Dual pairs and contragredients of irreducible representations

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 $π$ be an irreducible admissible smooth representation of $G$. It is well known that the contragredient of $π$ is isomorphic to a twist of $π$ by an automorphism of $G$. We prove a similar result for double covers of $G$ which occur in the study of local theta correspondences.