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Perspectives in Representation Theory

A Conference in Honor of Igor Frenkel’s 60th Birthday on Perspectives in Representation Theory May 12–17, 2012 Yale University, New Haven, CT

Pavel Etingof Mikhail Khovanov Alistair Savage Editors

American Mathematical Society Perspectives in Representation Theory

A Conference in Honor of Igor Frenkel’s 60th Birthday on Perspectives in Representation Theory May 12–17, 2012 Yale University, New Haven, CT

Pavel Etingof Mikhail Khovanov Alistair Savage Editors

610

Perspectives in Representation Theory

A Conference in Honor of Igor Frenkel’s 60th Birthday on Perspectives in Representation Theory May 12–17, 2012 Yale University, New Haven, CT

Pavel Etingof Mikhail Khovanov Alistair Savage Editors

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classification. Primary 17Bxx, 22E57.

Library of Congress Cataloging-in-Publication Data Perspectives in representation theory : a conference in honor of Igor Frenkel’s 60th birthday : May 12–17, 2012, Yale University, New Haven, CT / Pavel Etingof, Mikhail Khovanov, Alistair Savage, editors. pages cm. – (Contemporary mathematics ; volume 610) Includes bibliographical references. ISBN 978-0-8218-9170-4 (alk. paper) 1. Frenkel, Igor–Congresses. 2. Representations of algebras–Congresses. 3. Lie algebras– Congresses. 4. Representations of groups–Congresses. 5. Group theory–Congresses. I. Etingof, P. I. (Pavel I.), 1969- editor of compilation. II. Khovanov, Mikhail editor of compi- lation. III. Savage, Alistair editor of compilation.

QA176.P47 2014 512.22–dc23 2013035921

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/610

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. 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 http://www.ams.org/ 10987654321 191817161514 This volume is dedicated to Igor Frenkel on the occasion of his sixtieth birthday.

Contents

On the work of Igor Frenkel John Duncan, Pavel Etingof, Ivan Ip, Mikhail Khovanov, Matvei Libine, Anthony Licata, Alistair Savage, and Michael Schlosser 1 Macdonald polynomials, Laumon spaces and perverse coherent sheaves Alexander Braverman, Michael Finkelberg, and Jun’ichi Shiraishi 23 An affine Gindikin-Karpelevich formula Alexander Braverman, Howard Garland, David Kazhdan, and Manish Patnaik 43 On the discrete groups of Mathieu moonshine Miranda C. N. Cheng and John F. R. Duncan 65 The classical master equation Giovanni Felder and David Kazhdan, with an Appendix by Tomer M. Schlank 79 DG indschemes Dennis Gaitsgory and Nick Rozenblyum 139 Special functions in minimal representations Toshiyuki Kobayashi 253 Asymptotic Hecke algebras and involutions George Lusztig 267 Chiral differential operators on abelian varieties Fyodor Malikov and Vadim Schechtman 279 Refined Chern-Simons theory and Hilbert schemes of points on the plane Hiraku Nakajima 305 Categorified Jones-Wenzl projectors: a comparison Catharina Stroppel and Joshua Sussan 333 Weil representations and theta functionals on surfaces Yongchang Zhu 353

vii

Preface

This is the proceedings of the conference Perspectives in representation theory, held at Yale University May 12–17, 2012, in honor of the 60th birthday of Prof. Igor Frenkel. It contains papers by some of the speakers at the conference and their collaborators, as well as a summary of the work of Igor Frenkel, prepared by his former students and collaborators. The meeting featured talks by mathemati- cians working in representation theory, with an emphasis on its relations to other subjects (notably, topology, algebraic geometry, number theory, and mathemati- cal physics). The papers in this volume concern current progress on the following (interrelated) topics: vertex operator algebras and chiral algebras, conformal field theory, the (geometric) Langlands program, affine Lie algebras, Kac-Moody al- gebras, quantum groups, crystal bases and canonical bases, quantum cohomology and K-theory, geometric representation theory, categorification, higher-dimensional Kac-Moody theory, integrable systems, quiver varieties, representations of real and p-adic groups, quantum gauge theories. The conference was an occasion to discuss representation theory in the context of its connections with numerous other sub- jects, and to discuss some of the most recent advances in representation theory, including those which occurred thanks to application of techniques in other areas of mathematics, and of ideas of quantum field theory and string theory. Videos of most of the talks given at the conference, as well as photos taken during the event, can be found on the conference website at: www.math.yale.edu/frenkel60 The organizing committee consisted of P. Etingof (M.I.T.) M. Khovanov (Columbia University) A. Kirillov Jr. (SUNY Stony Brook) A. Lachowska (Yale University) A. Licata (Australian National University) A. Savage (University of Ottawa) G. Zuckerman (Yale University) The organizers of the conference would like to thank the National Science Foun- dation and Yale University for their support, without which the event would not have been possible.

Pavel Etingof Mikhail Khovanov Alistair Savage

ix

Selected Published Titles in This Series

610 Pavel Etingof, Mikhail Khovanov, and Alistair Savage, Editors, Perspectives in Representation Theory, 2014 607 Kiyoshi Igusa, Alex Martsinkovsky, and Gordana Todorov, Editors, Expository Lectures on Representation Theory, 2014 606 Chantal David, Matilde Lal´ın, and Michelle Manes, Editors, Women in Numbers 2, 2013 605 Omid Amini, Matthew Baker, and Xander Faber, Editors, Tropical and Non-Archimedean Geometry, 2013 604 Jos´eLuisMonta˜na and Luis M. Pardo, Editors, Recent Advances in Real Complexity and Computation, 2013 603 Azita Mayeli, Alex Iosevich, Palle E. T. Jorgensen, and Gestur Olafsson,´ Editors, Commutative and Noncommutative Harmonic Analysis and Applications, 2013 602 Vyjayanthi Chari, Jacob Greenstein, Kailash C. Misra, K. N. Raghavan, and Sankaran Viswanath, Editors, Recent Developments in Algebraic and Combinatorial Aspects of Representation Theory, 2013 601 David Carf`ı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel van Frankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II, 2013 600 David Carf`ı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel van Frankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I, 2013 599 Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, Fernando Schwartz, and Gilbert Weinstein, Editors, Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, 2013 598 Eric Todd Quinto, Fulton Gonzalez, and Jens Gerlach Christensen, Editors, Geometric Analysis and Integral Geometry, 2013 597 Craig D. Hodgson, William H. Jaco, Martin G. Scharlemann, and Stephan Tillmann, Editors, Geometry and Topology Down Under, 2013 596 Khodr Shamseddine, Editor, Advances in Ultrametric Analysis, 2013 595 James B. Serrin, Enzo L. Mitidieri, and Vicent¸iu D. R˘adulescu, Editors, Recent Trends in Nonlinear Partial Differential Equations II, 2013 594 James B. Serrin, Enzo L. Mitidieri, and Vicent¸iu D. R˘adulescu, Editors, Recent Trends in Nonlinear Partial Differential Equations I, 2013 593 Anton Dzhamay, Kenichi Maruno, and Virgil U. Pierce, Editors, Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, 2013 592 Arkady Berenstein and Vladimir Retakh, Editors, Noncommutative Birational Geometry, Representations and Combinatorics, 2013 591 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, Vladimir Maz’ya, Simeon Reich, David Shoikhet, Gilbert Weinstein, and Lawrence Zalcman, Editors, Complex Analysis and Dynamical Systems V, 2013 590 Ursula Hamenst¨adt, Alan W. Reid, Rub´ıRodr´ıguez, Steffen Rohde, and Michael Wolf, Editors, In the Tradition of Ahlfors-Bers, VI, 2013 589 Erwan Brugall´e, Mari´aAng´elica Cueto, Alicia Dickenstein, Eva-Maria Feichtner, and Ilia Itenberg, Editors, Algebraic and Combinatorial Aspects of Tropical Geometry, 2013 588 David A. Bader, Henning Meyerhenke, Peter Sanders, and Dorothea Wagner, Editors, Graph Partitioning and Graph Clustering, 2013

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/conmseries/. CONM 610 esetvsi ersnainTheory Representation in Perspectives

This volume contains the proceedings of the conference Perspectives in Representation Theory, held from May 12–17, 2012, at Yale University, in honor of Igor Frenkel’s 60th birthday. The aim of the conference was to present current progress on the following (interrelated) topics: vertex operator algebras and chiral algebras, conformal field theory, the (geometric) Langlands program, affine Lie algebras, Kac-Moody algebras, quantum groups, crystal bases and canonical bases, quantum cohomology and K-theory, geometric representation theory, categorification, higher-dimensional Kac-Moody theory, integrable systems, quiver varieties, representations of real and p-adic groups, and quantum gauge theories. The papers in this volume present representation theory connections to numerous other subjects, as well as some of the most recent advances in representation theory, including those which occurred thanks to the application of techniques in other areas of mathematics, and of ideas of quantum field theory and string theory. • tno ta. Editors al., et Etingof

ISBN 978-0-8218-9170-4

9 780821 891704

CONM/610 AMS