^-modules and Spherical Representations V-modules and Spherical Representations by Frederic V. Bien Mathematical Notes 39 PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1990 oC yp rihg t © 0991yb Princeton nU iev rsitPyr ess ALL RIGHTS RESERVED The PrincetaMotnh ematical oN tesa re edited yb Luis .A aC aff relli ,oJ hn .N aM ther , oJ hn iM lnor ,and lEieis .M tS ein PrincetonUni ev rsity Pressob osk arerp inted on acid-rf ee ap ep r ,and meetth e ug idelines of r ep rma nence and duraib loi ftyth e oC mmittee oPnr oduc tionuG idelines of rB oo kLoneg iv ty o f the oC mv cil on Lirb aryeR sources Printed in tnU hitee d tS ates o f mA erica yb Princeton nU iev rsitPyr ess , 14 iW lliam tS reet Princeton ,eN w eJ rsey Library of Congress Cataloging-in-PublicatioDn ata Bien ,rF ederic .V , —1691 -Dmodules ands hp erical rerp esentatio / ybn s rF ederic .V Bien. .p cm. — aM( thematical note ;s )93 Includes ib lb iorg ahp ical reef rences ainndd e.x IBS N-0-196-71520 7 a( cid-rf ee ap ep r ) .1 iD eff rentialb e maniof lds. .2 eR rp esentations o frg ousp . .3 Lie rg ousp . I. Title.I I. eS ries :aM thematical notes P( rincetonnU i ev rsity Press ) ;.93 .416AQ.3B 5 0991 .'415d—47 c 02 -09 6778 A mes parents Nadinee t Hubert et a ma muse Tesa. W.A.M. Acknowledgements The main part of this monograph consists of my Ph.D. thesis [Bien2]. I wish to express my sincere gratitude to my advisor Joseph Bernstein, for generously sharing with me his understanding of mathematics, and stimulat ing my work with inspiring conversations. I am also very grateful to David Vogan for helpful discussions, and for examining carefully early versions of the "SUNscript". Without him, this book would have appeared in a poor state of accuracy. He provided me with a list of errors and detailed comments which improved greatly the presentation. I apologize for the imperfections that undoubtedly remain. Specialthanks go to Mogens Flensted-Jensen, Pierre-Yves Gaillard, Bert ram Kostant, and Toshio Oshima for valuable pieces of advice and insightful comments. It is a pleasure to acknowledge the support of the following in stitutions during the preparation of this monograph: the Belgian American Educational Foundation, the Massachusetts Institute of Technology, the Al fred P. Sloan Foundation, the National Science Foundation, the Institute for Advanced Study, and Princeton University. Finally, I thank Becci Davies who typed the first version of this book using IATgX. The many succeeding versions were made by the author. Contents Introduction 1 I Localization Theory 9 1.1 Basic notions on V-modules .... . 9 1.2 V-modules with a group action .. . 10 1.3 Parabolic subgroups and flag spaces . 12 1.4 Differential operators on flag spaces 14 1.5 Global sections of V). ........ . 15 1.6 V-affine varieties . . . . . . . . . . 19 1. 7 Holonomic modules and Harish-Chandra modules 24 II Spherical V-modules 29 1I.l K-orbits in a flag space. ....... . 29 11.2 (V, K)-modules with a K-fixed vector. 33 11.3 Relations with the analytic theory. 34 II.4 Going from (V, K) modules to (V, H) modules. 39 II.5 H-spherical (V, K)-modules .... 45 11.6 V-modules and hyperfunctions. 52 II.7 Closed orbits and discrete series .. 57 II.8 An algebraic Poisson transform. 63 IIIMicrolocalization and Irreducibility 68 111.1 Microlocal Differential Operators 68 111.2 Microlocalization of a non-commutative ring 69 111.3 Microlocalization of Dx 111.4 Microlocalization of Ux 73 111.5 Microlocal study of the moment map 74 111.6 Associated variety of a submodule of TM. 76 111.7 Irreducibility criterion 78 111.8 Decomposable modules 80 IV Singularities and Multiplicities 84 IV. 1 Normality 84 IV.2 Unibranchness 86 IV.3 Invariant differential operators on a reductive symmetric space 89 IV.4 Multiplicity one within an eigenspace of D{G/H) 96 IV.5 Intertwining maps between distinct eigenspaces of T)[G/H) . . 108 IV.6 Multiplicities of principal series 116 IV.7 T-invariant of discrete series on exceptional symmetric spaces . 118 Bibliography 125 Index 130