Andres Fernandez Herrero, Rodrigo Horruitiner
In this note we give a classification of all affine normal $SL_2$-varieties containing an open dense orbit over an algebraically closed field of characteristic zero. Such a classification was first obtained by Popov. Here we provide an alternative approach.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
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.
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.
We study the Jordan-Hoelder series for nearby cycles on certain Shimura varieties and Rapoport-Zink local models, and on finite-dimensional pieces of Beilinson's deformation of the affine Grassmannian to the affine flag variety (and their p-adic analogues). We give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups, and we also provide an algorithm to compute multiplicities, in terms of the affine Hecke algebra.
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 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 construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew $(GL(m), GL(n))$ duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine…
We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.
We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar-Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces.
We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a "twisted case"; a consequence of our results is that our construction also includes the flag varieties for Kac-Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t…
We give a concrete description of the category of G-equivariant vector bundles on certain affine G-varieties (where G is a reductive linear algebraic group over an algebraically closed field of characteristic 0) in terms of linear algebra data.
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 $G$.
Normality of Schubert varieties in affine Grassmannians
本文分类了任意域上半单群仿射格拉斯曼流形中的正规Schubert簇，特别关注小正特征，通过切空间计算、Levi引理和极小退化分类给出初等证明。
We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either ${\mathbb P}^2$ or ${\mathbb P}^1\times {\mathbb P}^1$. In positive characteristic, a basic tool in the proof is a new generalization of Miyaoka's generic semipositivity theorem.
We study spherical Schubert varieties in the affine Grassmannian. These Schubert varieties have a natural conjectural modular description due to Finkelberg-Mirković. This modular description is easily seen to be set-theoretically correct, but it is not obviously scheme-theoretically correct. We prove that this modular description is correct in many cases. We also link this modular description to the reducedness conjecture from Kamnitzer-Webster-Weekes-Yacobi for tranverse slices in the affine Grassmannian.
Affine Deligne-Lusztig varieties via the double Bruhat graph I: Semi-infinite orbits
本文引入双Bruhat图新语言，通过半无限轨道研究仿射Deligne-Lusztig簇的几何，克服了正则性条件限制。
This paper studies affine Deligne-Lusztig varieties $X_{\tw}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{\tw}(b)$ for a minimal length element $\tw$ in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in \cite{HL} to the affine case. We then provide a reduction method that relates the structure of $X_{\tw}(b)$ for arbitrary elements $\tw$ in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the …
On the smooth locus of affine Schubert varieties
本文以切空间的组合分析这一初等方法，统一证明了刻画扭仿射 Grassmannian 中 Schubert 簇光滑轨迹的 Haines–Richarz 猜想。
The motivation of this work is to construct an analog of compactified moduli of abelian varieties and toric pairs in the case of non-commutative algebraic group G. We introduce a class of "stable reductive varieties" which contain connected reductive groups and their equivariant compactifications, and is closed under flat reduced degenerations. We classify them all, describe their degenerations, and establish a connection between these varieties and "reductive semigroups" which we also define. Finally, we construct a Hilbert scheme of embedded G-varieties by applying and generalizing a constru…
Let G be a connected complex reductive group. A well known theorem of I. Losev's says that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.
Affine Deligne-Lusztig varieties via the double Bruhat graph II: Iwahori-Hecke algebra
本文引入双 Bruhat 图语言，描述仿射旗簇中仿射 Deligne-Lusztig 簇的几何，并用于确定 Iwahori-Hecke 代数的结构常数，进而刻画大多数仿射 Weyl 群元素及任意 $\sigma$-共轭类对应的簇的非空性与维数。
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.
On affine Lusztig varieties
该研究建立仿射Lusztig簇与仿射Deligne–Lusztig簇的紧密关联，推导得出多数情况下仿射Lusztig簇明确的非空模式与维数公式，为仿射特征层猜想理论提供支撑。
We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $Γ$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_Γ$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_Γ$ to be irreducible and exhibit several examples where $\mathrm M_Γ$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_Γ$.
Affine Deligne-Lusztig Varieties of Positive Coxeter Type
本文提出正 Coxeter 类型的仿射 Deligne-Lusztig 变种，揭示其简单几何结构，并推广了 He-Nie-Yu 的工作，应用于 Shimura 变种。
We study Alexeev and Brion's moduli scheme $M_Γ$ of affine spherical varieties with weight monoid $Γ$ under the assumption that $Γ$ is free. We describe the tangent space to $M_Γ$ at its `most degenerate point' in terms of the combinatorial invariants of spherical varieties and deduce that the irreducible components of $M_Γ$, equipped with their reduced induced scheme structure, are affine spaces.
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 study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.
We define Schwartz functions, tempered functions and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.
To any connected reductive group G over a non-archimedean local field F (of characteristic p > 0) and to any maximal torus T of G, we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of F. This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G. For G = GL_2, we prov…
Our concern in this paper is the dimension and inclusion relations of Schubert varieties in twisted partial affine flag varieties. In the end we apply our results to some local models of certain Schubert varieties.
Higher nearby cycles and central sheaves on affine flag varieties
本文基于Beĭlinson的描述与Gaitsgory的思路，推广研究高维基上的幺幂近邻圈，其仿射格拉斯曼簇框架下的推广是相关学者近期工作的必要条件。
We determine the set of connected components of closed affine Deligne-Lusztig varieties for hyperspecial maximal parahoric subgroups of unramified connected reductive groups. This extends the work by Viehmann for split reductive groups, and the work by Chen-Kisin-Viehmann on minuscule affine Deligne-Lusztig varieties.
We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed as identical. In particular, we are able to give an alternative and much simpler geometric proof of a result of E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado on the construction of bases of affine Lie algebra representations. At the same time, we give a simple parametrization of the irreducible components of Nakajima quiver varieties asso…
In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group representations. Affine Stiefel algebraic varieties and in particular $S^{n}$ are two special cases of the generalised affine Stiefel algebraic varieties.
In this paper, we discuss some partitions of affine flag varieties. These partitions include as special cases the partition of affine flag variety into affine Deligne-Lusztig varieties and the affine analogue of the partition of flag varieties into $\cb_w(b)$ introduced by Lusztig in \cite{L1} as part of the definition of character sheaves. Among other things, we give a formula for the dimension of affine Deligne-Lusztig varieties for classical loop groups in terms of degrees of class polynomials of extended affine Hecke algebra. We also prove that any simple $GL_n(\FF_q((\e)))$-module occurs …
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.
We prove an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundles and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial $\tilde H_λ[X;q,t]$ is i…
Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke algebra with coefficients in the algebraic closure k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the simply connected simple case, that a torus-equivariant K'-theory of F (with coefficients in k) admits an H-action by Demazure operators and that this provides a model for the regular representation of H.
We compute motivic Donaldson-Thomas invariants for crepant resolutions of quotients of affine three-space by even dihedral groups in terms of an affine type D root system, using double dimensional reduction and the representation theory of affine type D quivers.
For split reductive algebraic groups, we determine the connected components of closed affine Deligne-Lusztig varieties of arbitrary parahoric level.
A classical result of Miyanishi-Sugie and Keel-McKernan asserts that for smooth affine surfaces, affine-uniruledness is equivalent to affine-ruledness, both properties being in fact equivalent to the negativity of the logarithmic Kodaira dimension. Here we show in contrast that starting from dimension three, there exists smooth affine varieties which are affine-uniruled but not affine-ruled.
This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus.
A dimension formula of closed affine Deligne-Lusztig varieties of parahoric level
本文在一般性假设下，给出了分裂约化群抛物水平仿射 Deligne-Lusztig 簇并集的维数公式。
Flexibility criterion for affine horospherical varieties
本文提出了一种仿射复杂度为零的球面变种的灵活性准则，扩展了之前关于正常球面变种和非正常典范变种的灵活性研究。
In this paper we calculate the dimension of affine Deligne-Lusztig varieties in the affine Grassmannian of unramified groups. We also examine the irreducible components of ADLVs in the superbasic case and the $J_b(F)$-action on them.
We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety is flexible.
In this paper, we consider affine Deligne-Lusztig varieties $X_w(b)$ and their certain union $X(μ,b)$ inside the affine flag variety of a reductive group. Several important results in the study of affine Deligne-Lusztig varieties have been established under the so-called superregularity hypothesis. Such results include a description of generic Newton points in Iwahori double cosets of loop groups, covering relation in associated Iwahori-Weyl group and dimension formula for $X(μ,b)$. We show that one can considerably weaken the superregularity hypothesis and sometimes completely eliminate it, t…
We determine the set of connected components of closed affine Deligne-Lusztig varieties for special maximal compact subgroups of split connected reductive groups. Especially, we show that such an affine Deligne-Lusztig variety has isolated points if and only if its dimension is 0. From the dimension formula for affine Deligne-Lusztig varieties we obtain a description of the set of these varieties that are zero-dimensional.
We prove the equality of the Makar-Limanov invariant and the modified Makar-Limanov invariant in the case of not necessary normal affine toric varieties. Also we give a combinatorial description of these invariants.
In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_3$ and $b$ basic. We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.
Let G be a complex reductive group. A normal G-variety X is called spherical if a Borel subgroup of G has a dense orbit in X. Of particular interest are spherical varieties which are smooth and affine since they form local models for multiplicity free Hamiltonian K-manifolds, K a maximal compact subgroup of G. In this paper, we classify all smooth affine spherical varieties up to coverings, central tori, and C*-fibrations.
Automorphism groups of affine varieties and their Lie algebras
该研究对仿射簇自同构群相关的精选章节内容进行了简要概述，并列出了部分待解决的开放问题。
We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.
Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of $G$-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness properties of affine Deligne-Lusztig varieties under minimal assumptions on the associated group. We show that affine Deligne-Lusztig varieties are locally of finite type, and prove a global finiteness result related to the natural group action. Similar results have previously been known for special situations.
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, …
We prove Rapoport's dimension conjecture for affine Deligne-Lusztig varieties for GL_h and superbasic b. From this case the general dimension formula for affine Deligne-Lusztig varieties for special maximal compact subgroups of split groups follows, as was shown in a recent paper by Goertz, Haines, Kottwitz, and Reuman.
(Quasi-)affineness of perverse character varieties
本文通过纯叠理论方法构造足够多的结构层截面，证明了反常特征簇是(拟)仿射的。
Let X be an irreducible affine T-variety. We consider families of affine stable toric T-varieties over X and give a description of the corresponding moduli space as the quotient stack of an open subscheme in a certain toric Hilbert scheme under the action of a torus.
This article is a survey on ind-varieties and ind-groups introduced by Shafarevich in 1965, with a special emphasis on automorphism groups of affine varieties and actions of ind-groups on ind-varieties. We give precise definitions and complete proofs, including several known results. The survey contains many examples and also some questions which came up during our work on the subject. Among the new results we show that for an affine variety X the automorphism group Aut(X) is always locally closed in the ind-semigroup End(X) of all endomorphisms, and we give an example of a strict closed subgr…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。