On the irreducibility of Deligne-Lusztig varieties

Parabolic Deligne-Lusztig varieties

Motivated by the Broué conjecture on blocks with abelian defect groups for finite reductive groups, we study "parabolic" Deligne-Lusztig varieties and construct on those which occur in the Broué conjecture an action of a braid monoid, whose action on their $\ell$-adic cohomology will conjecturally factor trough a cyclotomic Hecke algebra. In order to construct this action, we need to enlarge the set of varieties we consider to varieties attached to a "ribbon category"; this category has a {\em Garside family}, which plays an important role in our constructions, so we devote the first part of o…

Compactification des variétés de Deligne-Lusztig

We construct explicitly the normalisation of the Bott-Samelson-Demazure-Hansen compactification of Deligne-Lusztig varieties $X(w)$ in their covering $Y(w)$: we retrieve a result by Deligne-Lusztig about the local monodromy around the divisors of the compactification.

Stratifications of affine Deligne-Lusztig varieties

Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in the context of affine flag varieties and affine Grassmannians. They are closely related to moduli spaces of $p$-divisible groups in positive characteristic, and thus to arithmetic properties of Shimura varieties. We compare stratifications of affine Deligne-Lusztig varieties attached to a basic element $b$. In particular, we show that the stratification defined by Chen and Viehmann using the relative position to elements of the group $J_b$, the $σ$-centralizer of $b$, coincides with the Bruhat-Tits stratification in…

Cohomologie des variétés de Deligne-Lusztig

We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Broué, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element $w\_0$ (some indeterminacies remain for $G\_2$). We use this to construct Hec…

Affine Deligne-Lusztig varieties and folded galleries governed by chimneys

We characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety $X_x(b)$ in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of b has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements x by explicitly constructing such galleries.

关于Deligne-Lusztig特征标的p-有理性

On the $p$-rationality of Deligne--Lusztig characters

该研究针对有限单群中李型单群特征标p-无理性更高的现象，证明p-互素次数的Deligne-Lusztig特征标的p-有理性水平等于对应极大环面特征标的水平，并提出Lusztig诱导对p'-次数特征标保持p-有理性的猜想。