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

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 $L$.