# D-Modules and Spherical Representations. (MN-39):

Frédéric V. Bien
Pages: 142
https://www.jstor.org/stable/j.ctt7zvs32

1. Front Matter
(pp. [i]-[vii])
(pp. [viii]-[x])
3. Introduction
(pp. 1-8)

The goal of this work is to study the representations of reductive Lie groups which occur in the space of smooth functions on an indefinite symmetric space. The representations realized by square integrable functions were con structed by Flensted-Jensen, Oshima and Matsuki. We prove that the discrete series has multiplicity one, except possibly in a few exceptional cases, and we present a cohomological formula for the multiplicities of standard representations. We study these representations by Beilinson and Bernstein’s theory of differential operators on complex flag manifolds, and we present a summary of this theory. We also find a canonical map...

4. I Localization Theory
(pp. 9-28)

LetXbe a smooth complex algebraic manifold. Let$o = {o_x}$be the sheaf of regular functions onXand$D = {D_X}$be the sheaf of differential operators onX.AD-moduleMis a sheaf onXwhich is quasi-coherent eis amO-module and which has a structure of module overD. LetD-mod denote the category of leftD-modules. We shall deal with some sheaves of rings slightly more general thanD. Consider the category of pairs$(A,{i_A})$whereAis a sheaf onXofC-algebras and${i_A}:O \to A$is a morphism ofC-algebras. The pair$(D,i)$where$i:O \to A$is...

5. II Spherical D-modules
(pp. 29-67)

LetGbe a complex connected reductive linear algebraic group. LetBbe the variety of Borel subgroups ofG, andPthe variety of subgroups ofGconjugate to a fixed parabolic subgroupP. By analogy with the caseG=GL(n), we will also refer toBas the variety of complete flags, or the full flag variety, andPwill be called a partial flag variety. LetKbe an algebraic subgroup ofG; it acts onBandP. Recall thatKis called admissible if the number ofK-orbits inBis finite. In this...

6. III Microlocalization and Irreducibility
(pp. 68-83)

Consider a flag spaceX, a sheaf Dλof twisted differential operators onXand an irreducible Dλ-moduleM. Suppose λ is dominant, then we will show thatM= Г(X, M) is an irreducible module over Dλ=Г(X, Dλ) or it vanishes. However,U=U(g) may not generate all of Dλand henceMcould be reducible as aU-module.

In this chapter, we obtain a criterion for the irreducibility ofMas aU-module (Theorem 7.2). Its proof requires a microlocalization technique which is of independent interest and is explained at the beginning of the chapter. This result...

7. IV Singularities and Multiplicities
(pp. 84-124)

In III.7, we saw that the properties of the moment map π lead to results about the irreducibility of Dλ-modules considered as g-modules. In this chapter, we continue in this direction to prove our main result, namely, that the discrete series of a reductive symmetric space is multiplicity free, except possibly for a few excetional cases.

We first state some known results about the singularities of the image of π . According to III.6.2, if the moment map is birational and has a normal image, then the global section functor Г sends an irreducible Dλ-moduleMto an irreducible g-module...

8. Bibliography
(pp. 125-129)
9. Index
(pp. 130-131)