Hecke algebra of a finite group


The Hecke algebra of a finite group is the algebra spanned by the double cosets HgH of a subgroup H of a finite group G. It is a special case of a Hecke algebra of a locally compact group.

Definition

Let F be a field of characteristic zero, G a finite group and H a subgroup of G. Let denote the
group algebra of G: the space of F-valued functions on G with the multiplication given by convolution. We write for the space of F-valued functions on. An function on G/H determines and is determined by a function on G that is invariant under the right action of H. That is, there is the natural identification:
Similarly, there is the identification
given by sending a G-linear map f to the value of f evaluated at the characteristic function of H. For each double coset, let denote the characteristic function of it. Then those 's form a basis of R.