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Pure Mathematics Proceedings of Symposia in PURE MATHEMATICS Volume 101 Representations of Reductive Groups Conference in honor of Joseph Bernstein Representation Theory & Algebraic Geometry June 11–16, 2017 Weizmann Institute of Science, Rehovot, Israel and The Hebrew University of Jerusalem, Is- rael Avraham Aizenbud Dmitry Gourevitch David Kazhdan Erez M. Lapid Editors 10.1090/pspum/101 Volume 101 Representations of Reductive Groups Conference in honor of Joseph Bernstein Representation Theory & Algebraic Geometry June 11–16, 2017 Weizmann Institute of Science, Rehovot, Israel and The Hebrew University of Jerusalem, Is- rael Avraham Aizenbud Dmitry Gourevitch David Kazhdan Erez M. Lapid Editors Proceedings of Symposia in PURE MATHEMATICS Volume 101 Representations of Reductive Groups Conference in honor of Joseph Bernstein Representation Theory & Algebraic Geometry June 11–16, 2017 Weizmann Institute of Science, Rehovot, Israel and The Hebrew University of Jerusalem, Is- rael Avraham Aizenbud Dmitry Gourevitch David Kazhdan Erez M. Lapid Editors 2010 Mathematics Subject Classification. Primary 11F27, 11F70, 20C08, 20C33, 20G05, 20G20, 20G25, 22E47. Library of Congress Cataloging-in-Publication Data Names: Conference on Representation Theory and Algebraic Geometry in honor of Joseph Bern- stein (2017 : Jerusalem, Israel) | Aizenbud, Avraham, 1983- editor. | Gourevitch, Dmitry, 1981- editor. | Kazhdan, D. A. (David A.), 1946- editor. | Lapid, Erez, 1971- editor. | Bernstein, Joseph, 1945- honoree. Title: Representations of reductive groups : Conference on Representation Theory and Algebraic Geometry in honor of Joseph Bernstein, June 11-16, 2017, The Weizmann Institute of Sci- ence & The Hebrew University of Jerusalem, Jerusalem, Israel / Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, Erez M. Lapid, editors. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Pro- ceedings of symposia in pure mathematics ; Volume 101 | Includes bibliographical references. Identifiers: LCCN 2018041948 | ISBN 9781470442842 (alk. paper) Subjects: LCSH: Representations of groups–Congresses. | Representations of algebras–Congresses. Classification: LCC QA176 .C665 2019 | DDC 512/.22–dc23 LC record available at https://lccn.loc.gov/2018041948 DOI: http://dx.doi.org/10.1090/pspum/101 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2019 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 242322212019 To Joseph Bernstein, in gratitude Contents Preface ix Character values and Hochschild homology Roman Bezrukavnikov and David Kazhdan 1 Schwartz space of parabolic basic affine space and asymptotic Hecke algebras Alexander Braverman and David Kazhdan 31 Explicit local Jacquet-Langlands correspondence: The non-dyadic wild case Colin J. Bushnell and Guy Henniart 45 On the Casselman-Jacquet functor T.-H. Chen, D. Gaitsgory, and A. Yom Din 73 Periods and theta correspondence Wee Teck Gan 113 Generalized and degenerate Whittaker quotients and Fourier coefficients Dmitry Gourevitch and Siddhartha Sahi 133 Geometric approach to the fermionic Fock space, via flag varieties and representations of algebraic (super)groups Caroline Gruson and Vera Serganova 155 Representations of a p-adic group in characteristic p G. Henniart and M.-F. Vigneras´ 171 On the support of matrix coefficients of supercuspidal representations of the general linear group over a local non-archimedean field Erez Lapid 211 On the generalized Springer correspondence G. Lusztig 219 The modular pro-p Iwahori-Hecke Ext-algebra Rachel Ollivier and Peter Schneider 255 Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda on the Plancherel density Eric Opdam 309 Limiting cycles and periods of Maass forms Andre Reznikov 351 vii viii CONTENTS On vector-valued twisted conjugation invariant functions on a group; with an appendix by Stephen Donkin Liang Xiao and Xinwen Zhu 361 Local theta correspondence and nilpotent invariants Chen-Bo Zhu 427 Preface The conference “Representation Theory and Algebraic Geometry” took place on June 11-16, 2017 at the Weizmann Institute of Science and the Hebrew University of Jerusalem. The conference honored Joseph Bernstein upon his 72nd birthday and his recent retirement from Tel Aviv University. Representation theory in its broadest sense is one of the areas that Joseph Bernstein has fundamentally transformed through his seminal contributions, to- gether with many of his collaborators. An in-depth account of the Mathematics of Joseph Bernstein appeared recently in the special 2016 issue of Selecta Mathe- matica, which also contains some of the topics not covered by these proceedings. Suffice it to say that Bernstein has had a towering influence on Mathematics, in particular in Israel, through his writings, guidance of students, communication with colleagues and generously sharing his ideas. The papers in this volume are mostly authored by the conference speakers and their collaborators, although the contents do not necessarily match exactly with the lectures themselves. While the underlying theme is representation theory, the papers are quite diverse. The lion’s share of the papers concerns different aspects of representations of p-adic groups and Hecke algebras, including in characteristic p. Even here, the papers differ significantly in methodology and emphasis and rep- resent the full spectrum of techniques from the algebraic to the analytic. Other topics, such as representations of finite groups, real groups and supergroups, an- alytic aspects of automorphic forms and invariant theory are also covered. This diversity reflects the wide interests of Joseph Bernstein in representation theory and beyond. We would like to thank all the speakers of the conference, most of whom con- tributed to this volume. We are also very grateful to the many referees for this volume whose assiduous and selfless work was absolutely crucial. The conference was the pinnacle of a special trimester on representation theory of reductive groups over local fields and applications to automorphic forms, which took place at the Weizmann Institute in the spring of 2017. Altogether there were over 150 participants including more than 100 foreign visitors. We take this opportunity to thank them all, both lecturers and students who made this event so successful. The conference and the trimester were made possible by the generous support of the Chorafas Institute for Scientific Exchange, the Arthur and Rochelle Belfer Institute, the Israel Science Foundation and the European Research Council. We thank these organizations wholeheartedly. Last but not least, we thank the administrative staff of the Weizmann Institute, especially at the faculty of Mathematics and Computer Science, as well as that of ix xPREFACE the Institute of Mathematics of the Hebrew University of Jerusalem, who worked diligently towards the success of the conference. It is a great pleasure to dedicate this volume to Joseph Bernstein. The editors Published Titles in This Series 101 Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, and Erez M. Lapid, Editors, Representations of Reductive Groups, 2019 100 Chiu-Chu Melissa Liu and Motohico Mulase, Editors, Topological Recursion and its Influence in Analysis, Geometry, and Topology, 2018 99 Vicente Mu˜noz, Ivan Smith, and Richard P. Thomas, Editors, Modern Geometry, 2018 98 Amir-Kian Kashani-Poor, Ruben Minasian, Nikita Nekrasov, and Boris Pioline, Editors, String-Math 2016, 2018 97 Tommaso de Fernex, Brendan Hassett, Mircea Mustat¸˘a, Martin Olsson, Mihnea Popa, and Richard Thomas, Editors, Algebraic Geometry: Salt Lake City 2015 (Parts 1 and 2), 2018 96 Si Li, Bong H. Lian, Wei Song, and Shing-Tung Yau, Editors, String-Math 2015, 2017 95 Izzet Coskun, Tommaso de Fernex, and Angela Gibney, Editors, Surveys on Recent Developments in Algebraic Geometry, 2017 94 Mahir Bilen Can, Editor, Algebraic Groups: Structure and Actions, 2017 93 Vincent Bouchard, Charles Doran, Stefan M´endez-Diez, and Callum Quigley, Editors, String-Math 2014, 2016 92 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, Lie Algebras, Lie Superalgebras, Vertex Algebras and Related Topics, 2016 91 V. Sidoravicius and S. Smirnov, Editors, Probability and Statistical Physics in St.
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