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 ${\displaystyle F[G]}$ denote the group algebra of G: the space of F-valued functions on G with the multiplication given by convolution. We write ${\displaystyle F[G/H]}$ for the space of F-valued functions on ${\displaystyle G/H}$. 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:

${\displaystyle F[G/H]=F[G]^{H}.}$

Similarly, there is the identification

${\displaystyle R:=\operatorname {End} _{G}(F[G/H])=F[G]^{H\times H}}$

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

Application in representation theory

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