### 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.