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Questions tagged [deligne-lusztig-theory]

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Decomposition of cohomology of Deligne-Lusztig varieties

Colloquially I've often heard that “one can construct” all representations of simple groups of Lie type in the cohomology of Deligne–Lusztig varieties. The technical way I've seen this written is as ...
curious math guy's user avatar
2 votes
1 answer
285 views

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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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
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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
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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
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6 votes
1 answer
696 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
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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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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
310 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
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353 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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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-Zh's user avatar
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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
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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
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1 answer
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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-Zh's user avatar
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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
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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
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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
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6 votes
0 answers
315 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
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27 votes
3 answers
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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 (...
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5 votes
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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
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0 votes
1 answer
270 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
537 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
1k 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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