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!



description of an endomorphism algebra
I think Thiem's thesis [Unipotent Hecke algebras of GL\_n(F\_q)](http://arxiv.org/abs/math/0402383) 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.


