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 /math> denote the
group algebra of ''G'': the space of ''F''-valued functions on ''G'' with the multiplication given by convolution. We write F /H/math> 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 /H= F H.
Similarly, there is the identification
:R := \operatorname_G(F /H = F
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 $H = \operatorname_G(V)$ is the algebra of ''G''-equivariant endomorphisms of ''V''. For each irreducible representation $W$ of ''G'', the action of ''H'' on ''V'' preserves $\tilde$ – 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

Algebras
Representation theory of Lie groups