Self-dual representations with vectors fixed under an Iwahori subgroup

Let G be the group of F-points of a split connected reductive F-group over a non-Archimedean local field F of characteristic 0. Let π be an irreducible smooth self-dual representation of G. The space W of π carries a non-degenerate G  -invariant bilinear form (,) which is unique up to scaling. The form is easily seen to be symmetric or skew-symmetric and we set ε(π)=±1ε(π)=±1 accordingly. In this article, we show that ε(π)=1ε(π)=1 when π is a generic representation of G with non-zero vectors fixed under an Iwahori subgroup I.