Bernstein's presentation for the Hecke algebra Any one know of any good references for reading about Bernstein's presentation of the Iwahori Hecke algebra? I need some notes which has an example or two. It would really help.
 A: Like some of his other important ideas, Bernstein's presentation has mostly been disseminated through the papers of other people.   Probably the most influential is the 1989 JAMS paper by Lusztig, freely available from the AMS here.    Combinatorial work by Arun Ram and others involving affine Hecke algebras also depends on this viewpoint: see for example Parkinson-Ram
Two relevant recent papers with extensive references are also available on arXiv and would be worth looking at in any case: haines-pettit and goertz.
A: I found the paper of Chriss and Khuri-Makdisi (Chriss, Neil; Khuri-Makdisi, Kamal.
On the Iwahori-Hecke algebra of a $p$-adic group. Internat. Math. Res. Notices 1998, no. 2, 85--100.) quite helpful.
You may also look at Haines-Kottwitz-Prasad and Prasad.
A: In book form, the equivalence of the Coxeter and Bernstein presentations of the affine Hecke algebra appears in the first 4 chapters of Macdonald's book "Affine Hecke algebras and orthogonal polynomials".  It is very carefully written, but the notation can get a bit heavy.  When first reading it, I suggest you always assume that you are in case (1.4.1) in Macdonald's notation.
Chapter 6 does rank one examples.
A: MathSciNet search for Hecke and Bernstein Presentation gives
this ...
