Questions tagged [deligne-lusztig-theory]

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4
votes
1answer
200 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 ...
2
votes
1answer
150 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 ...
9
votes
0answers
259 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 ...
2
votes
1answer
143 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 ...
6
votes
0answers
218 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''...
14
votes
2answers
2k 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 (...
5
votes
0answers
284 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 $...
0
votes
1answer
218 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 \...
2
votes
1answer
387 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 ...
11
votes
3answers
825 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 ...