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Representations of Reductive Groups in Honor of the 60Th Birthday of David A Progress in Mathematics 312 Monica Nevins Peter E. Trapa Editors Representations of Reductive Groups In Honor of the 60th Birthday of David A. Vogan, Jr. Progress in Mathematics Volume 312 Series Editors Hyman Bass Jiang-Hua Lu Joseph Oesterle´ Yuri Tschinkel More information about this series at http://www.springer.com/series/4848 Monica Nevins Peter E. Trapa Editors Representations of Reductive Groups In Honor of the 60th Birthday of David A. Vogan, Jr. Editors Monica Nevins Peter E. Trapa Department of Mathematics and Statistics Department of Mathematics University of Ottawa University of Utah Ottawa, ON, Canada Salt Lake City, UT, USA ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-319-23442-7 ISBN 978-3-319-23443-4 (eBook) DOI 10.1007/978-3-319-23443-4 Library of Congress Control Number: 2015955838 Mathematics Subject Classification (2000): 11E72, 11F72, 14F05, 14F10, 14L24, 14M15, 14N15, 16RXX, 16S30, 20B35, 20C08, 20C20, 20F55, 20G05, 20G07, 20G20, 20G99, 22D10, 22E25, 22E30, 22E45, 22E46, 22E47, 22E50, 32C38, 53C35, 55N33, 58A14 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com) To David Vogan, with admiration Contents Preface ............................................................ ix The Mathematical Work of David A. Vogan, Jr. ::::::::::::::::::::::: 1 William M. McGovern and Peter E. Trapa On exotic and perverse-coherent sheaves :::::::::::::::::::::::::::: 11 Pramod N. Achar Parameters for twisted representations :::::::::::::::::::::::::::::: 51 Jeffrey Adams and David A. Vogan, Jr. Ladder representations of GL.n; Qp/::::::::::::::::::::::::::::::::117 Dan Barbasch and Dan Ciubotaru Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits :::::::::::::::::::::::::::::::::::::::::::139 Manjul Bhargava, Benedict H. Gross, and Xiaoheng Wang Hecke algebras with unequal parameters and Vogan’s left cell invariants :::::::::::::::::::::::::::::::::::::::::::::::::::::::173 Cedric´ Bonnafe´ and Meinolf Geck The smooth loci of spiral Schubert varieties of type Az2 :::::::::::::::::189 William Graham and Wenjing Li Centers and cocenters of 0-Hecke algebras :::::::::::::::::::::::::::227 Xuhua He Dirac cohomology, elliptic representations and endoscopy ::::::::::::::241 Jing-Song Huang A program for branching problems in the representation theory of real reductive groups :::::::::::::::::::::::::::::::::::::::::::::::::277 Toshiyuki Kobayashi vii viii Contents Equations for a filtration of sheets and the variety of singular elements of a complex semisimple Lie algebra ::::::::::::::::::::::::::::::::323 Bertram Kostant On conjugacy classes in a reductive group :::::::::::::::::::::::::::333 George Lusztig Hecke algebras and involutions in Coxeter groups:::::::::::::::::::::365 George Lusztig and David A. Vogan, Jr. Comparing and characterizing some constructions of canonical bases from Coxeter systems :::::::::::::::::::::::::::::::::::::::::::::399 Eric Marberg Upper semicontinuity of KLV polynomials for certain blocks of Harish-Chandra modules::::::::::::::::::::::::437 William M. McGovern Hodge theory and unitary representations :::::::::::::::::::::::::::443 Wilfried Schmid and Kari Vilonen On elliptic factors in real endoscopic transfer I :::::::::::::::::::::::455 Diana Shelstad On the Gelfand–Kirillov dimension of a discrete series representation ::::505 Nolan R. Wallach A reducible characteristic variety in type A ::::::::::::::::::::::::::517 Geordie Williamson Preface This volume is an outgrowth of the conference Representations of Reductive Groups: A conference dedicated to David Vogan on his 60th birthday, which took place at MIT from May 19–23, 2014. This celebratory conference showcased developments in the representation theory of reductive Lie groups and algebraic groups over finite and local fields, as well as connections of this theory with other subjects, such as number theory, automorphic forms, algebraic geometry and combinatorics. It was an occasion to mark the 60th birthday of David Vogan, who has inspired and shaped the development of this field for almost 40 years. A list of conference speakers ap- pears on page xiii. David Vogan has proven himself a leader in the field in many ways. The most evident of these are his vast and influential mathematical contributions, an overview of which appears in the chapter The Mathematical Work of David A. Vogan, Jr. in this volume. Also, of significant importance to so many — and to us personally — has been David’s mentorship, insight and guidance. We include a subset of those who have benefitted from working closely with David — namely, his PhD students to date, as well as some of his mathematical descendants who were able to attend the conference — on pages xvi to xviii. AMS Centennial Fellow (1977), Plenary Lecturer of the International Congress of Mathematicians (1986), Herman Weyl Lecturer (IAS, 1986), Member of the American Academy of Arts and Sciences (1996), Robert E. Collins Distinguished Scholar, MIT, (2007–2012), winner of the Levi L. Conant Prize (2011) for his paper “The character table of E8”, Member of the National Academy of Science (2013), and current Norbert Weiner Chair at MIT (2014–2019), David Vogan has been often been distinguished by his peers and by the mathematical community. Significantly, he has also given back through substantial work in academic service, including as Head of the MIT Department of Mathematics (1999–2004) and President of the American Mathematical Society (2013–2015). This volume represents a step towards expressing our thanks. ix x Preface Acknowledgements It is a pleasure to thank our conference co-organizers, Roman Bezrukavnikov, Pavel Etingof, and George Lusztig, as well as members of the MIT administrative staff — particularly Shirley Entzminger, Jonathan Harmon, and Dennis Porche, whose assistance and support were indispensable. We are espe- cially grateful for the editorial assistance of Ann Kostant in the preparation of this work. Ottawa, Canada Monica Nevins Salt Lake City, USA Peter E. Trapa May 2015 Preface xi David A. Vogan, Jr., 1999 Photo courtesy of Donna Coveney, MIT. David A. Vogan, Jr., 2011 Photo courtesy of Bryce Vickmark. Representations of reductive groups A conference dedicated to David Vogan on his 60th birthday M.I.T., May 19 – May 23, 2014 Conference Speakers Pramod Achar Equivariant coherent sheaves on the nilpotent cone of a reductive algebraic group Jeffrey Adams Galois and theta cohomology of real groups James Arthur On Langlands’ automorphic Galois group and Weil’s explicit formulas Joseph Bernstein Stacks in Representation Theory: What is a representation of an algebraic group? Dan Ciubotaru On formal degrees of unipotent discrete series representations of semisimple p-adic groups Stephen DeBacker Nilpotent orbits revisited Michel Duflo On Frobenius Lie subalgebras of simple Lie algebras Meinolf Geck Computing with left cells of type E8 Benedict Gross Newforms for odd orthogonal groups Xuhua He Cocenters and representations of affine Hecke algebras Jing-Song Huang Elliptic representations, Dirac cohomology and endoscopy Toshiyuki Kobayashi Branching problems of representations of real reductive Lie groups Bert Kostant Generalized Amitsur-Levitski Theorem and equations for sheets in a reductive complex Lie algebra George Lusztig Conjugacy classes in a reductive group xiii xiv Representations of reductive groups W. Monty McGovern Upper semicontinuity of KLV polynomials for certain blocks of Harish-Chandra modules Diana Shelstad Transfer results for real groups Wilfried Schmid On the n-cohomology of the limits of the discrete series Wolfgang Soergel Graded versions of categories of representations and true motives Peter Trapa Relationships between unitary representations of real and p-adic groups Akshay Venkatesh Analytic number theory and harmonic analysis on semisimple Lie groups Jean-Loup Waldspurger Stabilization of the twisted trace formula: the local theorem Nolan Wallach Gleason’s theorem and unentangled orthonormal bases Geordie Williamson Global and local Hodge theory of Soergel bimodules Representations of reductive groups xv Photo reprinted courtesy of Dennis Porche. Conference Participants, MIT, May 2014. xvi Ph.D. students of David A. Vogan,
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