Geometric analysis on small unitary representations of GL(N,R)

TL;DR

This paper provides explicit branching laws for small unitary representations of GL(N,R) when restricted to various symmetric subgroups, revealing their irreducibility and spectral composition through geometric analysis.

Contribution

It derives explicit formulas for the decomposition of small unitary representations of GL(N,R) upon restriction to symmetric subgroups, highlighting new spectral and irreducibility phenomena.

Findings

Irreducibility of restrictions for H=Sp(n,R) when λ≠0

Discrete spectrum for H=GL(n,C)

Mixed discrete and continuous spectra for H=O(p,q)

Abstract

The most degenerate unitary principal series representations {\pi}_{i{\lambda},{\delta}} (with {\lambda} \in R, \delta \in Z/2Z) of G = GL(N,R) attain the minimum of the Gelfand-Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction {\pi}_{i{\lambda},{\delta}}|_H (branching law) with respect to all symmetric pairs (G,H). For N=2n with n \geq 2, the restriction {\pi}_{i{\lambda},{\delta}}|_H remains irreducible for H=Sp(n,R) if {\lambda}\neq0 and splits into two irreducible representations if {\lambda}=0. The branching law of the restriction {\pi}_{i{\lambda},{\delta}}|_H is purely discrete for H = GL(n,C), consists only of continuous spectrum for H = GL(p,R) \times GL(q,R) (p+q=N), and contains both discrete and continuous spectra for H=O(p,q) (p>q\geq1). Our emphasis is laid on geometric analysis, which arises from the restriction of `small representations' to various subgroups.