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
of twisted algebraic
differential operators on X. This sheaf is acyclic for the global sections functor and
its algebra of global sections
is naturally isomorphic to U\(Q). These
observations allowed Beilinson and Bernstein to define the algebraic localization
from the category of U\($)-modules to the category of (quasicoherent) sheaves of
-modules. Beilinson and Bernstein proved that if A is regular and antidominant
then A^
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\
satisfying natural
compatibility conditions, called the category of Harish-Chandra sheaves and they
showed that the restriction of A ^
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.
Previous Page Next Page