Branching laws for small unitary representations of GL(n,C)

TL;DR

This paper explicitly determines how certain minimal Gelfand--Kirillov dimension unitary representations of GL(n,C) decompose when restricted to symmetric subgroups, advancing understanding of their branching laws.

Contribution

It provides explicit branching laws for principal series representations of GL(n,C) restricted to symmetric subgroups, a novel detailed analysis in this context.

Findings

Explicit branching laws for principal series representations

Decomposition patterns when restricted to symmetric subgroups

Enhanced understanding of minimal Gelfand--Kirillov dimension representations

Abstract

The unitary principal series representations of $G=GL(n,\mathbb{C})$ induced from a character of the maximal parabolic subgroup $P=(GL(1,\mathbb{C})\times GL(n-1,\mathbb{C}))\ltimes\mathbb{C}^{n-1}$ attain the minimal Gelfand--Kirillov dimension among all infinite-dimensional unitary representations of $G$. We find the explicit branching laws for the restriction of these representations to symmetric subgroups of $G$.