊕ ωψ− v

for each v,

we see that as an abstract representation, ωψ = ⊕ ωψ,S S

where

ωψ,S = ( ⊗ ωψ− v ) ⊗ ( ⊗ ωψ+ v ) v∈S

v∉S

for finite subsets S of places of k. Then Ker(θ) = ⊕ ωψ,S , ∣S∣ odd

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and we have an injective map (still denoted by θ) θ∶

⊕ ωψ,S ↪ A(Mp(W ))

∣S∣ even

upon restriction to the “even” subspace above. 3.5. Pulling back. Now let us return to our reductive dual pair U(V )×U(W ) over k. Recall that we have ι ∶ U(V )(A) × U(W )(A) → Sp(V ⊗E W )(A). If we fix a pair of automorphic characters (χV , χW ) of E × with dimE V χV ∣A× = ωE/k

dimE W and χW ∣A× = ωE/k ,

then as in the local case, one obtains an associated lifting ˜ιχV ,χW ,ψ ∶ U(V )(A) × U(W )(A) → Mp(V ⊗ W )(A). Using this lifting, we may pullback the global Weil representation ωψ to obtain the Weil representation Ω = ΩV,W,χV ,χW ,ψ

of U(V )(A) × U(W )(A).

Moreover, the lifting ˜ι sends the group U(V )(k) × U(W )(k) into Sp(V ⊗ W )(k) ⊂ Mp(V ⊗ W )(A). Hence, the pullback of a function in A(Mp(V ⊗ W )) by ˜ι gives a smooth function on [U(V )] × [U(W )]. Thus we have θ ∶ Ω → A(Mp(V ⊗ W )) → C ∞ ([U(V ) × U(W )]). 3.6. Global theta liftings. Now for f ∈ Acusp (U(V )) and ϕ ∈ Ω, we set: θ(ϕ, f )(g) = ∫

[U(V )]

θ(ϕ)(g, h) ⋅ f (h) dh.

The cuspidality of f implies that the integral above converges absolutely (because of the rapid decay of f modulo center). Then θ(ϕ, f ) is an automorphic form on U(W ). Suppose that π ⊂ Acusp (U(V )) is an irreducible cuspidal automorphic representation. Let Θ(π) = ⟨θ(ϕ, f ) ∶ f ∈ π, ϕ ∈ Ω⟩ ⊂ A(U(W )). This is a U(W )(A)-submodule of A(U(W )) and we call it the global theta lift of π. 3.7. Questions. The main questions concerning global theta lifting are analogs of those in the local case: ● Is Θ(π) nonzero? ● Is Θ(π) contained in the space of cusp forms? In addition, we can ask: ● For π = ⊗′v πv , how is the global Θ(π) related to the local theta liftings θ(πv ) for all v? We shall begin by addressing this issue of local-global compatibility.

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3.8. Compatibility with local theta lifts. How is the representation Θ(π) related to the abstract irreducible representation Θabs (π) ∶= ⊗v θ(πv )? We have: Proposition 3.1. Suppose that Θ(π) is non-zero and is contained in the space A2 (U(W )) of square-integrable automorphic forms on U(W ). Then Θ(π) ≅ Θabs (π). Proof. Since Θ(π) ⊂ A2 (U(W )), it is semisimple. Let σ be an irreducible summand of Θ(π). Then consider the linear map Ω ⊗ π ∨ ⊗ σ ∨ → C defined by: ϕ ⊗ f1 ⊗ f2 ↦ ∫

[U(W )]

θ(ϕ, f1 )(g) ⋅ f2 (g) dg.

This map is non-zero and U(V ) × U(W )-invariant. Thus it gives rise to a non-zero equivariant map Ω → π ⊗ σ, and thus for all v, a non-zero U(Vv ) × U(Wv )-equivariant map Ωv → πv ⊗ σv . In other words, we must have σv ≅ θ(πv ). Hence, Θ(π) must be an isotypic sum of Θabs (π). Moreover, the multiplicity-one statement in the Howe duality theorem implies that dim HomU(V )(A)×U(W )(A) (Ω, π ⊗ Θabs (π)) = 1. Thus Θ(π) is in fact irreducible and isomorphic to Θabs (π).



3.9. Cuspidality and Nonvanishing. As in the local case, it is useful to consider a Rallis tower of theta lifitngs, corresponding to a Witt tower Wr = W0 ⊕Hr of skew-Hermitian spaces, with W0 anisotropic. One has the analogous results: Proposition 3.2. Let π be a cuspidal automorphic representation of U(V ), and consider its global theta lift ΘV,Wr ,ψ (π) on U(Wr ) (relative to a fixed pair (χV , χW )). Then one has: (i) There is a smallest r0 = r0 (π) ≤ dim V such that ΘV,Wr0 ,ψ (π) ≠ 0. Moreover, ΘV,Wr0 ,ψ (π) is contained in the space of cusp forms. (ii) For all r > r0 , ΘV,Wr ,ψ (π) is nonzero and noncuspidal. (iii) For all r ≥ dim V , ΘV,Wr0 ,ψ (π) ⊂ A2 (U(W )), unless r = dim V , r0 = 0 and W0 = 0. As in the local case, we call r0 = r0 (π) the first occurrence of π in the relevant Witt tower, and we call the range where r ≥ dim V the stable range. Let us give some indication of the proof of parts (ii) and (iii) of this proposition, assuming the existence of r0 as in (i). The key input is Rallis’ computation of the constant term of a global theta lift [24, Thm. I.1.1]. More precisely, for a maximal parabolic subgroup Pj = Mj ⋅Nj of U(Wr ) with Levi factor Mj ≅ GLj (E)×U(Wr−j ) (for 1 ≤ j ≤ r), one considers the normalized constant term −1/2

RPj (ΘV,Wr ,ψ (π)) ∶= δPj

⋅ ⟨θ(ϕ, f )Nj ∶ f ∈ π, ϕ ∈ ΩV,Wr ,ψ ⟩

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of ΘV,Wr ,ψ (π) along Nj : this is a space of automorphic forms on Mj and is the global analog of the normalized Jacquet module in the local setting. Rallis’ result [24, Thm. I.1.1] states that RPj (ΘV,Wr ,ψ (π)) = ∣ det ∣

dim V −dim Wr +j 2

⊠ ΘV,Wr−j ,ψ (π)

where det refers to the determinant character of GLj (AE ). Since ΘV,Wr−j ,ψ (π) = 0 if r − j < r0 (by the definition of r0 ), we deduce: ● All constant terms of ΘV,Wr0 ,ψ (π) vanish, so that ΘV,Wr0 ,ψ (π) is cuspidal (and nonzero). ● For r > r0 , ΘV,Wr ,ψ (π) is noncuspidal (and hence nonzero), since its normalized constant term with respect to Pr−r0 is nonzero. Indeed, its normalized constant term along Pr−r0 is ∣ det ∣

dim V −dim W0 −(r+r0 ) 2

⊠ ΘV,Wr0 ,ψ (π).

● It follows that for r > r0 , the cuspidal support of ΘV,Wr ,ψ (π) is the parabolic subgroup Qr0 with Levi factor GL1 (E)r−r0 × U(Wr0 ). Further, ΘV,Wr ,ψ (π) has a unique cuspidal exponent with respect to Qr0 , given by the following character of GL1 (E)r−r0 : ∣−∣

dim V −dim W0 −(r+r0 ) 2

⋅ (∣ − ∣−

j−1 2

× ∣ − ∣−

j−3 2

×⋅⋅⋅×∣ −∣

j−1 2

).

When r ≥ dim V , the exponent dim V − dim W0 − (r + r0 ) < 0, unless r = dim V , r0 = 0 and W0 = 0. Since this exponent is negative, it follows from the square-integrability criterion of Jacquet [20, Lemma I.4.11] that ΘV,Wr ,ψ (π) is square-integrable. 3.10. The case of U1 × U1 . In parallel with the local setting, we may consider the theta lift for the basic case of U(V ) × U(W ) with dim V = dim W = 1. We have seen that the nonvanishing of local theta lifts is controlled by a local root number. Here is the global theorem [27]: Theorem 3.3. Let χ be an automorphic character of U(V ) = E 1 . Its global theta lift ΘχV ,χW ,ψ (π) on U(W ) is nonzero if and only if : (i) for each place v, the local theta lift θχV,v ,χW,v ,ψv (χ) is nonzero; (ii) the global L-value L(1/2, χE χ−1 W ) ≠ 0. Note that under condition (i), our local theorem for U1 × U1 implies that (1/2, χE,v χ−1 W,v , ψv (Tr(δ−))) = (Vv ) ⋅ (Wv ) for all v, where δ is a trace 0 element of E. On taking product over all places v, we see that (1/2, χE χ−1 W) = 1 since ∏v (Vv ) = 1 = ∏v (Wv ). Hence, there is a chance that the global L-value in (ii) is nonzero! The proof of this Theorem is a global analog of the local theorem in the U1 × U1 case, via the doubling seesaw and doubling zeta integral. One then gets the Rallis

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inner product formula, which relates the Petersson inner product of two global theta lifts and the special L-value. This result thus gives an interpretation for the nonvanishing of the central L-value. 3.11. The case of U1 × U3 . Let us now specialize to the case of interest, where dim V = 1 and dim W = dim W1 = 3. Let χ be an automorphic character of E 1 . Then χ is cuspidal since E 1 is anisotropic. We can thus apply the above general results to conclude: Corollary 3.4. ΘV,W1 (χ) is nonzero and square-integrable. It is cuspidal if and only if the global theta lift of χ to U(W0 ) = U1 is zero. 3.12. Counterexample to Ramanujan. Using all the results we have seen, we can now construct a counterexample to the naive Ramanujan conjecture: ● Consider the global theta correspondence for U(V ) × U(W1 ) with dim V = 1 and dim W1 = 3, and take χV = χW and χ = 1 (the trivial character of U(V )(A)). The global theta lift ΘV,W,ψ (1) is a nonzero irreducible square-integrable automorphic representation. ● If 1 has zero global theta lift to the lower step U(W0 ) = U1 of the Rallis tower, then Θ(1) is cuspidal irreducible. Moreover, by part (ii)(b) of the guided exercise in §2.12, one sees that for almost all v, Θ(1)v ≅ θ(1v ) is an unramified representation belonging to the principal series U(Wv )

IndBv

(χV,v ∣ − ∣−1/2 ⊗ 1v ) v

and thus is nontempered. Hence Θ(1) would be a counterexample to the naive Ramanujan conjecture. ● On the other hand, if 1 has nonzero theta lift to U(W0 ), then we may select two places v1 and v2 and replace each of Wv1 and Wv2 by the other local skew-Hermitian space. In other words, we can find a global skewHermitian space W ′ such that Wv′1 ≠ Wv1

and Wv′2 ≠ Wv2

but Wv′ ≅ Wv

for all v ≠ v1 , v2 .

Such a global skew-Hermitian space exists, by our classification of global Hermitian spaces. Now consider the global theta lift ΘV,W ′ (1) on U(W ′ ) and observe that θVv1 ,Wv′ 1 (1) = 0 = θVv2 ,Wv′ 2 (1) because of dichotomy. Hence we can repeat the above argument in replacing W by W ′ . 3.13. Guided exercise. As in the local case, it will be instructive to carry out some of the computations in proving some of the above results, at least in our setting of U1 × U3 . The guided exercise below is the global analog of the local guided exercise in Lecture 2, following the various notation there.

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To do the exercise, it is necessary to write down the theta function θ(φ) (for φ ∈ Ω) as explicitly as possible. Recall that Ω is realized on S(AE e∗ ⊗ v0 ) ⊗ ω where ω is the global Heisenberg-Weil representation of (U(V ) × U(W0 )) ⋉ H(V ⊗E W0 ). We have an automorphic functional θ0 ∶ ω → A(Mp2 ) → C where the last arrow is the evaluation at 1. The automorphic realization θ ∶ Ω = S(AE ) ⊗ ω → A(U(V ) × U(W )) is then given by θ(f )(g) = ∑ θ0 (Ω(g)(f )(x)) . x∈E

Now for the exercise: Exercise: ● For a cusp form f on U(V ) and φ ∈ Ω, compute: θ(φ, f )Z (g) ∶= ∫

[Z]

θ(φ, f )(zg) dz

and θ(φ, f )U (g) = ∫

[U]

θ(φ, f )(ug) du,

where Z = {u(0, z) ∶ z ∈ k} is the center of the unipotent radical U of B. ● Likewise, compute: θ(φ, f )Z,ψ (g) ∶= ∫

[Z]

ψ(z) ⋅ θ(φ, f )(zg) dz

● From these computations, deduce as much of Cor. 3.4 as possible. 4. Lecture 4: Arthur’s Conjecture In this final lecture, we will discuss an influential conjecture of Arthur [2] which explains and classifies all possible failures of the naive Ramanujan conjecture. We will then illustrate with some examples, including the Howe-PS example discussed earlier, the Saito-Kurokawa example for PGSp4 and the case of the exceptional group G2 . 4.1. A basic hypothesis. In the formulation of Arthur’s conjecture, one needs to make a (serious) assumption: (Basic Hypothesis): There is a topological group Lk (depending only on the number field k) satisfying the following properties: ● the identity component L0k of Lk is compact and the group of components Lk /L0k is isomorphic to the Weil group Wk of k; ● for each place v, there is a natural conjugacy class of embeddings Lkv ↪ Lk , where Lkv is the Weil group if kv is archimedean, and the Weil-Deligne group Wkv × SU2 (C) if kv is non-archimedean.

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● there is a natural bijection between the set of isomorphism classes of irreducible representations of Lk of dimension n and the set of cuspidal representations of GLn (A). This assumption is basically the main conjecture in the Langlands program for GLn . We can view it as a classification of the cuspidal representaitons of GLn in terms of irreducible n-dimensional Galois representations. 4.2. Arthur’s conjecuture. Arthur’s conjecture is a classification of the constituents of A2 (G), i.e. a classification of the square-integrable automorphic representations of G. This classification proceeds in two steps. The first step is approximately the classification of these constituents up to near equivalence (this is not entirely true, but for the groups discussed here, it is expected to be so). Here, we say that two representations π1 = ⊗′v π1,v and π2 = ⊗′v π2,v of G(A) are nearly equivalent if for almost all places v, π1,v and π2,v are isomorphic: this is an equivalence relation. More precisely, Arthur speculated that there is a decomposition A2 (G) = ⊕ A2,ψ , ψ

where each A2,ψ is a near equivalence class, and the direct sum runs over equivalence classes of discrete A-parameters ψ. For a split group G, an A-parameter is an admissible map ψ ∶ Lk × SL2 (C) → G∨ where G∨ is the complex Langlands dual group of G. One property of being admissible is that ψ(Lk ) should be bounded in G∨ . An A-parameter is discrete if the centralizer group Sψ ∶= ZG∨ (ψ)/Z(G∨ ) is finite. The second step of the classification is to describe the decomposition of each A2,ψ . This has a local-global structure. That is, for each place v, there will be a finite set of unitary representations of G(kv ) (depending on the A-parameter ψ). If we pick an element πv from each of these finite sets at each place v, we may form the (restricted) tensor product π ∶= ⊗v πv , which is a representation of G(A). Then A2,ψ is the sum of such representations, with appropriate multiplicities. Let us now be more precise. 4.3. Local A-packets. The global A-parameter ψ gives rise (by restriction) to a local A-parameter ψv ∶ Lkv × SL2 (C) → G∨ for each place v of k. Set Sψv = π0 (ZG∨ (ψv )/Z(G∨ )) . This is the local component group of ψv . To each irreducible representation ηv of Sψv , Arthur speculated that one can attach a unitarizable finite length (possibly reducible, possibly zero) representation πηv of G(kv ). The set Aψv = {πηv ∶ ηv ∈ Irr(Sψv )} is called a local A-packet. Among other things, it is required that

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● for almost all v, πηv is irreducible and unramified if ηv is the trivial character 1v . In fact, for almost all v, πηv is the unramified representation whose Satake parameter is: 1/2

sψv = ψv (Frobv × (

qv

−1/2

qv

)) ,

where Frobv is a Frobenius element at v and qv is the number of elements of the residue field at v. 4.4. Global A-packets. With the local packets Aψv at hand, we may define the global A-packet by: Aψ = {π = ⊗v πηv ∶ πηv ∈ Aψv ηv = 1v for almost all v} . It is a set of nearly equivalent representations of G(A), indexed by the irreducible representations of the compact group Sψ,A ∶= ∏ Sψ,v . v

If η = ⊗v ηv is an irreducible character of Sψ,A , then we may set πη = ⊗ πηv . v

This is possible because for almost all v, ηv = 1v and π1v is required to be unramified by the above. 4.5. Multiplicity formula. The space A2,ψ will be the sum of the elements of Aψ with some multiplicities. More precisely, note that there is a natural map Sψ → Sψ,A . Arthur attached to ψ a quadratic character ψ of Sψ (whose definition is given below). Now if η is an irreducible character of Sψ,A , we set mη =

⎛ ⎞ 1 ⋅ ∑ ψ (s) ⋅ η(s) . #Sψ ⎝s∈Sψ ⎠

Then Arthur conjectures that A ψ ≅ ⊕ mη π η . η

4.6. The character ψ . The definition of the quadratic character ψ is quite subtle. For a discrete ψ, one considers the adjoint action of Sψ × Lk × SL2 (C) on Lie(G∨ ) via ψ and decomposes it into the direct sum of irreducible summands, each of which has the form η ⊗ ρ ⊗ Sr where Sr denotes the r-dimensional irreducible representation of SL2 (C). Note that this is an orthogonal representation, since the adjoint representation has a nondegenerate invariant symmetric bilinear form (e.g. the Killing form if G is semisimple).

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We consider only those irreducible components τ = η ⊗ ρ ⊗ Sk satisfying the following properties: ● η ⊗ ρ ⊗ Sk is orthogonal; ● k is even, so that Sk is a symplectic representation. ● ρ is symplectic, and (1/2, ρ) = −1. The conditions above imply that η is orthogonal, so that det(η) is a quadratic character of Sψ . Let T be the set of irreducible summands satisfying these conditions. Then we set ψ (s) = ∏ det(η)(s) for s ∈ Sψ . τ ∈T

As an example, suppose that ψ is trivial on SL2 (C). Then the set T is empty (since the only Sk which occurs above is the trivial representation S1 ). 4.7. Tempered and non-tempered parameters. An A-parameter ψ is called tempered if ψ is trivial when restricted to SL2 (C). In this case, the representations in Aψ are conjectured to be tempered (this corresponds to the boundedness condition on ψ(Lk )). A non-tempered A-parameter is one for which ψ(SL2 (C)) is not trivial. In this case, for almost all v, the unramified representation π1v (which has Satake parameter sψv ) is nontempered. Thus, according to Arthur’s conjecture, the cuspidal representations in ⊕

Aψ

nontempered ψ

are precisely those which violate the naive Ramanujan conjecture. On the other hand, the cuspidal representations in Aψ for tempered ψ are all tempered and the representation π1 = ⊗v π1v should be globally generic. In this sense, Arthur’s conjecture provides an explanation and classification of nontempered cusp forms. Though the group Lk is conjectural, we really only need its irreducible representations in formulating Arthur’s conjecture for classical groups. As such, under our (basic hypothesis), we can replace all occurrences of “an irreducible n-dimensional representation of Lk by “an irreducible cuspidal representation of GLn ”. Then we may view Arthur’s conjecture as a description of A2 (G) in terms of cuspidal representations of GL’s. Understood in this way, when G is a quasi-split classical group, Arhtur’s conjecture has been verified in the works of Arthur [3] and Mok [21]. 4.8. Connection with theta correspondence. Here is a natural question one can ask concerning theta correspondence and Arthur’s conjecture. We have seen in Lecture 3 that when U(V ) × U(W ) is a dual pair in the stable range (with V the smaller space, and barring an unfortunate boundary case), then for any cuspidal representation π of U(V ), its global theta lift Θ(π) on U(W ) is a nonzero irreducible summand of A2 (U(W )). Since square-integrable automorphic representations have A-parameters, it is natural to ask how the A-parameters of Θ(π) and π are related. If we view A-parameters as representing near equivalence classes, answering this question is about identifying the local theta lifts of unramified representations and then detecting the automorphy of the family of unramified local theta lifts. This line of reasoning leads to:

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Conjecture 4.1 (Adam’s conjecuture). If π ⊂ Acusp (U(V )) has A-parameter Ψ (thought of as an dim V -dimensional representation of LE × SL2 (C)), then under the global theta lift with respect to (χV , χW , ψ), the global theta lift of π (which is a summand in A2 (U(W ))) has A-parameter χV ⋅ (χ−1 W Ψ ⊕ Sdim W −dim V ). Given Arthur’s conjecture, this is largely a local unramified issue. What is subtle about Adam’s conjecture is its local analog (which we don’t discuss here) 4.9. Examples. We shall illustrate Arthur’s conjecture with several families of examples in low rank in the rest of the lecture. For the groups SO5 , U3 and G2 , we shall write down a family of nontempered A-paremeters. For each such A-parameter ψ, we will examine the consequences of Arthur’s conjecture. This will involve determining: ● the local and global component groups associated to ψ; ● the quadratic character ψ ; ● the size of the local A-packets and the structure of the submodule Aψ . We will then see if the description of these A-parameters provide some clues to how the A-packets and Aψ may be constructed. 4.10. Saito-Kurokawa example. Let G = SO5 = PGSp4 , so that its Langlands dual group is G∨ = Sp4 (C). We have the subgroup SL2 (C) × SL2 (C) ⊂ Sp4 (C) = G∨ . These two commuting SL2 ’s play symmetrical roles, as they correspond to a pair of orthogonal long roots in the C2 root system. We will consider A-parameters of the form: ψ = ρ × Id ∶ Lk × SL2 (C) → SL2 (C) × SL2 (C) ⊂ G∨ = Sp4 (C). Such an A-parameter ψ is specficied by giving an (admissible) homomorphism ρ ∶ Lk → SL2 (C). Note also that ZG∨ (ψ) = ZSL2 (ρ) × ZSL2 Hence, the A-parameter ψ is discrete if and only if ZSL2 (ρ) is finite, or equivalently if the 2-dimensional representation of Lk afforded by ρ is irreducible. By our (basic hypothesis), to give such a ρ is the same as giving a cuspidal representation τ = τρ of GL2 with trivial central character, i.e. a cuspidal representation of PGL2 . A discrete A-parameter of G = SO5 of the above form is called a Saito-Kurokawa A-parameter. We have just seen that such A-parameters are parametrized by cuspidal representations of PGL2 . Given a parameter ψ = ψτ , let us compute the various quantities that appear in Arthur’s conjecture. As we saw above Sψ = (ZSL2 (ρτ ) × ZSL2 )/ZSp4 = (μ2 × μ2 )/Δμ2 = μ2 . Likewise the local component groups Sψτ are given by ⎧ ⎪ ⎪μ2 , if ρτv is irreducible; Sψτ,v = ⎨ ⎪ ⎪ ⎩1, if ρτv is reducible.

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The condition that ρτ,v be irreducible is equivalent to τv being a discrete series representation of PGL2 (Fv ). 4.10.1. Local Arthur packets. Now Arthur’s conjecture predicts that for each place v, the local A-packet Aψτ,v has the form: ⎧ ⎪ ⎪{π + , π − }, Aψτ,v = ⎨ τ+v τv ⎪ ⎪ ⎩{πτv },

if τv is discrete series, if τv is not discrete series

Here, πτ+v is indexed by the trivial character of Sψτ,v . Of course, we know what πv+ has to be for almost all v: it is irreducible unramified with Satake parameter sψτ,v . This unramifed representation πv+ is the unramfied constituent of the induced representation 1/2 IP (τv , 1/2) = IndG P ∣ − ∣v ⊗ τv .

where P = M N is the Siegel parabolic subgroup of SO5 with Levi factor GL1 × SO3 = GL1 × PGL2 . From this, we see that the representations in the global A-packet are G(A) nearly equivalent to the constitutents of IndP (A) ∣ − ∣1/2 ⊗ τ . Moreover, their local components are nontempered for almost all v. 4.10.2. Global A-packets. Let Sτ be the set of places v where τv is discrete series, so that the global A-packet has 2#Sτ elements. To compute the multiplicity mη of πη ∈ Aψτ we need to know the quadratic character ψτ of Sψτ . By a short computation (which you should do), ψτ is the non-trivial character of Sψτ ≅ μ2 if and only if (1/2, τ ) = −1. Here (s, τ ) is the global -factor of τ . Now if π = ⊗v πτvv ∈ Aψτ , then the multiplicity associated to π by Arthur’s conjecture is: ⎧ ⎪1, if π ∶= ∏v v = (1/2, τ ); ⎪ m(π) = ⎨ ⎪ ⎪ ⎩0, if π = −(1/2, τ ). Thus, we should have: A ψτ ≅ π. ⊕ π∈Aψτ ∶π =(1/2,τ )

4.10.3. Construction. How can one construct the A-packets Aψτ and the space Aψτ ? It seems that we need a lifting to go from τ to these square-integrable representations of PGSp4 . Since the theta correspondence is 1-to-1, one cannot hope to use theta correspondence to go from τ to Aψτ , never mind the fact that PGL2 × SO5 is not a dual pair in a symplectic group. We need an intermediate step: the Shimura correspondence, or rather its automorphic description by Waldspurger [28, 30]. Using the theta correspondence for Mp2 × SO3 , Waldspurger was able to provide a classification of the constituents of A2 (Mp2 ) in the style of Arthur’s conjecture. More precisely, τ gives rise to a packet of cuspidal representations on Mp2 , whose structure is exactly the same as that of the Saito-Kurokawa packets. Namely,

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for each place v, one has a local packet of irreducible representations of Mp2 (kv ): ⎧ ⎪ ⎪{σ + , σ − }, if τv is discrete series; A˜τv = ⎨ τ+v τv ⎪ {σ }, if τv is not discrete series. ⎪ ⎩ τv We call these the Waldspurger packets. One can form the global packet as a restricted tensor product of the local ones, and one gets a submodule A˜τ =

⊕

σ ⊂ Acusp (Mp2 ).

˜τ ∶σ =(1/2,τ ) π∈A

Observe the formal similarity between the structure of the Waldspurger packets and the Saito-Kurokawa ones. Given this, and taking note that one has a dual pair Mp2 × SO5 (which is the next step of the SO2n+1 Rallis tower), it is then not surprising that the local Saito-Kurokawa packets can be constructed as local theta lifts of the local Waldspurger packet: one sets πτvv = θψv (στvv ). These local theta lifts are nonzero because we are in the stable range. Likewise, the Saito-Kurokawa submodule Aψτ can be constructed as the global theta lift of the submodule A˜τ ⊂ Acusp (Mp2 ). This was first studied by Piatetski-Shapiro [23], but see [7] for a more refined discussion. 4.11. U3 : Howe-PS example. Now we carry out the same analysis for a family of nontempered A-parameters of G = U3 (relative to E/k) which will explain the Howe-PS example we discussed. The Langlands dual group of G = U3 is GL3 (C), but we need to work with the L-group L G = GL3 (C) ⋊ Gal(E/k). The A-parameters of G are then ψ ∶ Lk × SL2 (C) → L G. Thankfully, by [9, Thm. 8.1], the equivalence class of ψ is determined by the equivalence class of its restriction to LE , so we can simply consider ψ ∶ LE → G∨ = GL3 (C). In other words, an A-parameter of G = U3 is simply a 3-dimensional semisimple representation of LE . But this representation needs to satisfy an extra condition: it should be conjugate orthogonal. In addition, for it to be discrete, ψ should be multiplicity-free. Clearly, one has a subgroup GL1 (C) × GL2 (C) ⊂ GL3 (C). We are going to build a discrete A-parameter ψ ∶ LE × SL2 (C) → GL1 (C) × GL2 (C) ⊂ GL3 (C), so that ψ(SL2 (C)) = SL2 (C) ⊂ GL2 (C). As a 3-dimensional representation, ψ takes the form ψ = ψχ,μ ∶= μ ⊕ χ ⊗ S2

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where S2 denotes the 2-dimensional irreducible representation of SL2 (C). The conjugate-orthogonal condition amounts to requiring that ● μ is a conjugate-orthogonal 1-dimensional character of LE , which by our (basic hypothesis) corresponds to an automorphic character of A×E trivial on A×k ; ● χ is a conjugate-symplectic 1-dimensional character of LE , which corresponds by our (basic hypothesis) to an automorphic character of A×E whose restriction to A×k is ωE/k . Thus, such a ψ = ψχ,μ is specified by the pair (χ, μ) satisfying the above properties. 4.11.1. Component groups and A-packets. The global component group of ψ is S ψ = μ2 and the local component groups are: Sψv

⎧ ⎪ ⎪μ2 , if v is inert in E; =⎨ ⎪ ⎪ ⎩1, if v splits in E.

So the local A-packets have the form ⎧ ⎪ ⎪{π + , π − }, if v is inert in E; Aψv = ⎨ v+ v ⎪ {π }, if v splits in E. ⎪ ⎩ v Moreover, for almost all inert places v, the representation πv+ is the unramified representation contained in the principal series representation 1/2 IndG ⊗μ ˜, B χ∣ − ∣

where μ ˜ is μ regarded as a character of E 1 , via the standard isomorphism Ev× /kv× ≅ 1 Ev . Observe that the global A-packet Aψ has infinitely many elements in this case. 4.11.2. Multiplicity formula. To work out the multiplicity formula, we need to work out the quadratic character ψ . A short and instructive computation shows that ψ is the trivial character of Sψ = μ2 if and only if E (1/2, χμ−1 ) = 1. So Arthur’s conjecture predicts that Aψχ,μ ≅

⊕

π.

π∈Aψ ∶(π)=E (1/2,χμ−1 )

4.11.3. Construction via theta lifts. Adam’s conjecture suggests that the theta correspondence for U1 × U3 that we discussed in Lectures 2 and 3 can be used to construct the local A-packets and the submodule Aψχ,μ . Let us fix our skew-Hermitian space W over k and consider an inert place v of k. Recall that there are two rank 1 Hermitian spaces Vv+ and Vv− over kv for such an inert place v. Roughly speaking, the local A-packet Aψχv ,μv should be constructed as the local theta lift of a particular character of U(Vv+ ) and U(Vv− ), under the

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two theta correspondences for U(Vv+ ) × U(W ) and U(Vv− ) × U(W ). To make this precise, we need to answer a few questions: ● Which pair of splitting characters (χV , χW ) should we use for the theta correspondence? ● Having fixed (χV , χW ), which character of U(Vv+ ) = U(Vv− ) should we start with? ● How should we label the two representations in the local A-packet, i.e. which of these two theta lifts is πv+ , so as to achieve the predicted multiplicity formula? Based on our understanding of the theta correspondence from Lectures 2 and 3, can you answer these questions? 4.12. Example of G2 . We conclude this section by describing 2 families of A-parameters of the split exceptional group G2 . 4.12.1. Some structural facts. The Langlands dual group of G is G2 (C). We list a few relevant facts about the structure of G2 (C), referring to its root system here for justification:

6 Q k 3 ]  

Q J Q J

 Q J  Q  Q J

α  Q  J Q  J Q 

J QQ  

^ J +  s Q

β

?

● The root system of G2 contains a mutually orthogonal pair of long and short roots, giving rise to a commuting pair of SL2 ’s (as in the case of Sp4 (C)) (SL2,l × SL2,s )/μΔ 2 ⊂ G2 . The difference is that in the Sp4 case, the two roots involved have the same length (they are both long), whereas here they are of different length. Hence, these two SL2 ’s are not conjugate to each other. Further, the centralizer of one of these SL2 ’s is the other SL2 . ● The 6 long roots of the G2 root system gives an A2 root system, reflecting the fact that G2 (C) contains a subgroup SL3 (C). Observe that SL2,l (C)×μ2 T ⊂ SL3 (C) (where T is the diagonal torus of SL2,s ) but SL2,s is not contained in SL3 (even after conjugation). Moreover, the normalizer

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of SL3 (C) in G2 (C) contains SL3 (C) with index 2. Indeed, an element in the non-identity component is given by the longest Weyl group element of G2 . This is also the element (w, w) ∈ SL2,l ×μ2 SL2,s , where w is the standard Weyl element in SL2 . The conjugation action of this element on SL3 (C) is an outer automorphism. Hence one has containment SL2,l ×μ2 NSL2 (T ) ⊂ NG2 (SL3 ) = SL3 ⋊Z/2Z ⊂ G2 (C). ● The smallest faithful (irreducible) algebraic representation of G2 is 7dimensional; in fact one has G2 ↪ SO7 . The weights of this 7-dimensional representation are the short roots and the zero vector. When restricted to the subgroup SL3 , this 7-dimensional representations decomposes as : (std3 ) ⊕ 1 ⊕ (std3 )∨ where (std3 ) is a 3-dimensional irreducible representation of SL3 and (std3 )∨ is its dual. When restricted to the subgroup SL2,l ×μ2 SL2,s , it decomposes as: (std2 ) ⊗ (std2 ) ⊕ 1 ⊗ Sym2 (std2 ) where (std2 ) denotes the 2-dimensional representation of SL2 . ● consider the adjoint action of G2 on its Lie algebra g2 . When restricted to the subgroup SL3 , g2 = sl3 ⊕ (std3 ) ⊕ (std3 )∨ . When restricted to the subgroup SL2,l ×μ2 SL2,s , one has g2 = sl2,l ⊕ sl2,s ⊕ (std2 ) ⊗ Sym3 (std2 ). 4.12.2. Some A-parameters. Now suppose that τ is a cuspidal representation of P GL2 , which by our (basic hypothesis) corresponds to an irreducible representation ρτ ∶ LF → SL2 (C). Using ρτ , we can build 2 different nontempered A-parameters of G2 , depending on whether SL2 (C) is mapped to SL2,l or SL2,s : ψτ,s ∶ Lk × SL2 (C) → SL2,l × SL2,s ⊂ G2 (C) or ψτ,l ∶ Lk × SL2 (C) → SL2,s × SL2,l ⊂ G2 (C). We call ψτ,s the short root A-parameter and ψτ,l the long root A-parameter associated to τ .

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4.12.3. Short root A-parameters. Now let’s work out the consequences of Arthur’s conjecture for the short root A-parameter. We have: Sψτ,s ≅ μ2 and for a place v of k, ⎧ ⎪ ⎪μ2 , if τv is discrete series (i.e. ρτ,v is irreducible); Sψτ,s,v = ⎨ ⎪ ⎪ ⎩1 if τv is not discrete series (i.e. ρτ,v is reducible). 4.12.4. Local short root A-packets. Now Arthur’s conjecture predicts that for each place v, the local A-packet Aτ,s,v has the form: ⎧ ⎪ ⎪{π + , π − }, if τv is discrete series, Aτ,s,v = ⎨ τ+v τv ⎪ if τv is not discrete series ⎪ ⎩{πτv }, Here, πτ+v is indexed by the trivial character of Sτ,v . Moreover, we know what πv+ has to be for almost all v. Indeed, for almost all v where τv is unramified, πv+ is the unramified representation with Satake parameter sψτ,v , and this representation is the unramified constituent of 1/2 2 IP (τv , 1/2) = IndG P τv ⋅ ∣ det ∣

where P is the Heisenberg parabolic subgroup of G2 with Levi factor GL2 . 4.12.5. Global short root A-packets. Let Sτ be the set of places v where τv is discrete series, so that the global A-packet has 2#Sτ elements. To describe the multiplicity of πη ∈ Aτ,s in L2ψτ , we need to know the quadratic character ψτ,s of Sψτ,s . It turns out that ψτ,s is the non-trivial character of Sψτ ≅ μ2 if and only if (1/2, τ ) = −1. Now if π = ⊗v πvv ∈ Aτ,s , then the multiplicity associated to π by Arthur’s conjecture is: ⎧ ⎪ ⎪1, if π ∶= ∏v v = (1/2, τ ); m(π) = ⎨ ⎪ ⎪ ⎩0, if π = −(1/2, τ ). Thus, Arthur’s conjecture predicts that: Aψτ,s ≅

⊕

π.

π∈Aτ ∶π =(1/2,τ )

4.12.6. Construction of short root A-packets. Observe that the structure of these A-packets is thus entirely the same as that of the Saito-Kurakawa packets for SO5 . Since the Saito-Kurokawa packets were constructed as theta liftings of Waldspurger’s packets on Mp2 , one might guess that one can construct the short root A-packets of G2 by lifting from the corresponding packets on Mp2 . But is Mp2 ×G2 a reductive dual pair? Well, it turns out that one may consider the dual pair Mp2 × O7 . Recalling that G2 ↪ SO7 , we may consider theta lifts from Mp2 to O7 , followed by restriction of representations from O7 to G2 . Somewhat amazingly, this restriction does not lose much information. In other words, one may consider the commuting pair Mp2 ×G2

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and restrict the Weil representation of Mp2 × O7 to it. Such a construction was first conceived by Rallis and Schiffman, but the full analysis was completed in [8]. In this way, it was shown in [8] that one may construct the A-packets and the corresponding submodule in A2 (G2 ). 4.12.7. Long root A-parameters. The main project for this course is the analysis and construction of the long root A-packets of G2 . In particular, the first task of the project is to work out the prediction of Arthur’s conjecture for the long root A-parameter ψτ,l , and then specialize to the case when τ is dihedral. We list the expected answers here, leaving it as a series of guided exercises: ● the global and local component groups are the same as for the short root A-parameters; so the local A-packets have 2 or 1 elements depending on whether τv is discrete series or not. ● the quadratic character ψτ,l is trivial if and only if (1/2, τ, Sym3 ) = (1/2, Sym3 ρτ ) = 1. So we see that the Sym3 -epsilon factor appears in the Arthur multiplicity formula. 4.13. Dihedral long root A-parameters. We now suppose that τ is a dihedral cuspidal representation relative to a quadratic field extension E/k. This can be interpreted in one of the following equvialent ways: k ● ρτ ≅ IndW WE χ for some 1-dimensional character χ of the global Weil group WE (which is supposedly a quotient of LE ). ● τ ⊗ ωE/k ≅ τ .

The fact that det ρτ = 1 implies that when regarded as an automorphic character of A×E , χ∣A×k = ωE/k , i.e. χ is a conjugate-symplectic automorphic character. The image of ρτ is contained in the normalizer NSL2 (T ), where T is a maximal torus of SL2 . Now we observe: ● When τ is dihedral as above, the long root A-parameter ψτ,l factors as: ψτ,l ∶ Lk ↠ Wk → NSL2,s (T ) ×μ2 SL2,l (C) ⊂ SL3 (C) ⋊ Z/2Z ⊂ G2 (C). This follows from one of the structural facts we recall about G2 (C). ● SL3 (C) ⋊ Z/2Z ⊂ GL3 (C) ⋊ Z/2Z = L U3 . ● Hence the long root A-parameter ψτ,l of G2 factors through the L-group of U3 , thus giving rise to an A-parameter for U3 . Moreover, when restricted to WE , one obtains a 3-dimensional representation of WE × SL2 (C) of the form ψτ,l ∣WE = χ−2 ⊕ χ ⊗ S2 . In other words, one obtains a Howe-PS A-parameter for U3 . Said in another way, one could start with a Howe-PS A-parameter ψ for U3 with ψ(LE ) ⊂ SL3 (C) (or equivalently, giving rise to representations of PU3 ). The composition of ψ with the natural inclusion SL3 (C) ⋊ Z/2Z = NG2 (SL3 ) ↪ G2 (C)

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then gives a long root A-parameter whose associated τ is dihedral with respect to E/k. This suggests the following question: Question: Is it possible to construct the local and global long root A-packets of G2 by lifting from the Howe-PS packets for U3 , and then verify the Arthur multiplicity formula? Addressing this question is the main project for this course. 5. More Exercises In this final section, we give a list of extended exercises. 5.1. Lecture 1. (1) This exercise concerns the parametrization of unramified representations of G(F ) in terms of semisimple classes in the Langlands dual group G∨ . (i) Let T be a split torus over a p-adic field F and let T (OF ) ⊂ T (F ) be the maximal compact subgroup of T (F ). A character χ ∶ T (F ) → C× is unramified if χ is trivial on T (OF ). The set of unramified characters of T (F ) is thus Hom(T (F )/T (OF ), C× ). The dual torus of T is the complex torus defined by T ∨ ∶= X ∗ (T ) ⊗Z C× , where X ∗ (T ) = Hom(T, Gm ). Construct a natural bijection Hom(T (F )/T (OF ), C× ) ≅ T ∨ . (ii) Based on (i) and Proposition 1.1 of the lecture notes, deduce that unramified representations of G = GLn (F ) are paramertrized by semisimple conjugacy classes in G∨ = GLn (C). (2) This exercise gives you a chance to work with unitary groups in low rank. (i) In §1.9 of the lecture notes, we gave as examples the quasi-split unitary groups U(V + ) with dim V + = 2 and 3 and wrote down certain elements as matrices. Let B ⊂ U(V + ) be the upper triangular Borel subgroup. Compute the modulus character δB as a character of the diagonal torus T . (ii) At the end of §1.9 of the lecture notes, we introduced the element u(x, z) as a matrix, but one particular entry of the matrix is given as ∗, as it is determined by the others. Determine the entry ∗ explicitly. (iii) In §1.9 of the lecture notes, we described some elements of U(V ) where V is a 3-dimensional Hermitian space. In fact, from Lecture 2 onwards, we will be working with 3-dimensional skew-Hermitian spaces W . As what we did for the Hermitian case, write down elements in the Borel subgroup B = T U of U(W ), with respect to a Witt basis of W , i.e. a basis {e, w0 , e∗ } with ⟨e, e∗ ⟩ = 1, ⟨w0 , w0 ⟩ = δ (with δ trace 0) and ⟨e, w0 ⟩ = 0 = ⟨e∗ , w0 ⟩.

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5.2. Lecture 2. (3) This exercise is based on §2.3 which introduces the Heisenberg group H(W ) associated to a symplectic vector space W . (i) Let W be a 3-dimensional skew-Hermitian space as in Problem 2(iii) above, with isometry group U(W ) containing the Borel subgroup B = T U . Write down an isomorphism of U with the Heisenberg group associated to a 2-dimensional symplectic space. (ii) In §2.3, we introduced the representation H(W )

ωψ = indH(X) ψ of a Heisenberg group H(W ) on the space S(Y ) of Schwartz functions on Y . Prove that this representation is indeed irreducible. (iii) In the context of §2.3, suppose that W = W1 ⊕W2 is the sum of two smaller symplectic spaces, construct a natural surjective group homomorphism f ∶ H(W1 ) × H(W2 ) → H(W ). What is the kernel of your homomorphism f ? (iv) Let ωW,ψ be the irreducible representation of H(W ) with central character ψ. Show that the pullback f ∗ (ωW,ψ ) is isomorphic to ωW1 ,ψ ⊗ ωW2 ,ψ . (v) One has a natural embedding f ∶ Sp(W1 ) × Sp(W2 ) ↪ Sp(W ). Deduce that f ∗ induces a natural isomoprhism of projective representations: AW,ψ ○ f ≅ AW1 ,ψ ⊗ AW2 ,ψ where AW,ψ ∶ Sp(W ) → GL(S)/S 1 is as constructed in §2.3. Indeed, f can be lifted to f˜ ∶ Mp(W1 ) × Mp(W2 ) → Mp(W ) so that one has an isomoprhism f˜∗ (ωW,ψ ) ≅ ωW

1 ,ψ

⊗ ωW2 ,ψ2

of representations of Mp(W1 ) × Mp(W2 ). (4) This exercise asks you to reconcile the different ways that Θ(π) has been presented in the lecture notes. In §2.5, we have given two descriptions of Θ(π): ●

π ⊠ Θ(π) = Ω/

⋂

f ∈HomU(V ) (Ω,π)

Ker(f ).

● Θ(π) = (Ω ⊗ π ∨ )U(V ) . Show that these are equivalent, and prove the “universal property”: HomU(V )×U(W ) (Ω, π ⊗ σ) ≅ HomU(W ) (Θ(π), σ) for any smooth representation σ of U(W ).

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(5) Deduce Corollary 2.5 (dichotomy) from Theorem 2.4 (conservation relation). (6) This problem introduces the Schrodinger model of the Weil representation of a dual pair U(V ) × U(W ), when one of the spaces is split of even dimension. In §2.4, we wrote down some formulas in the Schrodinger model for the Weil representation ωψ of a metaplectic group; this model is based on a Witt decomposition of the symplectic space. In §2.5, we considered a dual pair U(V ) × U(W ) with its splitting ˜ι ∶ U(V ) × U(W ) → Mp(V ⊗E W ) associated to a pair (χV , χW ). One can ask if we can inherit the formulas in the Schrodinger model and write down the action of some elements of U(V ) × U(W ). For this to be possible, there must be some compatibility between the Witt decomposition we used on V ⊗ W and the map ι ∶ U(V ) × U(W ) → Sp(V ⊗E W ). More precisely, suppose V is a split Hermitian space and we fix a Witt decomposition V = X ⊕ Y . Then we inherit a Witt decomposition of V ⊗ W : V ⊗E W = (X ⊗E W ) ⊕ (Y ⊗E W ). Relative to this Witt decomposition, the Schrodinger model of the Weil representation is realized on S(Y ⊗ W ) and one can write down explicit formulas for the elements of U(W ) × P (X) ⊂ P (X ⊗ W ), where P (X) is the Siegel parabolic subgroup of U(V ) stabilzing X. Consider the case when W = E ⋅ w = ⟨δ⟩ (with T r(δ) = 0) is 1-dimensional and V = Ee ⊕ Ee∗ is the split skew-Hermitian space of dimension 2. From the formulas of the Schrodinger model, deduce (as much as you can) the following actions of U(W ) × B(E ⋅ e)) on S(Y ⊗ V ) = S(Ee∗ ⊗ w) (relative to the fixed (χV , χW )): ● for g ∈ U(W ) = E 1 , (g ⋅ f )(x) = χV (i(g))f (g −1 x), where i ∶ E 1 ≅ E × /F × is the inverse of the isomorphism i−1 ∶ x ↦ x/xc . ● For t(a) ∈ T , a ∈ E × , 1/2

(t(a) ⋅ f )(x) = χW (a) ⋅ ∣a∣E ⋅ f (ac ⋅ x). ● For u(z) ∈ U , with z ∈ E and T r(z) = 0, (u(z) ⋅ f )(x) = ψ(δ ⋅ z ⋅ N (x)) ⋅ f (x). To be honest, since we did not explicate the definition of the splitting associated to (χV , χW ), you could not really show the above formulas in full , but you can at least deduce those parts of the formula without χV or χW . The effects of the choice of (χV , χW ) can be seen from the first two formulas. Actually, this exercise is setting the scene for Problem 7 below, so you may take the formulas above as given. (7) This exercise continues from Problem 6. It is the first exercise that allows you to work with the Weil representation and to calculate some theta lifts.

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Consider the dual pair U(V ) × U(W ) as in Problem 6, so that W = E ⋅ w = ⟨δ⟩ and V the split 2-dimensional Hermitian space. Fix a pair of splitting characters (χV , χW ) and consider the associated Weil representation Ω = ΩχV ,χW ,ψ of U(V ) × U(W ). Because U(W ) is compact, Ω is semisimple as a U(W )-module and we can write: Ω= μ ⊗ Θ(μ). ⊕ μ∈Irr(U(W ))

The goal is to understand Θ(μ) as much as possible. Using the formulas for the Weil representation Ω = ΩχV ,χW ,ψ from Problem 6 above, (i) Compute ΩU (the U -coinvariants of Ω) as a module for U(W ) × T . (ii) For any nontrivial character ψ ′ of U ≅ F ⋅ δ −1 , compute ΩU,ψ′ as a module for U(W ) × Z(U(V )). (Note that there are two orbits of such nontrivial characters ψ ′ under the conjugation action of T (F )). (iii) Using your answers from (i) and (ii), show that Θ(μ) is nonzero irreducible for any irreducible character μ of U(W ) = E 1 . Moreover, show that Θ(μ) is supercuspidal if and only if μ ≠ χV ○ i (see Problem 6 for the definition of i). (iv) Show that U(V )

Θ(χV ○ i) ↪ I(χW ) ∶= IndB(E⋅e) χW . Indeed, your proof should suggest an explicit description of this embedding. More precisely, show that the natural map f ↦ (h ↦ (Ω(h)f )(0)) gives a nonzero equivariant map Ω → (χV ○ i) ⊗ I(χW ), thus inducing the embedding of Θ(χV ○ i) into I(χW ). The results of this exercise will be used in the next exercise. (8) The purpose of this exercise is to indicate a proof of the Howe duality conjecture and Theorem 2.6 for the dual pair U1 × U1 . It is very long, but is the highlight of the problem sheet! As mentioned in the notes, the proof makes use of the doubling seesaw argument (among other things). We will introduce some of these notions in turn. ● (Seesaw pairs) Suppose a group E contains two dual pairs G1 × H1 and G2 × H2 (so Gi is the centralizer of Hi in E and vice versa). Suppose that G1 ⊂ G2 . Then it follows that H1 ⊃ H2 . In this situation, we say that the two dual pairs form a seesaw pair, and we often represent this in the following seesaw diagram: H1 G2 B BB || BB|| || BB || B H2 G1

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In this diagram, the diagonal line represents a dual pair, and the vertical line denotes containment, with the group at the bottom contained in the group at the top. ● (Standard example) Here is the standard example of constructing seesaw pairs in the symplectic groups. Suppose that V1 ⊕ V2 is the orthogonal sum of two Hermtian spaces. Set W = (V1 ⊕ V2 ) ⊗ W (a symplectic space over F ), and note that W = (V1 ⊗ W ) ⊕ (V2 ⊗ W ). This gives the following two dual pairs in Sp(W): U(V1 + V2 ) × U(W Δ ) and

(U(V1 ) × U(V2 ))) × (U(W ) × U(W )).

Convince yourself that these form a seesaw pair and draw the relevant seesaw diagram. ● (Seesaw identity) Suppose one has a seesaw diagram as in the abstract situation above, and let Ω be a representation of E, which we may restrict to G1 × H1 and G2 × H2 . Deduce the following seesaw identity: for π ∈ Irr(G1 ) and σ ∈ Irr(H2 ), one has natural isomorphisms HomG1 (Θ(σ), π) ≅ HomG1 ×H2 (Ω, π ⊗ σ) ≅ HomH2 (Θ(π), σ). Note that in the above identity, Θ(σ) is a representation of G2 , whereas Θ(π) is a representation of H1 . This seesaw identity allows one to transfer a restriction problem from one side of the seesaw to the other. ● (Compatible splittings) To apply this seesaw identity to the standard example, there is an extra step, because we need to consider splittings of the dual pair into the metaplectic group Mp(W). To have a splitting of the metaplectic cover for the dual pair U(V1 ) × U(W ), we need to fix a pair (χV1 , χW ); likewise, we need to fix (χV2 , χ′W ) for U(V2 )×U(W ). Similarly, for the dual pair U(V1 + V2 ) × U(W Δ ), we may fix (χV1 +V2 , χW Δ ). So we see that we have 6 splitting characters to fix here. If we were to choose these randomly, then there is no reason for the resulting splittings to be compatbile with each other. What does being compatible with each other mean? From the viewpoint of the seesaw diagram, in order to have the seesaw identity, we need to ensure that when the splittings of a group at the top of the diagram is restricted to the subgroup below it, the restriction agrees with the splitting below. This is to ensure that we still have a seesaw situation in Mp(W). So for example, we fix the character χW Δ which determines the splitting of U(V1 + V2 ). When restricted to U(V1 ) × U(V2 ), the resulting splitting over U(V1 ) and U(V2 ) are both associated with χW Δ . This forces us to take χW Δ = χW = χ′W , and we shall denote this by χW . Likewise, we fix the characters χV1 and χV2 which determine a splitting over U(W ) × U(W ). When restricted to the diagonal U (W Δ ), the resulting splitting of the latter is associated with the character χV1 χV2 . This forces us to take χV1 +V2 = χV1 ⋅ χV2 .

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(i) (Doubling seesaw) Now we apply the above to the following concrete situation. We place ourselves in the setting of Theorem 2.6. Hence, let W = ⟨b ⋅ δ⟩ be a 1-dimensional skew-Hermitian space, and V = ⟨a⟩ a 1-dimensional Hermitian space, so that (W ) = ωE/F (b) and (V ) = ωE/F (a). Let V − = ⟨−a⟩ and apply the seesaw construction above with W as given, V1 = V,

and V2 = V −

so that V ◻ ∶= V1 ⊕ V2 = V ⊕ V − is a split 2-dimensional Hermitian space. We thus have the seesaw diagram: U(V ◻ )

U(W ) × U(W ) QQQ QQQ mmmmm QmQm mmm QQQQQ m m m U(W )Δ U(V ) × U(V − ) This is called the doubling seesaw, because we have doubled V (to yield V ◻ ), but note that we have introduced a negative sign in the second copy of V , so that the doubled-space V ◻ is split! Indeed, the diagonally embedded V Δ is a maximal isotropic subspace, and one has a Witt decomposition V◻ =VΔ⊕V∇ where V ∇ = {(v, −v) ∶ v ∈ V }. (ii) (Doubling seesaw identity) Choose splitting characters χV , χV − and χW as explained above. In fact, we insist further (as we may) that χV = χV ′ = χW = γ

(a conjugate-symplectic character of E × ).

Fix a χ ⊗ χ′ ∈ Irr(U(V ) × U(V − )) and the character χV ∣E 1 of U(W Δ ). Write down what the doubling seesaw identity gives. (iii) (Duality) We have the two decompositions ΩV,W,γ,ψ = ⊕ χ ⊗ ΘV,W (χ), χ

and ΩV − ,W,γ,ψ = ⊕ χ′ ⊗ ΘV − ,W (χ′ ). χ′

Express the Weil representation ΩV − ,W,γ,ψ in terms of ΩV,W,γ,ψ and deduce that ΘV − ,W (χ−1 χW ∣E 1 ) ≅ ΘV,W (χ) ⋅ χV ∣E 1 . Using this, show that −1

ΩV,W,γ,ψ ⊗ ΩV − ,W,γ,ψ = ⊕ χ ⊗ χ′ χW ∣E 1 ⊗ HomU(W ) (ΘV,W (χ) ⊗ ΘV,W (χ′ ), C) χ,χ′

as a module for U(V ) × U(V − ) × U(W Δ ). (Note that here and below, we could have replaced χV and χW by γ, but we have refrained from doing so, in order to make the dependence of (χV , χW ) more transparent).

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(iv) (Siegel-Weil) The LHS of the doubling seesaw is −1

HomU(V )×U(V − ) (ΘW Δ ,V ◻ (χV ∣E 1 ), χ ⊗ χ′ χW ∣E 1 ). To address this problem, we first need to understand ΘW Δ ,V ◻ (χV ∣E 1 ). This is the local version of the Siegel-Weil formula and is where Problem 7 comes in. Using your results in Problem 7(iv), show that U(V ◻ )

ΘW Δ ,V ◻ (χV ∣E 1 ) ↪ IndB(V Δ ) χW =∶ I(χW ) (v) (Principal series) Show that the principal series I(χW ) is reducible and in fact is the direct sum of two irreducible summands. How does one distinguish between those two summands? Which of these two summand is ΘW Δ ,V ◻ (χV ∣E 1 ) equal to? Show that ΘW Δ ,V ◻ (χV ∣E 1 ) is the unique summand of the induced representation whose (U, ψ ′ )-coinvariant is nonzero for ψ ′ (zδ −1 ) = ψ(bz). This shows that if W and W ′ are the two 1-dimensional skew-Hermitian spaces, then I(χW ) = ΘW,V ◻ (χV ∣E 1 ) ⊕ ΘW ′ ,V ◻ (χV ∣E 1 ). (vi) (Mackey theory) The previous part implies that it will be necessary to understand I(χW ) as a module for U(V ) × U(V − ). This can be approached by Mackey thoery, as it involves the restriction of an induced representation. Show that U(V ) × U(V − ) acts transitively on the flag variety B(V Δ )/ U(V ◻ ) with stabilizer of the identity coset given by U(V )Δ . From this, deduce that as a U(V ) × U(V − )-module, I(χW ) is isomorphic to Cc∞ (U(V )) ⊗ (1 ⊗ χW ∣E 1 ), −1

i.e. a twist of the regular representation. Hence, for χ ⊗ χ′ ∈ Irr(U(V ) × U(V − )), one has ⎧ ⎪ ⎪C, if χ′ = χ; −1 HomU(V )×U(V − ) (I(χW ), χ ⊗ χ′ ⋅ χW ∣E 1 ) = ⎨ ⎪ ⎪ ⎩0, otherwise. Moreover, as a U(V )×U(V − )-module, ΘW Δ ,V ◻ (χV ∣E 1 ) is a submodule of the above twisted regular representation. (vii) (Howe duality) Using (iii) and (vi), show the following: ● For each χ ∈ Irr(U(V )), ΘV,W (χ) is irreducible or 0; in particular it is either χ or 0. ● If χ ≠ χ′ , then ΘV,W (χ) and ΘV,W (χ′ ) are disjoint. This is the Howe duality theorem for U(V ) × U(W ). (viii) (Doubling zeta integral) The remaining issue is to decide for which χ is ΘV,W (χ) ≠ 0. This requires the use of the doubling zeta integral. In this context, the doubling zeta integral is an explicit integral which defines a nonzero element of HomU(V )×U(V − ) (I(χW ), χ ⊗ χ−1 χW ∣E 1 ) = HomU(V )×U(V − ) (Cc∞ (U(V )), χ ⊗ χ−1 ). More precisely, we define U(V ◻ )

Z(s, χ) ∶ I(s, χW ) = IndB(V Δ ) χW ⋅ ∣ − ∣sE → C

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by Z(s, χ)(fs ) = ∫

U(V )

fs (h, 1) ⋅ χ(h) dh.

Verify that the integral converges absolutely and defines a nonzero functional Z(s, χ) ∈ HomU(V )×U(V − ) (I(s, χW ), χ ⊗ χ−1 χW ∣E 1 ). Deduce also that θV,W (χ) is nonzero if and only if Z(0, χ) is nonzero on ΘW Δ ,V ◻ (χV ∣E 1 ). (ix) (Functional equation) There is a standard U(V ◻ )-intertwining operator M (s) ∶ I(s, χW ) → I(−s, χW ), whose precise definition need not concern us here. There is a normalization of this intertwining operator (which we will not go into here) with the following properties: ● one has M (−s) ○ M (s) = 1. ● At s = 0 (where I(χW ) is the sum of two irreducible summands), M (s) is holomorphic and M (0) acts as +1 on ΘW Δ ,V ◻ (χV ∣E 1 ) and as −1 on the other summand. A basic result in the theory of the doubling zeta integral is that there is a functional equation: Z(−s, χ−1 )(Ms (fs )) 1 Z(s, χ)(fs ) = E ( + s, χE χ−1 , W , ψ) ⋅ 1 −1 2 LE ( 12 − s, χ−1 χ ) L E ( 2 + s, χE χW ) E W where we recall that χE (x) = χ(x/xc ) for x ∈ E × . We shall take this as a given. (x) (Proof of Theorem 2.6) Using the functional equation and the properties of M (0) recalled in (ix), as well as the relevant results in earlier parts, prove Theorem 2.6. (9i) Consider the dual pair U(V ) × U(W ) as in §2.11. Over there, a model for the Weil representation is written down. Understand how the formulas there are deduced from Problems 3(v) and 4. (ii) Do the exercise formulated at the end of §2.11. 5.3. Lecture 3. (10) The purpose of this exercise is to do the global analog of Problem 7. Hence, we are considering the dual pair U(W ) × U(V ) over a number field k, with W = E ⋅ w = ⟨δ⟩ a 1-dimensional skew-Hermitian space (so δ ∈ E × is a trace 0 element) and V = Ee ⊕ Ee∗ = X ⊕ Y is a split 2-dimensional Hermitian space. As in Problem 6, the global Weil representation Ω is realized on S(YA ⊗ WA ) = S(AE e∗ ⊗ w). The automorphic realization θ ∶ S(YA ⊗ WA ) → A(U(V ) × U(W )) of Ω is defined by θ(φ)(g, h) = ∑ (Ω(g, h)φ)(v). v∈Vk

AUTOMORPHIC FORMS AND THE THETA CORRESPONDENCE

107

For an automorphic character χ of U(W ), its global theta lift Θ(χ) is spanned by the automorphic forms θ(φ, χ)(h) = ∫

[U(W )]

θ(φ)(g, h) ⋅ χ(g) dg

as φ ranges over elements of S(YA ⊗ WA ). (i) Recall the Borel subgroup B = T U of U(V ) which is the stabilizer of X = E ⋅ e. For any character ψ ′ of U (k)/U (A), compute the (U, ψ ′ )-Fourier coefficient θ(φ, χ)U,ψ′ (h) = ∫

[U]

θ(φ, χ)(uh) ⋅ ψ ′ (u) du.

(ii) From your computation in (i), deduce that the global theta lift Θ(χ) is nonzero for any χ, and is cuspidal if and only if χ ≠ χV ○ i (see Problem 6 for the definition of i). (iii) (Challenging) We would like to express the global theta lift θ(φ, χV ○ i) (which is noncuspidal) as an explicit Eisenstein series. Observe that the map φ ↦ θ(φ, χV ○ i) is an equivariant map Ω ↠ (χV ○ i) ⊗ Θ(χV ○ i) ⊂ (χV ○ i) ⊗ A(U(V )) and that dim HomU(W )×U(V ) (Ω, (χV ○ i) ⊗ Θ(χV ○ i)) = 1. Now we shall produce another element in this 1-dimensional vector space. Recall from Problem 7(iv) that the map φ ↦ (h ↦ Ω(h)φ(0)) defines an equivariant map j ∶ Ω → (χV ○ i) ⊗ I(χW ) whose image is isomorphic to (χV ○ i) ⊗ Θ(χV ○ i). Now the Eisenstein series is a U(V )-equivariant map E(s, −) ∶ I(s, χW ) → A(U(V )) defined by E(s, f )(h) =

∑

f (γg).

γ∈B/ U(V )

This converges only when Re(s) is sufficiently large (actually Re(s) > 1/2), but a basic theorem is that it admits a meromorphic continuation to C and that it is holomorphic at s = 0. Admitting this, we can thus consider E(−) ∶= E(0, −). Now we have the composite map E ○ j ∶ Ω → (χV ○ i) ⊗ A(U(V )). Show that this map is nonzero, so that its image is isomorphic to (χV ○i)⊗Θ(χV ○i). Note however that the image of E ○ j is not yet known to be equal to the submodule Θ(χV ○ i) ⊂ A(U(V )), but merely isomorphic to it. If we had known that the image is equal to Θ(χV ○ i), then we would have immediately deduce that there is a nonzro constant c ∈ C× such that θ(φ, χV ○ i) = c ⋅ E(j(φ)). Despite this, show that the above identity holds (so that the image is indeed equal to Θ(χV ○i)) (Hint: you may want to consider the Fourier expansion of both sides). This is the so-called global Siegel-Weil formula.

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(11) The purpose of this exercise is to do global analog of the very long Problem 8. We shall use (the global analog of) the notations from Problem 8, as well as the same seesaw setup. Before that, let us explain the global seesaw identity in the context of a general seesaw in an ambient group E: G2 B H1 BB || BB|| || BB || B H2 G1 We are now working over a number field k. Suppose Ω is the “Weil representation” of E and Ω is equipped with an automorphic realization θ ∶ Ω → A(E). For φ ∈ Ω and f ∈ A(G1 ), one has the global theta lift: θ(φ, f )(h) = ∫

[G1 ]

θ(φ)(g, h) ⋅ f (g) dg

so that θ(φ, f ) ∈ A(H1 ). Likewise for f ′ ∈ A(H2 ), one has θ(φ, f ′ ) ∈ A(G2 ) defined by θ(φ, f ′ ) = ∫

[H2 ]

θ(φ)(g, h) ⋅ f ′ (h) dh.

Now the global seesaw identity is simply: ⟨θ(φ, f ), f ′ ⟩H2 = ⟨θ(φ, f ′ ), f ⟩G1 , where we are using the Petersson inner products on G1 and H2 here. Indeed, from definition, it follows that both sides are given by the double integral ∫

[G1 ×H2 ]

θ(φ)(g, h) ⋅ f (g) ⋅ f ′ (h) dg dh.

Hence the global seesaw identity is simply an application of Fubini’s theroem: exchanging the order of integration. Now we place ourselves in the context of Problem 8, with the follwing seesaw: U(V ◻ )

U(W ) × U(W ) QQQ QQQ mmmmm QmQm mmm QQQQQ m m m U(V ) × U(V − ) U(W )Δ −1

(i) Taking the automorphic characters f = χ ⊗ χ′ χW ∣E 1 on U(V ) × U(V − ) and f ′ = χV ∣E 1 on U(W Δ ), write down the resulting global seesaw identity. (ii) We now examine the RHS of the seesaw identity (the side of W ’s). For φ ∈ ΩV − ,W,ψ , show using Problem 8(iii) that −1

θ(φ, χ′ χW ∣E 1 ) ∈ ΘV,W,ψ (χ′ ) ⋅ χV ∣E 1 . (iii) For φ1 ∈ ΩV,W,ψ and φ2 ∈ ΩV − ,W,ψ , show that ∫

−1

[U(W )]

θ(φ1 , χ)(g) ⋅ θ(φ2 , χ′ χW ∣E 1 )(g) ⋅ χV (g) dg

can be nonzero only if χ′ = χ. Moreover, θ(φ1 , χ) is nonzero if and only if the above integral is nonzero for some φ2 (and taking χ′ = χ).

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109

(iv) The LHS of the global seesaw identity (the side of V ’s) has the form: ∫

[U(V )×U(V − )]

θ(φ1 , ⊗φ2 , χV ∣E 1 ) ⋅ χ(h1 ) ⋅ χ(h2 ) ⋅ χW (h2 )−1 dh1 dh2 ,

where θ(φ1 , ⊗φ2 , χV ∣E 1 ) = ∫

[U(W Δ )]

θ(φ1 ⊗ φ2 )(gh) ⋅ χV (g) dg

is the global theta lift of χV ∣E 1 from U(W Δ ) to U(V ◻ ). As in Problem 8, we now need to explicate the theta lift θ(φ1 , ⊗φ2 , χV ∣E 1 ). This is provided by the global Siegel-Weil formula of Problem 10(iii), which expresses θ(φ1 , ⊗φ2 , χV ∣E 1 ) as an Eisenstein series E(j(φ1 ⊗ φ2 )) (where we recall that j(φ1 ⊗ φ2 ) ∈ I(χW )). (v) On replacing θ(φ1 , ⊗φ2 , χV ∣E 1 ) by the Eisenstein series E(j(φ1 ⊗ φ2 )), the integral in (iv) becomes (a special value of) the gloibal doubling zeta integral: Z(s, χ)(f ) = ∫

−1

[U(V )×U(V − )]

E(s, f )(h1 , h2 ) ⋅ χ(h1 )

−1

⋅ χ(h2 ) ⋅ χW (h2 )

dh1 dh2 ,

for fs ∈ I(s, χW ). The theory of this doubling zeta integral is discussed in Ellen Eischen’s lectures. As discussed there, this doubling zeta integral represents the L-value L(1/2 + s, χE ⋅ χ−1 W ). More precisely, for fs ∈ I(s, χW ), we have: Z(s, χ)(fs ) =

LE ( 12 + s, χE ⋅ χ−1 Zv (s, χv )(fs,v ) ⋅ L(1 + 2s, ωEv /kv ) W) ⋅∏ , L(1 + 2s, ωE/k ) LEv ( 12 + s, χEv ⋅ χ−1 v W,v )

where the product over v is finite (as almost all terns are equal to 1). Using this identity and the last assertion in Problem 8(viii), prove Theorem 3.3 in the lecture notes. (12) Do the guided exercise in §3.13 of the lecture notes. This is the global analog of Problem 9. 5.4. Lecture 4. (13) For an A-parameter ψ considered in Lecture 4, i.e. Saito-Kurokawa type for PGSp4 , Howe-PS type fop U3 and the short and long root type for G2 , compute the global component group Sψ and the quadratic character ψ . (14) Do the same for the original Howe-PS A-parameter on PGSp4 defined as follows: ψ = ρ × id ∶ Lk × SL2 (C) → O2 (C) × SL2 (C) → Sp4 (C), where the second arrow is defined by the natural map associated to the tensor product of a 2-dimensional quadratic space and a 2-dimensional symplectic space (which yields a 4-dimensional symplectic space). In addition, note that a homomorphism ρ ∶ Lk → O2 (C) determines: ● by composition with the determinant map on O2 (C) a quadratic ´etale k-algebra E ● an automorphic character χρ of E 1 ≅ E × /k× .

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Acknowledgments: I thank the organizers of the 2022 Arizona Winter School for their kind invitation to deliver a short course based on these lecture notes and for their hospitality and logistical support. I also thank the referees for their thorough work and pertinent questions. The author is partially supported by a Singapore government MOE Tier 1 grant R-146-000-320-114.

Bibliography [1] J. Adams, L-functoriality for dual pairs, Ast´ erisque 171-172 (1989), 85–129. Orbites unipotentes et repr´ esentations, II. MR1021501 [2] James Arthur, Unipotent automorphic representations: global motivation, Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 1–75. MR1044818 [3] James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups, DOI 10.1090/coll/061. MR3135650 [4] P. Baki´ c and G. Savin, Howe duality for a quasi-split exceptional dual pair, arXiv:2112.02760. ´ [5] Pierre Deligne, La conjecture de Weil. I (French), Inst. Hautes Etudes Sci. Publ. Math. 43 (1974), 273–307. MR340258 [6] Wee Teck Gan, Automorphic forms and automorphic representations, Automorphic forms and the Langlands program, Adv. Lect. Math. (ALM), vol. 9, Int. Press, Somerville, MA, 2010, pp. 68–134. MR2581948 [7] Wee Teck Gan, The Saito-Kurokawa space of PGSp4 and its transfer to inner forms, Eisenstein series and applications, Progr. Math., vol. 258, Birkh¨ auser Boston, Boston, MA, 2008, pp. 87–123, DOI 10.1007/978-0-8176-4639-4 3. MR2402681 ̃ 2 , Amer. [8] Wee Teck Gan and Nadya Gurevich, Nontempered A-packets of G2 : liftings from SL J. Math. 128 (2006), no. 5, 1105–1185. MR2262172 [9] Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups (English, with English and French summaries), Ast´erisque 346 (2012), 1–109. Sur les conjectures de Gross et Prasad. I. MR3202556 [10] Wee Teck Gan and Gordan Savin, On minimal representations definitions and properties, Represent. Theory 9 (2005), 46–93, DOI 10.1090/S1088-4165-05-00191-3. MR2123125 [11] Wee Teck Gan and Shuichiro Takeda, A proof of the Howe duality conjecture, J. Amer. Math. Soc. 29 (2016), no. 2, 473–493, DOI 10.1090/jams/839. MR3454380 [12] Stephen Gelbart, On theta-series liftings for unitary groups, Theta functions: from the classical to the modern, CRM Proc. Lecture Notes, vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 129–174, DOI 10.1090/crmp/001/04. MR1224053 [13] Stephen S. Gelbart and Jonathan D. Rogawski, L-functions and Fourier-Jacobi coefficients for the unitary group U(3), Invent. Math. 105 (1991), no. 3, 445–472, DOI 10.1007/BF01232276. MR1117148 [14] Michael Harris, Stephen S. Kudla, and William J. Sweet, Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9 (1996), no. 4, 941–1004, DOI 10.1090/S0894-0347-96-00198-1. MR1327161 [15] R. Howe and I. I. Piatetski-Shapiro, A counterexample to the “generalized Ramanujan conjecture” for (quasi-) split groups, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 315–322. MR546605 [16] Stephen S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), no. 2, 229–255, DOI 10.1007/BF01388961. MR818351 [17] Stephen S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), no. 1-3, 361–401, DOI 10.1007/BF02773003. MR1286835 [18] Alberto M´ınguez, Correspondance de Howe explicite: paires duales de type II (French, with ´ Norm. Sup´ English and French summaries), Ann. Sci. Ec. er. (4) 41 (2008), no. 5, 717–741, DOI 10.24033/asens.2080. MR2504432 111

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[19] Colette Mœglin, Marie-France Vign´ eras, and Jean-Loup Waldspurger, Correspondances de Howe sur un corps p-adique (French), Lecture Notes in Mathematics, vol. 1291, SpringerVerlag, Berlin, 1987, DOI 10.1007/BFb0082712. MR1041060 [20] C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. ´ Une paraphrase de l’Ecriture [A paraphrase of Scripture], DOI 10.1017/CBO9780511470905. MR1361168 [21] Chung Pang Mok, Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015), no. 1108, vi+248, DOI 10.1090/memo/1108. MR3338302 [22] Dipendra Prasad, Weil representation, Howe duality, and the theta correspondence, Theta functions: from the classical to the modern, CRM Proc. Lecture Notes, vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 105–127, DOI 10.1090/crmp/001/03. MR1224052 [23] I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math. 71 (1983), no. 2, 309– 338, DOI 10.1007/BF01389101. MR689647 [24] S. Rallis, On the Howe duality conjecture, Compositio Math. 51 (1984), no. 3, 333–399. MR743016 [25] Jonathan D. Rogawski, The multiplicity formula for A-packets, The zeta functions of Picard modular surfaces, Univ. Montr´ eal, Montreal, QC, 1992, pp. 395–419. MR1155235 [26] Binyong Sun and Chen-Bo Zhu, Conservation relations for local theta correspondence, J. Amer. Math. Soc. 28 (2015), no. 4, 939–983, DOI 10.1090/S0894-0347-2014-00817-1. MR3369906 [27] Tonghai Yang, Theta liftings and Hecke L-functions, J. Reine Angew. Math. 485 (1997), 25–53, DOI 10.1515/crll.1997.485.25. MR1442188 [28] J.-L. Waldspurger, Correspondance de Shimura (French), J. Math. Pures Appl. (9) 59 (1980), no. 1, 1–132. MR577010 [29] J.-L. Waldspurger, D´ emonstration d’une conjecture de dualit´ e de Howe dans le cas p-adique, p ≠ 2 (French), Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 267–324. MR1159105 [30] Jean-Loup Waldspurger, Correspondances de Shimura et quaternions (French), Forum Math. 3 (1991), no. 3, 219–307, DOI 10.1515/form.1991.3.219. MR1103429 National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 Email address: [email protected]

10.1090/surv/279/03

Modular forms on exceptional groups Aaron Pollack

1. Introduction This course is about quaternionic modular forms (QMFs). A QMF is a very special type of automorphic form, much like holomorphic modular forms (HMFs) are a special type of automorphic form. In fact, quaternionic modular forms appear to behave very similarly to holomorphic modular forms, and this is one reason to be interested in them. In this introduction, I will briefly explain what are QMFs and try to motivate why they are interesting objects to study. I begin by briefly reviewing holomorphic modular forms. 1.1. Holomorphic modular forms. Suppose G is a semisimple Q group, with symmetric space XG = G(R)0 /K 0 , where K 0 is a maximal compact subgroup of G(R)0 . For some groups G, the symmetric space XG has a structure of a complex manifold, for which the G(R)0 action is via biholomorphic maps. This is the case1 if G(R)0 is, for example, isogenous to one of the following groups: (1) Sp2n (R) (2) SO(2, n) (3) U (p, q) In these cases, one can make a classical definition of holomorphic modular forms. The holomorphic modular forms can be thought of sections of holomorphic line bundles on Γ/XG , where Γ ⊆ G(Q) is an arithmetic subgroup. Another way to think of the holomorphic modular forms is as very special automorphic forms ϕ ∶ G(Q)/G(A) → C. Compared to general automorphic forms, holomorphic modular forms are special for at least a few reasons. One reason is that, in many situations2 , they have a classical Fourier expansion and corresponding Fourier coefficients. General automorphic forms do not have as nice of a notion of Fourier The author has been supported by the Simons Foundation via grant 585147, by the NSF via grant 2101888 and by the American Mathematical Society via a Centennial Research Fellowship. He extends his sincere thanks to the organizers of the Arizona Winter School for putting on an exceptional workshop under difficult circumstances, to the other lecturers for their terrific lectures, and to all the participants for their engagement which makes the Winter School a fantastic experience. 1 This is not an exhaustive list 2 If X G is of tube type and G has an appropriate rational parabolic subgroup ©2024 American Mathematical Society

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coefficients. These Fourier coefficients can contain very interesting arithmetic in2 formation. For example, if θ(q) = ∑n∈Z q n is the classical θ-function, which is a modular form of weight 1/2, then θ(q)k = ∑n≥0 rk (n)q n where rk (n) is the number of ways of writing n as the sum of k squares. So the Fourier coefficients of θ(q)k , which is a special modular form of weight k/2, see the arithmetically interesting numbers rk (n). Another reason holomorphic modular forms are interesting is that they are frequently, although certainly not always, the objects for which we can say arithmetically interesting things about their L-values. To be more precise, automorphic representations π have associated L-functions, L(π, r, s). For certain automorphic representations π, the L-functions L(π, r, s) are conjectured to be motivic, i.e., equal to the L-functions of certain motives. One can then transfer the conjectures about motivic L-functions (e.g., the Deligne and Bloch-Kato conjectures) to the automorphic L-functions. For example, one can ask if the special L-values L(π, r, s = s0 ) are algebraic (after dividing by a certain period) and if they can be p-adically interpolated. Working with holomorphic modular forms enables one to prove statements of this sort. For example, there is the recent work of Eischen-Harris-Li-Skinner [5] who construct p-adic L-functions for holomorphic modular forms on unitary groups. Finally, because of their familiarity and extra structure, holomorphic modular forms are a great testing ground for potentially new phenomena. For theorems about automorphic forms–whether it be regarding L-values, periods, the trace formula, Galois representations–the first special cases that are proved are in the context of holomorphic modular forms. Trying to find and test new phenomena is yet another reason why holomorphic modular forms deserve special attention. 1.2. Quaternionic modular forms. For most semisimple groups G, the symmetric space XG does not have a G(R)0 -invariant complex structure. Consequently, there are no holomorphic modular forms on XG , and no obvious notion of very special3 automorphic forms on G(A). Phrased this way, it begs the question: Are there a class of groups G, and a class of very special automorphic on these groups G, that can in some ways take the place of holomorphic modular forms? Quaternionic modular forms are a potential answer to the above question. Their study was initiated by Gross-Wallach [13], Wallach [33], and Gan-Gross-Savin [9]. I have been studying them for the last few years, trying to provide evidence that they behave like holomorphic modular forms. In this course, I will define quaternionic modular forms and give some of this evidence that quaternionic modular forms are arithmetic in a way similar to how holomorphic modular forms are arithmetic. So, what are the groups G above, and what are the quaternionic modular forms? This will be described in detail below, but for now let me give a brief indication. Suppose G is a semisimple group for which XG has a G(R)-invariant complex structure. If ϕ is an automorphic form on G(A) that corresponds to a holomorphic modular form fϕ of some integer scalar weight w, then ϕ has two properties: (1) ϕ(gk) = z(k)−w ϕ(g) for all k ∈ K 0 , where z ∶ K 0 → U (1) is a certain fixed surjection; 3 Some notions of special automorphic forms for G, which we do not consider “very special” for this purpose, are as follows: 1) functorial lifts from smaller groups (not wide enough a class of automorphic forms); 2) cohomological automorphic forms (too broad a class of automorphic forms)

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(2) DCR,w ϕ(g) ≡ 0, where DCR,w is a certain linear differential operator corresponding to the fact fϕ satisfies the Cauchy-Riemann equation. And conversely, the automorphic forms ϕ with these properties correspond to holomorphic modular forms fϕ of weight w. One can also make a definition in terms of the representation theory of real group G(R): At least if w is sufficiently large, ϕ corresponds to a holomorphic modular form of weight w if the (g, K 0 )-module generated by ϕ is a holomorphic discrete series representation πw and ϕ spans a one-dimensional minimal K 0 -type in this representation. In other words, holomorphic modular forms correspond to special vectors in automorphic representations π = πf ⊗ π∞ , where π∞ is a special type of representation of the real group G(R)0 . Now, the groups G with holomorphic modular forms have K 0 such that K 0 possesses a surjection to the smallest nontrivial connected compact group, U (1). Gross and Wallach had the insight to ask the question, “What if G(R)0 is such that K 0 possesses a surjection to the next smallest compact group, SU(2)/μ2 = SO(3)?” The list of these groups includes (strictly) the following groups: ● split G2 ● split F4 ● En,4 , n = 6, 7, 8 (groups of type E6 , E7 , E8 with real rank four) ● SO(4, n)0 with n ≥ 3. The G for which G(R)0 is as above are called quaternionic groups; they possess quaternionic modular forms. Set Vw = Sym2w (C2 ), a representation of SU(2)/μ2 , or of K 0 via the surjection K 0 → SU(2)/μ2 . A quaternionic modular form of integer weight w is an automorphic form ϕ ∶ G(Q)/G(A) → Vw satisfying (1) ϕ(gk) = k−1 ⋅ ϕ(g) for all k ∈ K 0 ; (2) Dw ϕ(g) ≡ 0, for a certain specific linear differential operator Dw . Like with holomorphic modular forms, one can also make a definition of quaternionic modular forms in terms of certain (discrete series) representations of G(R)0 . In more detail, if w is sufficiently large, there is a discrete series representation πw ∨ of G(R)0 whose minimal K 0 -type is Vw ≃ Vw . quaternionic modular forms are automorphic forms that correspond to the (entire) minimal K 0 -type of πw , embedded in the space of automorphic forms on G. 1.3. Why study quaternionic modular forms. From my point of view, the primary purpose of this course is to convince you that quaternionic modular forms are interesting gadgets which deserve further study4 . Let me briefly indicate some ways in which quaternionic modular forms are interesting, leaving a more detailed description to the rest of the notes. 1.3.1. Quaternionic modular forms possess Fourier coefficients. One of the first things to say about quaternionic modular forms is that they have a Fourier expansion and corresponding Fourier coefficients, very similar to that of holomorphic modular forms. In other words, associated to a quaternionic modular form ϕ, one can define a list of complex numbers aϕ (λ), where λ varies in some lattice Λ. The proof of the existence of these Fourier coefficients began with work of Wallach [33], was used by Gan-Gross-Savin [9], and then was made more complete and explicit in [25]. The existence and properties of these Fourier coefficients are somewhat miraculous. Here is, in my mind, an important example: Suppose G1 ⊆ G2 are 4 The secondary purpose of this course is to convince you that exceptional groups are beautiful, and that you can work with them concretely.

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two quaternionic groups, embedded appropriately, and ϕ2 on G2 is a quaternionic modular form of weight w. Then the pullback ϕ1 of ϕ2 to G1 is again a quaternionic modular form of weight w. Moreover, one can show that the Fourier coefficients of ϕ1 are finite sums of the Fourier coefficients of ϕ2 . Note that, for a general automorphic form ϕ2 , one would not be able to say anything of content about a Fourier expansion of ϕ1 from that of ϕ2 . 1.3.2. The Fourier coefficients of quaternionic modular forms appear to be arithmetic. The Fourier coefficients of a quaternionic modular form are defined in a very transcendental way. Nevertheless, they appear to be highly arithmetic. Here are examples: (1) There are degenerate Eisenstein series E2k on G2 of even weight 2k. Using work of Jiang-Rallis [19], Gan-Gross-Savin [9] showed that if the nondegenerate Fourier coefficients of the E2k are nonzero5 , then they are essentially values of Zeta functions ζE (1 − 2k) of totally real cubic fields E. (2) In the works [28], [29], it is shown that for two very special quaternionic modular forms on E8,4 , their Fourier coefficients are (nonzero) rational numbers. (3) In the paper [28], I gave an example of a quaternionic modular forms on E6,4 that is distinguished. I will define this notion precisely below, but for now let me say that a distinguished modular form ϕ is one whose non-degenerate Fourier coefficients aϕ (λ) are 0 unless a certain arithmetic condition on λ is satisfied. (4) In the paper [26], I proved that in every even weight w ≥ 16, there is a nonzero cuspidal modular form on G2 with all Fourier coefficients algebraic numbers. (5) In work with Spencer Leslie [23], we define the notion of modular forms of half-integral weight on exceptional groups and prove that they have a similar notion of Fourier coefficients. Moreover, we construct a modular form of weight 1/2 on G2 whose Fourier coefficients see the 2-torsion in the narrow class groups of totally real cubic fields. Based on the above-mentioned evidence, I want to take this opportunity to make the following conjecture: Conjecture 1.1. Suppose G is a quaternionic exceptional group, and w ≥ 1 is an integer. Then there exists a basis {ϕi } of quaternionic modular forms on G of weight w, such that all the Fourier coefficients aϕi (λ) of the ϕi are algebaic numbers. Conjecture 1.1 says that quaternionic modular forms possess a very surprising arithmeticity. Remark 1.2. For the group GL2 , the Fourier coefficients are essentially the Satake parameters, and so algebraicity of Satake parameters implies that of the Fourier coefficients. For larger groups, the relationship between Satake parameters and Fourier coefficients is much more elaborate. In particular, algebraicity of Satake parameters does not in any clear way imply that of the Fourier coefficients. 5 I believe one still does not know if these Fourier coefficients are nonzero, although of course it is believed that they are nonzero. This would be a good project for someone!

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1.3.3. The representation theory of quaternionic real representations is particularly nice. I won’t have much to say about this in the course, but I did want to take this opportunity to reference work of Wallach [34], work of Gross-Wallach [13], work of Loke [22]. Moreover, I also want to mention the work [35] of Marty Weissman and the recent work [3] of Rahul Dalal, although they are more global in nature and don’t exactly fit into this category. The paper of Dalal gives a dimension formula for the space of level one even weight modular forms on G2 . 1.3.4. Quaternionic modular forms are potentially a testing ground for new phenomena. Because of their very rich structure, and because they appear to possess surprising arithmeticity, quaternionic modular forms behave very much like holomorphic modular forms. I suspect that they are ripe for study. In particular, like holomorphic modular forms, they may be a fertile testing ground for as-yetundiscovered phenomena. The rest of the notes will try to describe quaternionic modular forms in more detail, and explain some of the above-mentioned results. 2. The group G2 In this chapter, we describe the group G2 in ways that generalize to the other exceptional groups. For deeper reading on the group G2 and modular forms on G2 , we refer the reader to [26]. 2.1. The group G2 . We begin by defining the group G2 . We will define G2 in a way that will generalize to all the (quaternionic) exceptional Lie groups. We work over a field k of characteristic 0. Let sl3 be the Lie algebra of SL3 . We may identify sl3 with the trace zero 3 × 3 matrices. Let V3 denote the standard representation of sl3 and V3∨ its dual. Then End(V3 ) ≃ V3 ⊗ V3∨ . Note that if v ∈ V3 and δ ∈ V3∨ , then v ⊗ δ − δ(v) 13 has trace 0, so is an element of sl3 . Fix an identification ∧3 V3 ≃ k. 3 This gives rise to an identification ∧2 V3 ≃ V3∨ and ∧2 V3∨ ≃ V3 . We can describe this identification in bases. Let v1 , v2 , v3 be the standard basis of V3 and δ1 , δ2 , δ3 be the dual basis of V3∨ . Then we identify vi ∧ vi+1 with δi−1 (indices taken modulo 3) and δj ∧ δj+1 with vj−1 . Now we set g2 = sl3 ⊕ V3 ⊕ V3∨ , a k vector space of dimension 14. This is a Z/3-grading, with sl3 in degree 0, V3 in degree 1 and V3∨ in degree 2. One defines a Lie bracket on g2 as follows. First, if φ, φ′ ∈ sl3 , then [φ, φ′ ] is the usual Lie bracket on sl3 : φ○φ′ −φ′ ○φ. Next, if φ ∈ sl3 , v ∈ V3 and δ ∈ V3∨ , then [φ, v] = φ(v) ∈ V3 and [φ, δ] = φ(δ) ∈ V3∨ . Here recall that the action on the dual space V3 is defined as φ(δ)(v) = −δ(φ(v)), so that ⟨φ(v), δ⟩ + ⟨v, φ(δ)⟩ = 0, where ⟨ , ⟩ is the canonical pairing between V3 and V3∨ . If v, v ′ ∈ V3 , then [v, v ′ ] = 2v ∧ v ′ ∈ ∧2 V3 ≃ V3∨ , and similarly, if δ, δ ′ ∈ V3∨ , then [δ, δ ′ ] = 2δ ∧ δ ′ in ∧2 V3 ≃ V3 . Finally, if δ ∈ V3∨ and v ∈ V3 , then [δ, v] = 3v ⊗ δ − δ(v)13 . All other Lie brackets are determined by linearity and antisymmetry. Proposition 2.1. With these definitions, g2 is a Lie algebra, i.e., the Jacobi identity is satisfied. Proof. This is a fun exercise, which we leave to the reader.



Proposition 2.2. The Lie algebra g2 is simple. Proof. Any ideal I of g2 will be a representation of sl3 , and thus will be a  direct sum of irreducible sl3 pieces. The rest is an exercise.

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The algebraic group G2 is defined as the (identity component of) the automorphism group of g2 : For a Lie algebra g, define Aut(g) = {g ∈ GL(g) ∶ g[X, Y ] = [gX, gY ] ∀X, Y ∈ g} and G(g) = Aut(g)0 , the connected component of the identity. When g is semisimple, G(g) is a connected algebraic group with Lie algebra g, and of adjoint type. We define G2 = G(g2 ). As a Cartan subalgebra h, we may take the usual (diagonal) Cartan of sl3 . Indeed, it is clear that this h acts diagonally on g2 with distinct nonzero weights. 2.2. The Z/2-grading and the Heisenberg parabolic. The Lie algebra g2 possesses a 5-step Z-grading, and a closely related Z/2-grading. It will be useful to review these gradings on g2 , which we will do presently. Let Eij = vi ⊗ δj ∈ End(V3 ) ≃ V3 ⊗ V3∨ be the matrix with a 1 in the (ij) place and 0’s elsewhere. For the 5-step Z-grading: ● In degree 2, put kE13 ● In degree 1, put kE12 + kv1 + kδ3 + kE23 ● In degree 0, put kδ2 + h + kv2 ● In degree −1, put kE32 + kv3 + kδ1 + kE21 ● In degree −2, put kE31 . Exercise 2.3. Write Fi g2 for the degree i piece. Find an element h ∈ h so that [h, Fk g2 ] = kFk g2 . Deduce that [Fj g2 , Fk g2 ] ⊆ Fj+k g2 . The degree 0 piece is isomorphic to gl2 . Write W for the degree 1 piece. Exercise 2.4. Prove that W is isomorphic to the representation Sym3 ⊗det()−1 of gl2 ≃ F0 g2 . The Z/2-grading is defined as follows: Set g0 = F−2 g2 ⊕ F0 g2 ⊕ F2 g2 and g1 = F−1 g2 ⊕ F1 g2 . It is clear that this is a Z/2-grading. Exercise 2.5. Prove that g0 ≃ sl2 ⊕ sl2 and g1 ≃ V2 ⊗ W as a representation of g0 . Let P be the subgroup of G2 that stabilizes the line kE13 . One can show that P is a parabolic subgroup of G2 , with Lie algebra ⊕k≥0 Fk g2 , the part of g2 with non-negative components in the Z-grading. We call P the Heisenberg parabolic of G2 . The group P has a Levi decomposition P = M N with M ≃ GL2 . The Lie algebra of M is F0 g2 and the Lie algebra of N is F1 g2 ⊕ F2 g2 . Set Z = [N, N ]. Then one can identify Z with F2 g2 via the exponential map, and one can identify N ab = N /[N, N ] = N /Z with W = F1 g2 via the exponential map. 2.3. The Cartan involution. In this section the ground field is the field R of real numbers. We describe the Cartan involution θ on g2 , and the corresponding and p0 = gθ=−1 . To describe the involudecomposition g2 = k0 ⊕ p0 , where k0 = gθ=1 2 2 tion, we use the Z/3-model of g2 . Recall that v1 , v2 , v3 and δ1 , δ2 , δ3 are our fixed bases of V3 and V3∨ . ● On sl3 , which we identify with the trace 0 three-by-three matrices, we define θ(X) = −X t ● On V3 , θ is given by θ(vj ) = δj ● On V3∨ , θ is given by θ(δj ) = v3 .

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Exercise 2.6. Check that θ is a Lie algebra involution on g2 , i.e., θ[X, Y ] = [θX, θY ] for X, Y ∈ g2 . In fact, θ is a Cartan involution, i.e., the bilinear form Bθ (X, Y ) ∶= −B(X, θ(Y )) is positive definite on g2 , where B is the Killing form. The group G2 (R) has a corresponding maximal compact subgroup: Set KG2 = {k ∈ G2 (R) ∶ kθ = θk on g2 }. Equivalently, KG2 is the subgroup of G2 (R) that also preserves Bθ . In fact, KG2 ≃ (SU(2) × SU(2))/{±1}. Set k = k0 ⊗ C and p = p0 ⊗ C. Then k ≃ sl2 ⊕ sl2 and p ≃ V2 ⊗ W . For details, see [26, section 4.1]. In fact, there is an explicit c ∈ G2 (C) such that c(k) = g0 ⊗C and c(p) = g1 ⊗C. This is the exceptional Cayley transform, which can be found in [25]. 3. Modular forms on G2 In this chapter, we describe modular forms on G2 , and what is known about them. 3.1. Warm-up: Holomorphic modular forms. We warm up to the definition by first revisiting the definition of holomorphic modular forms for SL2 . 3.1.1. Classical definition. Let h denote the complex upper half-plane. For g = ( ac db ) ∈ SL2 (R) and z ∈ h set j(g, z) = cz + d. A function f ∶ h → C is a holomorphic modular form of weight  if (1) f is holomorphic (2) f (γz) = j(γ, z) f (z) for all γ ∈ Γ some congruence subgroup (3) The function on SL2 (R) defined by g ↦ j(g, i)− f (g ⋅ i) is of moderate growth. (See [2] for the notion of moderate growth.) Denote the above space of modular forms by M (Γ). Holomorphic modular forms have a classical Fourier expansion: Assume for simplicity that Γ contains the subgroup {( 10 n1 ) ∶ n ∈ Z}. Then f (z) = ∑n≥0 af (n)e2πinz with the Fourier coefficients af (n) ∈ C. 3.1.2. Semi-classical definition. We now give a semi-classical definition of holomorphic modular forms. We say a smooth function ϕ ∶ SL2 (R) → C is a holmorphic modular form of weight  if: (1) ϕ is of moderate growth (2) ϕ(γg) = ϕ(g) for all γ ∈ Γ, some congruence subgroup − sin(θ) (3) ϕ(gkθ ) = e−iθ ϕ(g) for all g ∈ SL2 (R) and kθ = ( cos(θ) sin(θ) cos(θ) ) ∈ SO(2). (4) DCR ϕ ≡ 0 where DCR is a linear differential operator described below. i ) and X = To describe the operator DCR , we proceed as follows. Let X+ = ( 1i −1 − 1 −i ( −i −1 ). Then sl2 ⊗C = k⊕p+ ⊕p− , where k is the complexified Lie algebra of SO(2), and p± is spanned by X± . One defines DCR ϕ = X− ϕ, the differential of the right regular action. Let A (Γ) denote the above space of modular forms. Exercise 3.1. Prove that the maps f (z) ↦ ϕf (g) ∶= j(g, i)− f (g⋅i) and ϕ(g) ↦ fϕ (z) ∶= j(gz , i) ϕ(gz ) define inverse bijections between M (Γ) and A (Γ). Here gz is any g ∈ SL2 (R) with g ⋅ i = z. Assume for simplicity that Γ contains the subgroup ( 10 Z1 ). Then ϕ ((

1 x y 1/2 )( 0 1

y −1/2

)) = y /2 ∑ aϕ (n)e2πinx e−2πny . n≥0

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We can re-express this Fourier expansion as follows. Define the function Wn ∶ SL2 (R) → C as Wn ((

1 x y 1/2 )( 0 1

y −1/2

) kθ ) = y /2 e−iθ e2πinx e−2πny .

Then ϕ(g) = ∑n≥0 aϕ (n)Wn (g) with Fourier coefficients aϕ (n) ∈ C. 3.1.3. Adelic definition. We now give the adelic definition of holomorphic modular forms and their Fourier expansion. A smooth function ϕ ∶ SL2 (A) → C is a holomorphic modular form of weight  if: (1) (2) (3) (4) (5)

ϕ is of moderate growth ϕ is right invariant under and open compact subgroup of SL2 (Af ) ϕ is left-invariant under SL2 (Q) cos(θ) − sin(θ) ϕ(gkθ ) = e−iθ ϕ(g) for all g ∈ SL2 (A) and kθ = ( sin(θ) cos(θ) ) ∈ SO(2). DCR ϕ ≡ 0 where DCR is the linear differential operator described above.

The Fourier expansion of ϕ is ϕ(gf g∞ ) =

∑

aϕ,n (gf )Wn (g∞ )

n∈Q,n≥0

for locally constant functions aϕ,n (gf ), the Fourier coefficients of ϕ. Here gf ∈ SL2 (Af ) and g∞ ∈ SL2 (R). 3.2. The definition. We now give the definition of modular forms on G2 , mimicking the adelic definition of holomorphic modular forms on SL2 . Let V = Sym2 (C2 ) ⊠ 1 as a representation of KG2 ≃ (SU(2) × SU(2))/±1. Here the SU(2) factors are ordered so that p ≃ C2 ⊠ W (as opposed to W ⊠ C2 ). To define modular forms on G2 , we need a certain differential operator D that will take the place of DCR above. Suppose ϕ ∶ G2 (R) → V is a smooth function, satisfying ϕ(gk) = k−1 ϕ(g) for all g ∈ G2 (R) and k ∈ KG2 . We define D ϕ ∶ G2 (R) → Sym2−1 (C2 )⊠W ̃  ϕ ∶ G2 (R) → V ⊗ p∨ as: as follows. First, we define D ̃  ϕ = ∑ Xj ϕ ⊗ X ∨ D j j

where {Xj } is a basis of p and {Xj∨ } is the dual basis of p∨ . Here Xj ϕ is the right ̃  ϕ is still KG -equivariant, regular action of p ⊆ g on ϕ. One checks easily that D 2 ̃  ϕ, then ϕ′ (gk) = k−1 ϕ(g) for all k ∈ K and g ∈ G. Now, because i.e., if ϕ′ = D p ≃ p∨ , we have V ⊗ p∨ ≃ Sym2−1 (C2 ) ⊠ W ⊕ Sym2+1 (C2 ) ⊠ W. Let pr ∶ V ⊗ p∨ → Sym2−1 (C2 ) ⊠ W be a KG2 equivariant projection (unique up ̃ . to scalar multiple). We define D = pr ○ D Definition 3.2. Suppose  ≥ 1 is a non-negative integer. A smooth function ϕ ∶ G2 (A) → V is a quaternionic modular form of weight  if (1) (2) (3) (4) (5)

ϕ is of moderate growth ϕ is right-invariant under an open compact subgroup of G2 (Af ) ϕ is left G(Q)-invariant, i.e., ϕ(γg) = ϕ(g) for all γ ∈ G(Q) ϕ is KG2 -equivariant, i.e., ϕ(gk) = k−1 ϕ(g) for all k ∈ KG2 and D ϕ ≡ 0.

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Quaternionic modular forms have a Fourier expansion and corresponding Fourier coefficients. We briefly state the result now, and will discuss it in more detail below. Denote ϕZ and ϕN the constant terms of ϕ along Z and N . We identify elements of W with four-tuples (a, b, c, d) so that (a, b, c, d) = aE12 +bv1 +cδ3 +dE23 . Define a symplectic form on W as ⟨(a, b, c, d), (a′ , b′ , c′ , d′ )⟩ = ad′ − 3bc′ + 3cb′ − da′ . This form is none other than the commutator of two elements of W , i.e., if w, w′ ∈ W then [w, w′ ] = ⟨w, w′ ⟩E13 . If w = (a, b, c, d) ∈ W (R), write w ≥ 0 if az 3 +3bz 2 +3cz +d is never 0 on the upper half plane in C; we say such elements of W (R) are positive semi-definite. Theorem 3.3 ([26],[25]). Suppose w ∈ W (R) is positive semi-definite. There is a completely explicit function Ww ∶ G2 (R) → V satisfying the following properties: (1) Ww (ng) = ei⟨w,n⟩ Ww (g) for all n ∈ N (R). Here n denotes the image of n in N ab ≃ W . (2) Ww (gk) = k−1 ⋅ Ww (g) for all k ∈ KG2 . (3) D Ww (g) ≡ 0. (4) Ww is of moderate growth. Moreover, if ϕ is a modular form on G2 of weight , then ϕZ (g) = ϕN (g) +

∑

w∈2πW (Q)∶w≥0

aϕ,w (gf )Ww (g∞ )

for locally constant functions aϕ,w on G2 (Af ). Additionally, the constant term ϕN is essentially a holomorphic modular form of weight 3 on M ≃ GL2 . The functions aϕ,w , or sometime their value at gf = 1, are the Fourier coefficients of ϕ. We say aϕ,w (1) is the Fourier coefficient of ϕ associated to w. When w is in the open orbit of GL2 (R) acting on W (R), these Fourier coefficients were defined by Gan-Gross-Savin [9], using a multiplicity one result of Wallach [33], even though these authors did not have the explicit functions Ww (g). Note that in the theorem, the constant terms ϕN are essentially modular forms of weight 3 on GL2 . So, the Ramanujan cusp form Δ can (and does) show up, but the cusp form of weight 16 does not. 3.3. Examples and theorems. We give some examples and theorem about modular forms on G2 . 3.3.1. Eisenstein series. The easiest family of examples is the degenerate Heisenberg Eisenstein series. To define these, let x2 , . . . , y 2 be a particular fixed basis of V , that we will specify in more detail later. Let P ⊆ G be the Heisenberg parabolic, with ν ∶ P → GL1 the character given by p⋅E13 = ν(p)E13 . If  > 2 is even, +1 there is a weight  modular form associated to inducing sections in IndG ). P (∣ν∣ G(R) +1 In more detail, let f (g) ∈ IndP (R) (∣ν∣ ) be the unique K-equivariant, V -valued G(A )

section whose value at g = 1 is x y  . Let now ff te ∈ IndP (Aff ) (∣ν∣+1 ) be an arbitrary element. Set f (g) = ff te (gf )f (g∞ ) and E(g, f ) = ∑γ∈P (Q)/G(Q) f (γg). The sum converges absolutely using that  > 2. Then it can be shown that the value E(g, f ) is a quaternionic modular form of weight . More generally, both for even and odd , one can make quaternionic modular forms by starting with inducing sections in G(A) IndP (A) ((χ ○ ν)∣ν∣+1 ), where χ ∶ GL1 (Q)/ GL1 (A) → C× is an automorphic char

acter satisfying χ∞ (t) = ( ∣t∣t ) = sgn(t) . Another way to create modular forms is

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to use Heisenberg Eisenstein series with nontrivial inducing data. One can take a classical holomorphic modular form Φ of weight 3 on M ≃ GL2 , and from it produce a weight  modular form E(g, Φ) on G2 if  is sufficiently large. This is spelled out in [25]. Here are some interesting open questions about Eisenstein series: (1) Do the degenerate Eisenstein series E(g, f ) have rational or algebraic Fourier coefficients, when ff te is spherical? (2) Say that w ∈ W (R) is non-degenerate if w is in the open orbit of M (C) on W (C). Are the non-degenerate Fourier coefficients of the Eisenstein series E(g, f ) nonzero? (3) If the modular form Φ on GL2 has algebraic Fourier coefficients, does the same occur for E(g, Φ)? In the setting of holomorphic modular forms on tube domains, an analogous result is due to Harris [15, 16]. Nothing is known in the quaternionic case. Recall (see [9]) that there is a correspondence between GL2 (Z) orbits of integral binary cubic forms and cubic rings. If ϕ is a level one modular form on G2 , and w = (a, b/3, c/3, d) corresponds to the integral binary cubic form au3 + bu2 v + cuv 2 + dv 3 , then we can consider the Fourier coefficient aϕ,w (1). If ϕ is of even weight, and γ ∈ GL2 (Z), it is easy to check that aϕ,w (1) = aϕ,γ⋅w (1). Let A(w) be the cubic ring corresponding to the orbit GL2 (Z) ⋅ w. Thus, following [9], we can define aϕ (A(w)) = aϕ,w (1) to be the Fourier coefficient of ϕ associated to the cubic ring A(w). We use this definition in the statement of the following theorem, and also in the subsection below on L-functions. Theorem 3.4 (Gan-Gross-Savin [9], Jiang-Rallis [19]). For  ≥ 4 even, let E (g) be the spherical weight  Eisenstein series on G2 . Assume the non-degenerate Fourier coefficients of E (g) are nonzero6 . Let ω = (a, b/3, c/3, d) ∈ W correspond to the totally real cubic ring A(ω), via the correspondence between binary cubic forms and cubic rings. If A(ω) is maximal, then the Fourier coefficient of E (g) corresponding to ω is ζA(ω) (1 − ), up to a nonzero constant independent of ω. We will have more to say later about some examples of modular forms constructed in [9]. 3.3.2. Cusp forms. We now explain what is known about cusp forms. The following is the main theorem of [27]. Theorem 3.5. [27] Let w ≥ 16 be even. Then there are nonzero cuspidal modular forms on G2 of weight w, all of whose Fourier coefficients are algebraic integers. In fact, in weight 20, there is nonzero cuspidal level one modular form on G2 , all of whose Fourier coefficients are integers. Recently, Dalal has given a dimension formula for the level one cuspidal quaternionic modular forms on G2 . Theorem 3.6 (Dalal [3]). There is an explicit formula for the dimension of level one cuspidal modular forms on G2 . In particular, the smallest nonzero level one cuspidal modular form on G2 is in weight 6. 6 For each , one only needs to assume that a certain purely archimedean integral (that depends on ) is nonzero

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Here are some questions: (1) Can the cuspidal weight 6, level one quaternionic modular form of Dalal be constructed explicitly? (2) Let w = (0, 1/3, −1/3, 0) so that A(w) ≃ Z×Z×Z. Do level one quaternionic modular forms with nonzero w Fourier coefficient exist? Do they exist in abundance? 3.3.3. L-functions. Nothing is known about L-functions of quaternionic modular forms on groups bigger than G2 . However, on G2 , one can say a bit about the standard L-function of quaternionic modular forms. Thus let π = πf ⊗ π,∞ be a cuspidal automorphic representation of G2 (A) for which π,∞ is the quaternionic discrete series having minimal KG2 -type V . Let ϕ be the level one cuspidal modular form associated to π. Then one can write the standard L-function of π as a Dirichlet series in the Fourier coefficients of ϕ. Theorem 3.7. [18] Let aϕ (T ) denote the Fourier coefficient of ϕ corresponding to the cubic ring T . Then aϕ (Z + nT ) L(π, Std, s − 2 + 1) . = aϕ (Z3 ) 3 s−+1 s n ζ(s − 2 + 2)2 ζ(2s − 4 + 2) T ⊆Z3 ,n≥1 [Z ∶ T ] ∑

Here the sum is over the subrings T of Z × Z × Z and integers n ≥ 1. It is also known that the completed L-function has a functional equation, if aϕ (Z3 ) ≠ 0. To state the result, define the archimedean L-factor as L∞ (π,∞ , s) = ΓC (s +  − 1)ΓC (s + )ΓC (s + 2 − 1)ΓR (s + 1). Here ΓR (s) = π −s/2 Γ(s/2) and

ΓC (s) = 2(2π)−s Γ(s),

where Γ is the usual gamma function. The completed L-function is given by Λ(π, Std, s) = L∞ (π,∞ , s)L(π, Std, s). Theorem 3.8. [18] Suppose that ϕ is a level one cuspidal modular form on G2 of positive even weight  that generates the cuspidal automorphic representation π. Further, assume that the Fourier coefficient of ϕ corresponding to the split cubic ring Z × Z × Z is nonzero. Then Λ(π, Std, s) = Λ(π, Std, 1 − s) for all s ∈ C. The proof of the theorems above is based on a refined analysis of the RankinSelberg integral in [12]. 4. Exceptional algebra In this chapter we will work our way up to defining the quaternionic exceptional group of type E8 . We begin with some exceptional algebra: composition algebras and cubic norm structures. Our aim is just to give the necessary definitions. For many of the omitted proofs in this chapter, the reader is encouraged to see my course notes [36, Chapters 1 and 2].

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4.1. Composition algebras. Suppose k is a field, not of characteristic 2. In this section we discuss composition algebras, which we are about to define. For more on composition algebras, one can see [32] in addition to [36]. Definition 4.1. Suppose C is a not-necessarily-associative k algebra with unit 1, and that C comes equipped with non-degenerate quadratic form nC ∶ C → k. Then C is said to be a composition algebra if nC is multiplicative, i.e., nC (xy) = nC (x)nC (y) for all x, y ∈ C. Composition algebras can be classified, and in fact are always dimension 1, 2, 4 or 8 over the ground field. Every dimension four composition algebra is a quaternion algebra. Example 4.2. C = k with nC (x) = x2 is a composition algebra. Example 4.3. C = E, an etale quadratic extension of k, with nC (x) = NE/k (x) the norm. Example 4.4. C = B, a quaternion k-algebra, with nC (x) the reduced norm. There is a way of defining an involution ∗ on a composition algebra, as follows. Let (x, y) = nC (x+y)−nC (x)−nC (y) be the non-degenerate bilinear form associated to nC . Note that nC (1) = 1 from the multiplicativity of the norm, and thus from the definition of (x, y), 1 satisfies (1, 1) = 2 ≠ 0. Let C 0 be the perpendicular space to 1 under the bilinear form. Define ∗ on C as (x1 + y)∗ = x − y if x ∈ k and y ∈ C 0 . In other words, z ∗ = (z, 1)1 − z for z ∈ C. Note that z + z ∗ ∈ k ⋅ 1 for all z ∈ C. Also note that nC (z) = nC (z ∗ ) for all z ∈ C. Theorem 4.5. The map ∗ satisfies (1) z ∗ z = nC (z) for all z ∈ C. (2) Moreover, ∗ is an algebra involution, i.e., (xy)∗ = y ∗ x∗ for all x, y ∈ C. Proof. See [36, Chapter 1, Section 2.1]. The proof there follows [32].



Definition 4.6. An octonion algebra Θ is an eight-dimensional composition algebra. Octonion algebras exist. We give two different constructions, called the Zorn model and the Cayley-Dickson construction. Definition 4.7 (The Zorn model). Denote by V3 the three-dimensional defining representation of SL3 and by V3∨ its dual representation. Recall that we identify ∧2 V3 ≃ V3∨ and ∧2 V3∨ ≃ V3 . Denote by Θ the set of two-by-two matrices ( φa vd ) with a, d ∈ k, v ∈ V3 and φ ∈ V3∨ with multiplication (

a φ

v a′ )( ′ φ d

aa′ + φ′ (v) v′ ′ )=( ′ d a φ + dφ′ + v ∧ v ′

a The involution ∗ is ( φ ad − φ(v).

∗

v d ) =( d −φ

av ′ + d′ v − φ ∧ φ′ ). φ(v ′ ) + dd′

−v a ) and the norm is nΘ (( a φ

v )) = d

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The Cayley-Dickson construction starts with a quaternion algebra B and an element γ ∈ k× , and defines Θ = B ⊕ B with multiplication as follows. Definition 4.8. Let ∗ denote the involution on the quaternion algebra B. Then the multiplicaton on Θ = B ⊕ B is (x1 , y1 )(x2 , y2 ) = (x1 x2 + γy2∗ y1 , y2 x1 + y1 x∗2 ). The involution ∗ on Θ is (x, y)∗ = (x∗ , −y) and the norm is nΘ ((x, y)) = nB (x) − γnB (y). The Cayley-Dickson construction and the Zorn model produces composition algebras. Proposition 4.9. The Zorn model is a special case of the Cayley-Dickson construction, with B = M2 (k) and γ = 1. Proof. See [36, Chapter 1, Proposition 2.0.6].



Proposition 4.10. The Zorn model and the Cayley-Dickson construction define octonion algebras, i.e., the norms are multiplicative. Proof. See [36, Chapter 1, Proposition 2.2.1].



4.2. Cubic norm structures. In this section we define another algebraic gadget, which is called a cubic norm structure. For more on cubic norm structures, one can see [24] in addition to [36]. You can think of cubic norm structures as generalizations the pair (3 × 3 matrices, the determinant map). We will jump straight into the definition, and then give the examples. Thus suppose k is a field of characteristic 0 and J is a finite dimensional k vector space. That J is a cubic norm structure means that it comes equipped with a cubic polynomial map N ∶ J → k, a quadratic polynomial map # ∶ J → J, an element 1J ∈ J, and a non-degenerate symmetric bilinear pairing ( , ) ∶ J ⊗J → k, called the trace pairing, that satisfy the following properties. For x, y ∈ J, set x × y = (x + y)# − x# − y # and denote ( , , ) ∶ J ⊗ J ⊗ J → k the unique symmetric trilinear form satisfying (x, x, x) = 6N (x) for all x ∈ J. Then (1) (2) (3) (4)

N (1J ) = 1, 1# J = 1J , and 1J × x = (1J , x) − x for all x ∈ J. (x# )# = N (x)x for all x ∈ J. The pairing (x, y) = 14 (1J , 1J , x)(1J , 1J , y) − (1J , x, y). One has N (x + y) = N (x) + (x# , y) + (x, y # ) + N (y) for all x, y ∈ J.

Instead of thinking of x# ∈ J, it is more natural to think of x# ∈ J ∨ , the linear dual of J, defined as (x# , y) = 12 (x, x, y). Identifying J with J ∨ via the pairing (x, y) gives x# ∈ J. This leads to a weaker notion of a cubic norm pair. In this case, the pairing ( , ) is between J and J ∨ , the linear dual of J, the adjoint map # takes J → J ∨ and J ∨ → J, and each J, J ∨ have a norm map NJ ∶ J → F and NJ ∨ ∶ J ∨ → F . The adjoints and norms on J and J ∨ satisfy the same compatibilities as above in items (2) and (4). Example 4.11. Let J = k with N (x) = x3 , x# = x2 , 1J = 1, and (x, y) = 3xy. Example 4.12. Let J = M3 (k) with N (x) = det(x), x# the adjoint matrix, (i.e., x# = det(x)x−1 for invertible x), (x, y) = tr(xy), 1J = 13 .

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Example 4.13. J = k × S with S a pointed quadratic space. In more detail, take 1S ∈ S with q(1S ) = 1. Define an involution ι on S fixing 1S and acting as minus the identity on (k ⋅ 1S )⊥ . The norm on J is NJ (β, s) = βqS (s), one has 1J = (1, 1S ), and the adjoint map is (β, s)# = (qS (s), βι(s)). Finally, the pairing is ((β, t), (β ′ , t′ )) = ββ ′ + (t, ι(t′ )). Proposition 4.14. With data as defined above J = k × S is a cubic norm structure. Proof. See [36, Chapter 2, Proposition 2.1.4].



Set J = H3 (C) the Hermitian 3×3 matrices with coefficients in the composition algebra C. We make J = H3 (C) into a cubic norm structure, with the following choice of data: ∗ ⎛ c 1 x3 x2 ⎞ ∗ (1) NJ (X) = NJ ⎜ x3 c2 x1 ⎟ = c1 c2 c3 −c1 nC (x1 )−c2 nC (x2 )−c3 nC (x3 )+ ⎝ x2 x∗1 c3 ⎠ trC (x1 x2 x3 ). ∗ ∗ x3 x1 − c2 x∗2 ⎞ ⎛ c2 c3 − nC (x1 ) x2 x1 − c3 x3 ∗ # c1 c3 − nC (x2 ) x∗3 x∗2 − c1 x1 ⎟ (2) X = ⎜ x1 x2 − c3 x3 ⎝ x∗1 x∗3 − c2 x2 x2 x3 − c1 x∗1 c1 c2 − nC (x3 ) ⎠ ′ (3) The pairing (X, X ), in obvious notation, is (X, X ′ ) = c1 c′1 + c2 c′2 + c3 c′3 + (x1 , x′1 ) + (x2 , x′2 ) + (x3 , x′3 ). Theorem 4.15. With data described above, J = H3 (C) is a cubic norm structure. Proof. See [36, Chapter 2, Theorem 2.1.1].



4.3. The group MJ . If J is a cubic norm structure, we define the algebraic k-group MJ to be the group of linear automorphisms of J that preserve the norm form N up to scaling. On k-points, one has MJ (k) = {(λ, g) ∈ GL1 (k) × GL(J) ∶ N (gX) = λN (X) for all X ∈ J}. We set MJ1 the subgroup of MJ consisting of those g with λ(g) = 1 and we set AJ the subgroup of MJ1 that also stabilizes the element 1J ∈ J. It follows that AJ preserves the bilinear pairing ( , ): if a ∈ AJ , then (ax, ay) = (x, y) for all x, y ∈ J. The group AJ is the automorphism group of J. If a ∈ AJ , then one also has (ax) × (ay) = a(x × y) for all x, y ∈ J. Let m(J) = {(μ, Φ) ∈ k × End(J) ∶ (Φ(z1 ), z2 , z3 ) + (z1 , Φ(z2 ), z3 ) + (z1 , z2 , Φ(z3 )) = μ(z1 , z2 , z3 )}. Then m(J) is the Lie algebra of MJ . We define an MJ -equivariant map J ⊗ J ∨ → m(J). See [31] and [30]. For γ ∈ J ∨ and x ∈ J, define the element Φγ,x ∈ End(J) as Φγ,x (z) = −γ × (x × z) + (γ, z)x + (γ, x)z. Proposition 4.16. [30, Equation (9)] One has (Φγ,x (z1 ), z2 , z3 ) + (z1 , Φγ,x (z2 ), z3 ) + (z1 , z2 , Φγ,x (z3 )) = 2(γ, x)(z1 , z2 , z3 ) for all z1 , z2 , z3 in J. In particular, Φγ,x ∈ m(J).

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Note that Φγ,x (z) = Φγ,z (x). One sets Φ′γ,x = Φγ,x − 23 (γ, x). Then Φ′γ,x ∈ m(J)0 , the Lie algebra of MJ1 . Proposition 4.17. The map Φ ∶ J ⊗ J ∨ → m(J) is equivariant, i.e., if g ∈ MJ then Ad(g)Φγ,x = Φ̃g(γ),g(x) . Proof. See [36, Chapter 2, section 3].



4.4. The Z/3 grading on g(J). In this subsection we recall elements from the paper [30] (in different notation). Rumelhart constructed a Lie algebra g(J) out of a cubic norm structure J, as follows. Denote by V3 the defining representation of sl3 , and by V3∨ the dual representation. One defines g(J) = sl3 ⊕ m(J)0 ⊕ V3 ⊗ J ⊕ V3∨ ⊕ J ∨ . We consider V3 , V3∨ as left modules for sl3 , and J, J ∨ as left modules for m(J)0 . This is a Z/3-grading, with sl3 ⊕ m(J)0 in degree 0, V3 ⊗ J in degree 1 and V3∨ ⊕ J ∨ in degree 2. Note that when J = k, m(J)0 = 0 and g(J) becomes g2 . 4.4.1. The bracket. Following [30], a Lie bracket on g(J) is given as follows. First, because V3 is considered as a representation of sl3 , there is an identification ∧2 V3 ≃ V3∨ , and similarly ∧3 V3∨ ≃ V3 . If v1 , v2 , v3 denotes the standard basis of V3 , and δ1 , δ2 , δ3 the dual basis of V3∨ , then v1 ∧ v2 = δ3 , δ1 ∧ δ2 = v3 , and cyclic permutations of these two identifications. Take φ3 ∈ sl3 , φJ ∈ m(J)0 , v, v ′ ∈ V3 , δ, δ ′ ∈ V3∨ , X, X ′ ∈ J and γ, γ ′ ∈ J ∨ . The Lie bracket on g(J) is defined as [φ3 , v ⊗ X + δ ⊗ γ] = φ3 (v) ⊗ X + φ3 (δ) ⊗ γ. [φJ , v ⊗ X + δ ⊗ γ] = v ⊗ φJ (X) + δ ⊗ φJ (γ) [v ⊗ X, v ′ ⊗ X ′ ] = (v ∧ v ′ ) ⊗ (X × X ′ ) [δ ⊗ γ, δ ′ ⊗ γ ′ ] = (δ ∧ δ ′ ) ⊗ (γ × γ ′ ) [δ ⊗ γ, v ⊗ X] = (X, γ)v ⊗ δ + δ(v)Φγ,X − δ(v)(X, γ) 1 2 = (X, γ) (v ⊗ δ − δ(v)) + δ(v) (Φγ,X − (X, γ)) . 3 3 1 2 ′ 0 Note that v ⊗ δ − 3 δ(v) ∈ sl3 and Φγ,X − 3 (X, γ) = Φγ,X ∈ m(J) . Theorem 4.18. [30] The vector space g(J) is a Lie algebra, i.e., the Jacobi identity is satisfied. Example 4.19. When J = k, g(J) = g2 . Example 4.20. When J = H3 (k), g(J) is of type f4 . Example 4.21. When J = H3 (E) with E a quadratic etale extension of k, g(J) is of type e6 . Example 4.22. When J = H3 (B) with B a quaternion algebra, g(J) is of type e7 . Example 4.23. When J = H3 (Θ) with Θ an octonion algebra, g(J) is of type e8 . Let GJ = Aut(g(J))0 be the identity component of the automorphisms of the Lie algebra g(J). Then GJ is a connected adjoint algebraic group of the above types. When k = R, the norm form on the composition algebra C is positive definite, and J = H3 (C), GJ (R) is the adjoint quaternionic exceptional group of the above types. You can take this to be the definition of the quaternionic exceptional groups.

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5. The Freudenthal construction In the previous section, we gave a construction of the exceptional Lie algebras g(J), via a Z/3Z-grading. In order to discuss quaternionic modular forms on GJ = Aut0 (g(J)), we will require a 5-step Z-grading on g(J). The purpose of this chapter is to give some preliminaries needed for the description of this 5-step Z-grading. Throughout this chapter, the omitted proofs can be found in [36, Chapter 3]. 5.1. The Freudenthal construction. Suppose that J is a cubic norm structure, or that J, J ∨ is a cubic norm pair, over a field k of characteristic 0. Define a vector space WJ = k ⊕ J ⊕ J ∨ ⊕ k. The space WJ comes equipped with a symplectic pairing ⟨ , ⟩ and a quartic form q, which are defined as follows. We write a typical element in WJ as v = (a, b, c, d), so that a, d ∈ k, b ∈ J and c ∈ J ∨ . Then ⟨(a, b, c, d), (a′ , b′ , c′ , d′ )⟩ = ad′ − (b, c′ ) + (c, b′ ) − da′ and q((a, b, c, d)) = (ad − (b, c))2 + 4aN (c) + 4dN (b) − 4(b# , c# ). The definition of this algebraic data goes back to Freudenthal. We now define a group HJ (k) = {(g, ν) ∈ GL(WJ ) × GL1 (k) ∶ ⟨gv, gv ′ ⟩ = ν⟨v, v ′ ⟩ ∀v, v ′ ∈ WJ and q(gv) = ν 2 q(v) ∀v ∈ WJ }. The element ν is called the similitude. More generally, we let HJ be the algebraic group of linear automorphisms of WJ that preserve the symplectic form ⟨ , , ⟩ and the quartic form q up to appropriate similitude. We set HJ1 = ker ν ∶ HJ → GL1 . In general, the identity component HJ0 of HJ will be a Levi subgroup of a maximal parabolic subgroup of GJ . Example 5.1. When J = k, HJ ≃ GL2 . Example 5.2. When J = H3 (k), HJ ≃ GSp6 . Example 5.3. When J = H3 (E) with E a quadratic etale extension of k, HJ is of type A5 . Example 5.4. When J = H3 (B) with B a quaternion algebra, HJ is of type D6 . Example 5.5. When J = H3 (Θ) with Θ an octonion algebra, HJ is of type E7 . Denote by ( , , , )WJ the unique symmetric four-linear form normalized so that (v, v, v, v)WJ = 2q(v). Define t ∶ WJ × WJ × WJ → WJ as (w, x, y, z) = ⟨w, t(x, y, z)⟩ and set v ♭ = t(v, v, v). 5.2. Rank one elements. If J is a cubic norm structure, there is a notion of rank of elements of J, which we now define. Definition 5.6. All elements of J are of rank at most 3. If N (x) = 0, then x has rank at most 2. If x# = 0, then x has rank at most one. If x = 0, then x has rank 0. If J = H3 (k), the above definition reduces to the usual notion of rank of a 3 × 3 symmetric matrix. There is a related notion of rank of elements of WJ .

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Definition 5.7. All elements of WJ are of rank at most 4. If q(v) = 0, equivalently if (v, v, v, v) = 0, then v has rank at most 3. If v ♭ = 0, equivalently if (v, v, v, w) = 0 for all w ∈ WJ , then v has rank at most two. If (v, v, w′ , w) = 0 for all w′ ∈ (kv)⊥ and all w ∈ WJ , then v has rank at most one. If v = 0, then v has rank 0. Example 5.8. If a, d ≠ 0, then (a, 0, 0, d) has rank 4. If x ∈ J has rank j = 0, 1, 2, 3, then (0, x, 0, 0) has rank j in in WJ . Example 5.9. The element (1, 0, 0, 0) of WJ has rank one. If v = (1, 0, c, d) has rank one, then c = d = 0. We will need a definition of certain elements of End(WJ ), constructed from two elements of WJ . Definition 5.10. For w, w′ ∈ WJ define Φw,w′ ∈ End(WJ ) as follows: Φw,w′ (x) = 6t(w, w′ , x) + ⟨w′ , x⟩w + ⟨w, x⟩w′ . One can show that v has rank at most one if and only if 12 Φv,v (x) ∶= 3t(v, v, x)+ ⟨v, x⟩v = 0 for all x ∈ WJ . (Note that this condition implies the rank one condition of the definition, but the converse is not at all obvious.) It is clear that the set of rank one elements is an HJ -set. In fact, Proposition 5.11. There is one HJ1 -orbit of rank one lines. Proof. See [36, Chapter 3, Proposition 3.0.6].



Let h(J)0 denote the Lie algebra of HJ1 . Proposition 5.12. For w, w′ ∈ WJ , the endomorphism Φw,w′ is in h(J)0 , i.e., it preserves the symplectic and quartic form on WJ . Furthermore, if φ ∈ h(J)0 , then [φ, Φw,w′ ] = Φφ(w),w′ + Φw,φ(w′ ) . Proof. See [36, Chapter 3, Proposition 4.0.3].



5.3. The exceptional upper half-space. When the ground field k = R and the pairing ( , ) on J is positive definite, the group HJ has an associated Hermitian symmetric space. This space is HJ = {Z = X + iY ∶ Y > 0}. Here Y > 0 means that Y = y 2 for some y ∈ J with N (y) ≠ 0. 5.3.1. The positive definite cone. Theorem 5.13. The following statement are equivalent: (1) Y ∈ J is positive-definite, i.e., Y = y 2 for some y ∈ J with N (y) ≠ 0. (2) There exists a ∈ AJ with aY diagonal with positive entries (3) tr(Y ), tr(Y # ) and N (Y ) are all positive. Proof. See [36, Chapter 3, Corollary 5.1.2].



Theorem 5.14. Let C denote the set of Y in J with Y > 0. Then C is connected and convex. Proof. See [36, Chapter 3, Exercise 5.1.4].



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5.3.2. The upper half space. We now define how HJ (R)0 acts on HJ . To do this, suppose Z ∈ JC . Define r0 (Z) = (1, −Z, Z # , −N (Z)). Then one has the following proposition. Proposition 5.15. Suppose Z ∈ HJ , so that Im(Z) is positive definite. Suppose moreover that g ∈ HJ (R)0 . Then there is j(g, Z) ∈ C× and gZ ∈ HJ so that gr0 (Z) = j(g, Z)r0 (gZ). This equality defines the factor of automorphy j(g, Z) and the action of HJ (R)0 simultaneously. 

Proof. See [36, Chapter 3, Proposition 5.2.1].

Suppose J is a cubic norm structure. Let ι ∶ J ↔ J ∨ be the identification given by the symmetric pairing on J. Define J2 ∶ WJ → WJ as J2 (a, b, c, d) = (d, −ι(c), ι(b), −a). One checks that J2 ∈ HJ1 . We will now explain the stabilizer of i1J inside of HJ1 . Proposition 5.16. Suppose g ∈ HJ1 (R) stabilizers i1J ∈ HJ . Then g commutes with J2 . 

Proof. See [36, Chapter 3, Proposition 5.2.3].

5.3.3. Modular forms. As an aside, we now define holomorphic modular forms on E7,3 . To do so, let J = H3 (Θ) be as above, where Θ is an octonion algebra with positive definite norm form. Define G = HJ1 (R). It turns out that the group G is connected, so it acts (transitively) on HJ . Following Baily [1], a discrete subgroup Γ ⊆ G is defined as follows. Let Θ0 ⊆ Θ be Coxeter’s ring of integral octonions; see, e.g., loc cit. Define J0 ⊆ J to be the integral lattice consisting of matrices ∗ ⎛ c 1 x3 x2 ⎞ ∗ X = ⎜ x3 c2 x1 ⎟ with c1 , c2 , c3 ∈ Z and x1 , x2 , x3 ∈ Θ0 . Let J0∨ be the integral ⎝ x2 x∗1 c3 ⎠ dual lattice to J0 , which can be identified with J0 via the trace pairing. Define WJ0 ⊆ WJ to be the lattice WJ0 = Z ⊕ J0 ⊕ J0∨ ⊕ Z. Then Γ is defined to be the subgroup of HJ1 (Q) that preserves WJ0 . A modular form for Γ of weight  > 0 is a holomorphic function f ∶ HJ → C satisfying (1) f (γZ) = j(γ, Z) f (Z) for all γ ∈ Γ and (2) the function φf ∶ Γ/G → C defined by φf (g) = j(g, i)− f (g⋅i) is of moderate growth. Some results about modular forms on G can be found in [1], [20], [10], [21]. 6. The quaternionic groups In this chapter we discuss more about the quaternionic exceptional groups. A reference for this material is [36, Chapter 4] and [25]. 6.1. The Z/2-grading. Previously, we defined the Lie algebra g(J) via a Z/3-grading. In this section, we redefine g(J) via a Z/2-grading. I believe the construction of the Lie algebra g(J) in this section essentially goes back to Freudenthal. Denote by V2 the defining two-dimensional representation of sl2 . We have an identification Sym2 (V2 ) ≃ sl2 as (v ⋅ v ′ )(x) = ⟨v ′ , x⟩v + ⟨v, x⟩v ′ . Here ⟨ , ⟩ is the standard symplectic pairing on V2 : ⟨(a, b)t , (c, d)t ⟩ = ( a

b )(

1 −1

)(

c ) = ad − bc. d

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131

We define g(J) = g(J)0 ⊕ g(J)1 ∶= (sl2 ⊕ h(J)0 ) ⊕ (V2 ⊗ WJ ) . Here g(J)0 = sl2 ⊕ h(J)0 is the zeroth graded piece of g(J), and g(J)1 = V2 ⊗ WJ is the first graded piece of g(J). 6.1.1. The bracket. We define a map [ , ] ∶ g(J) ⊗ g(J) → g(J) as follows: If φ, φ′ ∈ g(J)0 = sl2 ⊕ h(J)0 , v, v ′ ∈ V2 , and w, w′ ∈ WJ , then (6.1) [(φ, v ⊗ w), (φ′ , v ′ ⊗ w′ )] 1 1 = ([φ, φ′ ] + ⟨w, w′ ⟩(v ⋅ v ′ ) + ⟨v, v ′ ⟩Φw,w′ , φ(v ′ ⊗ w′ ) − φ′ (v ⊗ w)) . 2 2 With this definition, we have the following fact. Proposition 6.1. The bracket [ , , ] on g(J) satisfies the Jacobi identity. Proof. To check the Jacobi identity ∑cyc [X, [Y, Z]] = 0, by linearity it suffices to check it on the various Z/2-graded pieces. Then there are four types identities that must be checked. Namely, if 0, 1, 2 or 3 of the elements X, Y, Z are in g(J)1 = V2 ⊗ WJ . If all three of X, Y, Z are in g(J)0 = sl2 ⊕ h(J)0 , then the Jacobi identity is of course satisfied. If two of X, Y, Z are in g(J)0 , then the Jacobi identity is satisfied. This fact is equivalent to the fact that the bracket [ , ]α defines a Lie algebra action of g(J)0 on g(J)1 : [φ, φ′ ](x) = φ(φ′ (x)) − φ′ (φ(x)) for x ∈ g(J)1 and φ, φ′ ∈ g(J)0 . If one of X, Y, Z is in g(J)0 , then the Jacobi identity is satisfied by the equivariance of the map g(J)1 ⊗ g(J)1 → g(J)0 . Finally, when X, Y, Z are all in g(J)1 , a simple direct computation shows that ∑cyc [X, [Y, Z]] = 0. In more detail, suppose X1 = v1 ⊗ w1 , X2 = v2 ⊗ w2 and X3 = v3 ⊗ w3 . We must evaluate: −2 ∑ [X1 , [X2 , X3 ]] = 2 ∑ [v2 ⊗ w2 , v3 ⊗ w3 ](v1 ⊗ w1 ) cyc

cyc

= ∑ (⟨v2 , v3 ⟩Φw2 ,w3 + ⟨w2 , w3 ⟩v2 ⋅ v3 )(v1 ⊗ w1 ) cyc

= ∑ ⟨v2 , v3 ⟩v1 ⊗ (6t(w1 , w2 , w3 ) + ⟨w3 , w1 ⟩w2 + ⟨w2 , w1 ⟩w3 ) cyc

+ ∑ ⟨w2 , w3 ⟩(⟨v3 , v1 ⟩v2 + ⟨v2 , v1 ⟩v3 ) ⊗ w1 . cyc ′

′′

The term t(w, w , w ) drops out right away because it is symmetric by applying the identity ∑cyc ⟨v2 , v3 ⟩v1 = 0 for v1 , v2 , v3 ∈ V2 . The other cyclic sums cancel in pairs.  One can give an explicit identification between the Lie algebra g(J) defined in this section and the one defined via the Z/3-grading, which is why we have given both Lie algebras the same name. See [25, Proposition 4.2.1]. 6.2. The Heisenberg parabolic. We assume in this subsection that the Lie algebra g(J) is defined over a ground field F of characteristic 0. Recall that the group GJ is defined to be the connected component of the identity of the automorhpism group of g(J). This is a connected reductive adjoint group. For notation, we write e0 , h0 , f0 for the usual sl2 -triple inside sl2 ⊆ g(J)0 , so that e0 = ( 00 10 ), h0 = ( 1 −1 ), and f0 = ( 01 00 ). Set e = (1, 0)t and f = (0, 1)t the standard basis of V2 .

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6.2.1. The 5-grading. We now define the 5-grading on g(J). Namely, the components of g(J) in each graded piece are ● In degree −2: spanned by f0 ● In degree −1: f ⊗ WJ ● In degree 0: F h0 ⊕ h(J)0 ● In degree 1: e ⊗ WJ ● In degree 2: spanned by e0 . Note that this is the grading associated to the eigenvalues of h0 on g(J). The degree 0 piece F h0 ⊕ h(J)0 is the Lie algebra h(J), via the map αh0 + φ0 ↦ αIdWJ + φ0 , where φ0 ∈ h(J)0 and IdWJ denotes the identity on WJ . 6.2.2. The Heisenberg parabolic. We now define the Heisenberg parabolic of GJ . Define P ⊆ GJ to be the g ∈ GJ stabilizing the line F e0 generated by e0 . Then P is parabolic. For instance, the variety G/P is the subset of the projective space P(g(J)) consisting of those X with [X, [X, y]] + 2Bg (X, y)X = 0 for all y ∈ g(J). Thus G/P is cut out by closed conditions, so is projective. Equivalently, define pHeis ⊆ g(J) to consist of the elements X so that [X, e0 ] ∈ F e0 . Then the Heisenberg parabolic is equivalently defined to be the g ∈ GJ satisfying gpHeis = pHeis . Furthermore, pHeis consists exactly of the element of g(J) non-negative degree in the 5-grading. The Lie algebra of P is pHeis . Define a Levi subgroup M of P to be the subgroup of P that preserves the 5-grading. Equivalently, M is the subgroup of P that also fixes the line spanned by f0 . The Levi subgroup M is exactly the group HJ0 , as we now prove. Lemma 6.2. The map M → GL1 × GL(WJ ) defined by the conjugation action of M on the degree 2 and degree 1 pieces of the 5-grading defines an isomorphism M ≃ HJ . The GL1 -projection is the similitude. Proof. See [25, Lemma 4.3.1].



Let N denote the unipotent radical of the Heisenberg parabolic P . Then the center Z of N is the exponential of the one-dimensional space F e0 . The group Z is also the commutator subgroup [N, N ] of N . Consequently, the abelianization of N is naturally identified with WJ , via the map w ↦ exp(w) ↦ exp(w), where exp(w) ∈ N and for n ∈ N , n denotes the image of n in N /Z. In case G = G2 , the Heisenberg parabolic is the (standard) maximal parabolic with two-step unipotent radical. 6.3. The Cartan involution. In this section, we discuss the Cartan involutions on various relevant groups. 6.3.1. The Cartan involution on Sp2n . Suppose W = F 2n , Jn = ( −10n 10n ), and Sp(W ) = {g ∈ Aut(W ) ∶ t gJn g = Jn }. In other words, assume that the sympletic form on W is defined by ⟨w1 , w2 ⟩ = t w1 Jn w2 for w1 , w2 column vectors in W = F 2n . Then Jn ∈ Sp(W ). This induces an involution Θ on Sym2 (W ) via Θ(ww′ ) = (Jn w)(Jn w′ ). If the ground field F = R, this is a Cartan involution and (w1 , w2 ) ∶= ⟨Jn w1 , w2 ⟩ defines a symmetric positive definite form on W . 6.3.2. The Cartan involution on SO(V ). The Killing form is Bso (w ∧ x, y ∧ z) = (x, y)(w, z) − (w, y)(x, z). This is a symmetric so(V ) invariant form on so(V ); if the ground field F = R, it is a positive multiple of the Killing form. Suppose F = R. Suppose ι ∶ V → V is an

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involution, for which the quadratic form (v, v) is positive definite on the subspace of V for which ι is +1, and is negative definite where ι is −1. Further assume that ι defines an element of the orthogonal group O(V ). Then (v, ι(w)) is a positive definite symmetric bilinear form on V . Associated to ι, one can define a Cartan involution Θι on the Lie algebra so(V ) ≃ ∧2 V . Namely, one sets Θι ∶ ∧2 V → ∧2 V via Θι (v ∧ w) = ι(v) ∧ ι(w). 6.3.3. The Cartan involution on MJ . In the next several subsections, we assume the trace pairing on J is positive definite. This occurs if J = H3 (C), with C a composition algebra with positive-definite norm form nC . Recall that the pairing on J gives rise to an ι ∶ J → J ∨ and thus an involution Θm on m(J) via ̃ ○ ι, where φ ̃ denotes the action of φ on J ∨ . One computes immeΘm (φ) = ι−1 ○ φ diately that Θm (Φγ,x ) = −Φι(x),ι(γ) . If the ground field F = R, Θm is a Cartan involution on m(J). 6.3.4. The Cartan involution on HJ . Suppose the ground field F = R. Consider the map J2 on WJ , given by J2 (a, b, c, d) = (d, −ι(c), ι(b), −a). Define a symmetric pairing on WJ via (v1 , v2 ) ∶= ⟨J2 v1 , v2 ⟩. Since J2 is in HJ1 , there is an associated involution on hJ given by Θh (φ) = J2 φJ2−1 . One has Θh (Φw,w′ ) = ΦJ2 w,J2 w′ . Then Θh is a Cartan involution on h(J)0 . 6.3.5. The Cartan involution on GJ I. We abuse notation and also write J2 = ( −1 1 ) ∈ SL2 . (There is a natural map SL2 → HJ1 , and the image of J2 ∈ SL2 is the J2 ∈ HJ1 .) Using J2 , we define an involution Θg on g(J) as Θg (φ2 + φJ , v ⊗ w) = (J2 φ2 J2−1 + J2 φJ J2−1 , J2 v ⊗ J2 w). Here φ2 ∈ sl2 , φJ ∈ h(J)0 , v ∈ V2 and w ∈ WJ . It is clear that Θg is an involution on g(J). If the ground field F = R then Θg defines a Cartan involution on g(J). 6.3.6. The Cartan involution on GJ II. We now express the Cartan involution Θg on g(J) via the definition of g(J) in terms of its Z/3-grading. To do this, we endow V3 with the positive definite symmetric form given by (v, v ′ ) = t vv ′ . In other words, we make the standard basis v1 , v2 , v3 of V3 orthonormal. This induces an identification ι between V3 and V3∨ . Define an involution Θg on g(J) as follows: On sl3 it is X ↦ −X t . On m(J)0 it is Θm . On V3 ⊗J it is v ⊗X ↦ ι(v)⊗ι(X) ∈ V3∨ ⊗J ∨ and on V3∨ ⊗J ∨ it is δ ⊗γ ↦ ι(δ)⊗ι(γ) ∈ V3 ⊗J. The map Θg is a Cartan involution. 7. Quaternionic modular forms In this chapter, we explain a bit about what is known regarding quaternionic modular forms beyond the case of G2 . 7.1. The differential equation. We begin with the definition of quaternionic modular forms. Thus suppose J is a cubic norm structure with positive definite trace pairing, and GJ is the associated quaternionic group. Recall that we have identified a Cartan involution Θg on g(J) ⊗ R. Let K ⊆ GJ (R) be the associated maximal compact subgroup. We assume GJ is of an exceptional Dynkin type; this insures that GJ (R) is connected. Our assumptions let us uniformly describe the maximal compact subgroup K as K = (SU(2) × L)/μ2 for a certain group L (that depends on J). We write g = k⊕p for the Cartan decomposition. As a representation of K, one has p = V2 ⊠ W , where V2 is the standard representation of SU(2) and W is a certain symplectic representation of L. See [14] and [25]. For an integer  ≥ 1, let V = Sym2 (V2 ) ⊠ 1, as a representation of K. Now suppose ϕ ∶ GJ (R) → V is a smooth function, satisfying ϕ(gk) = k−1 ϕ(g) for all g ∈ GJ (R) and k ∈ K. We

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define a differential operator D ϕ ∶ G(R) → Sym2−1 (V2 ) ⊠ W as follows. First, we ̃  ϕ ∶ G(R) → V ⊗ p∨ as: define D ̃  ϕ = ∑ Xj ϕ ⊗ Xj∨ D j

where {Xj } is a basis of p and {Xj∨ } is the dual basis of p∨ . Here Xj ϕ is the right ̃  ϕ is still K-equivariant, i.e., regular action of p ⊆ g on ϕ. One checks easily that D ̃  ϕ, then ϕ′ (gk) = k−1 ϕ(g) for all k ∈ K and g ∈ GJ (R). Now, because if ϕ′ = D p ≃ p∨ , we have V ⊗ p∨ ≃ Sym2−1 (V2 ) ⊠ W ⊕ Sym2+1 (V2 ) ⊠ W. Let pr ∶ V ⊗ p∨ → Sym2−1 (V2 ) ⊠ W be a K equivariant projection (unique up to ̃ . scalar multiple). We define D = pr ○ D Definition 7.1. Suppose  ≥ 1 is a non-negative integer. A smooth function ϕ ∶ GJ (A) → V is a quaternionic modular form of weight  if (1) (2) (3) (4) (5)

ϕ is of moderate growth ϕ is right invariant under an open compact subgroup of GJ (Af ) ϕ is left G(Q)-invariant, i.e., ϕ(γg) = ϕ(g) for all γ ∈ G(Q) ϕ is K-equivariant, i.e., ϕ(gk) = k−1 ϕ(g) for all k ∈ K and D ϕ ≡ 0.

7.2. Representation theory. One can also define quaternionic modular forms from the lens of representation theory. In this section, we briefly explain this representation-theoretic approach and its relationship to the definition above. To begin, we start with the quaternionic representations of the group GJ (R). These were defined and studied by Gross-Wallach [13, 14]. We say7 an irreducible representation π of GJ (R) is quaternionic of weight  if (1) π contains the K-type V∨ ≃ V with multiplicity one (2) π does not contain the K-type Sym2−1 (V2 ) ⊠ W . Gross-Wallach [13, 14] constructed quaternionic representations π of GJ (R), which at least for  sufficiently large are moreover discrete series representations of GJ (R). Let S ≃ SU(2) ⊆ K be the normal subgroup that is the kernel of the map K → L/μ2 . Gross-Wallach also showed that the π are admissible when restricted to S. The above properties make the π analogous to the so-called holomorphic (discrete series) representations of the groups that have an associated Hermitian symmetric space. Suppose π is a quaternionic representation of GJ (R) of weight . Suppose Φπ ∶ π → A(GJ ) is homomorphism of (g, K)-modules, from π to the space of automorphic forms on GJ . In the parlance of [9], such a map is a modular form of weight . To relate the maps Φπ to the ϕ’s of the previous section, one proceeds as follows. Restricting Φπ to the K-type V∨ of π, one obtains a map Φπ ∶ V∨ → A(GJ ), or equivalently, a function ϕ ∶ GJ (Q)/GJ (A) → V . Because Φπ is K-equivariant, so is the function ϕ. Moreover, because π does not contain the K-type Sym2−1 (V2 ) ⊠ W , one can check that D ϕ = 0. Consequently, out of a map Φπ ∶ π → A(GJ ), one obtains a modular form ϕ of weight . 7

This is an ad-hoc definition that is suitable for our purposes

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7.3. The Fourier expansion. In this section, we describe the Fourier expansion of quaternionic modular forms. Suppose ϕ is a quaternionic modular form. One can take the constant term of ϕ along Z, and then Fourier expand the result along N /Z: (7.1)

ϕZ (g) = ϕN (g) +

∑

ϕω (g)

ω∈WJ ,ω≠0

where ϕω (g) = ∫N (Q)/N (A) ψ(⟨ω, n⟩)−1 ϕ(ng) dn. The Fourier expansion we explain in this section is a refinement of (7.1) for quaternionic modular forms. The existence of the Fourier expansion that we describe is based on the following theorem, which is the main result of [25]. To state the theorem, we need a couple definitions. Say a nonzero element ω ∈ WJ (R) is positive semi-definite, written ω ≥ 0, if ⟨ω, (1, −Z, Z # , −N (Z))⟩ is never 0 for Z in the upper half space hJ = {Z = X + iY ∶ X, Y ∈ J, Y > 0}. (It is mildly remarkable that such ω exist!) We now define generalized Whittaker functions. If χ is a character of N (R), we say a function F ∶ GJ (R) → V is a generalized Whittaker function of type χ if it satisfies ● F (ng) = χ(n)F (g) for all n ∈ N (R) ● F (gk) = k−1 F (g) for all k ∈ K ● D F ≡ 0 If ω ∈ WJ and χ(n) = χω (n) = ei⟨ω,n⟩ , we alternatively say that F is a generalized Whittaker function of type ω. Theorem 7.2. Suppose ω ∈ WJ (R) is nonzero. Then the space of moderate growth generalized Whittaker functions of type ω is at most one-dimensional. Moreover: (1) Suppose ω is not positive semi-definite. Then every moderate growth generalized Whittaker function of type ω is 0. (2) Suppose ω is positive semi-definite. Then there is a completely explicitly function Wω ∶ GJ → V satisfying: (a) Wω is a moderate growth generalized Whittaker function of type ω (b) If F is a moderate growth generalized Whittaker function of type ω then F = λF Wω for some λF ∈ C. When ω ∈ WJ (R) is rank four, the first two statements of the Theorem (i.e., the theorem without the explicit function Wω ) are due to Wallach [33]. An immediate corollary is the Fourier expansion of quaternionic modular forms: Corollary 7.3. Let ω ∈ WJ be nonzero, let ϕ be a quaternionic modular form of weight , and let ϕω be as in (7.1). Then (1) If ω is not positive semi-definite, ϕω (g) ≡ 0. (2) If ω is positive semi-definite, then there is locally constant function cω ∶ GJ (Af ) → C so that ϕω (gf g∞ ) = cω (gf )W2πω (g∞ ). The locally constant functions cω (gf ), or sometimes their values at gf = 1, are called the Fourier coefficients of ϕ. When ω is rank four, Gan-Gross-Savin [9] had defined these Fourier coefficients without the use of the explicit functions Wω , but instead using Wallach’s results in [33]. In case ϕ is of level one, one can prove that

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ϕ is determined by its constant term ϕN (g) and the numerical Fourier coefficients cω (1). We now describe the functions Wω . Because of the equivariance properties of Wω , to describe it completely it suffices to give a formula for Wω on the Levi HJ of the Heisenberg parabolic. Let e , h , f be the basis of the long root sl2 of k from [25]. Let x, y be a basis of its standard representation C2 , so that h x = x, h y = −y and f x = y. We write V for the 2th symmetric power representation of this sl2 , which we also consider as a representation of K. The space V has as basis the elements x+v y −v for − ≤ v ≤ . Let r0 (i) ∈ WJ ⊗ C be the element r0 (i) = (1, −i, −1, i). Denote by Kv the K-Bessel function, defined as Kv (y) =

1 ∞ v −y(t+t−1 )/2 dt . t e 2 ∫0 t

Then if m ∈ HJ , v

Wω (m) = ν(m) ∣ν(m)∣ ∑ ( v

∣⟨ω, m ⋅ r0 (i)⟩∣ x+v y −v ) Kv (∣⟨ω, m ⋅ r0 (i)⟩∣) . ⟨ω, m ⋅ r0 (i)⟩ ( + v)!( − v)!

7.4. Examples of quaternionic modular forms. In this section, we give results and examples about quaternionic modular forms and their Fourier coefficients that go beyond G2 . 7.4.1. Eisenstein series. The easiest family of examples of quaternionic modular forms is the degenerate Heisenberg Eisenstein series. Thus let P ⊆ GJ be the Heisenberg parabolic, with ν ∶ P → GL1 the character given by p ⋅ e0 = ν(p)e0 .  There is a weight  modular form associated to inducing sections in IndG P (ν ∣ν∣). G(R) s In more detail, suppose  > 0 is even, and let f (g, s) ∈ IndP (R) (∣ν∣ ) be the

unique K-equivariant, V -valued flat section whose value at g = 1 is x y  . Let now G(A ) ff te ∈ IndP (Aff ) (∣ν∣s ) be an arbitrary flat section. Set f (g, s) = ff te (gf , s)f (g∞ , s) and E(g, f, s) = ∑γ∈P (Q)/G(Q) f (γg, s). The sum converges absolutely if Re(s) > 1 + dim(WJ )/2. Suppose s =  + 1 is in the range of absolute convergence, so that  > dim(WJ )/2. Then it can be shown that the value E(g, f, s =  + 1) is a quaternionic modular form of weight . Another way to create modular forms is to use Heisenberg Eisenstein series with nontrivial inducing data. One can take a classical holomorphic modular form Φ of weight n on HJ (weight 3n on G2 ), and from it produce a weight n modular form E(g, Φ) on GJ , if n is sufficiently large. This is spelled out in [25]. 7.4.2. Small representations. Because absolutely convergent degenerate Eisenstein series always define modular forms, it makes sense to ask about the automorphic forms defined by these Eisenstein series outside the range of absolute convergence. In other words, suppose  ≤ dim(WJ )/2. Then one can ask: ● Is the Eisenstein series E(g, f, s) regular at s =  + 1? ● If so, is the resulting automorphic function a modular form of weight ? In some cases, these questions have been answered. Theorem 7.4. Let E (g, s) be the Eisenstein series on G = E8 with ff te spherical at every finite place. (1) (Gan [6], Gan-Savin [11]) Suppose  = 4. Then E4 (g, s) is regular at s = 5, and θmin (g) ∶= E4 (g, s = 5) defines a square integrable, non-cuspidal, modular form of weight 4.

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(2) (Pollack [29]) Suppose  = 8. Then E8 (g, s) is regular at s = 9, and θntm (g) ∶= E8 (g, s = 9) defines a square-integrable, non-cuspidal, modular form of weight 8. The choice of  = 4, 8 are inspired by the work [13]. About the modular forms θmin , θntm , one can prove that their Fourier coefficients are rational: Theorem 7.5 ([28],[29]). The modular forms θmin and θntm have rational Fourier coefficients. 7.4.3. Theta lifts. Using θmin , one can construct new, interesting modular forms. The first such examples were considered by Gan-Gross-Savin [9]. Recall that, via the correspondence between binary cubic forms and cubic rings, the Fourier coefficients of modular forms on G2 correspond to totally real cubic rings. Let I ∈ J be the three-by-three identity matrix. For a cubic ring A, and an element X ∈ J with NJ (X) = 1 (such as X = I), let N (A, X) be the number of maps f ∶ A → J satisfying (1) f (1) = X (2) NJ (f (x)) = NA (x) for all x ∈ A. Here NA is the cubic norm on A. This is the number of embeddings of pointed cubic spaces of A into J. Besides X = I, there another second element E ∈ J with NJ (E) = 1, with the property that the pointed cubic spaces (J, I) and (J, E) are not globally equivalent; see [9] for a discussion. See also [8], [4]. Theorem 7.6. [9] There are level one modular forms θI and θE on G2 of weight 4 whose Ath Fourier coefficient is N (A, I), respectively, N (A, E), if A is a non-degenerate totally real cubic ring. Let H be a certain group of type F4 , which is compact at the archimedean place and split at every finite place. Then there is a dual pair G2 ×H ⊆ E8 . It is a theorem ̂ has size two: see that the automorphic double quotient H(Q)/H(A)/H(R)H(Z) [4] and [8]. There is thus a two-dimensional space of special automorphic forms on H(A). The modular forms θI , θE are lifts from this two-dimensional space, using θmin as the kernel function. The linear combination 91θI + 600θE is the theta lift of the trivial function on H. A Siegel-Weil theorem of Gan identifies this lift with the Eisenstein series of weight 4 on G2 . Theorem 7.7 (Gan [7]). The linear combination 91θI + 600θE is the spherical weight 4 Eisenstein series on G2 . We mention that the archimedean theta correspondence between G2 (R) and H(R) has been determined in Huang-Pandzic-Savin [17]. 7.4.4. Distinguished modular forms. Suppose f (Z) = ∑T af (T )e2πi tr(T Z) is a Siegel modular form. Then f is said to be distinguished if it satisfies the following condition: (1) There exists T0 with det(T0 ) ≠ 0 so that af (T0 ) ≠ 0 (2) If T is such that det(T ) ≠ 0 and af (T ) ≠ 0, then det(T ) ≡ det(T0 ) mod (Q× )2 .

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One can make an analogous definition for quaternionic modular forms: Definition 7.8. A quaternionic modular form ϕ with Fourier coefficients aϕ (ω) is said to be distinguished if (1) There exists a rank four ω0 with aϕ (ω0 ) ≠ 0 (2) If ω is rank four and aϕ (ω) ≠ 0, then q(ω) ≡ q(ω0 ) modulo (Q× )2 . In [28], by restricting θmin to quaternionic groups of type E6 , we constructed distinguished modular forms: √ Theorem 7.9. [28] Let E = Q( −d) be an imaginary quadratic field, and GE the quaternionic group of type E6 defined from E. Then there is a weight 4 distinguished modular form ϕE on GE . The form ϕE satisfies: if ω is rank four and aϕ (ω) ≠ 0, then q(ω) ≡ −d modulo (Q× )2 . There is an embedding GE → E8 , and ϕE is defined as the pullback of θmin via this embedding. 8. The Fourier expansion on orthogonal groups In this chapter, we give an explicit Fourier expansion on SO(4, n + 2), to show how it may be done for a classical group. More precisely, we state and prove the formula for the generalized Whittaker function on SO(4, n + 2)(R)0 . To state the result, we use the notation of the following section. Throughout this chapter, we work over the ground field R of real numbers. 8.1. Setups. We begin by setting up notation. Set V2,n = V2 ⊕Vn . Here V2 is a positive definite quadratic space with orthonormal basis v1 , v2 and Vn is a negative definite quadratic space with basis v−j for 1 ≤ j ≤ n. Now set U = Span(b1 , b2 ) and U ∨ = Span(b−1 , b−2 ), i.e., U , respectively U ∨ , is a two-dimensional vector space with basis b1 , b2 , respectively b−1 , b−2 . Let V = U ⊕ V2,n ⊕ U ∨ . We put a symmetric bilinear form ( , ) on V in such a way that U, U ∨ are two-dimensional and isotropic with pairing (bi , b−j ) = δij and U ⊕ U ∨ is orthogonal to V2,n . We identify the Lie algebra of G ∶= SO(V ) with ∧2 V , so that v1 ∧ v2 (v) = (v2 , v)v1 − (v1 , v)v2 . We define the Heisenberg parabolic P = M N to be the stabilizer of U inside SO(V ). The Levi subgroup M is defined to be the subgroup of P that also stabilizes U ∨ . Then M ≃ GL2 × SO(V2,n ), and we write r = diag(m, h, t m−1 ) for a typical element algebra of M . Observe that the Lie algebra n of N is U ∧ (U + V2,n ) and the Lie √ ∨ 2 + ∧ V . Now, for j = 1, 2, one sets u = (b + b )/ 2 and m of M is U ∧ U 2,n j j −j √ u−j = (bj − b−j )/ 2. Thus u1 , u2 , u−1 , u−2 are orthonormal (up to sign). Define a Cartan involution on SO(V ) as conjugation by ι, where ι(bj ) = b−j , ι(b−j ) = bj , ι is +1 on V2 and ι is −1 on Vn . With this Cartan involution, p = Span{u1 , u2 , v1 , v2 } ∧ Span{u−1 , u−2 , v−1 , . . . , v−n } =∶ V4 ∧ Vn+2 and k = ∧2 V4 ⊕ ∧2 Vn+2 . We write K for the associated maximal compact subgroup of G = SO(V ) and K 0 for its identity component. Then K 0 = SO(V4 )×SO(Vn+2 ). Recall that SO(4) ≃ SU(2)×SU(2)/μ2 . Consequently, the complexified Lie algebra k of K has two sl2 pieces. The first sl2 is given as ● e+ = 12 (u1 − iu2 ) ∧ (v1 − iv2 ) ● h+ = i(u1 ∧ u2 + v1 ∧ v2 ) = 12 (u1 − iu2 ) ∧ (u1 + iu2 ) + 12 (v1 − iv2 ) ∧ (v1 + iv2 ) ● f + = − 21 (u1 + iu2 ) ∧ (v1 + iv2 ).

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The other sl2 is obtained by replacing v2 with −v2 in the above formulas: That is, it has basis ● e′+ = 12 (u1 − iu2 ) ∧ (v1 + iv2 ) ● h′+ = i(u1 ∧ u2 − v1 ∧ v2 ) ● f ′+ = − 12 (u1 + iu2 ) ∧ (v1 − iv2 ). Let V2 denote the standard representation of one of these sl2 ’s. We write x, y for a weight basis of it. We may identify V4 ⊗ C with V2 ⊗ V2 for these two sl2 ’s as follows: 1 u1 − iu2 x⊗x v1 + iv2 √ ( )=( x⊗y 2 −(v1 − iv2 ) u1 + iu2

y⊗x ). y⊗y

Identifying sl2 = Sym2 (V2 ), this leads to the identifications ● e+ = (x ⊗ x) ∧ (−x ⊗ y) ↦ −x2 ● h+ = (x ⊗ x) ∧ (y ⊗ y) − (x ⊗ y) ∧ (y ⊗ x) ↦ 2xy ● f + = −(y ⊗ y) ∧ (y ⊗ x) ↦ y 2 Finally, we set V = Sym2 (V2 ) ⊠ 1 ⊠ 1 as a representation of K 0 . It has basis x2 , ⋯, y 2 . 8.2. Statement of theorem. Our main result in this chapter is a formula for a generalized Whittaker function on SO(V )0 for the Heisenberg parabolic. In this section, we state that theorem. In more detail, suppose T1 , T2 are in V2,n . A generalized Whittaker function ϕ of type (T1 , T2 ) is a function on SO(V )(R)0 that satisfies: ● ● ● ●

ϕ is valued in V . ϕ(gk) = k−1 ⋅ ϕ(g) for all g ∈ SO(V )(R)0 and k ∈ K 0 . ϕ(exp(b1 ∧ y1 + b2 ∧ y2 ) exp(zb1 ∧ b2 )g) = ei(T1 ,y1 )+i(T2 ,y2 ) ϕ(g). D ϕ ≡ 0, where D is the so-called Schmid operator, defined exactly as it was in previous sections.

For r = (m, h, t m−1 ), and T1 , T2 ∈ V2,n define β(r) =

√ h(v1 + iv2 ) ). 2i(T1 , T2 )m ( ih(v1 + iv2 )

w1 ) means (T1 , w1 ) + (T2 , w2 ). Say that the pair (T1 , T2 ) w2 is positive semi-definite, written (T1 , T2 ) ≥ 0, if β(r) ≠ 0 for all r ∈ GL2 (R)+ × SO(V2,n )0 . Suppose ′ ∈ SO(V2,n ) takes v1 ↦ −v1 and v2 ↦ v2 and define  = diag(( −1 1 ) , ′ , ( −1 1 )). Then  ∈ SO(4, n + 2)(R)0 . Observe that β(r) = −β ∗ (r). With notation as above, and (T1 , T2 ) positive semi-definite, define

The notation (T1 , T2 ) (

v

∣β(r)∣ x+v y −v W(T1 ,T2 ) (r) = det(m) ∣ det(m)∣ ∑ ( . ) K (∣β(r)∣) v ∗ ( + v)!( − v)! −≤v≤ β(r) 

We will need the following result. Exercise 8.1. The function W(T1 ,T2 ) satisfies W(T1 ,T2 ) (rk) = k−1 W(T1 ,T2 ) (r) for all k ∈ K 0 ∩ M (R).

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Here is the theorem. Theorem 8.2. Suppose F is a moderate growth generalized Whittaker function for the pair (T1 , T2 ), and T1 , T2 are not both 0. Then, if (T1 , T2 ) is not positive semi definite, F = 0. If (T1 , T2 ) is positive semi-definite, then F is proportional to W(T1 ,T2 ) . 8.3. The Iwasawa decomposition. We need detailed forms of the Iwasawa decomposition of elements of p. Recall n = U ∧ (U + V2,n ) ⊆ ∧2 V ≃ g and m = U ∧ U ∨ + ∧2 V2,n . Then we have g = n + m + k, and we will write elements of p in terms of this decomposition: ● (u1 + iu2 ) ∧ (u−1 + iu−2 ) = (b−1 + ib−2 ) ∧ (b1 + ib2 ) is in m. ● (u1 − iu2 ) ∧ (u−1 − iu−2 ) = (b−1 − ib−2 ) ∧ (b1 − ib2 ) is in m. ● (u1 +iu2 )∧(u−1 −iu−2 ) = (b−1 +ib2 )∧(b1 +ib−2 ) = (b−1 ∧b1 +b−2 ∧b2 )+2ib2 ∧ b1 + i(b−1 ∧ b−2 + b1 ∧ b2 ). Moreover, b−1 ∧ b−2 + b1 ∧ b2 = u1 ∧ u2 + u−1 ∧ u−2 so that i(b−1 ∧ b−2 + b1 ∧ b2 ) has projection into ∧2 V4 equal to 12 h+ + 12 h′+ . ● (u1 −iu2 )∧(u−1 +iu−2 ) = (b−1 −ib2 )∧(b1 −ib−2 ) = (b−1 ∧b1 +b−2 ∧b2 )−2ib2 ∧ b1 − i(b−1 ∧ b−2 + b1 ∧ b2 ). Moreover, b−1 ∧ b−2 + b1 ∧ b2 = u1 ∧ u2 + u−1 ∧ u−2 so that −i(b−1 ∧ b−2 + b1 ∧ b2 ) has projection into ∧2 V4 equal to − 12 h+ − 12 h′+ . Some more Iwasawa decompositions: √ ● ui ∧v−j = 2bi ∧v−j −u−i ∧v−j is in n+k. Consequently (b1 + ib2 ) ∧ v−j −

√1 (u−1 2

+ iu−2 ) ∧ v−j and

√1 (u1 2

√1 (u1 +iu2 )∧v−j 2

=

− iu2 ) ∧ v−j = (b1 − ib2 ) ∧

v−j − √12 (u−1 − iu−2 ) ∧ v−j . These decompositions are in n + k. ● vi ∧ v−j is in m √ One has vi ∧ u−j = 2vi ∧ bj − vi ∧ uj is in n + k. This leads to the following decompositions: ● ● ● ●

1 (v +iv2 )∧(u−1 +iu−2 ) = √12 (v1 +iv2 )∧(b1 +ib2 )+ 2 1 √1 (v1 + iv2 ) ∧ (b1 + ib2 ) − f + . 2 1 (v +iv2 )∧(u−1 −iu−2 ) = √12 (v1 +iv2 )∧(b1 −ib2 )+ 2 1 √1 (v1 + iv2 ) ∧ (b1 − ib2 ) + e′+ 2 1 (v −iv2 )∧(u−1 +iu−2 ) = √12 (v1 −iv2 )∧(b1 +ib2 )+ 2 1 √1 (v1 − iv2 ) ∧ (b1 + ib2 ) − f ′+ 2 1 (v −iv2 )∧(u−1 −iu−2 ) = √12 (v1 −iv2 )∧(b1 −ib2 )+ 2 1 1 √ (v1 − iv2 ) ∧ (b1 − ib2 ) + e+ . 2

1 (u1 +iu2 )∧(v1 +iv2 ) 2

=

1 (u1 −iu2 )∧(v1 +iv2 ) 2

=

1 (u1 +iu2 )∧(v1 −iv2 ) 2

=

1 (u1 −iu2 )∧(v1 −iv2 ) 2

=

8.4. The differential equations. We assume ϕ is a generalized Whittaker function of type (T1 , T2 ). Suppose r = (m, h, t m−1 ) is in the Heisenberg Levi, so 11 m12 that m = ( m m21 m22 ) ∈ GL2 (R) and h ∈ SO(V2,n ). Then if z1 , z2 ∈ V2,n , (b1 ∧ z1 + b2 ∧ z2 )ϕ(M ) = i((T1 , m11 h(z1 ) + m12 h(z2 )) + (T2 , m21 h(z1 ) + m22 h(z2 )))ϕ(M ) = i(T1 , T2 )m(h(z1 ), h(z2 ))t ϕ(M ).

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141

̃ with D = pr ○ D. ̃ We now have: Recall the operator D ̃ = 1 (u1 + iu2 ) ∧ (u−1 + iu−2 )ϕ ⊗ x ⊠ x ⊗ √1 (u−1 − iu−2 ) Dϕ 2 2 1 1 + (u1 + iu2 ) ∧ (u−1 − iu−2 )ϕ ⊗ x ⊠ x ⊗ √ (u−1 + iu−2 ) 2 2 1 1 + (u1 − iu2 ) ∧ (u−1 + iu−2 )ϕ ⊗ y ⊠ y ⊗ √ (u−1 − iu−2 ) 2 2 1 1 + (u1 − iu2 ) ∧ (u−1 − iu−2 )ϕ ⊗ y ⊠ y ⊗ √ (u−1 + iu−2 ) 2 2 + ∑ ((b1 + ib2 ) ∧ v−j ϕ) ⊗ x ⊠ x ⊗ v−j 1≤j≤n

+ ∑ ((b1 − ib2 ) ∧ v−j ϕ) ⊗ y ⊠ y ⊗ v−j 1≤j≤n

1 + ∑ √ ((v1 + iv2 ) ∧ v−j ϕ) ⊗ −x ⊠ y ⊗ v−j 2 1≤j≤n 1 + ∑ √ ((v1 − iv2 ) ∧ v−j ϕ) ⊗ y ⊠ x ⊗ v−j 2 1≤j≤n 1 1 + ( √ (v1 + iv2 ) ∧ (b1 + ib2 ) − f + )ϕ ⊗ −x ⊠ y ⊗ √ (u−1 − iu−2 ) 2 2 1 1 + ( √ (v1 + iv2 ) ∧ (b1 − ib2 ) + e′+ )ϕ ⊗ −x ⊠ y ⊗ √ (u−1 + iu−2 ) 2 2 1 1 ′+ + ( √ (v1 − iv2 ) ∧ (b1 + ib2 ) − f )ϕ ⊗ y ⊠ x ⊗ √ (u−1 − iu−2 ) 2 2 1 1 + ( √ (v1 − iv2 ) ∧ (b1 − ib2 ) + e+ )ϕ ⊗ y ⊠ x ⊗ √ (u−1 + iu−2 ) 2 2 Simplifying and contracting, one finds that in D ϕ the coefficient of the various b a terms are as follows: Define [xa ] = xa! and similarly [y b ] = yb! . We write ϕ = ∑−≤v≤ ϕv [x+v ][y −v ]. Theorem 8.3. Let the notation be as above. Then (1) The coefficient of [x+v ][y −v−1 ] ⊗ x ⊗ √12 (u−1 − iu−2 ): i 1 − (u1 + iu2 ) ∧ (u−1 + iu−2 )ϕv − √ (T1 , T2 )m(h(v1 ) − ih(v2 ), ih(v1 ) + h(v2 ))t ϕv+1 2 2 (2) The coefficient of [x+v−1 ][y −v ] ⊗ y ⊗

√1 (u−1 2

+ iu−2 ):

i 1 (u1 − iu2 ) ∧ (u−1 − iu−2 )ϕv − √ (T1 , T2 )m(h(v1 ) + ih(v2 ), −ih(v1 ) + h(v2 ))t ϕv−1 2 2 (3) The coefficient of [x+v ][y −v−1 ] ⊗ x ⊗ v−j : 1 −i(T1 , T2 )m(h(v−j ), ih(v−j ))t ϕv + √ ((v1 − iv2 ) ∧ v−j )ϕv+1 2 (4) The coefficient of [x+v−1 ][y −v ] ⊗ y ⊗ v−j : 1 i(T1 , T2 )m(h(v−j ), −ih(v−j ))t ϕv + √ ((v1 + iv2 ) ∧ v−j )ϕv−1 2

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(5) The coefficient of [x+v ][y −v−1 ] ⊗ x ⊗

√1 (u−1 2

+ iu−2 ):

1 i (b1 ∧b−1 +b2 ∧b−2 −2(+1)−v)ϕv − √ (T1 , T2 )m(h(v1 )−ih(v2 ), −ih(v1 )−h(v2 ))t ϕv+1 2 2 (6) The coefficient of [x+v−1 ][y −v ] ⊗ y ⊗

√1 (u−1 2

− iu−2 ):

1 i − (b1 ∧b−1 +b2 ∧b−2 −2(+1)+v)ϕv − √ (T1 , T2 )m(h(v1 )+ih(v2 ), ih(v1 )−h(v2 ))t ϕv−1 2 2 8.5. Solving the differential equations. Abusing notation, write ϕv (w, x, y, h) = ϕv (r) where r = diag(m, h, t m−1 ) and m = ( w w ) ( 1 x1 ) ( y

1/2

). Define fv (w, x, y, h)

y −1/2 SO(V2,n )0 . We

R×>0 ,

fv . Here w, y ∈ x ∈ R and h ∈ will find the formula as ϕv = w ϕ by solving the differential equations above in terms of the explicit coordinates w, x, y, h. Recall √ h(v1 ) + ih(v2 ) β = 2i(T1 , T2 )m ( ). ih(v1 ) − h(v2 ) 2+2

Note that b1 ∧ b−1 + b2 ∧ b−2 becomes the differential operator w∂w . Also note that w∂w (wA f ) = wA (w∂w + A)f . With these definitions, the final two differential equations of Theorem 8.3 become ● (w∂w − v)fv + β ∗ fv+1 = 0 and ● (w∂w + v)fv + βfv−1 = 0. Note that β depends linearly on w. Solving these two equations on a domain where β ≠ 0 gives that, as a function of w, φv is proportional to w2+2 Kv (∣β∣). (This uses that φv is of moderate growth.) Define Yv (x, y, h) so that fv = Yv Kv (∣β∣); i.e., Yv is independent of w. Then the differential equations imply βYv−1 Kv−1 (∣β∣) = βfv−1 = −(w∂w + v)fv = Yv (−(w∂w + v)Kv (∣β∣)) = ∣β∣Yv Kv−1 (∣β)∣. One obtains that Yv =

β Y ∣β∣ v−1

=

∣β∣ Y . β ∗ v−1

It follows that v

φv (w, x, y, h) = Y0 (x, y, h)w2+2 (

∣β∣ ) Kv (∣β∣). β∗

We will now use the other differential equations of Theorem 8.3 to prove that Y0 is 1/2 constant. Suppose m = ( 10 x1 ) ( y −1/2 ) w. Set z = x + iy. Then one computes y

√ β = − 2y −1/2 w(z ∗ T1 + T2 , h(v1 + iv2 )). We will now prove the following proposition, which will be used in deducing that Y0 is constant. Proposition 8.4. One has y(∂x − i∂y )(∣β∣) =

i β (− √ wy −1/2 (z ∗ T1 + T2 , h(v1 − iv2 ))) . ∣β∣ 2

MODULAR FORMS ON EXCEPTIONAL GROUPS

143

√ ′ β Proof. Set β ′ =(z ∗ T1 +T2 , h(v1 +iv2 )), so that β = − 2wy −1/2 β ′ and − ∣ββ ′ ∣ = ∣β∣ . We first compute that √ i 2 −1/2 ′ √ wy ∣β ∣ + 2wy 1/2 (∂x − i∂y )(∣β ′ ∣). y(∂x − i∂y )(∣β∣) = 2 Now 1 (∂x − i∂y )(∣β ′ ∣) = (β ′∗ (∂x − i∂y )(β ′ ) + β ′ (∂x − i∂y )(β ′∗ )). 2∣β ′ ∣ One computes (∂x − i∂y )(β ′ ) = 0 and (∂x − i∂y )(β ′∗ ) = 2(T1 , h(v1 − iv2 )). Thus √ i y y(∂x − i∂y )(∣β∣) = ( 2wy −1/2 ) ( ∣β ′ ∣ + ′ β ′ (T1 , h(v1 − iv2 ))) 2 ∣β ∣ √ 2wy −1/2 β ′ i ( (zT1 + T2 , h(v1 − iv2 )) + y(T1 , h(v1 − iv2 ))) = ∣β∣ 2 √ −1/2 ′ i 2wy β (zT1 + T2 − 2iyT1 , h(v1 − iv2 )) = 2 ∣β ′ ∣ β′ i = (− √ wy −1/2 (z ∗ T1 + T2 , h(v1 − iv2 ))) (− ′ ) . ∣β ∣ 2 The proposition follows. Note that if X ∈ g and v ∈ V then Xh(v) = Lemma 8.5. One has (1) (v1 − iv2 ) ∧ v−j (∣β∣) = (2) (v1 + iv2 ) ∧ v−j (∣β∣) =

d (hetX )(v)∣t=0 dt



= h(X(v)).

√

β∗ ( 2y −1/2 w(z ∗ T1 + T2 , h(v−j ))) ∣β∣ √ β ( 2y −1/2 w(zT1 + T2 , h(v−j ))). ∣β∣

Proof. We have (v1 − iv2 ) ∧ v−j (∣β∣) =

1 (β ∗ ((v1 − iv2 ) ∧ v−j )(β) + β((v1 − iv2 ) ∧ v−j )(β ∗ )) 2∣β∣

and similarly for (v1 + iv2 ) ∧ v−j . Now, one has √ ● (v1 − iv2 ) ∧ v−j (β) = 2 2y −1/2 w(z ∗ T1 + T2 , h(v−j )) ● (v1 + iv2 ) ∧ v−j (β) = 0 ● (v1 − iv2 ) ∧ v−j (β ∗ ) = 0√ ● (v1 + iv2 ) ∧ v−j (β ∗ ) = 2 2y −1/2 w(zT1 + T2 , h(v−j )). The lemma follows.



Proposition 8.6. The function Y0 (x, y, h) above is constant. Proof. We first consider Y0 as a function of x, y the coordinates on the complex upper half-plane. Note that 1 1 − (u1 + iu2 ) ∧ (u−1 + iu−2 ) = (b1 + ib2 ) ∧ (b−1 + ib−2 ). 2 2 By acting on b1 and b2 , one sees that this is the element of gl2 (via our usual i ). As a differential operator, this Lie algebra isomorphism) with matrix 12 ( 1i −1 element gives iy(∂x − i∂y ) on functions that are right invariant under SO(2). Using the proposition above and the differential equations of the theorem, one obtains that y(∂x − i∂y )Y0 = 0. Similarly, one obtains y(∂x + i∂y )Y0 = 0. Thus Y0 is constant as a function of x, y. Using the lemma above, and again the differential equations

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of the theorem, one obtains XY0 = 0 for all X ∈ p ∩ ∧2 V2,n = V2 ∧ Vn . Because Y0 is  right invariant under K 0 ∩ SO(V2,n )0 , we see that Y0 is constant. We now put everything together. From the work above, one obtains that on a domain where β ≠ 0, there is a unique line of generalized Whittaker functions of type (T1 , T2 ), spanned by W(T1 ,T2 ) . Indeed, we solved the differential equations on GL2 (R)+ × SO(V2,n )0 , and then one must observe that W(T1 ,T2 ) is appropriately equivariant by . One can now argue as in [25] to prove that if (T1 , T2 ) is not positive semi-definite, then the only generalized Whittaker function is 0. The idea is that one first solves the differential equations on a domain where β ≠ 0, and then observes that this unique solution blows up as M approaches a point where β = 0. Finally, one can check that the W(T1 ,T2 ) satisfy the Schmid equations of the theorem above; we omit this aspect. This completes our work.

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10.1090/surv/279/04

Rigidity method for automorphic forms over function fields Zhiwei Yun

These are lectures given at the 2022 Arizona Winter School. It gives an introduction to the rigidity method for constructing automorphic forms for semisimple groups over function fields. The rigidity method leads to explicit constructions of local systems that are Langlands parameters of automorphic forms. The examples of local systems produced in this way have applications to algebraic geometry and number theory. We will emphasize on the “engineering” aspect of the theory, leaving out most of the proofs: we will give principles on how to design rigid automorphic data and methods for computing the resulting local systems. For more details and proofs, we refer to [43]. Acknowledgment I thank the organizers of the 2022 Arizona Winter School for inviting me to lecture, and the University of Arizona and the Clay Institute of Mathematics for supporting my visit. I thank the participants, especially those in my project group, for attending my lectures and giving feedbacks. I am grateful to Benedict Gross and Konstantin Jakob for helpful comments. 1. Automorphic forms over function fields In this section we recall some basic concepts for automorphic forms over a function field. 1.1. The setting. 1.1.1. Function field. Let k be a finite field. Let X be a projective, smooth and geometrically connected curve over k. Let F = k(X) be the field of rational functions on X. Let ∣X∣ be the set of closed points of X (places of F ). For each x ∈ ∣X∣, let Fx denote the completion of F at the place x. The valuation ring and residue field of Fx are denoted by Ox and kx . The maximal ideal of Ox is denoted by mx . The ring of ad`eles is the restricted product ′

AF ∶= ∏ Fx x∈∣X∣

2020 Mathematics Subject Classification. Primary 11F70, 14D24, 14F05. Supported by the Simons Investigatorship and Packard Fellowship. ©2024 American Mathematical Society

147

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where for almost all x, the x-component of an element a ∈ AF lies in Ox . There is a natural topology on AF making it a locally compact topological ring. 1.1.2. Groups. For simplicity of the presentation, we restrict ourselves to the following situation: G is a connected split reductive group over k; In addition, when talking about automorphic forms, G is assumed to be semisimple. We will fix a split maximal torus T and a Borel subgroup B ⊃ T . They give a based root system (X∗ (T ), Φ(G, T ), ΔB , X∗ (T ), Φ∨ (G, T ), Δ∨B ) where ΔB are the simple roots and Δ∨B the simple coroots. It makes sense to talk about G(R) whenever R is a k-algebra. For example, G(F ), G(Ox ) and G(Fx ), etc. ̂ to G is a connected reductive group over Q , The Langlands dual group G ̂ ⊃ T̂, such that the equipped with a maximal torus T̂ and a Borel subgroup B corresponding based root system ̂ T̂), Δ ̂ , X∗ (T̂), Φ∨ (G, ̂ T̂), Δ∨̂ ) (X∗ (T̂), Φ(G, B B is identified with the based root system (X∗ (T ), Φ∨ (G, T ), Δ∨B , X∗ (T ), Φ(G, T ), ΔB ) obtained from that of G by switching roots and coroots. 1.1.3. Adelic and level groups. The group G(AF ) of AF -points of G can also be expressed as the restricted product ′

G(AF ) = ∏ G(Fx ) x∈∣X∣

where most components lie in G(Ox ). This is a locally compact topological group. The diagonally embedded G(F ) inside G(AF ) is a discrete subgroup. The quotient G(F )/G(AF ) is a locally compact space. Let Kx ⊂ G(Fx ) be a compact open subgroup, one for each x ∈ ∣X∣, such that for almost all x, Kx = G(Ox ). Let K = ∏x Kx be the compact open subgroup of G(AF ). The double coset space G(F )/G(AF )/K has the discrete topology. 1.1.4. Automorphic forms. Let AK = C(G(F )/G(AF )/K, Q ) be the vector space of Q -valued functions on G(AF ) that are left invariant under G(F ) and right invariant under K. Let AK,c = Cc (G(F )/G(AF )/K, Q ) ⊂ AK be the subspace that are supported on finitely many double cosets. Elements in AK,c will be our working definition of (compactly supported) automorphic forms for G(AF ) with level K. For more precise definition, we refer to [3, Definition 5.8]. 1.1.5. Action of the Hecke algebra. Let HK be the vector space of Q -valued, compactly supported functions on G(AF ) that are left and right invariant under K. Similarly one can define the local Hecke algebra HKx . The vector space HK (resp. HKx ) has an algebra structure under convolution, such that the characteristic function of 1K (resp. 1Kx ) is the unit element. Then HK is the restricted tensor

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product of the local Hecke algebras HKx with respect to their algebra units. The algebra HK acts on AK and AK,c by the rule: (f ⋆ h)(x) =

∑

f (xgK)h(Kg −1 K),

∀f ∈ AK , h ∈ HK .

g∈G(AF )/K

Exercise 1.1.6. Give a formula for the algebra structure of HK . 1.1.7. Cusp forms. As our working definition, a cusp form for G(AF ) with level K is an element f ∈ AK,c that generates a finite-dimensional HK -submodule. The official definition of a cusp form uses the vanishing of constant terms and looks quite different. It is shown in [28, Proposition 8.23] that these two notions are equivalent. A cusp form f ∈ AK,c is called a Hecke eigenform if it is an eigenfunction for HKx , for almost all x ∈ ∣X∣. 1.2. Weil’s interpretation. We allow G to be reductive in this subsection. Let K ♮ = ∏x G(Ox ) in the following discussion. 1.2.1. Case of GLn . When G = GLn , Weil has given a geometric interpretation of the double coset G(F )/G(AF )/K ♮ in terms of vector bundles of rank n over X. Let Vecn (X) be the groupoid of rank n vector bundles over X. Let us give the map e ∶ G(AF ) → Vecn (X). First, for any finite subset S ⊂ ∣X∣, we shall define a map eS ∶ ∏x∈S G(Fx ) → Vecn (X) as follows. Let j ∶ X − S ↪ X be the open inclusion. Let (gx )x∈S ∈ ∏x∈S G(Fx ) and let Λx ⊂ Fx⊕n be the Ox -submodule gx Ox⊕n . Define ⊕n such that, for any affine eS ((gx )x∈S ) to be the quasi-coherent subsheaf V ⊂ j∗ OX−S open U ⊂ X, ⊕n ) ∩ ( ∏ Λx ). Γ(U, V) = Γ(U − S, OX x∈∣U∣∩S ⊕n ) is diagonally Here the intersection is taken inside ∏x∈∣U∣∩S Fx⊕n , and Γ(U − S, OX embedded into it by taking completion at each x ∈ ∣U ∣ ∩ S.

Exercise 1.2.2 (See [13, Lemma 3.1]). (1) Show that V constructed above is a vector bundle of rank n. (2) Suppose S ⊂ S ′ , gx ∈ G(Fx ) for each x ∈ S, and gx ∈ G(Ox ) for each x ∈ S ′ − S. Let gS = (gx )x∈S and gS ′ = (gx )x∈S ′ . Then there is a canonical isomorphism eS (gS ) = eS ′ (gS ′ ). Using this to show the eS for various S give a well-defined map e ∶ G(AF ) → Vecn (X). (3) Show that e is left invariant under G(F ) and right invariant under K ♮ . (4) Show that e is an equivalence of groupoids. 1.2.3. General G. For general G, there is a similar interpretation of G(F )/G(AF )/K ♮ in terms of G-torsors over X. Recall that a (right) G-torsor over X is a scheme Y → X together with a fiberwise action of G that looks like G × X (with G acting on itself by right translation) ´etale locally over X. An isomorphism between G-torsors Y and Y ′ is a G-equivariant isomorphism Y ≅ Y ′ over X. Exercise 1.2.4. When G = GLn , show that there is an equivalence of categories between G-torsors over X and vector bundles of rank n on X. You will need descent theory to show that ´etale local triviality is the same as Zariski local triviality for sheaves of OX -modules. Similarly, SLn -bundles are the same data as pairs (V, ι) where V is a vector ∼ bundle of rank n over X and ι ∶ ∧n V → OX is a trivialization of the determinant of V.

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Exercise 1.2.5. Show that PGLn -torsors over X are the same as rank n vector bundles over X modulo tensoring with line bundles. For this, you will need the fact that the Brauer group of X is trivial. Exercise 1.2.6. For G = Sp2n and SOn , give an interpretation of G-torsors using vector bundles with bilinear forms. Let BunG (k) be the groupoid of G-torsors over X: this is a category whose objects are G-torsors over X and morphisms are isomorphisms between G-torsors. The groupoid BunG (k) is in fact the groupoid of k-points of an algebraic stack BunG . For a k-algebra R the groupoid BunG (R) is the groupoid of G-torsors over X ⊗k R. Weil observed that there is a natural equivalence of groupoids (1.1)

∼

e ∶ G(F )/G(AF )/K ♮ → BunG (k).

So this is not just a bijection of sets, but for any double coset [g] = G(F )gK ♮ , the automorphism group of e([g]) (as a G-torsor) is isomorphic to G(F ) ∩ gK ♮ g −1 . The construction of e is similar to the case of G = GLn , using modification of the trivial G-torsor at finitely many points S given by (gx )x∈S ∈ ∏x∈S G(Fx ). 1.2.7. Birkhoff decomposition. We consider the case X = P1 . Grothendieck proves that every vector bundle on P1 is isomorphic to a direct sum of line bundles ⊕i O(ni ), and the multiset {ni } is well-defined. This implies that the underlying set of BunGLn (k) is in bijection with the Zn /Sn . More generally, there is a canonical bijection of sets ∣BunG (k)∣ ≅ ∣X∗ (T )/W ∣. By the bijection (1.1), we see that ∣G(F )/G(AF )/K ♮ ∣ is in bijection with X∗ (T )/W . We can construct the bijection as follows. Exercise 1.2.8. Let t be an affine coordinate of A1 ⊂ P1 . (1) Let G = T and λ ∈ X∗ (T ). Viewing λ as a homomorphism λ ∶ Gm → T , and t ∈ F0 (local field at 0 ∈ ∣P1 ∣), let tλ ∶= λ(t) ∈ T (F0 ). Now view tλ as an element in T (AF ) that is tλ at the place 0 and 1 elsewhere. The assignment λ → tλ defines a homomorphism X∗ (T ) → T (AF ). Show that ∼ it induces a bijection X∗ (T ) → ∣T (F )/T (AF )/KT♮ ∣. (2) For general G, the above construction gives a map X∗ (T ) → ∣T (F )/T (AF )/KT♮ ∣ → ∣G(F )/G(AF )/K ♮ ∣. Show that this map is W -invariant, and it induces a bijection between the set of W -orbits on X∗ (T ) and the underlying set of G(F )/G(AF )/K ♮ . 1.2.9. Interpretation of Hecke operators as modifications. By Weil’s interpretation, AK ♮ ,c can be identified with the space of Q -valued points on the set of isomorphisms classes of G-torsors over X, i.e., AK ♮ = C(BunG (k), Q ). Fix x ∈ ∣X∣ and let Kx = G(Ox ). The algebra HKx is called the spherical Hecke algebra for the group G(Fx ). It has a Q -basis consisting of characteristic functions 1Kx gKx of double cosets. We know that HKx acts on AK ♮ . We give an interpretation of the action of 1Kx gKx on AK ♮ = C(BunG (k), Q ) in terms of G-torsors.

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Take the example G = GLn and gi,x = diag(tx , ⋯, tx , 1, ⋯, 1) rss s s s s st s s s s s s u rss s tss s u i

n−i

(tx is a uniformizer at x). For two vector bundles E and E ′ of rank n on X, we use i

→ E ′ to mean that E ⊂ E ′ ⊂ E(x) and dimkx (E ′ /E) = i. Then for any the notation E  x

f ∈ AK ♮ , viewed as a function on Vecn (X) ≅ BunGLn (k), we have (f ⋆ 1Kx gi,x Kx )(E) = ∑ f (E ′ ). i E → E′ x 1.3. Level structures. 1.3.1. Parahoric subgroups. Let x ∈ ∣X∣. Let I ⊂ G(Ox ) be the preimage of a Borel subgroup B(kx ) ⊂ G(kx ) under the reduction map G(Ox ) → G(kx ). This is an example of an Iwahori subgroup of G(Fx ). General Iwahori subgroups are conjugates of I in G(Fx ). Parahoric subgroups of G(Fx ) always contain an Iwahori subgroup with finite index. A precise definition of parahoric subgroups involves a fair amount of Bruhat-Tits theory [5]. The conjugacy classes of parahoric subgroups under Gsc (Fx ) (Gsc is the simply-connected cover of G) are in bijection with proper ̃ subsets of the vertices of the extended Dynkin diagram Dyn(G) of G. Empty subset corresponds to Iwahori subgroups. The set of vertices of the finite Dynkin diagram corresponds to conjugates of G(Ox ). Extended Dynkin diagram of G is obtained from the Dynkin diagram of G by adding one vertex corresponding to the affine simple root α0 , which we mark as a black dot: ̃ n) Dyn(A ●N ppp NNNNN p p NNN pp NNN ppp p NN p pp ○ ○ ⋯ ○ ○ ̃ n) Dyn(B

○< λi ) with the restriction of ρx to the wild inertia. 3.3. More examples. Example 3.3.1 (Hypergeometric automorphic data). In [27], Kamgarpour and Yi constructed automorphic data realizing Katz’s hypergeometric local systems. Let G = PGLn , X = P1 . There are two kinds of such data corresponding to tame and wild hypergeometric sheaves. The tame case generalizes Example 2.1.4. In this case, S = {0, 1, ∞}. Both K0 and K∞ are taken to be an Iwahori subgroup. Let K0 and K∞ be arbitrary Kummer sheaves on the quotient tori of K0 and K∞ . Take K1 ⊂ L+1 G to be parahoric subgroup that is the preimage of a parabolic subgroup with block sizes (n−1, 1) under the evaluation map L+1 G → G (there are two conjugacy classes of such parabolic subgroups, but their preimages in L+1 G are conjugate under PGLn (F1 )). Let K1 be any Kummer sheaf pulled back from the abelianization map K1 → Gm . In the wild case, take S = {0, ∞}. Let K0 be an Iwahori subgroup and K0 be a Kummer sheaf pulled back from its quotient torus. To define (K∞ , K∞ ), we recall that the hypergeometric sheaf of rank n we are trying to get under the Langlands correspondence will have d slopes equal to 1/d (for some 1 ≤ d ≤ n), and n − d slopes equal to 0. Therefore it is natural to ask K∞ to contain P+ = P1/d for some parahoric subgroup with the first nontrivial filtration step 1/d, together with a bit n more to accommodate a Kummer sheaf on Gn−d m for the tame part. Let V = k and think of PGLn as PGL(V ). Choose a partial flag V● ∶

0 = V0 ⊂ V≤1 ⊂ V≤2 ⊂ ⋯V≤d = V

with Vi = V≤i /V≤i−1 nonzero. Fix a decomposition Vi = i ⊕ Vi′ (1 ≤ i ≤ d). Let P ⊂ L∞ G be the preimage of the parabolic PV● ⊂ PGL(V ) stabilizing the above partial flag. Then its Levi quotient LP ≅ (∏di=1 GL(Vi ))/ΔGm . Let P++ = P2/d . Then the next subquotient in the Moy-Prasad filtration has the form d

P+ /P++ ≅ ⊕ Hom(Vi , Vi−1 ) i=1

as an LP -module. Here V0 is understood to be Vd . Let ϕ ∶ P+ /P++ → Ga to be a linear function whose component ϕi ∶ Vi → Vi−1 sends the line i isomorphically to i−1 , and is zero on Vi′ . Let Lϕ be the stabilizer of ϕ under LP ; then Lϕ ≅ ∏i GL(Vi′ ). Let Bϕ ⊂ Lϕ be a Borel subgroup. Then its quotient torus Tϕ has dimension n − d. Finally, let K∞ be the preimage of Bϕ under the quotient P → LP . To define K∞ , we first extend ϕ∗ ASψ ∈ CS1 (P+ ) to a rank one character sheaf Kϕ on K∞ (which is possible because Bϕ fixes ϕ). Let Kρ be the pullback of any Kummer sheaf on Tn−d via K∞ → Bϕ → Tϕ . Finally take K∞ = Kϕ ⊗Kρ ∈ CS1 (K∞ ). The construction of (K∞ , K∞ ) in the wild case can be generalized to other types of groups, see Example 3.3.4. For a specific choice of the partial flag V● where the dimensions of Vi are distributed as evenly as possible, it was proved in [27] that (KS , KS ) is geometrically rigid.

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Example 3.3.2 (Airy automorphic datum). In the work of Kamgarpour, Jakob and Yi [22], a series of geometrically rigid examples with X = P1 and S = {∞} are introduced. When G = SL2 the corresponding local system, or rather its de Rham version, is the D-module given by the Airy equation. We explain the construction for G = SL2 . Let K∞ be the subgroup of I∞ consisting of the following matrices (where ai , bi , ci ∈ k) (

1 + a1 τ + ⋅ b0 + b1 τ + ⋅ ) b0 τ + c2 τ 2 + ⋯ 1 − a1 τ + ⋯

There is a homomorphism ϕ ∶ K∞ → k sending the above matrix to b1 + c2 (check this is a group homomorphism!). Let χ∞ be the pullback of a nontrivial additive character via ϕ. It is clear how to turn (K∞ , χ∞ ) into a geometric automorphic datum (K∞ , K∞ ). A more conceptual way of defining K∞ is the following. First take the MoyPrasad filtration I3/2 of I∞ and consider the homomorphism ϕ3/2 ∶ I3/2 → I3/2 /I2 ≅ sum k2 → k. We may ask if ϕ3/2 can be extended to a larger subgroup of I + = I1/2 . First ϕ3/2 can be extended to ϕ1 ∶ I1 → k by letting it to be trivial on the diagonal matrices. Next, we may view ϕ1 as an element (not uniquely determined) in the dual of the Lie algebra g((τ )), which can be identified with g((τ )) dτ using the Killing τ −2 0 τ form and the residue pairing. We choose ϕ ̃1 = ( −1 ) dτ . Let H(F∞ ) be the τ τ 0 centralizer of ϕ ̃1 under the adjoint action of G(F∞ ). We see that H(F∞ ) = {(

a bτ

b ) ∣a2 − b2 τ = 1, a, b ∈ F∞ } . a

In fact b ∈ O∞ and a ∈ 1 + τ O∞ . Finally K∞ = H(F∞ )I1 . Example 3.3.3 (Epipelagic automorphic data). In [42] we give generalizations of Kloosterman sheaves. Let X = P1 and S = {0, 1}. Let P∞ ⊂ L∞ G be a parahoric subgroup of specific types. These parahoric subgroups are singled out by Reeder and Yu [34] to define their epipelagic representations. Their conjugacy classes are in bijection with regular elliptic conjugacy classes in the Weyl group W . Consider + ++ the next step P++ ∞ = P∞,2/m in the Moy-Prasad filtration of P∞ , and VP ∶= P∞ /P∞ is a finite-dimensional representation of the Levi LP . We take K∞ = P+∞ . To define K∞ , we pick a linear function ϕ ∶ VP → k satisfying the stability condition: it has closed orbit and finite stabilizer under LP∞ . Define K∞ to be the pullback of ASψ under the composition ϕ

K∞ = P+∞ ↠ VP  → Ga . For K0 , we take it to be the parahoric subgroup P0 ⊂ L0 G opposite to P∞ : the intersection P0 ∩ P∞ is a common Levi subgroup LP of both. We can take any K0 ∈ CS1 (LP ). The geometric automorphic datum (KS , KS ) is geometrically rigid. To check this, one has to use the stability property of ϕ. One can let φ varies in an open subset of the dual space of VP and get a family ̂ version of geometric automorphic datum. The resulting G-local systems “glue” ̂ together to give a G-local system over an open subset of VP∗ × (P1 − {0, ∞}). Example 3.3.4 (Euphotic automorphic data, [23]). This is a further generalization of the epipelagic automorphic data that weakens the assumption that ϕ ∈ VP∗ is stable. Let X = P1 and S = {0, ∞}. Assume G is simply-connected.

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We start with any parahoric P∞ and any linear function ϕ on VP = P+∞ /P++ ∞ with a closed orbit under LP . Let Lϕ be the stabilizer of ϕ in LP , and let Bϕ ⊂ Lϕ be a Borel subgroup. Then K∞ = P+∞ Bϕ . We take K∞ to be the tensor product of ϕ∗ ASψ (extended to a character sheaf on K∞ by letting it be trivial on Bϕ ) and a Kummer sheaf Kχ from the quotient torus of Bϕ . At 0 we take K0 to be a parahoric subgroup contained in the parahoric P0 opposite to P∞ . It corresponds to the choice of a parabolic subgroup Q ⊂ LP . We require that (3.2)

The group Bϕ acts on LP /Q with an open orbit with finite stabilizer.

This is to match the numerical condition dim BunG (KS ) = 0. Let K0 be trivial. Such automorphic datum (KS , KS ) is not guaranteed to be weakly rigid. In [23], for P∞ = G(O∞ ), we give a complete list of pairs (ϕ, Q) (where ϕ ∈ g∗ is semisimple, and Q ⊂ G is a parabolic subgroup) satisfying the condition (3.2). In [23] we prove that in these cases, for a generic choice of the Kummer sheaf Kχ on Bϕ , (KS , KS ) is weakly rigid, but not always geometrically rigid. Example 3.3.5. We resume the setup of Example 3.3.4. We give a series of examples of euphotic automorphic data that are weakly rigid but not geometrically rigid. Consider the case G = SO(V ) with dimk V = 2n + 1 and P∞ = G(O∞ ). Write V = I ⊕ k ⋅ e0 ⊕ I ∗ where I is a maximal isotropic subspace of V and (e0 , e0 ) = 1. Let ϕ ∈ VP∗∞ ≅ g be the element idI ⊕0⊕(−idI ∗ ). Then the centralizer Gϕ = Lϕ ≅ GL(I). Choose a Borel subgroup Bϕ ⊂ Gϕ = GL(I) and denote it by BI , so that K∞ = G(O∞ )+ BI . Let K0 ⊂ G(O0 ) be the preimage of the parabolic subgroup QI ⊂ G, the stabilizer of I. Then BunG (K∞ , K0 ) contains an open substack isomorphic to BI /G/QI (this is the locus where the underlying G-bundle is trivial). For E in this open substack, the local system ϕ∗ ASψ is always trivial on the image of ev∞,E ∶ Aut(E) → K∞ . So E is relevant if and only if Kχ restricts to the trivial local system on the image of ev∞,E . In particular, E is relevant if Aut(E)○ is unipotent. If E corresponds to a point gQI ∈ G/QI , then Aut(E) is the stabilizer of gQI under the left translation by BI . So we should look for points on G/QI with unipotent stabilizers under BI . Assume n = 2m is even in the following discussion. The partial flag variety G/QI classifies n-dimensional isotropic subspaces of V . Consider the “big cell” Y ⊂ G/QI consisting of isotropic J ⊂ V such that J projects isomorphically to I. Then Y ≅ I ∗ ⊕ ∧2 (I ∗ ). There is an open BI -orbit O ⊂ ∧2 (I ∗ ). We will exhibit 2m BI -orbits on I ∗ × O with unipotent stabilizers, and hence all of them are relevant for the euphotic automorphic datum (KS , χS ). The Borel subgroup BI is the stabilizer of a flag 0 ⊂ I1 ⊂ ⋯ ⊂ In−1 ⊂ In = I. A point ω ∈ O ⊂ ∧2 (I ∗ ) is a symplectic form on I that is non-degenerate on I2i for each i = 1, ⋯, m. Fix such an ω, and we get an orthogonal decomposition I = N1 ⊕ N2 ⊕ ⋯ ⊕ Nm under ω with dim Ni = 2 such that I2i = N1 ⊕ ⋯ ⊕ Ni . Each Ni has a line i ⊂ Ni that is the image of I2i−1 . We have StabBI (ω) ≅ ∏m i=1 SL(Ni ). ∗ ∗ SL(N )-orbits on I = ⊕m The BI -orbits on I ∗ × O are in bijection with ∏m i i=1 i=1 Ni . ∗ ∗ Let  ∶ {1, ⋯, m} → {0, 1} be any map. Let I ⊂ I be the set of vectors (ξi )1≤i≤m with ξi ∈ Ni∗ − {0}, such that ξi (i ) = 0 if (i) = 0 or ξi (i ) ≠ 0 if (i) = 1. Then I∗ is a ∏m i=1 SL(Ni )-orbit with unipotent stabilizer, which corresponds to a BI -orbit on I ∗ × O with unipotent stabilizer.

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Example 3.3.6. In [41], we constructed a geometrically rigid automorphic datum for X = P1k and S = {0, 1, ∞} for groups G of type A1 , D2n , E7 , E8 , G2 that only involve mutiplicative characters. One interesting observations is that the construction makes sense over any field k with char(k) ≠ 2. In particular, in [41] it was applied to k = Q to obtain the first examples of motives with E7 and E8 as motivic Galois groups, and to solve the inverse Galois problem for E8 (F ) for sufficiently large prime . Assume G is simply-connected, and the longest element w0 in the Weyl group W of G acts by inversion on T . Up to conjugacy, there is a unique parahoric subgroup P ⊂ L0 G such that its reductive quotient LP is isomorphic to the fixed point subgroup Gτ of a Cartan involution corresponding to the split real form of G. For example, we can take P to be the parahoric subgroup corresponding to the facet containing the element ρ∨ /2 in the T -apartment of the building of L0 G (ρ∨ is half the sum of positive coroots of G). The Dynkin diagram of the reductive quotient LP ≅ Gτ of P is obtained by removing one or two nodes from the extended Dynkin diagram of G. We tabulate the type of LP and the nodes to be removed in each case. G B2n B2n+1 Cn D2n E7 E8 F4 G2

LP Bn × Dn Bn × Dn+1 An−1 × Gm Dn × Dn A7 D8 A1 × C3 A1 × A1

nodes to be removed the (n + 1)-th counting from the short node the (n + 1)-th counting from the short node the two ends the middle node the end of the leg of length 1 the end of the leg of length 2 second from the long node end middle node

Fact: If G is not of type C, then Lsc P → LP is a double cover. Even if G is of type Cn , LP ≅ GLn still admits a unique nontrivial double cover. ̃ P → LP . Therefore, in all cases, there is a canonical nontrivial double cover v ∶ L In particular, (3.3)

(v! Q )sgn ∈ CS1 (LP )

(here sgn denotes the nontrivial character of ker(v)(k) = {±1}). Let K0 = P ⊂ L0 G be a parahoric subgroup of the type defined above. Let K0 be the pullback of the local system (3.3). Let K∞ = P∞ ⊂ L∞ G be the parahoric subgroup of the same type as P. Let K1 = I1 ⊂ L1 G be an Iwahori subgroup. Let K∞ and K1 be trivial. The central character compatible with (KS , χS ) above is unique, and it exists if and only if χ∣Z(k) = 1, which can always be achieved by passing to a quadratic extension k′ /k. In [41], it is shown that if G is either simply-laced or of type G2 (again assume w0 = −1, so G is of type A1 , D2n , E7 , E8 and G2 ), then the geometric automorphic datum (KS , KS ) defined above is weakly rigid. Indeed, there is a unique relevant k-point in BunG (KS ) which is the unique open point. The moduli stack BunG (P0 , P∞ ) contains an open substack isomorphic to pt/LP . The preimage of pt/LP in BunG (KS ) is isomorphic to LP /G/B. Since

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LP is a symmetric subgroup of G, it acts on the flag variety of G with an open orbit O ⊂ G/B. We thus get an open point j ∶ LP /O ↪ BunG (KS ). When G is either simply-laced or of type G2 , this turns out to be the unique relevant point. The stabilizer Au of LP on any point u ∈ O is canonically isomorphic to T [2]. The category Dc (KS , KS ) is more interesting. Taking the preimage of Au in the ̃P we get a central extension double cover L ̃u → T [2] → 1. 1 → μ2 → A ̃u is a finite Heisenberg 2-group with For simplicity let’s assume G = E8 . Then A μ2 as the center. It has a unique irreducible Q -representation V with nontrivial central character. This representation gives an irreducible local system on the preimage of the open point LP /O in BunG (K+S ), whose extension to BunG (K+S ) by zero gives an object F ∈ Dc (k; KS , KS ). It turns out F is the unique simple perverse sheaf in Dc (k; KS , KS ). ̃u is larger. One can construct a When G is not of type E8 , the center of A ̃u whose restriction to μ2 is simple perverse sheaf for each central character of A nontrivial. 4. Computing eigen local systems ̂ In this section we explain how to get G-local systems out of rigid automorphic data. 4.1. From eigenforms to local systems. We give a very brief review of how Langlands correspondence works over function fields in the automorphic to Galois direction. 4.1.1. Eigenvalues. Given a Hecke eigenform f , then for x ∈ ∣X∣ − S (where S is a finite set of places), we have a character σx ∶ HG(Ox ) → Q of the spherical Hecke algebra HG(Ox ) such that f ⋆ hx = σx (hx )f,

∀hx ∈ HG(Ox ) .

We recall the Satake isomorphism: HG(Ox ) ≅ Q [X∗ (T )]W . Each homomorphism σx ∶ HG(Ox ) → Q gives rise to a W -orbit on T̂, which is the ̂ What property does the same datum as a semisimple conjugacy class [φx ] ∈ G. collection {φx }x∈∣X∣−S satisfy? ̂ = SL2 , σx is determined by its value In the simplest case G = PGL2 and G t 0 at the characteristic function of G(Ox ) ( x ) G(Ox ); this value is equal to the 0 1 number ax = Tr(φx ). The {ax } is the function field analogues of Fourier coefficients ap for a Hecke eigen modular form for PGL2 over Q. Can we get arbitrary collection of traces {ax }x∈∣X∣−S from eigenforms? The Langlands correspondence predicts that the collection {φx } of conjugacy ̂ must come from a single global object, namely a G-local ̂ classes in G system on X − S.

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4.1.2. Local systems. A rank n Q -local system on U for the ´etale topology is slightly technical to define: it arises from an inverse system of locally constant Z/m Z sheaves on U free of rank n. Let Loc(U, Q ) be the tensor category of Q -local systems on U of varying rank. This is a tensor category linear over Q . Let u ∈ U be a geometric point u ∈ U . We have the ´etale fundamental group π1 (U, u), which is a profinite group. A rank n local system E on U with Q coefficients gives rise to a continuous representation of π1 (U, u) on a n-dimensional Q -vector space (with -adic topology), called the monodromy representation of E. This representation is constructed by looking at the action of π1 (U, u) on the stalk Eu . The above discussion gives an equivalence of tensor categories (4.1)

∼

ωu ∶ Loc(U, Q ) → Repcont (π1 (U, u), Q ).

Note that π1 (U, u) is a quotient of the absolute Galois group Gal(F sep /F ) of the function field F = k(X). Therefore, a rank n local system on U gives rise to an n-dimensional continuous representation of Gal(F sep /F ). ̂ ̂ 4.1.3. G-local systems. With the choice of a geometric point u ∈ U , a G-local system on U is a continuous homomorphism (4.2)

̂  ). ρ ∶ π1 (U, u) → G(Q

Therefore, a GLn -local system is the same datum as a rank n local system on U . A more canonical definition can be given using the Tannakian formalism. Let ̂ Q ) be the tensor category of algebraic representations of G ̂ on finiteRep(G, ̂ dimensional Q -vector spaces. Then a G-local system on U is the same datum as a tensor functor (4.3)

̂ Q ) → Loc(U, Q ). E ∶ Rep(G,

̂ The two notions of G-local systems are equivalent. Given a representation ρ ̂ Q ) the local as in (4.2), we define a tensor functor E by assigning to V ∈ Rep(G, system with monodromy representation ρ ̂  ) → GL(V ). ρV ∶ π1 (U, u)  → G(Q

Conversely, given a tensor functor E as in (4.3), using the equivalence (4.1), E ̂ Q ) → Repcont (π1 (U, u), Q ). Tannakian can be viewed as a tensor functor Rep(G, formalism [8] then implies that such a tensor functor comes from a group homomorphism ρ as in (4.2), well-defined up to conjugacy. ̂ ̂ = SOn , a G-local system on U is the same thing as a Example 4.1.4. For G ∼ rank n local system E on U together with a self-adjoint isomorphism b ∶ E → E ∨ . Below we will get more technical because we will be using sheaf theory on ̂ algebraic stacks such as BunG and its Hecke correspondences to compute the G-local systems attached to an eigenform. The ideas come from the geometric Langlands program originated from the works of Drinfeld, Deligne, Laumon, etc. The main result is Theorem 4.3.3, which roughly says that for rigid automorphic datum, one can construct the corresponding local system explicitly.

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4.2. The Satake category. The Satake category is an upgraded version of the spherical Hecke algebra HG(Ox ) under the sheaf-to-function correspondence. We refer to Zhu’s article [45] for a comprehensive introduction. Let LG and L+ G be the group objects over k whose R-points are G(R((t))) and G(Rt) respectively. The quotient Gr = LG/L+ G is called the affine Grassmannian of G, and it is an infinite union of projective schemes. Then L+ G acts on Gr via left translation. The L+ G-orbits on Gr are indexed by dominant coweights λ ∈ X∗ (T )+ . The orbit containing the element tλ ∈ T (k((t))) is denoted by Grλ and its closure is denoted by Gr≤λ . We have dim Grλ = ⟨2ρ, λ⟩, where 2ρ is the sum of positive roots in G. Each Gr≤λ is a projective but usually singular variety over k. We denote the intersection complex of Gr≤λ by ICλ . Let Satk = Perv(L+ G)k (Grk ) be the category of (L+ G)k -equivariant perverse sheaves on Grk = Gr ⊗k k (with Q -coefficients) supported on finitely many (L+ G)k orbits. In [31], [16] and [33], it was shown that when k is algebraically closed, Satk carries a natural tensor structure, such that the global cohomology functor h = H ∗ (Grk , −) ∶ Satk → Vec(Q ) is a fiber functor. It is also shown that the ̂ The Tannakian group of the tensor category Satk is the Langlands dual group G. Tannakian formalism gives the geometric Satake equivalence of tensor categories ̂ Q ). Sat ≅ Rep(G, k

Similarly we define Satk = PervL+ G (Gr); this is also a tensor category with ̂ (one can show fiber functor H ∗ (Grk , −), but its Tannakian group is larger than G ̂ × Gal(k/k)). Define Sat0k ⊂ Satk its Tannakian group is the algebraic envelope of G to be the full subcategory consisting of direct sums of the normalized ICλ ’s whose restriction to Grλ is the Tate twisted sheaf Q [⟨2ρ, λ⟩](⟨ρ, λ⟩) (it requires choosing × a square root of q in Q if ρ ∉ X∗ (T )). It turns out that Sat0k is closed under the tensor structure, and we have an equivalence of tensor categories ̂ Q ). Sat0 ≅ Rep(G, k

̂ Q ), we denote the corresponding object in Sat0k by ICV . For V ∈ Rep(G, Example 4.2.1. For G = GLn , Gr(k) parametrizes O = kt-lattices in k((t))n . For λ = (1, ⋯, 1, 0, ⋯, 0) with i 1’s and n − i 0’s, Grλ = Gr≤λ ≅ Gr(n, i). In fact, Grλ (k) consists of lattices Λ such that t−1 O n ⊃ Λ ⊃ O n and dimk (Λ/O n ) = i. Example 4.2.2. For G = Sp(V ), V a symplectic space of dimension 2n, Gr(k) parametrizes O = kt-lattices Λ in V ⊗k((t)) such that the symplectic form restricts to a perfect pairing on Λ (these are called self-dual lattices). Let Λ0 = V ⊗ O be the standard self-dual lattice. For λ = (1, 0, ⋯, 0) ∈ Zn , Grλ (k) consists of self-dual lattices Λ such that dimk (Λ/Λ ∩ Λ0 ) = 1. The orbit closure Gr≤λ = Grλ ∪ {Λ0 }. There is a map Grλ → P(V ) sending Λ to the line in V that is the image of Λ mod t−1 ⊂ t−1 V . The fiber over a point  ∈ P(V ) is the line Hom(, V /- ) ≅ ⊗−2 (where - is defined using the symplectic form on V ). Therefore Grλ is isomorphic to the total space of O(2) over P2n−1 . Exercise 4.2.3. For G = SO(V ) with dim V = 2n or 2n + 1, describe Grλ for λ = (1, 0, ⋯, 0) ∈ Zn . 4.3. Geometric Hecke operators. We are going to apply a sheaf-theoretic version of Hecke operators. The starting case is when G = GLn and the automorphic forms are everywhere unramified, for which we refer to [13, 3.2].

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We consider the situation of §2.5.1. In particular, we have a geometric automorphic datum (KS , KS ), and moduli stacks BunG (KS ) and BunG (K+S ). 4.3.1. Hecke correspondence. Consider the following diagram (4.4)

HkG (K+S ) NNN / NNN→ h NNN NN' BunG (K+S ) π

← / ppp h pp

ppp xppp BunG (K+S )

/ U ∶= X − S

Here, the stack HkG (K+S ) classifies the data (x, E, E ′ , τ ) where x ∈ U ∶= X − S, ∼ E, E ′ ∈ BunG (K+S ) and τ ∶ E∣X−{x} → E ′ ∣X−{x} is an isomorphism of G-torsors over ←   → X − {x} preserving the K+x -level structures at each x ∈ S. The morphisms h , h and π send (x, E, E ′ , τ ) to E, E ′ and x respectively. For x ∈ U , we denote its preimage under π by HkG,x (K+S ). We have an evaluation morphism (4.5)

evx ∶ HkG,x (K+S ) → L+x G/Lx G/L+x G.

In fact, for a point (x, E, E ′ , τ ) ∈ HkG,x (K+S ), if we fix trivializations of E and E ′ over SpecOx , the isomorphism τ restricted to SpecFx is an isomorphism between the trivial G-torsors over SpecFx , hence given by a point gτ ∈ Lx G. Changing the trivializations of E∣SpecOx and E ′ ∣SpecOx will result in left and right multiplication of gτ by elements in L+x G. Therefore we have a well-defined morphism evx as above between stacks. As x moves along U , we may identify the target of (4.5) as L+ G/LG/L+ G by choosing a local coordinate t at x. Modulo the ambiguity caused by the choice of the local coordinates, we obtain a well defined morphism (4.6)

ev ∶ HkG (K+S ) → [

L+ G/LG/L+ G ], Aut+

where Aut+ is the group scheme over k of continuous ring automorphisms of kt, and it acts on LG and L+ G via its action on kt. ̂ Q ), the corre4.3.2. Geometric Hecke operators. For each object V ∈ Rep(G, sponding object ICV ∈ Sat under the geometric Satake equivalence defines a complex + + G ]. The geometric Hecke operator associated with on the quotient stack [ L G/LG/L Aut+ V is the functor TV ∶ D(LS ,KS ) (BunG (K+S ) × U ) → D(LS ,KS ) (BunG (K+S ) × U )  → ←  F ↦ ( h × π)! (( h × π)∗ F ⊗ ev∗ ICV )) . The composition of these functors are compatible with the tensor structure of Sat: there is a natural isomorphism of functors ̂ Q ) TV ○ TW ≅ TV ⊗W , ∀V, W ∈ Rep(G, which is compatible with the associativity constraint of the tensor product in ̂ Q ) and the associativity of composition of functors TV ○ TV ○ TV in the Rep(G, 1 2 3 obvious sense. Theorem 4.3.3 ([23, Corollary A.4.2], imprecise statement). Let (KS , KS ) be a geometrically rigid geometric automorphic datum. Then one can decompose Pervc (k; KS , KS ) (perverse sheaves in Dc (k; KS , KS )) into a finite direct sum of

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subcategories stable under the geometric Hecke operators, such that each direct sum̂ mand gives rise to a G-local system on U . ̂ on Pervc (k; KS , KS ) The proof uses the structure of an E2 -action of Rep(G) that “spreads over” the curve U . Such a structure comes from versions of geometric Hecke operators modifying at two or more points on U . A special case is when Pervc (k; KS , KS ) has a single simple object F (which we may assume is defined over k). In this case, Theorem 4.3.3 says the following: ̂ Q ), there is F is a Hecke eigensheaf in the following sense. For every V ∈ Rep(G, an isomorphism ϕV ∶ TV (F ⊠ Q ) ≅ F ⊠ EV for some local system EV ∈ Loc(U, Q ). Moreover the assignment V ↦ EV gives a ̂ ̂ Q ) → Loc(U ), hence a G-local system E. tensor functor E ∶ Rep(G, 4.4. Computing local systems – a simple case. Fix a geometric automorphic datum (KS , KS ). 4.4.1. Assumptions. We assume G is simply-connected and (KS , KS ) is geometrically rigid. Since BunG (KS ) is connected, it has a unique relevant point E. We further assume that Aut(E) is trivial. This implies that E is an open point in BunG (KS ). 4.4.2. Let G be the group ind-scheme over U whose fiber over x ∈ U is the automorphism group of E∣X−{x} as a G-torsor with KS -level structures. The evaluation map along S (well-defined up to conjugacy) can be extended to G: evS,E ∶ G → ∏ Kx ⊗ k → LS ⊗ k. x∈S

For any dominant coweight λ ∈ X∗ (T )+ , let G≤λ ⊂ G be the closed subscheme whose fiber over x ∈ U are those automorphisms of E∣X−{x} with modification type ≤ λ at x. Thus G≤λ x can be identified with an open subscheme of Gr≤λ . Proposition 4.4.3. Under the assumption that G is simply-connected and Aut(E) is trivial, Pervc (k; KS , KS ) contains a unique simple object F, and it is ̂ a Hecke eigensheaf. The corresponding G-local system E is described as follows. For a dominant coweight λ ∈ X∗ (T )+ and the corresponding irreducible reprê we have sentation Vλ of G, EVλ ≅ π!λ (ev∗S,E KS ⊗ ICG≤λ [−1]) where π λ ∶ G≤λ → U is the projection (it contains the implicit statement that the right side is concentrated in degree zero). In particular, if λ is minuscule so that G≤λ = Gλ is smooth of relative dimension ⟨λ, 2ρ⟩ over U , then EVλ ≅ R⟨λ,2ρ⟩ π!λ ev∗S,E KS . 4.5. Computing local systems – general case. Now we drop the condition that G be simply-connected (still assumed to be split semisimple) and the automorphism groups of the relevant points be trivial. The discussion in the general case is a bit technical.

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×

4.5.1. Twisted representations. Let Γ be a group and ξ ∈ Z 2 (Γ, Q ) a cocycle such that ξ1,γ = ξγ,1 = 1 for all γ ∈ Γ. A ξ-twisted representation of Γ is a finitedimensional Q -vector space V with automorphisms Tγ ∶ V → V , one for each γ ∈ Γ, such that T1 = idV and Tγδ = ξγ,δ Tγ Tδ , ∀γ, δ ∈ Γ. Let Repξ (Γ, Q ) be the category of ξ-twisted representations of Γ. This is a Q linear abelian category which, up to equivalence, only depends on the cohomology × class [ξ] ∈ H2 (Γ, Q ). A natural source of 2-cocycles on Γ come from rank one character sheaves. We did not define rank one character sheaves for disconnected groups but the same definition works, except now they may have nontrivial automorphisms. Isomorphism × classes in CS1 (Γ) are in bijection with H2 (Γ, Q ). Suppose E ∈ BunG (KS )(k) be a relevant point for (KS , KS ). Let AE = Aut(E) and ΓE = π0 (Aut(E)). Since ev∗S, E KS is trivial on Aut○ (E), it descends to a rank ×

one character sheaf on ΓE and gives a cocycle ξ ∈ Z 2 (ΓE , Q ) satisfying ξ1,γ = ξγ,1 = 1 for all γ ∈ ΓE , whose cohomology class is well-defined. Lemma 4.5.2. Let Pervc (k; KS , KS )E be the category of perverse sheaves in Pervc (k; KS , KS ) that have vanishing stalks outside the preimage of the relevant point E. Then Pervc (k; KS , KS )E ≅ Repξ (ΓE , Q ). Sketch of proof. We base change the spaces to k without changing notãS → LS with discrete kernel C, and a tion. We can find a finite isogeny ν ∶ L × character χC ∶ C → Q such that the local system KS on LS is of the form KS ≅ (ν∗ Q )χC ∈ CS1 (LS ). ̃E = π0 (A ̃E ). ̃E be the pullback of the cover ν along evS,E ∶ AE → LS . Let Γ Let A Then ΓE fits into an exact sequence ̃E → ΓE → 1. 1→C →Γ ×

×

The pushout of the sequence along χC ∶ C → Q gives a 2-cocycle ξ ∈ Z 2 (ΓE , Q ) ̃E , so well-defined up to cobound(upon choosing a set-theoretic splitting s ∶ ΓE → Γ χC ̃ aries). Let Rep (ΓE , Q ) be the category of finite-dimensional Q -representations ̃E whose restriction to C is χC .. We have an equivalence of Γ ∼

̃E , Q ) → Repξ (ΓE , Q ) RepχC (Γ by restriction along s. ̃S /A ̃E where AE The preimage of E in BunG (K+S ) is isomorphic to LS /AE ≅ L acts on LS via evS,E and right translation. For V ∈ Repξ (ΓE , Q ) viewed as a ̃E with restriction χC to C, we get a L ̃S -equivariant local system representation of Γ ̃ ̃ ̃ ̃ FV on LS /AE ≅ LS /AE (since ΓE = π0 (AE )). Let i ∶ LS /AE ↪ BunG (KS , KS ) be the inclusion, then up to a shift i! FV is the object in Pervc (k; KS , KS )E corresponding to V .  4.5.3. General case. Let BunG (KS ) = ∐ BunG (KS )α α∈Ω

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be the decomposition into connected components. Here Ω = X∗ (T )/ZΦ∨ is the algebraic π1 of G. Let Dc (k; KS , KS )α ⊂ Dc (k; KS , KS ) be the full subcategory of sheaves supported on BunG (KS )α . Similarly define the abelian category of perverse sheaves Pervc (k; KS , KS )α . Let Eα be the unique relevant k-point in BunG (KS )α . Let Aα = Aut(Eα ) (an algebraic group over k), and Γα = π0 (Aα ) as a discrete group. Define β Gα → U whose fiber over x ∈ U parametrize isomorphisms of G-torsors ∼ Eα ∣X−{x} → Eβ ∣X−{x} preserving the KS -level structures. This is a right torsor under the group ind-scheme α Gα and a left torsor under β Gβ . In particular, Aα acts on β Gα from the right and Aβ acts from the left. Upon choosing trivializations of Eα and Eβ near S, we have the evaluation map α evS,β

∶ α Gβ → ∏ Kx ⊗ k → LS ⊗ k. x∈S

̃S , χC ) as in the proof of Lemma 4.5.2. Pulling back the covering Choose (L ̃ β → α Gβ and central extensions ̃ ν ∶ LS → LS along α evS,β we get C-torsors α G ̃α → Γα → 1. ̃α → Aα → 1 and 1 → C → Γ 1→C →A Then by Lemma 4.5.2 we have an equivalence of categories ̃α , Q ). (4.7) Pervc (k; KS , KS )α ≅ Repξ (Γα , Q ) ≅ RepχC (Γ α

̃α where the last category consists of finite-dimensional Q -representations of Γ whose restriction to C is χC . We have an evaluation map α evU,β

∶ α Gβ → [

L+ G/LG/L+ G ] Aut+

+ recording the modification at x ∈ U . Let α G≤λ β be the preimage of L G/Gr≤λ under the above map, then α G≤λ β ≠ ∅ only when λ has image β − α ∈ Ω. The pullback ∗ ≤λ ̃ U,β be the precomposition of α evU,β with α evU,β ICλ is supported on α Gβ . Let α ev ∗ ̃ ̃ ≤λ . → G . We also have ev ̃ IC supported on α G G α β α β α U,β λ β ̃ β → U be the projection. Let Σα be the set of irreducible object πβλ ∶ α G Let α ̃ ̃α , Q ). For α ∈ Ω, let Fη ∈ Pervc (k; KS , KS )α corresponding to an in RepχC (Γ irreducible object η ∈ Σα under (4.7).

Proposition 4.5.4. Let (KS , KS ) be geometrically rigid. Then for a dominant coweight λ with image β − α in Ω, we have TVλ (Fη ⊠ Q ) ≅ ⊕ζ∈Σβ Fζ ⊠ ζ Eηλ where ζ Eηλ is the local system on U given by (4.8)

λ ζ Eη

λ ≅ HomΓ̃α ×Γ̃β (η ⊠ ζ ∨ , α ̃ πβ! ̃ ∗β ICλ )[−1]. α ev

If Aut(Eα ) is trivial for all α, there is a unique simple perverse sheaf Fα ∈ ̂ Dc (k; KS , KS )α . The sum of Fα is a Hecke eigensheaf whose eigen G-local system E is described as follows. For a dominant coweight λ with image β − α in Ω, we have λ (α ev∗S,β KS ⊗ α ev∗β ICλ )[−1]. EVλ ≅ α πβ! When λ is minuscule we have (4.9)

λ αev∗S,β KS . EVλ ≅ R⟨λ,2ρ⟩ α πβ!

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4.6. Kloosterman local systems. We apply the method above to the Kloosterman automorphic datum in Example 2.1.5 in the case G = PGLn . We con̂ = SLn . sider only the case λ = (1, 0, ⋯, 0), so Vλ is the standard representation of G ̂ = SLn -local system on U = P1 − {0, ∞} ≅ Gm . The So we are looking for an G answer turns out to be a classical object. 4.6.1. Calculation of G≤λ . The moduli stack BunGLn (KS ) classifies equivalence classes of (E, F ∗ , F∗ , ) where (1) E is a rank n vector bundle over P1 ; (2) F ∗ ∶ (0 = F n ⊂ F n−1 ⊂ ⋯ ⊂ F 0 = E∣0 ) is a complete flag of the fiber of E at 0; (3) F∗ ∶ (0 = F0 ⊂ F1 ⊂ ⋯ ⊂ Fn = E∣∞ ) is a complete flag of the fiber of E at ∞; (4) For 1 ≤ i ≤ n, vi ∈ Fi /Fi−1 is a nonzero vector. Let Pic∞ be the moduli space of line bundles on P1 together with a trivialization at ∞. We have a canonical isomorphism Pic∞ ≅ Z by taking degrees. Then Pic∞ acts on BunGLn (KS ) by tensoring, and BunG (KS ) = BunGLn (KS )/Pic∞ . The components of BunG (KS ) are parametrized by Z/nZ by taking deg E mod n. Let Ei be the unique relevant point of degree i ∈ ZnZ . Then we have the following description: (1) E0 = Oe1 ⊕ Oe2 ⊕ ⋯ ⊕ Oen , F i = Span{ei+1 , ⋯, en }, Fi = Span{e1 , ⋯, ei }, vi = ei mod Fi−1 . (2) E1 = Oe1 ⊕ Oe2 ⊕ ⋯ ⊕ O(1)en . Here O(1) is equipped with a trivialization at ∞, so we identify it with O({0}). Take F i = Span{ei , ⋯, en−1 } if i ≥ 1, Fi = Span{e1 , ⋯, ei }, and vi = ei mod Fi−1 . A point in G≤λ is an injective map of coherent sheaves f ∶ E0 → E1 preserving F ∗ , F∗ and {vi }. We may write f as a matrix (aij ) where aij ∈ k if i < n, and anj ∈ Γ(P1 , O(1)) = k ⊕ kτ . The fact that f preserve F ∗ , F∗ and {vi } imply (1) (aij )∣τ =0 is upper triangular with 1 on the diagonal. This implies (aij ) has the form ⎛ 1 ⎜ 0 ⎜ ⎜ 0 ⎝ a′n1 τ

∗ ∗ ⋱ ∗ 0 1 ⋯ a′n,n−1 τ

∗ ∗ ∗ 1 + a′nn τ

⎞ ⎟ ⎟ ⎟ ⎠

(2) Let a′ij = aij if i < n and let a′nj be as above. Then (a′ij ) sends the flag ei to Span{ei−1 , ⋯, en−1 } for i = 2, ⋯, n. This implies (aij ) has the form ⎛ 1 ⎜ 0 ⎜ ⎜ 0 ⎝ a0 τ

a1 ⋱ 0 ⋯

0 0 ∗ 0 1 an−1 0 1

⎞ ⎟ ⎟ ⎟ ⎠

Moreover, a0 , ⋯, an−1 are nonzero since they are the action of f on the associated graded of the filtrations F ∗ at 0. We get an isomorphism G≤λ ≅ Gnm with coordinates (a0 , ⋯, an−1 ). The map π ∶ G≤λ → U sends f to the support of the cokernel of f . Since det(f ) = 1 + (−1)n−1 a0 ⋯an−1 t−1 ∈ Γ(P1 , O(1)), we see that π(f ) = (−1)n a0 a1 ⋯an−1 .

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For simplicity assume K0 is trivial. By (4.9), we get that (St = Vλ is the ̂ = SLn ) standard representation of G ESt ≅ Rn−1 π! φ∗ ASψ

(4.10)

evS

ϕ

where we use φ to denote the composition G≤λ ≅ Gnm → ∏n−1 → Ga ; it is i=0 Uαi  linear in each coordinate ai with nonzero coefficient. 4.6.2. The classical Kloosterman sheaf. We first recall the definition of Kloosterman sums. Let p be a prime number. Fix a nontrivial additive character × ψ ∶ Fp → Q . Let n ≥ 2 be an integer. Then the n-variable Kloosterman sum over Fp is a function on F×p whose value at a ∈ F×p is Kln (p; a) =

∑

x1 ,⋯,xn ∈F× p ;x1 x2 ⋯xn =a

ψ(x1 + ⋯ + xn ).

These exponential sums arise naturally in the study of automorphic forms for GLn . Deligne [7] gave a geometric interpretation of the Kloosterman sum. He considered the following diagram of schemes over Fp

Gm

{ π {{{ { { }{ {

Gnm

BB BBϕ BB B A1

Here π is the morphism of taking the product and ϕ is the morphism of taking the sum. He defines the Kloosterman sheaf to be Kln ∶= Rn−1 π! ϕ∗ ASψ , over Gm = P1Fp − {0, ∞}. Up to a change of coordinates, this is essentially (4.10). The relationship between the local system Kln and the Kloosterman sum Kln (p; a) is explained by the following identity Kln (p; a) = (−1)n−1 Tr(Froba , (Kln )a ). Here Froba is the geometric Frobenius operator acting on the geometric stalk (Kln )a of Kln at a ∈ Gm (Fp ) = F×p . 4.6.3. Properties of Kloosterman local systems. The following properties of Kln were proved by Deligne. (0) Kln is a local system of rank n. (1) Kln is tamely ramified at 0, and the monodromy is unipotent with a single Jordan block. (2) Kln is totally wild at ∞ (i.e., the wild inertia at ∞ has no nonzero fixed vector on the stalk of Kln ), and the Swan conductor Sw∞ (Kln ) = 1. Applying a special case of Theorem 4.3.3 to the Kloosterman automorphic data, ̂ we get G-local systems KlĜ (χ, ϕ). In [21], we show that KlĜ (χ, ϕ) enjoy analogous properties as Kln . These properties were predicted by Gross [17], Frenkel–Gross [14]. For example, when χ = 1 we prove: (1) KlĜ (1, ϕ) is tame at 0, and a generator of the tame inertia maps to a ̂ regular unipotent element in G. (2) The local monodromy of KlĜ (1, ϕ) at ∞ is a simple wild parameter in the sense of Gross and Reeder [19, §5].

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5. Rigidity for local systems; Applications This section is likely not to be covered in the lectures. We compare the notion of rigidity for automorphic data and for local systems. We also mention some applications of the rigid automorphic data and some open problems. ̂ ̂ 5.1. Rigid G-local systems. We shall review the notion of rigidity for Ĝ = GLn . We assume the base field k to local systems, introduced by Katz [26] for G be algebraically closed in this subsection. Let X be a complete smooth connected algebraic curve over k. Fix an open subset U ⊂ X with finite complement S. 5.1.1. Physical rigidity. ̂ Definition 5.1.2 (extending Katz [26, §1.0.3]). Let E be an G-local system on ̂ U . Then E is called physically rigid if, for any other G-local system E ′ , E ′ ∣SpecFx ≅ E∣SpecFx for all x ∈ S implies E ′ ≅ E. Although the definition uses U as an input, the notion of physical rigidity is in fact independent of the open subset U : for any nonempty open subset V ⊂ U , ρ is rigid over U if and only if E∣V is rigid over V . Therefore, physical rigidity ̂  ) obtained by is a property of the Galois representation ρE ∶ Gal(F sep /F ) → G(Q restricting E to a geometric generic point η of the X. 5.1.3. Cohomological rigidity. Next we introduce cohomological rigidity. Let ̂ g ̂ and let Ad(E) be local system Êg , viewing ̂ be the Lie algebra of G, g as the adjoint ̂ Let j ∶ U ↪ X be the open embedding and j!∗ Ad(E) be the representation of G. non-derived direct image of Ad(E) along j. Concretely, the stalk of j!∗ Ad(E) at g. x ∈ S is the Ix -invariants on ̂ ̂ Definition 5.1.4 (extending Katz [26, §5.0.1]). A G-local system E on U is called cohomogically rigid, if τ (E) ∶= H1 (X, j!∗ Ad(E)) = 0. The vector space τ (E) does not change if we shrink U to a smaller open subset. Therefore cohomological rigidity is also a property of the Galois representation ̂  ). ρE ∶ Gal(F sep /F ) → G(Q Remark 5.1.5. When we work over the base field C and view U as a topological ̂ surface, one can define a moduli stack M of G-local systems over U with prescribed local monodromy around the punctures S. Then τ (E) is the Zariski tangent space of M at E. The condition τ (E) = 0 in this topological setting says that E does not admit infinitesimal deformations with prescribed local monodromy around S. This interpretation is the motivation for Definition 5.1.4. Remark 5.1.6. An alternative approach to define the notion of rigidity for a local system E over U over a finite field k is by requiring that the adjoint L-function of E to be trivial (constant function 1). This is the approach taken by Gross in [18]. When H0 (Uk , Ad(E)) = 0, triviality of the adjoint L-function of E is equivalent to cohomological rigidity of E. Using the Grothendieck-Ogg-Shafarevich formula, it is easy to give the following numerical criterion for cohomological rigidity: E is cohomologically rigid if and only if 1 ̂ − dim H0 (U, Ad(E)). (5.1) ∑ ax (Ad(E)) = (1 − gX ) dim G 2 x∈S

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Here ax (Ad(E)) is the Artin conductor of Ad(E) at x, see §3.2.3, and gX is the genus of X. ̂ From (5.1) we see that cohomologically rigid G-local systems exist only when gX ≤ 1. When gX = 1, rigid examples are very limited (see [26, §1.4]). Most examples of rigid local systems are over open subsets of X = P1 . In this case, if we further assume H0 (U, Ad(E)) = 0, then 1 ̂ ∑ ax (Ad(E)) = dim G. 2 x∈S Compare with (3.1), it is natural to expect that if E comes as the Hecke eigen local system of a weakly rigid geometric automorphic datum (Kx , Kx ), then 1 ? ax (Ad(E)) = [g(Ox ) ∶ kx ], ∀x ∈ S. 2 ̂ = SLn and the local system is irreducible, the two notions of rigidity When G are equivalent. Theorem 5.1.7. Let E be an irreducible rank n Q -local system on U = X − S with trivial determinant, viewed as an SLn -local system. (1) (Katz [26, Theorem 5.0.2]) If E is cohomologically rigid, then it is physically rigid. (2) (L.Fu [15, Theorem 0.1]) If E is physically rigid, then it is cohomologically rigid. 5.2. Applications of rigid automorphic datum. We shall give three applications of the rigid objects in the Langlands correspondence. 5.2.1. Local systems with exceptional monodromy groups. Katz [25] has constructed an example of a local system over P1Fp − {0, ∞} whose geometric monodromy lies in the exceptional group G2 and is Zariski dense there. This G2 -local system comes from a rank 7 rigid local system which is an example of Katz’s hypergeometric sheaves. The work [21], inspired by the work of Gross [17] and Frenkel–Gross [14], gives the first examples of local systems (coming from geometry) with Zariski dense monodromy in other exceptional groups F4 , E7 and E8 in a uniform way. The Zariski closure of the geometric monodromy of KlĜ (1, ϕ) is a connected simple ̂ of types given by the following table (assuming p ≠ 2, 3) subgroup of G ̂ G A2n A2n−1 , Cn Bn , Dn+1 (n ≥ 4) E7 E8 E 6 , F4 B3 , D4 , G2

geometric monodromy A2n Cn Bn E7 E8 F4 G2

5.2.2. Motives over number fields with exceptional motivic Galois groups. Although we have been working with a finite base field k in the previous discussions, the set up in Example 3.3.6 (where no additive character is used) in fact makes sense for any base field of characteristic not 2, and in particular for k = Q! This

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simple observation yields applications of rigid automorphic data to questions over number fields. In early 1990s, Serre asked the following question [36]: Is there a motive over a number field whose motivic Galois group is of exceptional type such as G2 or E8 ? A motive M over a number field K is, roughly speaking, part of the cohomology Hi (X) for some (smooth projective) algebraic variety X over K and some integer i, which is cut out by geometric operations (such as group actions). For each prime , the motive M has the associated -adic cohomology H (M ) ⊂ Hi (XK , Q ), which admits a Galois action: ρM, ∶ Gal(K/K) → GL(H (M )) The -adic motivic Galois group GM, of M is the Zariski closure of the image of ρM, . This is an algebraic group over Q . Since the motivic Galois groups that appear in the original question of Serre are only well-defined assuming the standard conjectures in algebraic geometry, we shall use the -adic motivic Galois group as a working substitute for the actual motivic Galois group (conjecturally they should be isomorphic to each other). Classical groups appear as motivic Galois groups of abelian varieties. This is why Serre raised the question for exceptional groups only. The G2 case was answered affirmatively by Dettweiler and Reiter [9]. In [41], we give an affirmative answer to the -adic version of Serre’s question for E7 , E8 and G2 in a uniform way. The key input is the rigid automorphic data in Example 3.3.6. We have remarked that the set up of Example 3.3.6 makese sense when the base field k is taken to be Q. The Hecke eigen local system attached to ̂ the rigid automorphic datum then gives a motivic G-local system on P1Q − {0, 1, ∞} ̂ = E7 , E8 and G2 . One proves that its monodromy is Zariski dense in G. ̂ for G To get motives over Q, we simply take a sufficiently general rational point ̂ x of P1 , and take the fiber of the motivic G-local system at x. Concretely, the resulting motive Mx is part of the intersection cohomology of the open subset of the subvariety G≤λ x of the affine Grassmannian introduced in §4.4. With a bit more work, one can also realize F4 as a motivic Galois group over Q (unpublished). We remark that in the case of E8 , no connections are known between E8 -motives and Shimura varieties. The example given by the rigidity method in [41] seems to be the only approach, even if one assumes knowledge about cohomology of Shimura varieties. Recently, E6 has also been realized as a motivic Galois group by Boxer– Calegari–Emerton–Levin–Madapusi-Pera–Patrikis [4]. 5.2.3. Inverse Galois Problem. The inverse Galois problem over Q asks whether every finite group can be realized as the Galois group of some Galois extension K/Q. The problem is still open for many finite simple groups, especially those of Lie type. The same rigid local systems over P1Q − {0, 1, ∞} constructed to answer Serre’s question can be used to solve new cases of the inverse Galois problem. We show in [41] that for sufficiently large primes , the finite simple groups G2 (F ) and E8 (F ) can be realized as Galois groups over Q. With a bit more work, one can show that F4 (F ) is also a Galois group over Q. In inverse Galois theory, people use the “rigidity method” to prove certain finite groups H are Galois groups over Q. This will be reviewed in the next subsection. In particular, the case of G2 (F ) for all primes  ≥ 5 was known by the work of Thompson [38] and Feit and Fong [12]. However, the case of E8 (F ) was known

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previously only for primes  satisfying a certain congruence condition modulo 31 (see the book of Malle and Matzat [32, II.10] for a summary of what was known before). The E8 -local system contructed in [41] also sheds some light to the rigidity method in inverse Galois theory. In fact, in [41, Conjecture 5.16] we suggest a rigid triple for E8 (F ), which was subsequently proved by Guralnick and Malle [20]. Their result shows that E8 (F ) is a Galois group over Q for all primes  > 7. 5.3. Rigidity in the inverse Galois theory. It is instructive to compare the notion of rigidity for local systems with the notion of a rigid tuple in inverse Galois theory. We give a quick review following [37, Chapter 8]. Let H be a finite group with trivial center. Definition 5.3.1. A tuple of conjugacy classes (C1 , C2 , ⋯, Cn ) in H is called (strictly) rigid, if ● The equation (5.2)

g1 g2 ⋯gn = 1 has a solution with gi ∈ Ci , and the solution is unique up to simultaneous H-conjugacy; ● For any solutions (g1 , ⋯, gn ) of (5.2), {gi }i=1,⋯,n generate H.

The connection between rigid tuples and local systems is given by the following theorem. Let S = {P1 , ⋯, Pn } ⊂ P1 (Q), and let U = P1Q − S. Theorem 5.3.2 (Belyi, Fried, Matzat, Shih, and Thompson). Let (C1 , ⋯, Cn ) be a rigid tuple in H. Then up to isomorphism there is a unique connected unramified Galois H-cover π ∶ Y → U ⊗Q Q such that a topological generator of the (tame) inertia group at Pi acts on Y as an element in Ci . Furthermore, if each Ci is rational (i.e., Ci takes rational values for all irreducible characters of H), then the H-cover Y → U ⊗Q Q is defined over Q. From the above theorem we see that the notion of a rigid tuple is an analog of physical rigidity for H-local systems when the algebraic group H is a finite group. Rigid tuples combined with the Hilbert irreducibility theorem solves the inverse Galois problem for H. Corollary 5.3.3. Suppose there exists a rational rigid tuple in H, then H can be realized as Gal(K/Q) for some Galois number field K/Q. For a comprehensive survey of finite simple groups that are realized as Galois groups over Q using rigidity tuples, we refer the readers to the book [32] by Malle and Matzat. 5.4. Further directions. 5.4.1. De Rham point of view. There is a well-known analogy between -adic local systems and connections (i.e., de Rham version of local systems) on algebraic curves. The rigidity method can be adapted to the de Rham setting by working ̂ with D-modules on various moduli stacks over C, and one gets G-connections on X − S as eigen-connections from Hecke eigen D-modules. In [14], the connection version of the Kloosterman local system was first constructed. In [44], it is shown that the D-module analogue of Hecke eigensheaf construction in [21] indeed yields the Frenkel-Gross connection. There are further works [6] and [39] in this direction.

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5.4.2. Checking rigidity. Typically, checking an automorphic datum is weakly rigid involves going over certain double cosets (indexed by the affine Weyl group or its cosets) and computing Aut(E) for many E. In [24], we will give a simple criterion for weak rigidity inspired by the microlocal geometry of BunG (KS ). Using this criterion, one can check that most of the euphotic automorphic data are weakly rigid. 5.4.3. Classification of rigid automorphic datum. Katz [26] gave an algorithmic classification of tamely ramified rigid rank n local systems on a punctured P1 for all n. Arinkin [1] extended Katz’s algorithm to rigid connections with irregular singularities (de Rham version of wildly ramified rigid local systems). It would be desirable to give a classification of rigid automorphic datum for G = PGLn . 5.4.4. Computing monodromy. Give a rigid automorphic datum one can make predictions on the local monodromy of its Langlands parameter following the principles in §3.2.4. Verifying these predictions is only done in a small number of cases such as the Kloosterman automorphic data [21], and the tame ramification of the epipelagic automorphic data [42]. For example, proving the prediction on the slopes of the epipelagic automorphic data is an open question. On the other hand, the global monodromy of the Langlands parameters ̂ is unknown for many rigid (Zariski closure of the image of π1geom (X − S) in G) automorphic data.

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Selected Published Titles in This Series 279 Ellen Elizabeth Eischen, Wee Teck Gan, Aaron Pollack, and Zhiwei Yun, Automorphic Forms Beyond GL2 , 2024 277 George Metcalfe, Francesco Paoli, and Constantine Tsinakis, Residuated Structures in Algebra and Logic, 2023 276 Bal´ azs B´ ar´ any, K´ aroly Simon, and Boris Solomyak, Self-similar and Self-affine Sets and Measures, 2023 275 Alekos Vidras and Alain Yger, Multidimensional Residue Theory and Applications, 2023 274 Willi Freeden and M. Zuhair Nashed, Recovery Methodologies: Regularization and Sampling, 2023 273 Gennadiy Feldman, Characterization of Probability Distributions on Locally Compact Abelian Groups, 2023 272 Tadashi Ochiai, Iwasawa Theory and Its Perspective, Volume 1, 2023 271 Hideto Asashiba, Categories and Representation Theory, 2022 270 Leslie Hogben, Jephian C.-H. Lin, and Bryan L. Shader, Inverse Problems and Zero Forcing for Graphs, 2022 269 Ralph S. Freese, Ralph N. McKenzie, George F. McNulty, and Walter F. Taylor, Algebras, Lattices, Varieties, 2022 268 Ralph S. Freese, Ralph N. McKenzie, George F. McNulty, and Walter F. Taylor, Algebras, Lattices, Varieties, 2022 267 Dragana S. Cvetkovi´ c Ili´ c, Completion Problems on Operator Matrices, 2022 266 Kate Juschenko, Amenability of Discrete Groups by Examples, 2022 265 Nabil H. Mustafa, Sampling in Combinatorial and Geometric Set Systems, 2022 264 J. Scott Carter and Seiichi Kamada, Diagrammatic Algebra, 2021 263 Vugar E. Ismailov, Ridge Functions and Applications in Neural Networks, 2021 262 Ragnar-Olaf Buchweitz, Maximal Cohen–Macaulay Modules and Tate Cohomology, 2021 261 Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman, Asymptotic Geometric Analysis, Part II, 2021 260 Lindsay N. Childs, Cornelius Greither, Kevin P. Keating, Alan Koch, Timothy Kohl, Paul J. Truman, and Robert G. Underwood, Hopf Algebras and Galois Module Theory, 2021 259 William Heinzer, Christel Rotthaus, and Sylvia Wiegand, Integral Domains Inside Noetherian Power Series Rings, 2021 258 Pramod N. Achar, Perverse Sheaves and Applications to Representation Theory, 2021 257 Juha Kinnunen, Juha Lehrb¨ ack, and Antti V¨ ah¨ akangas, Maximal Function Methods for Sobolev Spaces, 2021 256 Michio Jimbo, Tetsuji Miwa, and Fedor Smirnov, Local Operators in Integrable Models I, 2021 255 Alexandre Boritchev and Sergei Kuksin, One-Dimensional Turbulence and the Stochastic Burgers Equation, 2021 254 Karim Belabas and Henri Cohen, Numerical Algorithms for Number Theory, 2021 253 Robert R. Bruner and John Rognes, The Adams Spectral Sequence for Topological Modular Forms, 2021 252 Julie D´ eserti, The Cremona Group and Its Subgroups, 2021 251 David Hoff, Linear and Quasilinear Parabolic Systems, 2020 250 Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters, 2020

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The Langlands program has been a very active and central field in mathematics ever since its conception over 50 years ago. It connects number theory, representation theory and arithmetic geometry, and other fields in a profound way. There are nevertheless very few expository accounts beyond the GL2 case. This book features expository accounts of several topics on automorphic forms on higher rank groups, including rationality questions on unitary group, theta lifts and their applications to Arthur’s conjectures, quaternionic modular forms, and automorphic forms over functions fields and their applications to inverse Galois problems. It is based on the lecture notes prepared for the twenty-fifth Arizona Winter School on “Automorphic Forms Beyond GL2”, held March 5–9, 2022, at the University of Arizona in Tucson. The speakers were Ellen Eischen, Wee Teck Gan, Aaron Pollack, and Zhiwei Yun. The exposition of the book is in a style accessible to students entering the field. Advanced graduate students as well as researchers will find this a valuable introduction to various important and very active research areas.

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