Iwahori-Spherical Representations

Abstract

Let F be a nonarchimedean local field of characteristic zero, and let $(\pi ,V)$ be an irreducible admissible representation of $\mathrm {GSp}(4,F)$ with trivial central character. We will say that $\pi$ is Iwahori-spherical if the space $V^I$ of vectors in V  fixed by I is non-zero, where I is the Iwahori subgroup of $\mathrm {GSp}(4,F)$. The goal of this chapter is to describe the actions of the stable Hecke operators on $V_s(1)$ when $\pi$ is Iwahori-spherical. Since $\mathrm {K}_s(\mathfrak {p})=\mathrm {Kl}(\mathfrak {p})$, $V_s(1)$ is simply the space $V^{\mathrm {Kl}(\mathfrak {p})}$ of vectors fixed by the Klingen congruence subgroup $\mathrm {Kl}(\mathfrak {p})$ of level $\mathfrak {p}$.