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Gelfand Pairs and Invariant Distributions Weizmann Institute of Science Faculty of Mathematics and Computer Science Gelfand pairs and Invariant Distributions Thesis submitted for the degree \Doctor of Philosophy" by Avraham Aizenbud Submitted to the Scientific Council of the Weizmann Institute of Science Rehovot, Israel Prepared at the Mathematics Department under the supervision of Professor Vladimir Berkovich and Professor Joseph Bernstein July 2010 To my beloved wife This work was carried out under the supervision of Professor Vladimir Berkovich and Professor Joseph Bernstein 3 Acknowledgements First of all, I would like to thank my advisors Vladimir Berkovich and Joseph Bern- stein for guiding me through my PhD studies. I have been learning from Joseph Bern- stein since the age of 15. During this period, I have learned from him most of the mathematics I know. In addition, he has shown me his special approach to mathemat- ics. I hope that I have managed to comprehend at least part of it. When I started my PhD studies in the Weizmann Institute I have gained an additional supervisor Vladimir Berkovich. He has taken a great care of me and taught me his unique and fascinating approach to mathematics. I am deeply grateful to Dmitry Gourevitch who has been my friend and collab- orator for already more than a decade. The overwhelming majority of my mathematical studies and research were done in a close collaboration with him. In particular, a big part of this thesis is based on joint work with him. I would also like to thank the rest of my collaborators Nir Avni, Herve Jacquet, Erez Lapid, Stephen Rallis, Eitan Sayag, Gerard Schiffmann, Oded Yacobi, Frol Zapolsky from whom I have learned a lot. During the years of my mathematical studies I have benefitted from talks with many people. I cannot list all of them, but I would like to thank Semyon Alesker, Moshe Baruch, Paul Biran, Lev Buhovsky, Patrick Delorme, Gerrit van Dijk, Yuval Flicker, Stephen Gelbart, Maria Gorelik, Antony Joseph ,David Kazh- dan, Vadim Kosoy, Gady Kozma, Matthias Kreck, Bernhard Kroetz, Omer Offen, Leonid Polterovich, Lev Radzivilovsky, Andrey Reznikov, Siddhartha Sahi, Binyong Sun, Yiannis Sakellaridis, David Soudry, Ilya Tyomkin, Yakov Varshavsky and Oksana Yakimova for useful discussions. I am very grateful to the Department of Mathematics of the Weizmann Institute that has provided warm and fruitful environment for my PhD studies. In particular, I wish to thank Stephen Gelbart, Maria Gorelik, Dmitry Novikov and Sergei Yakovenko for their encouragement, guidance and support. I would also like to thank the secretary of the department, Gizel Maymon, and the previous secretary Terry Debesh for making the bureaucracy disappear for me. During my PhD studies, I have visited the Hausdorff Center of Mathematics and Max Planck Institute fur Mathematik in Bonn. I am thankful to both institutions for inspiring environment and working conditions. Finally, I wish to thank my entire family. In particular, I am deeply grateful to my parents, Ben-Zion and Hannah, who raised me and initiated my interest in science, and my wife, Inna, for being the light and the driving force of my life. Support information During my PhD studies I was supported in part by a BSF grant, a GIF grant, and an ISF center of excellency grant. 4 Abstract A Gelfand pair is a pair (G; H) consisting of a group and a subgroup such that the quazi- regular representation of G acting on the space F (G=H) of functions on G=H \includes" any irreducible representation of G with multiplicity at most 1. The theory of Gelfand pairs have various applications in representation theory, harmonic analysis and the theory of automorphic forms. The main tool for proving the Gelfand property of a given pair is the Gelfand-Kazhdan method which is based on the analysis of distributions on G which are invariant with respect to the two-sided action of H. In this thesis we present the proof of the Gelfand property for various pairs. Most of these pairs are symmetric pairs. In particular, we proved that the following pairs are Gelfand pairs: (GLn+k(F ); GLn(F ) GLk(F )) for F = R; C: The non-archimedean case was proven • in [JR]. × (GLn(C); GLn(R)): The non-archimedean case was proven in [Fli]. • (On+k(C);On(C) Ok(C)). • × (GLn(C);On(C)). • (GL2n(F ); Sp2n(F )) for F = R; C: The non-archimedean case was proven in [HR]. • (GL (F ); (GL (F ))) is a strong Gelfand pair (i.e. (GL (F ) • n+1 n n+1 × (GLn(F )); ∆(GLn(F ))) is a Gelfand pair) . The proof is based on various tools that we have developed in order to work with invariant distributions. There are other methods for proving the Gelfand property apart from the Gelfand- Kazhdan method. Most of them are based on deducing the Gelfand property of one pair from the Gelfand property of another pair. In this thesis there are two examples of such methods. The theory of invariant distributions has also different applications in representation theory, and we will discuss some of these applications, too. Content of the thesis In the introduction we discuss the results that appear in the thesis in more details. Chap- ter 2 consists of the papers which I wrote during my Ph.D. Studies that are discussed in the introduction. Chapter 3 contains a discussion about possible extensions of the results discussed in the thesis. 5 6 Contents 1 Introduction 9 1.1 Gelfand pairs . .9 1.1.1 Strong Gelfand pairs . .9 1.1.2 Gelfand-Kazhdan criterion . 10 1.1.3 Symmetric pairs . 10 1.1.4 The strong Gelfand property of the pair (GLn+1(F ); GLn(F )) . 10 1.1.5 General strategy for proving Gelfand property . 11 1.2 Other methods for proving Gelfand property . 11 1.2.1 Positive characteristic versus zero characteristic . 11 1.2.2 Integration of invariant functionals . 12 1.3 Invariant distributions . 12 1.3.1 Integrability theorem . 13 1.3.2 Matching problems . 13 2 Papers 15 3 Discussion 231 3.1 Gelfand pairs and Spherical pairs . 231 3.2 Invariant distributions . 231 7 8 Chapter 1 Introduction 1.1 Gelfand pairs Most of the problems in modern harmonic analysis are closely related to the following class of problems: let X be some space, and F (X) a space of (complex valued) functions on X of a certain type. Suppose that X possesses some symmetries. Could one find a basis for F (X) that behaves in a \good" way with respect to those symmetries? As a first step for solving this class of problems one can look at its following con- cretization: suppose that there exists a group of symmetries G that acts transitively on X. How does F (X) decomposes into irreducible representation of G? This question gives rise to the following notion: a pair (G; H) consisting of a group and a subgroup is said to be a Gelfand pair if any irreducible representation of G is \included" in F (G=H) with multiplicity at most 1. In any given case one should specify what kind of representations and functions we consider, and what does one mean by \included". For the case of reductive groups over local fields, there are several ways to do this (section 2 in [AGS] { see Chapter 2 below). An overview of the theory of Gelfand pairs can be found, for example, in [vD]. Note that if a pair (G; H) is a Gelfand pair then the \decomposition" of F (G=H) is unique. By Frobenius reciprocity the Gelfand property is equivalent to the fact that any irreducible representation of G, when restricted to H, \includes" the trivial representation with multiplicity at most 1. One can consider the representation theory of the group G as the harmonic analysis of the space F (G) with respect to the two-sided action of G G. From this point of view, Schur's lemma is equivalent to the Gelfand property of (G ×G; G): × Gelfand pairs have various applications to classical questions of representation theory and harmonic analysis. These include the classification of representations, and construct- ing canonical bases for irreducible representations and spaces of functions on homogenous spaces. More recent applications of Gelfand pairs are in the theory of automorphic forms, for instance in splitting of automorphic periods and in the relative trace formula. Some of these applications are described in [Gro]. 1.1.1 Strong Gelfand pairs A stronger version of the notion of a Gelfand pair is the following one: 9 A pair (G; H) is said to be a strong Gelfand pair if any irreducible representation of G, when restricted to H, \includes" any irreducible representation of H with multiplicity at most 1. The notion of Gelfand pair and strong Gelfand pair are connected in the following way: a pair (G; H) is a strong Gelfand pair if and only if the pair (G H; ∆H) is a Gelfand pair (here ∆H means the diagonal embedding of H in G H). × × In this thesis we prove the Gelfand property and the strong Gelfand property for various pairs. 1.1.2 Gelfand-Kazhdan criterion The main tool to prove that a pair (G; H) is a Gelfand pair is the following criterion by Gelfand and Kazhdan ([GK]). Suppose that there exists an involutive anti-automorphism σ of G such that any distribution on G which is invariant with respect to the H H two-sided action is invariant with respect to σ. Then the pair (G; H) is a Gelfand pair.× This criterion implies an analogous criterion for strong Gelfand pairs. 1.1.3 Symmetric pairs Most of the results in this thesis are about symmetric pairs.
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