Refined Anisotropic K-types and Supercuspidal Representations

Let G be any connected reductive group defined over a nonarchimedean local field F of residual characteristic p. Under some tameness assumptions on G, we construct families of positive-depth supercuspidal representations of G = G(F ). In particular, we classify (§2.7) the representations of G that contain any anisotropic unrefined minimal K-type (in the sense of MoyPrasad [28]) that satisfies a tameness condition. These representations are induced (§2.5), and from the inducing data (§2.6) we construct corresponding refined minimal K-types (§2.8), which are just types in the sense of BushnellKutzko [4]. One feature of this construction is that the resulting families of representations need not be associated to maximal anisotropic tori. Instead, they may arise from centralizers of singular anisotropic elements, or, depending on one’s point of view, certain non-maximal anisotropic tori (those that have compact centralizers). Such “singular” supercuspidal representations are implicit in the work of Moy [24, 26] and Jabon [17] on U(2, 1) and GSp4, but this is apparently the first general construction that produces them. (Another construction, using a different approach, is due to Kim [19].) Otherwise, the families that arise here are analogous to those constructed by Carayol [5] for GLn, those that arise from generic characters in the Howe construction [13] for GLn, and those that arise from “cuspidal data of rank 1” (and sometimes higher rank) in the work of Morris on classical groups [21, 22, 23]. They also include the families constructed by Gérardin [10] for Chevalley groups.