A note on self-dual representations of Sp(4,F)

Abstract

Let F be a non-Archimedean local field of characteristic 0 and G=Sp(4,F). Let (π,W) be an irreducible smooth self-dual representation of G. The space W of π admits a non-degenerate G-invariant bilinear form (,) which is unique up to scaling. It can be shown that (,) is either symmetric or skew-symmetric and we set ε(π)=±1 accordingly. In this paper, we show that ε(π)=1 when π is an Iwahori-spherical representation of G.