A Geometric Embedding for Standard Analytic Modules

Abstract

In this manuscript we make a general study of the representations realized, for a reductive Lie group of Harish-Chandra class, on the compactly supported sheaf cohomology groups of an irreducible finite-rank polarized homogeneous vector bundle defined in a generalized complex flag space. In particular, we show that the representations obtained are minimal globalizations of Harish-Chandra modules and that there exists a whole number q, depending only on the orbit, such that all cohomologies vanish in degree less than q. The representation realized on the q-th cohomology group is called a standard analytic module. Our main result is a geometric proof that a standard analytic module embeds naturally in an associated standard module defined on the full flag space of Borel subalgebras. As an application, we give geometric realizations for irreducible submodules of some principal series representations in case the group is complex.