description of an endomorphism algebra 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!
 A: 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.
A: 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.
