Let (G, K) be a pair consisting of a unimodular locally compact topological group G and a closed subgroup K of G. Then the space of bi-K-invariant continuous functions of compact support can be endowed with a structure of an associative algebra under the operation of convolution.
[3][4] When G is a finite group and K is any subgroup of G, then the Hecke algebra is spanned by double cosets of H\G/H.
For the special linear group over the p-adic numbers, the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald.
Iwahori showed that the Hecke ring is obtained from the generic Hecke algebra Hq of the Weyl group W of G by specializing the indeterminate q of the latter algebra to pk, the cardinality of the finite field.
George Lusztig remarked in 1984:[5] I think it would be most appropriate to call it the Iwahori algebra, but the name Hecke ring (or algebra) given by Iwahori himself has been in use for almost 20 years and it is probably too late to change it now.Iwahori and Matsumoto (1965) considered the case when G is a group of points of a reductive algebraic group over a non-archimedean local field F, such as Qp, and K is what is now called an Iwahori subgroup of G. The resulting Hecke ring is isomorphic to the Hecke algebra of the affine Weyl group of G, or the affine Hecke algebra, where the indeterminate q has been specialized to the cardinality of the residue field of F.