all open compact subgroups K, and a left Haar measure on . Let denote the space of locally constant functions on with compact support. With the multiplicative structure given by becomes not necessarily unital associative -algebra. It is called the Hecke algebra of G and is denoted by . The algebra plays an important role in the study of smooth representations of locally profinite groups. Indeed, one has the following: given a smooth representation of G, we define a new action on V: Thus, we have the functor from the category of smooth representations of to the category of non-degenerate -modules. Here, "non-degenerate" means . Then the fact is that the functor is an equivalence. Notes References Corinne Blondel, Basic representation theory of reductive p-adic groups Topological groups