Yael Fregier
In this note we show that the theory of non abelian extensions of a Lie algebra $\mathfrak{g}$ by a Lie algebra $\mathfrak{h}$ can be understood in terms of a differential graded Lie algebra $L$. More precisely we show that the non-abelian cohomology $H^2_{nab}(\mathfrak{g},\mathfrak{h})$ is $\mathcal{MC}(L)$, the $π_0$ of the Deligne groupoid of $L$.
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We determine the cohomology of Deligne-Lusztig varieties associated to some short-length regular elements for split groups of type F4 and En. As a byproduct, we obtain conjectural Brauer trees for the principal Phi_{14}-block of E7 and the principal Phi_{24}-block of E8.
We propose two inductive approaches for determining the cohomology of Deligne-Lusztig varieties in the case of the general linear group
We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of GLn(q). We show that the geometric version of Broué's conjecture over Q_\ell, together with Craven's formula, holds for any unipotent block whenever it holds for the principal Phi_1-block, that is for the variety X(π).
In this article, we study the mod $\ell$ cohomology of some Deligne-Lusztig varieties for $\operatorname{GL}_n(q)$. We prove that the cohomology groups of these varieties are torsion-free under some conditions on the characteristic. Under the torsion-free assumption we can compute the cohomology groups explicitly and we prove that the cohomology complex satisfies partial-tilting condition, which is one of the necessary conditions in the geometric version of Broué's abelian defect group conjecture.
We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual of the intersection cohomology of Deligne-Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent $\ell$-blocks of finite reductive groups.
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of GL_2. At first we determine all such varieties up to isomorphy. After this we investigate the representations of the sigma-stabilizer of an element b of the group on the etale cohomology of the affine Deligne-Lusztig variety X_w(b). We describe such representations as inductions from compact subgroups and in terms of noncuspidal representations.
Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we als…
We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.
Geometry and cohomology of compactified Deligne--Lusztig varieties
本文研究了紧化Deligne-Lusztig簇的几何与上同调性质，证明其上同调与Weyl群中最小长度元素的上同调同构。
On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $GL_n$
本文针对特征为正且ℓ≠p的系数情形，首次研究GL_n的p进Deligne-Lusztig簇的模ℓ上同调，将其与Vignéras模局部朗兰兹对应建立关联，推进p进李群表示论相关研究。
Let $X\to Y^0$ be an abelian prime-to-$p$ Galois covering of smooth schemes over a perfect field $k$ of characteristic $p>0$. Let $Y$ be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on $Y$. We describe a logarithmic $F$-crystal on $Y$ whose rational crystalline cohomology is the rigid cohomology of $X$, in particular provides a natural $W[F]$-lattice inside the latter; here $W$ is the Witt vector ring of $k$. If a finite group $G$ acts compatibly on $X$, $Y^0$ and $Y$ then our construction is $G$-equivariant. As an example we apply it to Deligne-Lusztig va…
We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne--Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne--Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes $X_h$. Boyarchenko's two conjectures are on the maximality of $X_h$ and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invarian…
We prove several conjectures about the cohomology of Deligne-Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type $A$, and using previous results of the second author, this implies Broué-Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broué's abelian defect group conjecture and the specific form Broué conjectu…
The cohomology of $p$-adic Deligne-Luszitg schemes of Coxeter type
本文确定了局部非阿基米德域上 Coxeter 型 $p$-Deligne-Lusztig 概形闭 Drinfeld 层的上同调，证明环面权空间集中于单一上同调度数且为成对非同构不可约表示，并探讨了 Moy-Prasad 商的极大性与 Kirillov 轨道方法的关系。
Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.
Convex elements and cohomology of deep level Deligne-Lusztig varieties
本文完整描述了深层 Deligne-Lusztig 簇的上同调，证明其 φ-权部分集中于单一次数且由 Yu 型子群诱导，并应用于表示分解、Fargues-Scholze 参数及验证 Chan-Oi 猜想。
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…
These notes are based on a series of lectures given by the author at the Centre Bernoulli (EPFL) in July 2016. They aim at illustrating the importance of the mod-$\ell$ cohomology of Deligne--Lusztig varieties in the modular representation theory of finite reductive groups.
In this paper, we present a conjecture on the degree of unipotent characters in the cohomology of particular Deligne-Lusztig varieties for groups of Lie type, and derive consequences of it. These degrees are a necessary piece of data in the geometric version of Broué's abelian defect group conjecture, and can be used to verify this geometric conjecture in new cases. The geometric version of Broué's conjecture should produce a more combinatorially defined derived equivalence, called a perverse equivalence. We prove that our conjectural degree is an integer (which is not obvious) and has the cor…
In this article, we study the étale cohomology of the compactification of Deligne-Lusztig varieties associated to a Coxeter element. We prove a result for the integral coefficients in the case of general linear group $GL_d$, and we conjecture that the similar result holds for general reductive groups.
Cohomology and geometry of Deligne--Lusztig varieties for ${\rm GL}_n$
本文描述了 ${\rm GL}_n$ 的 Deligne--Lusztig 簇的光滑紧化上结构层的上同调，进而得到其模 $p^m$ 与整 $p$-进 étale 上同调，并证明 Zariski 闭包具有伪有理奇点。
On the cohomology of differentiable stacks
本文证明李群胚以同伦意义下的表示为系数的上同调是Morita不变量，从而成为可微栈的不变量，并附带证明高阶李群胚的上同调下降性质。
This article is the $\mathrm{Z}_l$-version of my paper "Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples" in Invent. Math. 2009 vol 177 pp. 239-280, where we study the vanishing cycles of some unitary Shimura variety. The aim is to prove that the cohomology sheaves of this complexe are free so that, thanks to the main theorem of Berkovich on vanishing cycles, we can deduce that the $\mathrm{Z}_l$-cohomology of the model of Deligne-Carayol is free. There will be a second article which will be the $\mathrm{Z}_l$ version of my paper "Conjecture de mono…
This paper is a following to math.RT/0410454. For a finite group of Lie type we study the endomorphisms, commuting with the group action, of a Deligne-Lusztig variety associated to a regular element of the Weyl group. We state some general conjectures, in particular that the endomorphism algebra induced on the cohomology is a cyclotomic Hecke algebra, and prove them in some cases. On the way we also prove results on centralizers in braid groups.
By a ring groupoid we mean an animated ring whose i-th homotopy groups are zero for all i>1. In this expository note we give an elementary treatment of the (2,1)-category of ring groupoids (i.e., without referring to general animated rings and without using n-categories for n>2). The note is motivated by the fact that ring stacks play a central role in the Bhatt-Lurie approach to prismatic cohomology.
This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group representations, 2-vector spaces and the tangent and the cotangent algebroid (groupoid) to a Lie algebroid (groupoid). Moreover, they are geometric models for some kind of representations of Lie algebroids (groupoids), namely 2-term representations up to homotopy. Finally, it is well known that Lie groupoids are "concrete" incarnations of differentiable stack…
This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic group; though proof of the general statement in this case does not require new ideas, it provides examples (such as sheaves on the nilpotent cone of a semi-simple group equivariant under the adjoint action) where construction of coherent "intersection cohomology" sheaves works.
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