383:. 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
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369:, 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.
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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
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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,
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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
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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.
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at the start of the definitions altogether. The problem arises again in the fifth listed property of
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clearly have to be indeterminates for this to be true in the stated generality (i.e. without explicitly excluding some roots of unity).
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Iwahori-Hecke algebra is becoming more widespread, there is still a significant body of literature that refers to them simply as
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Lusztig gave in non-modular cases an explicit isomorphism between generic Hecke algebra and the group algebra of
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I don't know enough about Hecke algebras of modular forms to determine if they merit an article separate from
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and I seem to remember that one half is only necessary for exceptional cases, maybe even only in type
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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of two bi-invariant functions is left-invariant (using a left-invariant Haar measure), but
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link. I am going to write a separate article on affine Hecke algebras soon.
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I am not quite happy with the way the definition starts out: Are the
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becomes a particular number) of Hecke algebra is isomorphic to
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being unimodular. Feel free to undo this, if I'm wrong here.
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Note that this is a lot weaker result than saying that the
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The main other uses are the Hecke algebras generated by
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618:{\displaystyle (f\ast g)(xh^{-1})}
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659:{\displaystyle (f\ast g)(x)}
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