Siegel Vectors for Nongeneric Depth Zero Supercuspidals of $GSp(4)$

TL;DR

This paper computes the dimensions of Siegel-invariant vectors in certain depth zero supercuspidal representations of $GSp(4)$ over non-archimedean local fields, providing explicit results for all levels.

Contribution

It introduces a method to explicitly calculate Siegel-invariant vector spaces in nongeneric depth zero supercuspidals of $GSp(4)$, extending understanding of their structure.

Findings

Dimensions of Siegel-invariant vectors are explicitly determined for all levels.

Results apply to nongeneric supercuspidal representations over fields with even residual characteristic.

Provides new insights into the structure of depth zero supercuspidals in symplectic groups.

Abstract

Let $F$ be a non-archimedean local field of characteristic zero. If $F$ has even residual characteristic, we assume $F/\mathbb{Q}_2$ is unramified. Let $V$ be a depth zero, irreducible, nongeneric supercuspidal representation of $GSp(4, F)$. We calculate the dimensions of the spaces of Siegel-invariant vectors in $V$ of level $\mathfrak{p}^n$ for all $n\geq0$.