Dualizing involutions on the metaplectic GL(2)

Abstract

Let F be a non-Archimedean local field of characteristic zero. Let G=GL(2,F) and G˜=GL˜(2,F) be the metaplectic group. Let τ be the standard involution on G. A well known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we show that any lift of the standard involution to G˜ is also a dualizing involution.