D-modules and spherical representations

142 Pages·2014·4.949 MB·English

by  Frédéric V Bien

^-modules and Spherical
Representations

V-modules and Spherical
Representations
by
Frederic V. Bien
Mathematical Notes 39
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
1990
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ALL RIGHTS RESERVED
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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