Contragredients and a multiplicity one theorem for general Spin groups

TL;DR

This paper proves a multiplicity one theorem for restrictions of irreducible admissible representations of general Spin groups over nonarchimedean fields, extending known results for orthogonal and special orthogonal groups.

Contribution

It establishes the multiplicity free restriction property for $ ext{GPin}(n)$ and $ ext{GSpin}(n)$, and provides explicit descriptions of their contragredient representations.

Findings

Restriction to $ ext{GPin}(n-1)$ is multiplicity free.

Analogous theorem proven for $ ext{GSpin}(n)$.

Explicit description of contragredients provided.

Abstract

Each orthogonal group $\OO(n)$ has a nontrivial $\GL(1)$-extension, which we call $\GPin(n)$. The identity component of $\GPin(n)$ is the more familiar $\GSpin(n)$, the general Spin group. We prove that the restriction to $\GPin(n-1)$ of an irreducible admissible representation of $\GPin(n)$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $\GSpin(n)$. Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $\OO(n)$, and Waldspurger, who proved that for $\SO(n)$. We also give an explicit description of the contragredient of an irreducible admissible representation of $\GPin(n)$ and $\GSpin(n)$, which is needed to apply their method to our situations.