Hecke algebras of classical groups over p -adic fields and supercuspidal representations

Abstract

Let k be a p -adic field with involution σ 0 and let k 0 be its fixed subfield of k . Let V be a finite dimensional vector space defined over k equipped with ε-Hermitian form < , >. Let G be the connected component of the group of isometries on ( V , < , >). In order to understand the admissible representations of G, we construct a large family of Hecke algebras on G. In fact, we conjecture that all admissible representations of G arise in our construction. As a corollary of this construction, we also get many (possibly all) supercuspidal representations. Our constructions are valid when the residue characteristic of k 0 is large.