On the restriction of Zuckerman's derived functor modules A_q(\lambda) to reductive subgroups

TL;DR

This paper investigates how Zuckerman's derived functor modules decompose when restricted to reductive subgroups, providing bounds and structural insights using D-modules on flag varieties.

Contribution

It establishes upper bounds for the branching laws of A_q() modules upon restriction and characterizes the resulting modules as submodules of similar type, with associated varieties identified.

Findings

Provides upper bounds for branching laws

Shows restricted modules are submodules of A_q'(')

Determines associated varieties of restricted modules

Abstract

In this article, we study the restriction of Zuckerman's derived functor (g,K)-modules A_q(\lambda) to g' for symmetric pairs of reductive Lie algebras (g,g'). When the restriction decomposes into irreducible (g',K')-modules, we give an upper bound for the branching law. In particular, we prove that each (g',K')-module occurring in the restriction is isomorphic to a submodule of A_q'(\lambda') for a parabolic subalgebra q' of g', and determine their associated varieties. For the proof, we construct A_q(\lambda) on complex partial flag varieties by using D-modules.