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 $\varepsilon(π)=\pm 1$ accordingly. In this article, we show that $\varepsilon{(π)}=1$ when $π$ is a generic representation of $G$ with non-zero vectors fixed under an Iwahori subgroup $I$.

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