Typical representations for $\mathrm{Sp}_4(F)$

TL;DR

This paper classifies typical representations of the maximal compact subgroup in the p-adic symplectic group Sp_4(F) using types and covers, advancing the understanding of the Bernstein decomposition in this setting.

Contribution

It proves that the -s-typical irreducible representations are precisely the components of induced representations from specific types, linking types for G to those for Levi subgroups.

Findings

Classification of -s-typical representations in Sp_4(F)

Identification of these representations as induced from types

Connection between types for G and Levi subgroups

Abstract

Let $F$ be a non Archimedean local field with odd residual characteristic, and let $K$ be a hyperspecial maximal compact subgroup of the $p$-adic symplectic group $G=\mathrm{Sp}_4(F)$. Let $\mathfrak{s}$ be an inertial class for $G$ in the Bernstein decomposition of the category of smooth representations of $G$, which is attached to a proper Levi subgroup $L$ of $G$. We prove that the $\mathfrak{s}$-typical irreducible representations of $K$ are the irreducible components of $\mathrm{Ind}_{J_{\mathfrak{s}}}^{K}(\lambda_{\mathfrak{s}})$, where $(J_{\mathfrak{s}},\lambda_{\mathfrak{s}})$ is an $\mathfrak{s}$-type for $G$ such that $J_{\mathfrak{s}}\subset K$, and $(J_{\mathfrak{s}},\lambda_{\mathfrak{s}})$ is a $G$-cover of a Bushnell-Kutzko maximal simple type for $L$.