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

TL;DR

This paper tackles the explicit direct integral decomposition of the tensor product of irreducible unitary representations for the rank-one spin group G=Spin(m+1,1), focusing on the case where the first factor is a unitary principal series. The authors reduce the problem to parabolic restriction to P and apply a Mackey-type approach to express the tensor product as a finite sum of induced representations Ind_P^G(T_delta); they then decompose each Ind_P^G(T_delta) via Anh-Mackey reciprocity together with the Plancherel formula for G. The main contribution is an explicit, parity-dependent description of Ind_P^G(T_delta) into discrete and continuous spectrum components, parameterized by combinatorial sets derived from the restricted root data. This yields a complete, computable direct integral decomposition for pi1 ⊗ pi2 in this setting, with potential applications in harmonic analysis on G and theoretical physics contexts where Spin(m+1,1) symmetries arise.

Abstract

We provide an explicit direct integral decomposition for the tensor product representation $π_1\widehat{\otimes}π_2$ of the rank one spin group $\mathrm{Spin}(n,1)$ whenever $π_1$ is a unitary principal series representation and $π_2$ is an arbitrary irreducible unitary representation of $\mathrm{Spin}(n,1)$.

Theorems & Definitions (7)

• Remark 2.2
• Lemma 3.1
• proof : Proof of Lemma
• proof : Proof of Proposition
• proof
• Theorem 3.5
• proof