Equivariant Derived Category and Representation of Real Semisimple Lie Groups
This note is based on five lectures on the geometry of flag manifolds and the representation theory of real semisimple Lie groups, delivered at the CIME summer school “Representation theory and Complex Analysis”, June 10-17, 2004, Venezia.
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2
Derived Categories of Quasi-abelian Categories . . . . . . . . . . . 152
3
Quasi-equivariant D-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4
Equivariant Derived Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5
Holomorphic Solution Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6
Whitney Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7
Twisted Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8
Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
9
Application to the Representation Theory . . . . . . . . . . . . . . . . 210
10 Vanishing Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
1 Introduction This note is based on five lectures on the geometry of flag manifolds and the representation theory of real semisimple Lie groups, delivered at the CIME summer school “Representation theory and Complex Analysis”, June 10–17, 2004, Venezia. The study of the relation between the geometry of flag manifolds and the representation theory of complex algebraic groups has a long history. However, it was rather recently that we realized the close relation between the representation theory of real semisimple Lie groups and the geometry of the flag manifold and its cotangent bundle. In these relations, there are two facets, complex geometry and real geometry. The Matsuki correspondence is an example: it is a correspondence between the orbits of the real semisimple group
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M. Kashiwara
on the flag manifold and the orbits of the complexification of its maximal compact subgroup. Among these relations, we focus on the diagram below.
Complex World
Real World Representations of GR o
Harish-Chandra / correspondence
O
Harish-Chandra modules
O
B-B correspondence
(D X , K)-modules
O
Riemann-Hilbert correspondence GR -equivariant sheaves o
Matsuki correspondence
/ K-equivariant sheaves
Fig. 1. Correspondences
The purpose of this note is to explain this diagram. In the Introduction, we give an overview of this diagram, and we will explain more details in the subsequent sections. In order to simplify the arguments, in the Introduction we restrict ourselves to the case of the trivial infinitesimal character. In order to treat the general case, we need the “twisting” of sheaves and of the ring of differential operators. For them, see the subsequent sections. Considerable parts of this note are joint work with W. Schmid, and were announced in [21]. Acknowledgement. The author would like to thank Andrea D’Agnolo for the o
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