Hecke algebra

Summary

The name Hecke algebra covers several related algebras; the most common is the Iwahori–Hecke algebra of a Coxeter group. The Iwahori–Hecke algebra is a deformation of the group algebra of the Coxeter group, depending on a parameter $q$. At $q = 1$ it is the group algebra. Let $G$ be a finite group of Lie type over a field with $q$ elements. Iwahori showed that the Hecke algebra is the endomorphism algebra of the permutation module of $G$ on the cosets of a Borel subgroup. Other Hecke algebras are built from double cosets of a subgroup, as for the Hecke operators of number theory, or from locally compact groups.