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 Hecke algebra actions on the cohomology of varieties associated to $w\_0$ or its square, for groups of arbitrary rank. In a subsequent work, we construct actions associated to more general regular elements and we study their trace on cohomology.
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.
The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory.
In this article, we study the De Rham cohomology of the first cover in the Lubin-Tate tower. In particular, we get a purely local proof that the supercuspidal part realizes the local Jacquet-Langlands correspondence for ${\rm GL}_n$ by comparing it to the rigid cohomology of some Deligne-Lusztig varieties. The representations obtained are analogous to the ones appearing in the $\ell$-adic cohomology if we forget the action of the Weil group. The proof relies on the generalization of an excision result of Grosse-Klönne and on the existence of a semi-stable model constructed by Yoshida for which…
We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we p…
Lusztig defined an abelian category ${\mathscr{C}}_{k}$ of a class of representations of a multi-loop algebra and asked various questions connecting it to the modular representation theory of simple algebraic groups in char. p>0. The aim of this paper is to show that some of these questions have negative answer.
The Borel-Weil-Bott theorem describes the cohomology of line bundles over flag varieties. Here, one generalizes this theorem to a wider class of projective varieties : the wonderful varieties of minimal rank.
We prove that the Deligne-Lusztig variety associated to minimal length elements in any $\d$-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in \cite{OR}.
We give a generalisation of Deligne-Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne-Lusztig varieties to all tamely ramified maximal tori of the group. Moreover, we analyse the structure of various generalised Deligne-Lusztig varieties, and show that the "unramified" varieties, including a certain natural general…
Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic $σ$-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic $σ$-conjugacy class is the class of $b=1$, we also prove the dimension formula predicted in op. cit. in almost all cases.
We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they may be used to describe the components of Lusztig quiver varieties.
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 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).
We discuss some connections between the closure $\bar F$ of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne-Lusztig varieties $X_w(1)$ in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of $X_w(1)$ if the translation part of $w$ is quasi-regular. As a by-product, we give a new proof of the explicit description of $\bar F$, first obtained in \cite{H1}.
In this paper we give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne-Lusztig variety if we replace Frobenius morphism with conjugation by an element for $GL_n$. The main theorem states that it only depends on the unipotent part of the Jordan decomposition of an element and the conjugacy class in the Weyl group. Also it generalizes the formula of the Euler characteristic of a Springer fiber for type A.
This paper gives an explicit formula of the dimension of affine Deligne-Lusztig varieties associated with generic Newton point in terms of Demazure product of Iwahori-Weyl groups.
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.
In this paper we introduce a family of Deligne--Lusztig type varieties attached to connected reductive groups over quotients of discrete valuation rings, naturally generalising the higher Deligne--Lusztig varieties and some constructions related to the algebraisation problem raised by Lusztig. We establish the inner product formula between the representations associated to these varieties and the higher Deligne--Lusztig representations.
We use $G$-stable pieces to construct some equidimensional varieties and as a consequence, obtain Lusztig's dimension estimates \cite[section 4]{L2}. This is a generalization of \cite{HL}.
We study the quotient of parabolic Deligne-Lusztig varieties by a finite unipotent group U^F where U is the unipotent radical of a rational parabolic subgroup P = L U. We show that in some particular cases the cohomology of this quotient can be expressed in terms of "smaller" parabolic Deligne-Lusztig varieties associated to the Levi subgroup L.
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 prove a conjecture of Broue about the Jordan decomposition of blocks of finite reductive groups. We show that a block of a finite connected reductive group, in non-describing characteristic, is Morita-equivalent to a quasi-isolated block of a Levi subgroup. This involves showing that some local system over a Deligne-Lusztig variety has its mod l cohomology concentrated in one degree. We reduce this question to a question about tamely ramified local systems by proving that the category of perfect complexes for the group is generated by the images of the Deligne-Lusztig functors. Then, we des…
This note is based on my talk at ICCM 2013, Taipei. We give an exposition of the group-theoretic method and recent results on the questions of non-emptiness and dimension of affine Deligne-Lusztig varieties in affine flag varieties.
We shall give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac-Moody Lie algebras in terms of parabolic Kazhdan-Lusztig polynomials introduced by Deodhar.
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…
We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.
This paper is a continuation and a completion of [BoRo1]. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups. We also show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of [BoRo1], …
We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in Görtz et al. (2010). The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in p…
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.
The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.
We define a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. We study commutativity, cancellativity and associativity of this operation. We focus on rigid irreducible components and discuss inductive ways to construct them.
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.
Let $G$ be the Weil restriction of a general linear group. By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we obtain a stratification of the affine Deligne-Lusztig varieties for $G$ (in the affine Grassmannian) attached to a minuscule coweight and a basic element. As an application, we verify a conjecture by Chen and Zhu on irreducible components of affine Deligne-Lusztig varieties for $G$.
In a previous paper it was shown that a certain family of varieties suggested by Lusztig, is not enough to construct all irreducible complex representations of reductive groups over finite rings coming from the ring of integers in a local field, modulo a power of the maximal ideal. In this paper we define a generalisation of Lusztig's varieties, corresponding to an extension of the maximal unramified extension of the local field. We show in a particular case that all irreducible representations appear in the cohomology of some extended variety. We conclude with a discussion about reformulation…
We initiate the study of affine Deligne-Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for GLn and its inner forms, Lusztig's semi-infinite Deligne-Lusztig construction is isomorphic to an affine Deligne-Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet--Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig's 1979 conjecture in this set…
We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Broué's conjectures.
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.
By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we introduce a stratification for the affne Deligne-Lusztig variety (in the affne Grassmannian) attached to attached to a minuscule cocharacter and a basic element. As an application, we complete the proof of a conjecture on the irreducible components of affne Deligne-Lusztig varieties, due to Miaofen Chen and Xinwen Zhu.
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.
Rapoport and Kottwitz defined the affine Deligne-Lusztig varieties $X_{\tilde{w}}^P(bσ)$ of a quasisplit connected reductive group $G$ over $F = \mathbb{F}_q((t))$ for a parahoric subgroup $P$. They asked which pairs $(b, \tilde{w})$ give non-empty varieties, and in these cases what dimensions do these varieties have. This paper answers these questions for $P=I$ an Iwahori subgroup, in the cases $b=1$, $G=SL_2$, $SL_3$, $Sp_4$. This information is used to get a formula for the dimensions of the $X_{\tilde{w}}^K(σ)$ (all shown to be non-empty by Rapoport and Kottwitz) for the above $G$ that sup…
In this paper we consider Deligne-Lusztig varieties and their analogues when the Frobenius endomorphism is replaced with conjugation by an element in a group, especially a regular semisimple or regular unipotent one. We calculate their classes in the Chow group of the flag variety in terms of Schubert classes. Also we give some sufficient criteria when different elements in the Weyl group result in the same class.
In this paper we prove a direct geometric relation between Deligne--Lusztig varieties and Springer fibres in type $\mathsf{A}$: For any rational unipotent element, the Springer fibre cuts out a unique component of a specific Deligne--Lusztig variety; moreover, this component forms an open dense subset of a component of the Springer fibre. This boils down to a map from the unipotent variety to the Weyl group, and combines several constructions with a combinatorial flavour (like Weyr normal forms, Robinson--Schensted correspondence, and Spaltenstein's and Steinberg's labellings); it also provide…
Let $T$ be a maximal torus of a semisimple complex algebraic group, $\mathrm{BS}(s)$ be the Bott-Samelson variety for a sequence of simple reflections $s$ and $\mathrm{BS}(s)^T$ be the set of $T$-fixed points of $\mathrm{BS}(s)$. We prove the tensor product decompositions for the image of the restriction $H^\bullet_T(\mathrm{BS}(s),k)\to H_T^\bullet(X,k)$, where $X\subset\mathrm{BS}(s)^T$ is defined by some special not overlapping equations $γ_iγ_{i+1}\cdotsγ_j=w_{i,j}$ with right-hand sides belonging to the Weyl group.
The study of certain union $X(μ, b)$ of affine Deligne-Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit formula of $\dim X(μ, b)$ for sufficiently large dominant coweight $μ$.
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.
In the present article we define coverings of affine Deligne-Lusztig varieties attached to a connected reductive group over a local field of characteristic $p > 0$. In the case of $\GL_2$, the unramified part of the local Langlands correspondence is realized in the $\ell$-adic cohomology of these varieties. We show this by giving a detailed comparison with the realization of local Langlands via cuspidal types by Bushnell-Henniart. All proofs are purely local.
We prove that the Drinfeld halfspace is essentially the only Deligne-Lusztig variety which is at the same time a period domain over a finite field. This is done by comparing a cohomology vanishing theorem for DL-varieties, due to Digne, Michel, and Rouquier, with a non-vanishing theorem for PD, due to the first author. We also discuss an affineness criterion for DL-varieties.
A Kazhdan-Lusztig atlas, introduced by He, Knutson and Lu, on a stratified variety (V,Y) is a way of modeling the stratification Y of V locally on the stratification of Kazhdan-Lusztig varieties X^w_o \cap X_v by their intersection with opposite Schubert varieties X_u. We are interested in classifying smooth toric surfaces with Kazhdan-Lusztig atlases. This involves finding a degeneration of V to a union of Richardson varieties in the flag variety H/B_H of some Kac-Moody group H. We determine which toric surfaces have a chance at having a Kazhdan-Lusztig atlas by looking at their moment polyto…
We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Bialynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial description of the support of each class, which enables us to compute the symmetric group actions on the classes in our bases. We then successfully apply the results to the permutohedral varieties to explicitly write down each class and to construct permutation submodules that constitute summands of a decomposition of cohomology space of each degree. This reso…
In [Lu6] Lusztig defined a certain algebra $H,$ which is a direct sum of various algebras $H_{\mathfrak{o}}.$ We establish an explicit algebra isomorphism between each algebra $H_{\mathfrak{o}}$ and some matrix algebra with coefficients in the tensor product of an (affine or finite) Hecke algebra and a group algebra. We give an application.
