INTRODUCTION

One of the most fundamental problems in representation theory is to under-

stand the admissible representations of a real connected semisimple Lie group Go

with finite center. One approach to this problem, the geometric realizations ap-

proach, involves constructing the representations of Go as sheaf cohomology mod-

ules of Go-equivariant sheaves on a Go-space. The simplest construction of this

type is the famous Borel-Weil theorem which exhibits each finite dimensional irre-

ducible representation of Co as the space of global sections of a certain line bundle

on the flag variety X of the complexified Lie algebra g of Go-

The study of infinite dimensional representations of Go and their geometric re-

alizations involves many technical difficulties that do not appear in the Borel-Weil

theorem. One strategy for addressing these difficulties is to avoid them and instead

study the related but technically simpler problem of constructing Harish-Chandra

modules. This is, inside each admissible Go representation is a dense, countably

dimensioned subspace called its underlying Harish-Chandra module (with respect

to a fixed maximal compact subgroup Ko of Go). A Harish-Chandra module is not

a Go module but it does have compatible actions of the complexified Lie algebra g

and the complexification of Ko, K. In 1981, Alexander Beilinson and Joseph Bern-

stein introduced an elegant method for geometrically constructing Harish-Chandra

modules called algebraic localization. Let us briefly describe their method.

To each A in the dual ()* of a Cartan subalgebra of g there is associated a

character j)\ on the center of the universal enveloping algebra U(Q) and an algebra

U\(Q) which, by definition, is the quotient of U(&) by the ideal generated by the

kernel of (j)\. A U\(Q)-module is just a g-module with infinitesimal character (j)\.

Beilinson and Bernstein associated to each A a sheaf

V^g

of twisted algebraic

differential operators on X. This sheaf is acyclic for the global sections functor and

its algebra of global sections

T(X,V^g)

is naturally isomorphic to U\(Q). These

observations allowed Beilinson and Bernstein to define the algebraic localization

functor

^e(V)=Vf*

®UA5)V

from the category of U\($)-modules to the category of (quasicoherent) sheaves of

V\

s

-modules. Beilinson and Bernstein proved that if A is regular and antidominant

then A^

g

is an equivalence of categories and its quasi-inverse is the global sections

functor T(X, •). Their main theorem is a refinement of this equivalence. Namely,

they defined a category of iiT-equivariant coherent V\

g-modules

satisfying natural

compatibility conditions, called the category of Harish-Chandra sheaves and they

showed that the restriction of A ^

g

to the subcategory of Harish-Chandra U\(Q)-

modules is an equivalence of this category with the category of Harish-Chandra

Received by editor August 1, 1997.