Dual pairs and contragredients of irreducible representations

TL;DR

This paper extends known results about the contragredient of irreducible representations from classical groups to their double covers, relevant in local theta correspondence, revealing a similar twist relation.

Contribution

It proves that the contragredient of an irreducible admissible smooth representation of double covers of classical groups is isomorphic to a twist of the original representation by an automorphism.

Findings

Contragradients of representations on double covers relate to twists by automorphisms.

Extends classical results to non-linear covers in local representation theory.

Provides tools for understanding local theta correspondences.

Abstract

Let $G$ be a classical group $\GL(n)$, $\oU(n)$, $\oO(n)$ or $\Sp(2n)$, over a non-archimedean local field of characteristic zero. Let $\pi$ be an irreducible admissible smooth representation of $G$. It is well known that the contragredient of $\pi$ is isomorphic to a twist of $\pi$ by an automorphism of $G$. We prove a similar result for double covers of $G$ which occur in the study of local theta correspondences.