On typical representations for depth-zero components of split classical groups

Let $\mathbf{G}$ be a split classical group over a non-Archimedean local field $F$ with the cardinality of the residue field $q_F>5$. Let $M$ be the group of $F$-points of a Levi factor of a proper $F$-parabolic subgroup of $\mathbf{G}$. Let $[M,{\sigma }_M{]}_M$ be an inertial class such that ${\sigma }_M$ contains a depth-zero Moy–Prasad type of the form $(K_M,{\tau }_M)$, where $K_M$ is a hyperspecial maximal compact subgroup of $M$. Let $K$ be a hyperspecial maximal compact subgroup of $\mathbf{G}(F)$ such that $K$ contains $K_M$. In this article, we classify $\mathfrak{s}$-typical representations of $K$. In particular, we show that the $\mathfrak{s}$-typical representations of $K$ are precisely the irreducible subrepresentations of ${\text{ind}}_J^K\lambda$, where $(J,\lambda )$ is a level-zero $G$-cover of $(K\cap M,{\tau }_M)$.