Tensor product decomposition for rank-one spin groups I : unitary principal series representations

TL;DR

This paper explicitly decomposes tensor products of certain unitary representations of the rank-one spin group, advancing understanding of their structure and spectral properties.

Contribution

It provides a direct integral decomposition for tensor products involving principal series and arbitrary irreducible unitary representations of Spin(n,1).

Findings

Explicit decomposition formulas derived.

Applicable to a broad class of unitary representations.

Enhances understanding of representation theory for spin groups.

Abstract

We provide an explicit direct integral decomposition for the tensor product representation $\pi_1\widehat{\otimes}\pi_2$ of the rank one spin group $\mathrm{Spin}(n,1)$ whenever $\pi_1$ is a unitary principal series representation and $\pi_2$ is an arbitrary irreducible unitary representation of $\mathrm{Spin}(n,1)$.