Self-dual representations with vectors fixed under an Iwahori subgroup

TL;DR

This paper proves that generic irreducible smooth self-dual representations of split reductive groups over non-Archimedean fields, with non-zero Iwahori-fixed vectors, have a symmetric invariant bilinear form.

Contribution

It establishes that such representations always have a symmetric form, confirming a specific case of a broader conjecture in representation theory.

Findings

The invariant bilinear form is symmetric for generic Iwahori-fixed representations.

Self-dual representations with Iwahori-fixed vectors have a symmetric invariant form.

The result applies to split reductive groups over non-Archimedean local fields.

Abstract

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 $\pi$ be an irreducible smooth self-dual representation of $G$. The space $W$ of $\pi$ 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 $\varepsilon({\pi})=\pm 1$ accordingly. In this article, we show that $\varepsilon{(\pi)}=1$ when $\pi$ is a generic representation of $G$ with non-zero vectors fixed under an Iwahori subgroup $I$.