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 $$F[G]$$ denote the group algebra of G: the space of F-valued functions on G with the multiplication given by convolution. We write $$F[G/H]$$ for the space of F-valued functions on $$G/H$$. 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:
 * $$F[G/H] = F[G]^H.$$

Similarly, there is the identification
 * $$R := \operatorname{End}_G(F[G/H]) = F[G]^{H \times H}$$

given by sending a G-linear map f to the value of f evaluated at the characteristic function of H. For each double coset $$HgH$$, let $$T_g$$ denote the characteristic function of it. Then those $$T_g$$'s form a basis of R.

Application in representation theory
Let $$\varphi : G \rightarrow GL(V)$$ be any finite-dimensional complex representation of a finite group G, the Hecke algebra $$H = \operatorname{End}_G(V)$$ is the algebra of G-equivariant endomorphisms of V. For each irreducible representation $$W$$ of G, the action of H on V preserves $$\tilde{W}$$ – the isotypic component of $$W$$ – and commutes with $$W$$ as a G action.