Self-dual representations of Sp(4,F)

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 $G$. The space $W$ of $π$ admits 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 paper, we show that $\varepsilon{(π)}=1$ when $π$ is an Iwahori spherical representation of $G$.

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