Branching problems of Zuckerman derived functor modules

TL;DR

This paper explores the complex branching behaviors of Zuckerman's derived functor modules when restricted to subgroups, revealing diverse phenomena like infinite multiplicities and multiplicity-free cases, and proposing new conjectures.

Contribution

It provides a detailed analysis of branching laws for Zuckerman modules, highlighting new phenomena and formulating conjectures in the representation theory of real reductive groups.

Findings

Infinite multiplicities in branching laws

Existence of irreducible and multiplicity-free restrictions

Formulation of new conjectures in the field

Abstract

We discuss recent developments on branching problems of irreducible unitary representations $\pi$ of real reductive groups when restricted to reductive subgroups. Highlighting the case where the underlying $(g,K)$-modules of $\pi$ are isomorphic to Zuckerman's derived functor modules $A_q(\lambda)$, we show various and rich features of branching laws such as infinite multiplicities, irreducible restrictions, multiplicity-free restrictions, and discrete decomposable restrictions. We also formulate a number of conjectures.