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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