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Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).

I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd like to relate it with a structure of Hecke algebra, which is usually defined as $End_{G^F}( \mathbb{C}[G^F/B^F] ) := End_{G^F} ( Ind_{B^F}^{G^F} 1 )$. I hope to find that the endomorphism algebra is isomorphic to some kind of extension of the Hecke algebra by the torus $T$.

Thank you!

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2 Answers 2

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I think Thiem's thesis Unipotent Hecke algebras of GL_n(F_q) discusses this in detail -- if I'm not mistaken the Hecke algebra you're asking about goes by the name Yokonuma Hecke algebra and there's a fair amount of literature on it.

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The work Yokonuma did (influenced by Iwahori), which is referenced by Thiem as well as by Carter, extends somewhat the construction by Gelfand and Graev of a character induced from a regular character of the unipotent subgroup. (See Carter's book, 8.1.) This is at the opposite extreme from the question posed here, where induction starts with the trivial character. I'm doubtful about applying the Iwahori-Hecke approach directly in the latter case. –  Jim Humphreys Jun 18 '12 at 13:30
    
It's my understanding that the phrase "Yokonuma Hecke algebra" is precisely the case asked about here, the case of the trivial character of U, or functions on G/U (as opposed to the Whittaker case studied by Gelfand-Graev). That's in any case what Thiem writes in that paper and in greater detail in his thesis, available <a href="math.colorado.edu/~thiemn/preprints.html>here</a>; –  David Ben-Zvi Jun 21 '12 at 1:56
    
Oops link didn't convey: math.colorado.edu/~thiemn/preprints.html –  David Ben-Zvi Jun 21 '12 at 1:56
    
Thank you for this reference, I'll read i too, it could be useful too. –  th.ng Jun 21 '12 at 7:44
    
Now that I've read what you mention, the generators/relations presentation of the Yokonuma-Hecke algebra is (almost) precisely what I needed. Though, I'll also have to see how what Jim said could fit in. Thank you again. –  th.ng Jun 25 '12 at 8:11

Here you are working over $\mathbb{C}$ (or perhaps any other splitting field of characteristic 0 for $G$). So the representation you are starting with is just the direct sum over all characters $\chi$ of $T^F$ of the various induced characters from $B^F$ to $G^F$ obtained by lifting $\chi$ first to a character of $B^F$ and then inducing. All of these induced characters of $G^F$ have the same degree, but some are irreducible and others not (as in the extreme case $\chi =1$). So the resulting endomorphism algebra of the large direct sum will be cumbersome to study. It's helpful to consult Chapter 10 of Roger Carter's 1985 book for a more precisely organized program along these lines, due largely to Howlett and Lehrer. Naturally the usual Hecke algebra for $\chi=1$ plays a role here, as do analogous endomorphism algebras for other $\chi$.

But your expressed hope seems too loosely formulated in this extremely complicated situation. Have you tried to work this out explicitly when $G^F = \mathrm{SL}(2,p)$? In that case all the induced representations are easily identified.

By the way, there is a version of all this worked out in the defining characteristic $p$ by Carter and Lusztig in their old paper Modular representations of finite groups of Lie type, Proc. London Math. Soc. (3) 32 (1976), no. 2, 347–384. They use BN-pairs as a framework and develop intertwining operators in the spirit of Hecke algebras, but with some degeneracy.

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Thank you for your answer, actually, I didn't expect the permutation module $\mathbb{C}[G^F/U^F]$ so be the direct sum of all the induced representations, can you tell me why ? or give me a reference for it ? At the moment, what I want to do isn't clear : I was looking for something to replace the (usual) Hecke algebra, which is related to the unipotent representations of $G^F$, in the case where one considers the other representations (all this is related to Deligne-Lusztig theory and more precisely, instead of studying the varieties $X_w$, I want to have something for $Y_w$). –  th.ng Jun 20 '12 at 8:33
    
Concerning the permutation module (which is an induced module from a trivial representation), I was just relying on standard tools in characteristic 0: transitivity of induction, Frobenius reciprocity, complete reducibility. Concerning the study of Deligne-Lusztig representations, I'm doubtful that there is a simpler way to study them than found in the existing literature; but of course that's always an open question. –  Jim Humphreys Jun 20 '12 at 16:13

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