CHAPTE R 1

Preliminaries

Throughout this paper let Go be a real connected semisimple Lie group with

finite center and denote its complexified Lie algebra by g. Let X be the complex

manifold of all Borel subalgebras of g (i.e., the flag variety of g). Let Ox denote

the sheaf of holomorphic functions on X and let Tx denote the sheaf of differential

operators with holomorphic coefficients on X. Denote by UQ(Q) the quotient of the

enveloping algebra U(g) by the ideal generated by the kernel of the trivial character

on the center Z(g) oiU(g).

We are interested in studying Go-representations (as opposed to representa-

tions of Harish-Chandra modules) and so we must use homological algebra in the

context of topological vector spaces. In particular, throughout this paper we work

with topological modules over topological algebras. The theory of homological al-

gebra in the topological context is explained in detail in [T] but most of the results

from this theory which are needed in this paper are outlined in [HT]. The theory

of topological homological algebra closely parallels ordinary homological algebra,

with a few important exceptions. Among these exceptions are : (1) a short exact

sequence of topological vector spaces does not necessarily split; (2) since modules

and algebras are topological, ordinary tensor product, ®, must be replaced by com-

pleted projective topological tensor product, 0 and (3) the category of topological

vector spaces is additive but it is not abelian. This is true whether we insist that

topological vector spaces be Hausdorff or not. Indeed given any continuous mor-

phism / : W — Y between two topological vector spaces, ker(/) exists in either

case and is the ordinary vector space kernel of / with the induced topology. If

non Hausdorff spaces are not allowed in our category then coker(/) = Y/im(/) . If

non-Hausdorff spaces are allowed in our category then coker(/) = Y / i m ( / ) . The

problem is that, in either case, it can happen that ker(/) = coker(/) = 0 and / is

not an isomorphism. In this paper we shall not insist that topological vector spaces

be Hausdorff. Thus, Y / i m ( / ) is the appropriate notion of coker(/) for a morphism

/ : W — Y. Note, however, that we shall endeavor to always work in a class of

topological vector spaces (described below) whose members are Hausdorff. It's just

that some of our constructions a priori take us outside this class.

The operation of completed projective topological tensor product does not gen-

erally behave well and so it is necessary to work, whenever possible, in a class of

topological vector spaces in which this operation is nice. The category of topo-

logical vector spaces most suitable for our purposes is the category of DNF spaces

(duals of nuclear Frechet spaces). The main point is that the class of DNF spaces

not only includes all the objects we want to study but it also has some very special

properties. Among these are:

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