TITLE: Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?
QUESTION [11 upvotes]: Deligne-Lusztig theory
is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X_T$$^*$, take etale cohomology, get a virtual character $R_T^\theta$, maybe it's reducible, so you try to decompose it.
Gelfand-Graev character
is awesome. You take a maximal unipotent subgroup in some maximal split Borel subgroup, take a generic character, induce the character to the whole group, and you get many interesting subrepresentations.
My question

Is the Gelfand-Graev character equal to the character of the cohomology of some sheaf on some nice variety, similar to a Deligne-Lusztig character?

Why is this interesting?
Say you have an $R_T^\theta$ that is reducible. Before trying to find explicitly all constituents, let's try to decompose it first into constituents of the Gelfand-Graev character (generic), and the rest (not generic). If $R_T^\theta$ has exactly two subrepresentations, one generic and one not generic, then we needn't look further.
What am I looking for?
The best thing would be if there was a sheaf $F_{GG}$ on $X_T$ with cohomology, in the $\ell(w)$-degree, with character equal to the Gelfand-Graev character. Then we might have a sequece of sheaves
$$0\rightarrow F' \rightarrow F_\theta \rightarrow F_{GG} \rightarrow F^{''} \rightarrow 0$$
and we might get that the cohomologies of $F'$ and $F^{''}$ will break our $R_T^\theta$ into two parts.
So, in essence, what I'm looking for, is a geometric way to break a Deligne-Lusztig character into its genereic and non-generic parts.
This might not be possible, at least not in the way I described, which is very naive and wishful. The sentence before last should be regarded as the real question.
(*) Non-standard notation, I know. Fix some maximal $F$-stable torus, let $w$ be the Weyl element that twists the torus to desired $T$, and let $X_T=X(w)$, where $X(w)$ is the standard notation Deligne-Lusztig variety.

REPLY [3 votes]: This is not precisely what you're looking for, but Bonnafé and Rouquier have made in
"Coxeter orbits and modular representations." Nagoya Math. J., 183 :1–34, 2006,
the conjecture  that the DL-restriction of a GG-module is a (suitably shifted) GG-module. 
In particular it is concentrated in one (explicit) degree. This is even stated for integral coefficients and they give a geometric proof for T=Coxeter torus. Olivier Dudas has proved this statement for any torus in "Deligne-Lusztig restriction of a Gelfand-Graev character",
Annales scientifiques de l'ENS (42), 2009, pp 653-674. He also explains a couple of consequences of the statement, such as the fact that generic constituents of the cohomology of $X(w)$ only occurs in degree $l(w)$, etc.
As for the possibility of realizing the full GG module in the cohomology of a single DL variety, I am as skeptical as Jim Humphreys.