Self-dual representations of Sp(4,F)

TL;DR

This paper investigates self-dual representations of the symplectic group Sp(4,F) over a non-Archimedean local field, establishing that Iwahori spherical representations always have a symmetric invariant bilinear form.

Contribution

It proves that for Sp(4,F), all Iwahori spherical self-dual representations admit a symmetric invariant bilinear form.

Findings

() = 1 for Iwahori spherical representations

Self-dual representations have a unique G-invariant bilinear form

The form is symmetric or skew-symmetric, with symmetry determined for specific cases

Abstract

Let $F$ be a non-Archimedean local field of characteristic $0$ and $G=Sp(4,F)$. Let $(\pi,W)$ be an irreducible smooth self-dual representation $G$. The space $W$ of $\pi$ 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 $\varepsilon({\pi})=\pm 1$ accordingly. In this paper, we show that $\varepsilon{(\pi)}=1$ when $\pi$ is an Iwahori spherical representation of $G$.