Branching Laws for Some Unitary Representations of SL(4,R)

TL;DR

This paper studies how certain unitary representations of SL(4,R) decompose when restricted to specific subgroups, providing explicit formulas and constructing new representations in the spectrum.

Contribution

It introduces formulas for restriction of $A_{rak q}(rak extlambda)$ representations to subgroups and the concept of pseudo dual pairs, advancing understanding of branching laws.

Findings

Derived restriction formulas for symplectic and complex groups

Constructed representations in the cuspidal spectrum

Introduced the concept of pseudo dual pairs

Abstract

In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(\lambda)$ of $GL(4,{\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to $GL(2,{\mathbb C})$, and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseudo dual pairs of subgroups in a reductive Lie group.