Classification of symmetric pairs with discretely decomposable restrictions of (g,K)-modules

TL;DR

This paper classifies symmetric pairs (g,h) where certain infinite-dimensional (g,K)-modules decompose discretely when restricted to (h,H∩K), with implications for tensor products of highest weight modules.

Contribution

It provides a complete classification of symmetric pairs with discretely decomposable restrictions for a broad class of (g,K)-modules, including minimal and unitarizable modules.

Findings

Classification of symmetric pairs with discretely decomposable restrictions.

Tensor product of two irreducible modules is discretely decomposable iff both are highest weight modules.

Analysis of conditions for minimal and unitarizable modules to have discretely decomposable restrictions.

Abstract

We give a complete classification of reductive symmetric pairs (g, h) with the following property: there exists at least one infinite-dimensional irreducible (g,K)-module X that is discretely decomposable as an (h,H \cap K)-module. We investigate further if such X can be taken to be a minimal representation, a Zuckerman derived functor module A_q(\lambda), or some other unitarizable (g,K)-module. The tensor product $\pi_1 \otimes \pi_2$ of two infinite-dimensional irreducible (g,K)-modules arises as a very special case of our setting. In this case, we prove that $\pi_1 \otimes \pi_2$ is discretely decomposable if and only if they are simultaneously highest weight modules.