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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.

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3 Answers

In the 35 years following the Deligne-Lusztig construction of generalized characters of finite groups of Lie type, a considerable amount of work by Lusztig and others has led to a reasonably detailed understanding of the irreducible characters. This is usually quite difficult to make explicit, however, since a lot of recursive steps are involved. Looking at all this work in hindsight, it's definitely a worthwhile project to find shorter or more transparent pathways to the end results. Even so, these results are usually so complicated that one can't expect miracles, especially when the ambient algebraic group has a disconnected center (as in the case of special linear groups).

I'm somewhat doubtful about the parallel you suggest between D-L and G-G characters, since the two constructions have strongly contrasting features. The longstanding problem for finite group theorists was the lack of an elementary procedure like induction that would come close to producing all irreducible characters. Parabolic induction is successful up to a point, but fails to yield the "cuspidal" characters. The more sophisticated cohomological construction in the 1976 Annals paper of Deligne and Lusztig, followed by Lusztig's further refinements, produced a family of generalized characters having all irreducible characters as constituents. In fact, "most" of the D-L characters are (up to sign) already irreducible. But it still takes considerable ingenuity to decompose all D-L characters in a systematic way, compute the degrees of the irreducibles, etc.

In contrast, the G-G construction of a single large degree character uses standard induction methods. When the center of the ambient algebraic group is connected, the case treated carefully in Chapter 8 of Carter's book, the G-G character is independent of which "nondegenerate" (or "generic") linear character of the finite unipotent group was used in the induction. Moreover, the G-G character is multiplicity-free and has as constituents "almost all" of the irreducible characters being sought. In other words, it approximates a model for the character theory of the finite group.

Further study shows that the constituents of the G-G character are precisely the "regular" characters, one for each geometric conjugacy class in the D-L set-up. Extracting these constituents systematically is then an important part of the agenda following the D-L construction. All of this is extremely challenging. Even a well-behaved family of finite groups such as those of type $G_2$ over finite fields illustrates well the complexity of the problem. (Even though the character table was constructed for odd primes by Chang-Ree using more ad hoc methods based on Green's work for general linear groups, it's a useful exercise to reproduce the table in the spirit of D-L theory.)

Will further methods from algebraic geometry beyond those already exploited by Deligne and Lusztig provide better insights? This is impossible to predict, but at the moment I'm skeptical about the program sketched in this question.

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I think the relevant references are Kawanaka's papers:

Kawanaka, N. Generalized Gelʹfand-Graev representations and Ennola duality. Algebraic groups and related topics (Kyoto/Nagoya, 1983), 175–206, Adv. Stud. Pure Math., 6, North-Holland, Amsterdam, 1985.

Kawanaka, Noriaki Shintani lifting and Gelʹfand-Graev representations. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 147–163, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987.

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These papers are part of the formidable literature arising from the Deligne-Lusztig construction and Lusztig's further work, but don't directly address the sheaf-theoretic questions raised by Dror. On the other hand, Kawanaka's work points toward interesting connections with representations over local fields. –  Jim Humphreys Nov 8 '11 at 13:51
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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.

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