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 formula_1 denote the
group algebra of "G": the space of "F"-valued functions on "G" with the multiplication given by convolution. We write formula_2 for the space of "F"-valued functions on formula_3. An ("F"-valued) 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 formula_6, let formula_7 denote the characteristic function of it. Then those formula_7's form a basis of "R".