Talk:Iwahori–Hecke algebra

Definition
I am not quite happy with the way the definition starts out: Are the q_s elements of some ring R or are they supposed to be indeterminates over Z? I feel like two different definitions got mixed up in this. In the following definition of the multiparameter algebras the q_s are indeed elements of R so maybe one should remove the definition of A at the start of the definitions altogether. The problem arises again in the fifth listed property of Iwahori-Hecke algebras, because here the q_s clearly have to be indeterminates for this to be true in the stated generality (i.e. without explicitly excluding some roots of unity).134.61.173.178 (talk) 15:09, 24 August 2017 (UTC)

Benson-Curtis theorem and issues of accuracy
I feel that it is important (although, admittedly difficult) to maintain a balance between accuracy of the statements and encyclopaedic nature of the article. Obviously, we don't want to make incorrect claims, as I accidentally had done earlier (forgot characteristic 0!) and was corrected by 76.201.143.150. On the other hand, even for a representation theorist there are many subtle points in the theory, and sometimes it's best to just make a correct statement and not delve too much into the fine points. Maybe even that is excessive: for the case at hand, the important thing is that for R=Q, the generic Hecke algebra is isomorphic to the group algebra of W over the field of rational functions in $$q^{1/2},$$ and I seem to remember that one half is only necessary for exceptional cases, maybe even only in type $$E_8.$$ Thus it may suffice to say that
 * Lusztig gave in non-modular cases an explicit isomorphism between generic Hecke algebra and the group algebra of W.

Note that this is a lot weaker result than saying that the specialization (q becomes a particular number) of Hecke algebra is isomorphic to Q[W]. If you want to discuss roots of unity, go ahead and write a separate section (or article)!

The literature on Hecke algebras is enormous, just important contributions from Lusztig count in more than 20 papers. This is why I left the reference to Lusztig's paper establishing the isomorphism with the group algebra of W in the comments. My favorite list of references will include many more, but is hardly appropriate for Wikipedia. Arcfrk 02:42, 26 March 2007 (UTC)

Move?
Given the number of different uses of "Hecke algebra", it might be worth moving this article to (say) Iwahori–Hecke algebra, and using this page to link to the other uses. R.e.b. 00:38, 27 March 2007 (UTC)


 * One possible solution is to keep the Hecke algebras of locally compact groups on this page and move the bulk of the article to "Iwahori-Hecke algebra". What other uses of "Hecke algebra" were you thinking of? My impression is that although, contrary to Lusztig's prediction quoted in the article, the term Iwahori-Hecke algebra is becoming more widespread, there is still a significant body of literature that refers to them simply as Hecke algebras. For example, all applications mentioned in the second paragraph of the lead involve (Iwahori-)Hecke algebras of finite Weyl groups or even symmetric group. Arcfrk 01:44, 27 March 2007 (UTC)

The main other uses are the Hecke algebras generated by Hecke operators, and the Hecke algebra of K-finite distributions supported on a maximal compact subgroup K of a semisimple group, and there are some other minor uses such as  the double coset algebras of a finite index subgroup of a  group, or the algebras giving the decomposition of induced representations. All closely related, but sufficiently different to have their own articles. If you dont want to move it the article will need a lot of dablinks at the top. R.e.b. 02:20, 27 March 2007 (UTC)


 * I don't know enough about Hecke algebras of modular forms to determine if they merit an article separate from Hecke operators. As I indicated above, it's possible to move the Iwahori-Hecke algebras to a separate article, but it won't completely solve the problem because of the ambiguity of the current use in representation theory. Other algebras that you mention fall under "Hecke algebras of locally compact groups" and do not need separate articles, with the exception of Hecke algebra of a Lie group. The latter is a fairly peculiar construction, and not very popular at the moment, except in the proof of Flath's theorem on automorphic representations. My natural reaction would be to make a separate article for it (if appropriate), but not worry about it here, except a see also link. I am going to write a separate article on affine Hecke algebras soon. Arcfrk 03:08, 27 March 2007 (UTC)

The case of locally compact groups
If I see this correctly then the statement that $$\lbrace f\in C_c(G) | \forall h,h'\in H: f(hxh')=f(x)\rbrace$$ is a algebra under convolution, if $$H\leq G$$, is false for groups that are not uni-modular. The convolution $$f\ast g$$ of two bi-invariant functions is left-invariant (using a left-invariant Haar measure), but $$(f\ast g)(xh^{-1})$$ differs from $$(f\ast g)(x)$$ by $$\Delta(h)$$ if my calculations are correct.

Therefore I'm adding the requirement of $$G$$ being unimodular. Feel free to undo this, if I'm wrong here.141.35.40.141 (talk) 13:28, 2 December 2010 (UTC)