Dualizing involutions on the $n$-fold metaplectic cover of $\GL(2)$

TL;DR

This paper investigates dualizing involutions on the $n$-fold metaplectic cover of $ ext{GL}(2)$ over a non-Archimedean local field, establishing that such involutions exist only when $n=2$, extending classical results.

Contribution

It characterizes when the standard involution lifts to a dualizing involution on the metaplectic cover, showing this occurs exclusively for $n=2$, thus generalizing Gelfand-Kazhdan's theorem.

Findings

Dualizing involutions exist on the double cover ($n=2$).

No such involutions exist for $n eq 2$.

Extension of classical involution properties to metaplectic covers.

Abstract

Let $F$ be a non-Archimedean local field of characteristic zero and $G=\GL(2,F)$. Let $n\geq 2$ be a positive integer and $\widetilde{G}=\widetilde{\GL}(2,F)$ be the $n$-fold metaplectic cover of $G$. Let $\pi$ be an irreducible smooth representation of $G$ and $\pi^{\vee}$ be the contragredient of $\pi$. Let $\tau$ be an involutive anti-automorphism of $G$ satisfying $\pi^{\tau}\simeq \pi^{\vee}$. In this case, we say that $\tau$ is a dualizing involution. A well known theorem of Gelfand and Kazhdan says that the standard involution $\tau$ on $G$ is a dualizing involution. In this paper, we show that any lift of the standard involution to $\widetilde{G}$ is a dualizing involution if and only if $n=2$.