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SOVIET MATHEMATICS Volume 16, Part 2 Advances in SOVIET MATHEMATICS Volume 16, Part 2 I. M. Gelfand Seminar Sergei Gelfand Simon Gindikin Editors American Mathematical Society Titles in This Series 16 Sergei Gelfand and Simon Gindikin, editors, I. M. Gelfand seminar, Parts 1 and 2, 1993 15 A. T. Fomenko, editor, Minimal surfaces, 1993 14 Yu. S. Il'yashenko, editor, Nonlinear Stokes phenomena, 1992 13 V. P. Maslov and S. N. Samborskil, editors, Idempotent analysis, 1992 12 R. Z. Khasminskii, editor, Topics in nonparametric estimation, 1992 11 B. Ya. Levin, editor, Entire and subharmonic functions, 1992 10 A. V. Babin and M. I. Yishik, editors, Properties of global attractors of partial differential equations, 1992 9 A. M. Vershik, editor, Representation theory and dynamical systems, 1992 8 E. B. Vinberg, editor, Lie groups, their discrete subgroups, and invariant theory, 1992 7 M. Sh. Birman, editor, Estimates and asymptotics for discrete spectra of integral and differential equations, 1991 6 A. T. Fomenko, editor, Topological classification of integrable systems, 1991 5 R. A. Minlos, editor, Many-particle Hamiltonians: spectra and scattering, 1991 4 A. A. Suslin, editor, Algebraic A-theory, 1991 3 Ya. G. Sinai, editor, Dynamical systems and statistical mechanics, 1991 2 A. A. Kirillov, editor, Topics in representation theory, 1991 1 Y. I. Arnold, editor, Theory of singularities and its applications, 1990 Part 2 Photograph courtesy of C. MacPherson Advances in 10.1090/advsov/016.2 S oviet Mathematics Volume 16, Part 2 I. M. Gelfand Seminar Sergei Gelfand Simon Gindikin Editors American Mathematical Society Providence, Rhode Island A d v a n c e s i n S o v i e t M a t h e m a t i c s E d it o r ia l C o m m it t e e V. I. ARNOLD S. G. GINDIKIN V. P. MASLOV 1991 Mathematics Subject Classification. Primary 00B15. Library of Congress Catalog Card Number: 91-640741 ISBN 0-8218-4118-1 (Part 1) ISBN 0-8218-4119-X (Part 2) ISBN 0-8218-4117-3 (Set) ISSN 1051-8037 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 an article 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 publi­ cation (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance of the code on the first page of an article in this book indicates the copyright owner’s consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright ©1993 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. © This publication was typeset using ^jqS-TgX, the American Mathematical Society’s TgX macro system. 10 9 8 7 6 5 4 3 2 1 98 97 96 95 94 93 Contents Part 1 Foreword xi A Proof of Jantzen Conjectures A. BEILINSON and J. BERNSTEIN 1 String Bases for Quantum Groups of Type Ar ARKADY BERENSTEIN and ANDREI ZELEVINSKY 51 Limit Distributions of Orbits of Unipotent Flows and Values of Quadratic Forms S. G. DANI and G. A. MARGULIS 91 Coinvariants of Nilpotent Subalgebras of the Virasoro Algebra and Partition Identities BORIS FEIGIN and EDWARD FRENKEL 139 Induction and Restriction of Character Sheaves VICTOR GINZBURG 149 Explicit Construction of Characteristic Classes A. B. GONCHAROV 169 The Riemann-Roch Theorem for Elliptic Operators MIKHAEL GROMOV and MIKHAIL A. SHUBIN 211 Part 2 Foreword xi Homotopy Lie Algebras VLADIMIR HINICH and VADIM SCHECHTMAN 1 Chow Quotients of Grassmannians. I M. M. KAPRANOV 29 Reconstructing Monoidal Categories DAVID KAZHDAN and HANS WENZL 111 vii viii CONTENTS Vassiliev’s Knot Invariants MAXIM KONTSEVICH 137 Kazhdan-Lusztig Polynomials for Lie Superalgebra gl(m | n) VERA SERGANOVA 151 Hodge Filtration of Hypergeometric Integrals Associated with an Affine Configuration of General Position and a Local Torelli Theorem A. N. VARCHENKO 167 The Second Gelfand-Dickey Bracket as a Bracket on a Poisson-Lie Grassmannian ILYA ZAKHAREVICH 179 Photographs courtesy of C. MacPherson Foreword September 2, 1993 is Israel Moiseevich Gelfand’s 80th birthday. This date practically coincides with the 50th anniversary of the Gelfand Seminar in Functional Analysis at Moscow University. The present volume consists of papers written by “young” participants of this seminar. One of the reasons for introducing an age limit was to keep the volume’s size within reasonable bounds, another was Gelfand’s constant orientation to the younger partic­ ipants of his seminar. This collection is intended to be a surprise to the man whose birthday we are celebrating, and I hope that he will learn of the book’s existence only after its publication. However, we have tried to imitate Gelfand’s own preferences as much as possible. All of the invited authors were participants and welcome speakers at the seminar; if one imagines its Golden Jubilee Session, it may be safely conjectured that these mathemati­ cians would have been invited to participate. I hope that Gelfand will approve our choice and will enjoy seeing articles written by these remarkable mathe­ maticians. The invited authors were free to choose their topics and, if they so desired, their coauthors. The problem of defining the notion of “young” scientist, in particular, of “young mathematician”, is one of the most difficult unsolved problems, and is also of the utmost importance for the applications. In the given case its solution was formalized in the following way. For the upper bound of the age of an invited author to this collection, we chose the age of Serezha, Gelfand’s older son, who incidentally played a key role in the appearance of this book. As paradoxal as this may sound, there are serious grounds for this choice. When Serezha first appeared at the seminar in 1961, significant changes in it took place. The orientation to the younger participants that had always been important, but concerned only Ph.D.-track students (occasionally younger graduate students) until then, was now drastically amended to include fresh­ men and sophomores, and later even some high school students. Among the students of this “first draft” one should note Dima Kazhdan and, somewhat later, Ossya Bernstein. Both became important participants in the seminar for many years. Progressively the other authors of this volume also appeared there. By 1962 the seminar left the relatively small auditorium 13-11 to move to the roomier 14-08. By then no less than half the participants were under­ graduates. At that time Gelfand liked to repeat that the seminar was open to all students of the lower courses and to the most talented professors. XI xii FOREWORD I think that it is only natural to include in this foreword some informa­ tion about the seminar itself. I understand that it was well known in the West. Usually mathematical voyagers from Western Europe or the US felt compelled to visit it (to do that, they had to overcome the vigilance of the Moscow University guards, by no means an easy task even for Muscovites). It is difficult to explain to the Western reader what the seminar meant to “Soviet mathematical life”. Surprisingly, that life, in many respects, was not at all so bad, despite the almost unwavering antisemitism and the constantly increasing control by the mathematical rabble with communist party back­ ground over the key positions in mathematics. Mathematics was an Oasis of sorts, very attractive to young people with strong interest in science and without career aspirations in the communist hierarchy. If one were lucky, you were able to live an intense intellectual life and write articles free of any references to the classics of Marxism-Leninism. Fortunately for mathemat­ ics, Stalin did not find time for the subject (unlike economics, biology, and linguistics). The Gelfand seminar was always an important event in the very vivid mathematical life in Moscow, and, doubtless, one of its leading centers. A considerable number of the best Moscow mathematicians participated in it at one time or another. Mathematicians from other cities used all possible pretexts to visit it. I recall how a group of Leningrad students agreed to take turns to come to Moscow on Mondays (the day of the seminar, to which other events were linked), and then would retell their friends what they heard there. There were several excellent and very popular seminars in Moscow, but nevertheless the Gelfand seminar was always an event. I would like to point out that, on the other hand, the seminar was very important in Gelfand’s own personal mathematical life. Many of us wit­ nessed how strongly his activities were focused on the seminar. When, in the early fifties, at the peak of antisemitism, Gelfand was chased out of Moscow University, he applied all his efforts to save the seminar.
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