Localization of cohomological induction

TL;DR

This paper provides a geometric realization of cohomologically induced modules for real semisimple Lie groups, extending duality theorems to more general pairs and connecting algebraic induction with sheaf cohomology of D-modules.

Contribution

It extends the duality theorem for cohomological induction to broader pairs (g,K) and (h,L), linking algebraic and geometric perspectives.

Findings

Sheaf cohomology groups of tensor products are isomorphic to cohomologically induced modules.

Extended duality theorem applies to more general pairs (g,K) and (h,L).

Provides a geometric realization of cohomologically induced (g,K)-modules.

Abstract

We give a geometric realization of cohomologically induced (g,K)-modules. Let (h,L) be a subpair of (g,K). The cohomological induction is an algebraic construction of (g,K)-modules from a (h,L)-module V. For a real semisimple Lie group, the duality theorem by Hecht, Milicic, Schmid, and Wolf relates (g,K)-modules cohomologically induced from a Borel subalgebra with D-modules on the flag variety of g. In this article we extend the theorem for more general pairs (g,K) and (h,L). We consider the tensor product of a D-module and a certain module associated with V, and prove that its sheaf cohomology groups are isomorphic to cohomologically induced modules.