Some Hecke Algebras Associated to the P-adic Group Gl(v )

Let G be a locally compact, totally disconnected, Hausdorff topological group with a neighborhood basis {Ki}i of the identity consisting of compact, open, normal subgroups. Often K ⊂ G will denote a compact, open subgroup of G. We consider representations (ρ, V ) of G, where V is often infinite-dimensional. We say the representation is smooth if V = ∪KV K , i.e., if every v ∈ V is fixed by some compact, open subgroup of G. We say the representation is admissible if dimC(V K) <∞, for all open, compact subgroups K. We consider the set C∞ c (G) := {f : G→ C : f is locally constant and compactly supported}. It is an associative algebra over C under convolution. The important property of this algebra (sometimes denoted the Hecke algebra H(G)) is that there is an equivalence of categories between the smooth representations of G and the representations of H(G).