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.

See also

• Gelfand pair

References

• Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, ISBN 9780387260402.
• Mark Reeder (2011) Notes on representations of finite groups, notes.