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

Abstract

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