Multiplicity One Theorem for General Spin Groups: The Archimedean Case

TL;DR

This paper proves a multiplicity-one theorem for restrictions of irreducible representations of general spin and Pin groups over Archimedean fields, establishing that such restrictions are multiplicity free.

Contribution

It establishes the multiplicity-one property for restrictions of irreducible Casselman-Wallach representations of general spin and Pin groups over Archimedean fields.

Findings

Proves multiplicity-free restriction for GSpin(V) to GSpin(W).

Proves multiplicity-free restriction for GPin(V) to GPin(W).

Extends multiplicity-one results to the Archimedean setting.

Abstract

Let $\GSpin(V)$ (resp. $\GPin(V)$) be a general spin group (resp. a general Pin group) associated with a nondegenerate quadratic space $V$ of dimension $n$ over an Archimedean local field $F$. For a nondegenerate quadratic space $W$ of dimension $n-1$ over $F$, we also consider $\GSpin(W)$ and $\GPin(W)$. We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups ($\GSpin(V), \GSpin(W)$) and also for a pair of groups ($\GPin(V), \GPin(W)$); namely, we prove that the restriction to $\GSpin(W)$ (resp. $\GPin(W)$) of an irreducible Casselman-Wallach representation of $\GSpin(V)$ (resp. $\GPin(V)$) is multiplicity free.