Cohomology of Standard Modules on Partial Flag Varieties

TL;DR

This paper generalizes a duality theorem linking cohomological induction and geometric methods to partial flag varieties, broadening the construction of representations for real reductive Lie groups.

Contribution

It extends the duality theorem to partial flag varieties, enabling the construction of cohomologically induced modules from nonminimal parabolics.

Findings

Generalization of duality theorem to partial flag varieties

Connection between cohomological induction and D-module cohomology

Construction of new classes of representations from nonminimal parabolics

Abstract

Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group $G$ from irreducible representations of reductive subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for $G$ from certain representations of a Cartan subgroup. The duality theorem of Hecht, Mili\vci\'c, Schmid and Wolf establishes a relationship between modules cohomologically induced from minimal parabolics and the cohomology of the $\ms{D}$-modules on the complex flag variety for $G$ determined by the Beilinson-Bernstein construction. The main results of this paper give a generalization of the duality theorem to partial flag varieties, which recovers cohomologically induced modules arising from nonminimal parabolics.