Restriction of irreducible unitary representations of Spin(N,1) to parabolic subgroups

TL;DR

This paper explicitly describes how irreducible unitary representations of Spin(N,1) decompose when restricted to parabolic subgroups, confirming Duflo's conjecture for tempered representations using advanced harmonic analysis tools.

Contribution

It provides explicit branching laws for all irreducible unitary representations of Spin(N,1) restricted to parabolic subgroups and verifies Duflo's conjecture in this context.

Findings

Restriction decomposes into finite sums of irreducible representations

Duflo's conjecture holds for tempered representations of Spin(N,1)

Branching laws are determined by the orbit method and moment map behavior

Abstract

In this paper, we obtain explicit branching laws for all irreducible unitary representations of $\Spin(N,1)$ restricted to a parabolic subgroup $P$. The restriction turns out to be a finite direct sum of irreducible unitary representations of $P$. We also verify Duflo's conjecture for the branching law of tempered representations of $\Spin(N,1)$ with respect to a parabolic subgroup $P$. That is to show: in the framework of the orbit method, the branching law of a tempered representation is determined by the behavior of the moment map from the corresponding coadjoint orbit. A few key tools used in this work include: Fourier transform, Knapp-Stein intertwining operator, Casselman-Wallach globalization, Zuckerman translation principle, du Cloux's results for smooth representations of semi-algebraic groups.