Supercuspidal Representations: an Exhaustion Theorem

Let k be a p-adic field of characteristic zero and residue characteristic p. Let G be the group of k-points of a connected reductive group G defined over k. In [38], Yu gives a fairly general construction of supercuspidal representations of G in a certain tame situation. In this paper, subject to some hypotheses on G and k, we prove that all supercuspidal representations arise through his construction. While there have been numerous constructions of supercuspidal representations, the question of whether they are exhaustive is resolved only for depth zero representations [31, 28] and for groups of type An such as GLn [5, 22, 29], SLn [6, 7]. In [29], Moy proves the exhaustiveness for Howe’s construction of supercuspidal representations via the generalized Jacquet-Langlands correspondence in the tame case. In [22], Howe and Moy prove it by analyzing Hecke algebras when p > n. In [5], Bushnell and Kutzko construct supercuspidal representations and prove their exhaustiveness by analyzing simple types and split types with no assumption on k. Recently, Stevens showed that any supercuspidal representation of classical groups of positive depth contains a certain semisimple character [34]. However, since no analogue of the Jacquet-Langlands correspondence for general groups has been developed yet, and since types, or Hecke algebras for general groups, are far less understood than for GLn, it is not easy to extend their methods to other groups. In this paper, we approach this problem via harmonic analysis on G. We now briefly describe the main idea of the proof. From now on, we assume that the residue characteristic p of k is sufficiently large (see §3.4 for the precise condition). We first prove that any supercuspidal representation is either of depth zero or otherwise contains a K-type constructed in [27]. This we do by relating the Plancherel formulas on G and on its Lie algebra g and by using some results on asymptotic expansions [27]. We relate the K-type further to a supercuspidal type constructed in [38] by analyzing appropriate Hecke algebras and Jacquet modules. Before expanding our account of the main strategy of the proof, we first recall some results on Γ-asymptotic expansions.