# Bernstein's presentation for the Hecke Algebra

Any one know of any good references for reading about the Bernsteins Presentation of the Iwahori Hecke Algebra? I need some notes which has an example or two. It would really help.

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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.

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Do you know of a reference that treats the nonsplit case as well? Even the case of a quasisplit group that splits over an unramified extension (a so-called "unramified group" by some) would be helpful to me. –  Marty Sep 28 '10 at 16:27
Marty: The short answer is no. My knowledge of p-adic groups and such is fairly superficial. The older work of Iwahori-Matsumoto on "Hecke" algebras involved just split groups, but since then the subject has been developed (by a lot of people) far beyond my attention span. –  Jim Humphreys Sep 28 '10 at 17:53
Marty: Lusztig sets everything up to work in the non-split case. (Would you expect anything less?) He does not mention algebraic groups in the JAMS article, and so in order to connect his algebra to an algebraic group, you'd have to dig through Bruhat-Tits. For the unramified case, that digging is (as you probably know) not too painful. –  user2490 Sep 29 '10 at 3:03
Yes, the affine Hecke algebra is not tied directly to a specific application though it was originally studied in the context of split $p$-adic groups (where the affine Weyl group can replace the finite Weyl group to give an interesting BN-pair structure in the algebraic group). –  Jim Humphreys Sep 29 '10 at 17:46
@Marty: I think if you look at Lusztig's second paper on unipotent representations of p-adic groups he discusses the unramified case, possibly quite tersely, but it might be worth a look. –  Kevin McGerty Oct 1 '10 at 9:19

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.

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Dear Amri, Kottwitz once gave me a short set of notes (4 pages, I think) about Bernstein's presentation (maybe for split groups?), which I found very helpful. Did you ever see those? Do you know if they ever appeared somewhere (or are available online)? –  Emerton Oct 1 '10 at 5:45
He gave some lectures on Iwahori-Hecke algebras in 2000 or 2001. The notes for those lectures were [hugely] expanded into our paper by Tom. Maybe the notes that you are talking of were the original notes for those lectures? I do not know of anything else that got published. Tom is probably a better person to ask. –  Amritanshu Prasad Oct 1 '10 at 10:09

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.

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Thanks. I will take a look at it. –  user8974 Mar 9 '11 at 3:02

MathSciNet search for Hecke and Bernstein Presentation gives this ...

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