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

Let k be a p-adic field with involution σ₀ and let k₀ 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 k₀ is large.