Classification of discretely decomposable A_q(\lambda) with respect to reductive symmetric pairs

TL;DR

This paper classifies when certain derived functor modules associated with reductive Lie algebras are discretely decomposable upon restriction to symmetric pairs, using criteria and classifications from prior work.

Contribution

It provides a complete classification of triples (g, g', q) for which the modules are discretely decomposable, extending understanding of module restrictions in representation theory.

Findings

Classification of triples (g, g', q) for discretely decomposable modules

Application of criteria for discretely decomposable restrictions

Utilization of Berger's classification of symmetric pairs

Abstract

We give a classification of the triples (g,g',q) such that Zuckerman's derived functor (g,K)-module A_q(\lambda) for a \theta-stable parabolic subalgebra q is discretely decomposable with respect to a reductive symmetric pair (g,g'). The proof is based on the criterion for discretely decomposable restrictions by the first author and on Berger's classification of reductive symmetric pairs.