New Developments in Lie Theory and Their Applications PDF
Preview New Developments in Lie Theory and Their Applications
Progress in Mathematics Volume 105 Series Editors J. Oesterle A. Weinstein New Developments in Lie Theory and Their Applications Juan Tirao Nolan Wallach Editors Birkhauser Boston· Basel . Berlin Juan Tirao Nolan R. Wallach Facultad de Matematica Dept. of Mathematics Astronomia y Fisica Umversity of California Universidad Nacional de C6rdoba at SanDiego 5000 C6rdoba La Jolla, CA 92093 Argentina USA Library of Congress in-Publication Data New Developments in Lie theory and their applications I edited by Juan Tirao, Nolan Wallach. p. cm. - (Progress in mathematics: 105) Papers from the Third Workshop on Representation Theory of Lie Groups and Their Applications, held Aug.-Sept. 1989 in C6rdoba, Argentina. Includes bibliographical references. ISBN-13:978-1-4612-7743-9 1. Lie groups--Congresses. 2. Representations of groups- -Congresses. I. Tirao, Juan 1942 - . II. Wallach, Nolan R. ID. Series: Progress in mathematics (Boston, Mass.) ; vol. 105. QA387.N48 92-1255 512'.55--dc20 CIP Printed on acid-free paper © Birkhiiuser Boston 1992 Softcover reprint of the hardcover 1st edition 1992 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 retrievalsystem, or-transmitted, in any form or by any means, electronic, mechanical, photo-copying, 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 $5.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Birkhiiuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN-13:978-1-4612-7743-9 e-ISBN-13:978-1-4612-2978-0 DOl: 10.1007/978-1-4612-2978-0 Camera-ready copy prepared by the Authors in TeX. 987 6 5 432 1 Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . vii Automorphic Forms Nolan R. Wallach . 1 Analytic and Geometric Realization of Representations Wilfried Schmid .. . . . . . . . . . . . . . . . . . . . . . 27 Introduction to Quantized Enveloping Algebras George Lusztig . . . . . . . . . . . . . . . . . . . . . . . . 49 The Vanishing of Scalar Curvature, Einstein's Equation and Representation Theory Bertram Kostant . . . . . . . . . .. . ......... 67 Unitary Representations of Reductive Lie Groups and the Orbit Method David A. Vogan, Jr. ...................... 87 Twistor Theory for Riemannian Manifolds John Rawnsley . . . . . . . . . . . 115 You Can't Hear the Shape of a Manifold Carolyn Gordon . . . . . . . . 129 Kuznetsov Formulas R. J. Miatello 147 Lefschetz Numbers and Cyclic Base Change for Purely Imaginary Extensions Jurgen Rohlfs and Birgit Speh ..... 155 Some Zeta Functions Attached to r\G / K Floyd Williams . . . . . . . . . . . . 163 v On the Centralizer of K in the Universal Enveloping Algebra of SO(n, 1) and SU(n, 1) Juan Timo. . . . . . . . . . . . . . . . . . . . . . 179 On Spherical Modules Nicolas Andruskiewitsch 187 Generalized Weil Representations for Sl(n, k), n odd, k a Finite Field Jose Pantoja and Jorge Soto-Andmde . . . . 199 Local Multiplicity of Intersection of Lagrangian Cycles and the Index of Holonomic Modules Alberto S. Dubson. . . . . . . . . . . . . 207 vi Introduction Representation theory, and more generally Lie theory, has played a very important role in many of the recent developments of mathematics and in the interaction of mathematics with physics. In August-September 1989, a workshop (Third Workshop on Representation Theory of Lie Groups and its Applications) was held in the environs of C6rdoba, Argentina to present expositions of important recent developments in the field that would be accessible to graduate students and researchers in related fields. This volume contains articles that are edited versions of the lectures (and short courses) given at the workshop. Within representation theory, one of the main open problems is to determine the unitary dual of a real reductive group. Although this prob lem is as yet unsolved, the recent work of Barbasch, Vogan, Arthur as well as others has shed new light on the structure of the problem. The article of D. Vogan presents an exposition of some aspects of this prob lem, emphasizing an extension of the orbit method of Kostant, Kirillov. Several examples are given that explain why the orbit method should be extended and how this extension should be implemented. Another active direction of research is the (not related) problem of giving geometric realizations of admissible (and hopefully unitary) rep resentations of real reductive groups. This is the subject of the article by w. Schmid. In light of the various classifications of admissible represen tations of real reductive groups (Langlands, Beilinson-Bemstein, Vogan Zuckerman, Vogan) the structure of these representations is well under stood from the perspective of analysis and algebra «g, K)-modules). The main algebraic constructions are cohomological induction and V-module theoretic induction. The article of W. Schmid describes his work (and that of his coworkers) on a geometric approach to the construction of admissible representations of real reductive groups (the Hecht, Milich~, Schmid, Wolf construction) which is used to relate the various classifica tions. The article also contains an introduction to hyperfunction theory and a survey of results on functorial globalizations of (g, K)-modules. The representation theory of reductive groups has been intimately connected with number theory (in particular the theory of automorphic forms) since the basic work of Gelfand, Piatetski-Shapiro and Selberg in the 1950s and 1960s. In the form of the Langlands program, number theory has suggested important new approaches to problems in represen- vii tation theory (the flavor of the influence of the Langlands philosophy can be seen in the article of Vogan). The article of N. Wallach is an introduction to the theory of automorphic forms with emphasis on the classical theory (from the perspective of representation theory). Wallach emphasizes the theory for SL(2) and shows how the classical theory can be reformulated into a representation theoretic language. The article also contains some exposition of the general theory. Lie theory has almost from it earliest beginnings been ultimately connected with physics. Certainly, general relativity and quantum me chanics have made substantial use of Lie theory. The article of B. Kostant contains research that he and his coworkers have done on a remarkable representation of SO(4 , 4) which is analogous to the metaplectic (or Wei!) representation of the two-fold covering group of the symplectic group. In his article Kostant shows how this representation is intimately connected with general relativity on the one hand and to the triality associated with SO(8) on the other. The Penrose twistor theory relates complex analysis to general rela tivity and Maxwell's equations. This theory has influenced and has been influenced by the geometric methods of constructing representations of reductive groups (see the article of W. Schmid). Twistor theory is related to the group SU(2,2) and is a four-dimensional theory. In the article of J. Rawnsley, an extension of this theory is given to arbitrary even dimensions. The twistor space for a compact symmetric space of inner type is determined. Since the late 1960s there have been several important generaliza tions of (finite-dimensional) reductive groups and their representations. Although Kac-Moody algebras were first studied for purely mathematical reasons (work of Kac on finite order automorphisms of simple Lie alge bras over C, the proof of the Dyson-McDonald identities using the Kac character formula, the use of the normal operators of Lepowsky-Wilson to construct affine Kac-Moody algebras), the physics of the 1970s and 1980s influenced and was influenced by the development of this theory (notably through the theory of vortex operators and string theory). Related to this development is the so-called theory of quantum groups (called quantum deformations of enveloping algebras) in the article of G. Lusztig. These generalizations of reductive Lie groups are not even groups; they are ac tually Hopf algebras that deform in a parameter the universal enveloping algebra of a simple Lie algebra (or even a Kac-Moody algebra). These objects were initially studied to analyze explicitly solvable Hamiltonian viii systems. The article of G. Lusztig gives an introduction to this theory and in particular to the important problem of constructing "good" bases for these algebras and analyzing the specializations of the deformation parameter to a root of unity and descent to a finite characteristic. One of the most fertile areas in which representation theory has played an important role is the study of eigenfunction expansions of elliptic differential operators. The classical functions of analysis are for the most part derived by separation of variables along the orbits of an action of a reductive Lie group from an appropriate Laplacian. Also, the theory of partial differential equations has played a basic role in the analytic theory of representations (see the article of Schmid). The article of C. Gordon is more differential geometric in nature than the others and gives an exposition of the extent to which the spectrum of the Laplace-Beltrami operator determines the topology and or geometry of a Riemannian manifold. Since the early example of Milnor on isospectral manifolds, a virtual "zoo" of examples has been found. A significant number involve homogenous spaces or Lie theory. In Gordon's article, there is a survey of some of the most fruitful techniques of constructing isospectral manifolds. Several shorter contributions have been also included in this volume. For the theory of automorphic fonns, there are articles by R. Miatello and F. Williams, respectively, on the Kuznetsov fonnulas and the Selberg zeta function for discrete subgroups of reductive groups; J. Rohlfs and B. Speh discuss the use of the Lefschetz fixed point fonnula to study automorphic representations. The article of J. Tirao studies the important question of determining the structure of the centralizer of a maximal compact subgroup in the enveloping algebra of a real reductive group, and N. Andruskiewitsch considers the related question of polynomial invariants. The article of J. Pantoja and J. Soto-Andrade examines a generalization of the Weil representation to SL(n, k), where k is a finite field. The contribution of A. S. Dubson gives an exposition of several results on local intersection multiplicities for holonomic modules. Automorphic Forms Nolan R. Wallach Notes by Roberto Miatello Introduction. The purpose of these lectures is to give a relatively easily accessible introduction to the theory of automorphic forms. For most of the lectures we will confine our attention to the least complicated (but still important in its own right) case of 8L(2, R). The lectures begin with a discussion of the relationships between the 1]-function, the O-function and classical Eisenstein series. The emphasis here is on their q-expansions which we will see have interesting group and representation theoretic interpretations. Here we give some implications of a combinatorial nature to the explicit calcwations of the expansions. These functions were singled out for study for two reasons. The first is that they are the simplest automorphic forms that can be explicitly written. The second is that they are intimately related to the representation theory of loop groups and the Virasoro algebra (precisely through their q-expansions). In the second section we relate the explicit material in the first with the classical theory of holomorphic automorphic forms. The main con structions are (holomorphic) Eisenstein series and Poincare series. The modern theory of automorphic forms begins with the introduction of the Maass wave forms (now just called Eisenstein series). The general notion of automorphic form (in the case of 8L(2, R)) is given in section 3. The rest of the section involves first showing that classical automorphic forms are automorphic forms in the general sense and showing that Eisenstein series are automorphic forms. In section 4, the advertised interpretation of q-expansions is given. Also, the notion of cusp form and a basic prop erty of cusp forms is given. In section five, we leave the cozy environs of 8L(2, R) for the general arithmetic case describing how the concepts for 8L(2, R) generalize. Here we give a description of the basic theorems of Langlands on the decomposition of L2(r\G). We end the section with a rapid discussion of the trace formula. Most of the 'emphasis in the modern theory of automorphic forms is to its relationship with the Langlands program (see the article [AJ and the related material in the volume that contains it). Also, the general structure theory of arithmetic groups is alluded to in section 5. A good reference for this material is [ZJ.
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