Siegel ${\mathfrak{p}}^2$ Vectors for Representations of $GSp(4)$

Abstract

Let $F$ be a $p$-adic field and $(\ddot{I}€,V)$ an irreducible complex representation of $G=GSp(4,F)$ with trivial central character. Let $\mathrm{Si}({\mathfrak{p}}^2)\subset G$ denote the Siegel congruence subgroup of level ${\mathfrak{p}}^2$ and $u\in N_G(\mathrm{Si}({\mathfrak{p}}^2))$ the Atkin-Lehner element. We compute the dimension of the space of $\mathrm{Si}({\mathfrak{p}}^2)$-fixed vectors in $V$ as well as the signatures of the involutions $\ddot{I}€(u)$ acting on these spaces.