Dual pairs and contragredients of irreducible representations

Abstract

Let G be one of the classical groups GL(n), U(n), O(n) or Sp(2n), over a nonarchimedean local field of characteristic zero. It is well known that the contragredient of an irreducible admissible smooth representation of G is isomorphic to a twist of it by an automorphism of G. We prove that similar results hold for double covers of G that occur in the study of local theta correspondences.