Hecke algebra of a locally compact group
can be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted
and called the Hecke ring of the pair (G,K). If we start with a Gelfand pair then the resulting algebra turns out to be commutative. In particular, this holds when
- G = SLn(Qp) and K = SLn(Zp)
and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald.
On the other hand, in the case
- G = GL2(Q) and K = GL2(Z)
The case leading to the Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk elements, and B is its Borel subgroup. 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 (Characters of reductive groups over a finite field, xi, footnote):
- 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 K, 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 K.
- Shimura (1971). Introduction to the Arithmetic Theory of Automorphic Functions (Paperback ed.). Princeton University Press. ISBN 978-0-691-08092-5.