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.
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