TITLE: Definitions of Hecke algebras
QUESTION [34 upvotes]: There is a definition of Iwahori-Hecke algebras for Coxeter groups in terms of generators and relations and there is a definition of Hecke algebras involving functions on locally compact groups. Are these two concepts somehow related? I think I read somewhere that the Hecke algebra with functions includes Iwahori-Hecke algebras, is that correct? Is there a good motivation for introducing and studying either type of algebras? How much is known about their representations?

REPLY [5 votes]: The motivation for studying Iwahori-Hecke algebras (the convolution type) is the study of the unramified principal series representations. The details can get lengthy, but it is all in a paper of Borel's (Armand Borel. Admissible representations of a semi-simple group over a local ﬁeld with vectors ﬁxed under an Iwahori subgroup. Invent. Math., 35:233–259, 1976.)
The relationship between the convolution algebra and the algebra using Coxeter groups was due to Iwahori and Matsumoto (N. Iwahori and H. Matsumoto. On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Inst. Hautes Etudes Sci. Publ. Math., (25):5–48, 1965.)