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Questions tagged [deligne-lusztig-theory]

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Deligne-Lustzig varieties locally closed schemes

I have a couple of questions about basic properties of of Deligne-Lustzig varieties introduced in the seminal paper "Representations of Reductive Groups Over Finite Fields" [DL76]. The ...
user267839's user avatar
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4 votes
1 answer
116 views

Cohomology of Deligne-Lusztig variety associated to Coxeter element

Determining the individual ($l$-adic) cohomology groups of Deligne-Lusztig varieties has only been done for the general linear group and for some other very specific cases (as far as I know). However, ...
EJB's user avatar
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2 votes
0 answers
73 views

Relative position of Borel subgroups for the symplectic group

Background Let $n$ be a positive integer, let $W$ be the Weyl group of $\text{GL}_n$. Its set of Borel subgroups is isomorphic to the full flag variety $\mathcal{F}_n$. In this question, I studied ...
EJB's user avatar
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5 votes
1 answer
399 views

An example of a Deligne–Lusztig variety for a general linear group

Take $G=\operatorname{GL}_3$, defined over the algebraic closure of a finite field $\mathbb{F}_q$ and let $X$ be the set of Borel subgroups of $G$. The Frobenius morphism $F:G\to G$ induces a map $F:...
EJB's user avatar
  • 153
6 votes
1 answer
652 views

(Why) are Deligne-Lusztig varieties nonempty?

Background: Let $G$ be a reductive $\mathbb F_q$-group and let $X$ be the variety of Borel subgroups of $G$. By the Bruhat decomposition, the $G$-orbits in the space $X\times X$ (with diagonal action) ...
David Schwein's user avatar
3 votes
0 answers
66 views

Duals of unipotent characters of classical finite groups of Lie type in terms of Lusztig's symbols

The irreducible unipotent characters of classical finite groups of Lie type have been classified by Lusztig using the combinatorical notion of "symbols", see "Irreducible ...
Suzet's user avatar
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8 votes
3 answers
495 views

Intuitive reason that the regular representation is a uniform function

Corollary 12.14 of Digne-Michel's book Representations of finite groups of Lie type gives various decompositions of the regular representation $\operatorname{reg}_G$ in terms of the Deligne-Lusztig ...
Martin Skilleter's user avatar
3 votes
2 answers
293 views

Frobenius reciprocity for Deligne-Lusztig induction/restriction

I am currently trying to understand the properties of Deligne-Lusztig induction, following Carter's Finite groups of Lie type and Digne-Michel's Representations of finite groups of Lie type. I am ...
Martin Skilleter's user avatar
4 votes
0 answers
324 views

$\ell$-adic cohomology of a quotient by group action

Suppose $Y \to Y/G$ is the Galois cover induced from a finite group $G$ acting on a scheme $Y$ and that this is indeed a Galois cover with $Y/G$ a scheme. In my case $Y$ is the Drinfeld curve $\mathrm{...
TCiur's user avatar
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6 votes
0 answers
142 views

cuspidal unipotent representation in small characteristic

Let $\mathbb{F}_q$ be a finite field with $q=p^r$ and $p$ prime. Let $G$ be a connected reductive group over $\mathbb{F}_q$. Is there a difference between the theory of unipotent cuspidal ...
Q. Zhang's user avatar
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1 answer
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Representations of $\text{SL}_n(q)$ of degree $q^{O(n)}$

Every representation of $A_n$ of degree $n^{O(1)}$ is contained in a $O(1)$ tensor power of the defining permutation representation. Is there an analogous result for classical groups of Lie type, say $...
Sean Eberhard's user avatar
4 votes
0 answers
93 views

Rank and unipotent support

Let $G$ be a finite group of Lie type. I would like to be able to compute the rank (introduced by Howe and Gurevich in "Small representations of finite classical groups") of an irreducible ...
J. Epequin's user avatar
4 votes
1 answer
396 views

a question on Deligne-Lusztig characters

Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be ...
Q. Zhang's user avatar
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2 votes
1 answer
249 views

semisimple support of character sheaves

So the essential question is: How should we think about, or if possible compute, the semisimple support of a cuspidal character sheaf? For example, let $G=SL_2$. We have the cuspidal character ...
Cheng-Chiang Tsai's user avatar
9 votes
0 answers
495 views

Naive question about classification of unipotent character sheaves

Let $G$ be a connected reductive algebraic group over (say) $\mathbb{C}$. The set $\hat{G}_u$ of isomorphism classes of unipotent irreducible character sheaves has some complicated classification in ...
Sasha's user avatar
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2 votes
1 answer
210 views

On cuspidal maximal tori of a connected reductive group

Let $G$ be the base change of a connected reductive group over ${\mathbb{F}}_q$ to $\overline{\mathbb{F}}_q$, and let $F$ be the associated geometric Frobenius on $G$. Let $W$ be the Weyl group of a ...
user148212's user avatar
  • 1,606
6 votes
0 answers
305 views

An analogue of Deligne--Lusztig theory for real groups?

I am considering the following analogue of Deligne--Lusztig theory: Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have $G^F=GL_n(\mathbb{R})$. Consider the ``Lang map''...
user148212's user avatar
  • 1,606
25 votes
3 answers
5k views

learning Deligne-Lusztig theory

Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig) Edit: You may assume knowledge of representation theory of finite groups (...
user avatar
5 votes
0 answers
366 views

Deligne-Lusztig and Character sheaves

Consider: $G$ - a nice group ($GL_n$) over a finite field $F$. $X$ - the flag variety. Consider a nice $G$-equivariant $l$-adic sheaf $\mathcal{M}$ on $X \times X$, equipped with Weil structure. Fix $...
Sasha's user avatar
  • 5,522
0 votes
1 answer
263 views

A bijection between Lusztig series induced by inflation

Context: Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F \...
Matthias Klupsch's user avatar
3 votes
1 answer
517 views

Points on Deligne-Lusztig varieties: Interpreting Borels in relative position as flags with conditions

Background I am studying the paper "On the Green polynomials of classical groups" by Lusztig, in which he computes the values of the Deligne-Lusztig representation, corresponding to a Coxeter element ...
Dror Speiser's user avatar
  • 4,593
12 votes
3 answers
985 views

Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?

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 ...
Dror Speiser's user avatar
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