Progress in Mathematics Volume 132 Series Editors Hyman Bass Joseph Oesterle Alan Weinstein Functional Analysis on the Eve of the 21st Century Volume II In Honor of the Eightieth Birthday of I. M. Gelfand Simon Gindikin James Lepowsky Robert L. Wilson Editors Birkhauser Boston • Basel • Berlin Simon Gindikin James Lepowsky Department of Mathematics Department of Mathematics Rutgers University Rutgers University New Brunswick, NJ 08903 New Brunswick, NJ 08903 Robert L. Wilson Department of Mathematics Rutgers University New Brunswick, NJ 08903 Library of Congress Cataloging-in-Publication Data Functional analysis on the eve of the 21 st century in honor of the 80th birthday 0fI. M. Gelfand / [edited] by S. Gindikin, 1. Lepowsky, R. Wilson. p. cm. --(Progress in mathematics; vol. 132) Includes bibliographical references. ISBN-13: 978-1-4612-8651-6 e-ISBN-13: 978-1-4612-4098-3 001: I 0.1 007/978-1-4612-4098-3 1. Functional analysis. I. Gel'fand, I. M. (Izrail' Moiseevich) !I._Gindikin, S. G. (Semen Grigor'evich) III. Lepowsky, J. (James) 1. Functional analysis. I. Gel'fand, I. M. (Izrail' Moiseevich) II. Gindikin, S. G. (Semen Grigor'evich) III. Lepowsky, J. (James) IV. Wilson, R. (Robert), 1946- . V. Series: Progress in mathematics (Boston, Mass.) ; vol. 132. QA321.F856 1995 95-20760 515'.7--dc20 CIP Printed on acid-free paper O))® Birkhiiuser H{l?J © 1996 Birkhiiuser Boston Softcover reprint of the hadrcover I st edition 1996 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhiiuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhiiuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. Typeset and reformatted from disk by T EXniques, Inc., Boston, MA 9 8 7 6 543 2 1 I. M. Gelfand Contents Volume II Preface ................. . ix Speakers at Conference in Honor of I. M. Gelfand xi Curriculum Vitae of I. M. Gelfand xii List of Publications: 1987-1995 . xiii A Tribute to I. M. Gelfand by 1. M. Singer . . . xix Positive Curvature, Macroscopic Dimension, Spectral Gaps, and Higher Signatures by M. Gromov . . . . . . . . . . . 1 Geometric Construction of Polylogarithms, II by Masaki Hanamura and Robert MacPherson 215 A Note on Localization and the Riemann-Roch Formula Lisa C. Jeffrey and Prances C. Kirwan ..... 283 A Note on ODEs from Mirror Symmetry by A. Klemm, B. H. Lian, S. S. Roan and S. T. Yau 301 viii Contents Volume I Connection Formulas in the q-analog de Rham Cohomology by Kazuhiko Aomoto .............. . 1 Lagrangian Models of Minimal Representations of E6, E7 and E8 by Ranee Brylinski and Bertram Kostant ...... ....... 13 Trigonometric Solutions of the Yang-Baxter Equation, Nets, and Hypergeometric Functions by Igor B. Frenkel and Vladimir G. Turaev . . . . . . . . . . . . . . . 65 Analogies between the Langlands Correspondence and Topological Quantum Field Theory by M. M. Kapranov ...... . 119 "Forms" of the Principal Series for G Ln by David Kazhdan ....... . 153 Geometry of Determinants of Elliptic Operators by Maxim Kontsevich and Simeon Vishik . . 173 Quantum Groups at v = 00 by George Lusztig . 199 The Symplectic Operad by Olivier Mathieu. 223 Quadratic Unipotent Representations of p-adic Groups by C. Moeglin . . . . . . . . . . . . . . . . . 245 On the Master Field in Two Dimensions by I. M. Singer . . . . . . . . . 263 Physical Methods Applied to Donaldson Theory by Edward Witten . . . . . . . . . . . . 283 Preface A four-day conference, "Functional Analysis on the Eve of the Twenty First Century," was held at Rutgers University, New Brunswick, New Jersey, from October 24 to 27, 1993, in honor of the eightieth birthday of Professor Israel Moiseyevich Gelfand. He was born in Krasnye Okna, near Odessa, on September 2, 1913. Israel Gelfand has played a crucial role in the development of functional analysis during the last half-century. His work and his philosophy have in fact helped to shape our understanding of the term "functional analysis" itself, as has the celebrated journal Functional Analysis and Its Applications, which he edited for many years. Functional analysis appeared at the beginning of the century in the classic papers of Hilbert on integral operators. Its crucial aspect was the geometric interpretation of families of functions as infinite-dimensional spaces, and of op erators (particularly differential and integral operators) as infinite-dimensional analogues of matrices, directly leading to the geometrization of spectral theory. This view of functional analysis as infinite-dimensional geometry organically included many facets of nineteenth-century classical analysis, such as power series, Fourier series and integrals, and other integral transforms. Quantum mechanics provided a further strong stimulus and source of new ideas for the development of functional analysis. Several brilliant new directions in functional analysis appeared: Banach algebras (Gelfand), operator algebras (von Neumann), infinite-dimensional representations of semisimple Lie groups (Gelfand-Naimark, Bargmann, Harish-Chandra), and the theory of distribu tions or generalized functions (Sobolev, 1. Schwartz). The continued develop ment of quantum physics stimulated the creation of one of the most remarkable ideas in functional analysis, the idea of integration over spaces of functions: Feynman integrals. Today we can observe new horizons of functional analysis. Dramatic recent developments in theoretical physics - string theory, conformal field theory and topological field theory - are again supplying new problems. Israel Gelfand's own sense of the relative importance of research directions has played a major role in the development of these new areas. We mention for example his instructive and influential lectures prepared for the International Congresses in Amsterdam, Edinburgh, Stockholm and Nice; in these lectures, he formulated many important problems concerning functional analysis in the broad sense. His personal scientific activity has been distinguished by the coexistence of a very broad spectrum of mathematical interests and also by faithfulness to the ideology of functional analysis. Two remarkable examples are the application of the ideas of infinite-dimensional representations to the study of representations of finite groups, and the development of combinatorics with "infinite-dimensional" background. It was very difficult to make a selection of topics for this conference, and x Preface we decided to follow Gelfand's taste and choose topics in which he has been working actively in recent years or in which he has a very strong interest to day. For instance, we have chosen not to include subjects in which Gelfand made fundamental contributions but in which he does not actively work now. For these reasons, we chose the following list of (interrelated) topics for this conference: (1) Mathematical physics, especially geometric quantum field theory; (2) Representation theory, particularly, certain problems concerning repre sentations of groups over local fields; (3) Combinatorics and hypergeometric functions, with emphasis on combi natorial structures underlying various "continuous" constructions; (4) Noncommutative geometry, quantum groups and geometry. Support for the conference was generously provided by the National Sci ence Foundation, the A. P. Sloan Foundation and Rutgers University. Eighteen invited mathematical talks were presented at the conference. In addition, Pres ident Francis L. Lawrence of Rutgers University awarded Professor Gelfand the honorary degree of Doctor of Science, and on this occasion, Professor Israel M. Singer delivered a tribute to Professor Gelfand. Professor Singer's tribute and the program of the conference are included here. There was great interest in the talks throughout the entire conference, and the lecture hall was constantly filled with people from many countries, including many graduate students and young researchers who were able to attend due to support from the National Science Foundation and the Sloan Foundation. These two volumes contain papers contibuted by most of the invited speak ers. The second of the two volumes contains the somewhat more "geometric" papers, although such a designation is to a certain extent arbitrary, because of the breadth of the papers. The organizing committee for the conference consisted of: Sir Michael At iyah , Felix Browder, Alain Connes, Simon Gindikin, Phillip Griffiths, Friedrich Hirzebruch, David Kazhdan, Bertram Kostant, James Lepowsky, George Daniel Mostow, Ilya Piatetski-Shapiro, Mikio Sato, Isadore Singer, Robert Wilson and Edward Witten. Special thanks are due to Mary Anne Jablonski, who expertly coordinated the conference arrangements at Rutgers University, and to Ann Kostant and the entire staff of Birkhiiuser, who have displayed untiring efforts in bringing these volumes to completion. We believe that this conference gave the mathematical community the op portunity to honor one of the most remarkable mathematicians of our time. We are very happy to see that, as he enters his ninth decade, Israel Gelfand continues his brilliant mathematical life as a young mathematician. Simon Gindikin James Lepowsky Robert Lee Wilson Functional Analysis on the Eve of the Twenty-First Century A Conference in Honor of the Eightieth Birthday of Israel M. Gelfand Rutgers University, New Brunswick, New Jersey October 24-27, 1993 Sunday, October 24 D. Kazhdan, Quantization and series of representations of reductive groups G. Lusztig, Prom modular representations to combinatorics I. Frenkel, A representation-theoretic approach to four-dimensional topology C. Moeglin, Wave front set and unipotent representations for p-adic groups S.-T. Yau, Variational problems of differential geometry Monday, October 25 1. Singer, A tribute to Israel Gelfand Presentation of honorary degree of Doctor of Science to Israel Gelfand B. Kostant, Minimal unitary representations and the generalized Capelli iden tity (joint work with R. Brylinski) A. Polyakov, Gravitational dressing M. Kontsevich, Linear algebra of elliptic operators (joint work with S. Vishik) 1. Singer, On the quantization of two-dimensional gauge theories Tuesday, October 26 M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory L. Jeffrey, Equivariant cohomology and pairings in the cohomology of symplectic quotients (joint work with F. Kirwan) K. Aomoto, Connection problem in the q-analog of de Rham cohomology A. Zamolodchikov, Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory (joint work with S. Ghoshal) R. MacPherson, Combinatorial differential manifolds Wednesday, October 27 M. Jimbo, Algebraic analysis of solvable lattice models O. Mathieu, On the cohomology of the Lie algebra of hamiltonian vector fields (joint work with 1. M. Gelfand) M. Gromov, Almost fiat bundles and applications E. Witten, Physical methods applied to Donaldson theory