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

TL;DR

This paper computes the dimension and involution signatures of fixed vectors under Siegel congruence subgroups in certain representations of GSp(4) over p-adic fields, advancing understanding of local representation theory.

Contribution

It provides explicit calculations of fixed vector dimensions and involution signatures for GSp(4) representations at level p^2, a novel contribution to local representation theory.

Findings

Dimension formulas for fixed vectors under Siegel level p^2

Signatures of Atkin-Lehner involutions on these spaces

Enhanced understanding of GSp(4) local representations

Abstract

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