Nicolas Perrin, Evgeny Smirnov
We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N^2=0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen-Macaulay, and have rational singularities.
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The purpose of this paper is to show that, for each unipotent element in type E7, the corresponding Springer fiber can be paved by affine spaces.
We determine the Betti numbers of the Springer fibers in type A. To do this, we construct a cell decomposition of the Springer fibers. The codimension of the cells is given by an analogue of the Coxeter length. This makes our cell decomposition well suited for the calculation of Betti numbers.
On the cohomological purity of the affine Springer fibers
本文研究仿射斯普林纤维的上同调纯度问题，证明其纯度等价于其ξ-稳定商的纯度，并进一步等价于特定截断仿射斯普林纤维序列的纯度，推导出新的上同调纯度描述并比较了新旧标准。
The Springer Geometry of Specht Polynomials and Schubert cycle positivity for two row Springer fiber components
研究 A 型施普林格表示在完全旗簇同调中的几何实现，通过斯佩特多项式等进行一系列正展开，解决两行纤维的施普林格问题，证明相关猜想并识别施普林格基与网基，还推导了旗簇泊松退化轨迹分量的舒伯特圈展开。
We prove the "Gluing Conjecture" on the spectral side of the categorical geometric Langlands correspondence. The key tool is the structure of crystal on the category of singularities, which allows to reduce the conjecture to the question of homological triviality of certain homotopy types. These homotopy types are obtained by gluing from a global version of Springer fibers.
We explain an algorithm to calculate Arthur's weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur-Kottwitz reduction and by the Harder-Narasimhan reduction. A comparison of results obtained from these two approaches gives us recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur's weighted orbital integrals. As an example…
Equivalued affine springer fibers in mixed characteristic
研究温和分支群中温和等值共轭类的Witt向量仿射Springer纤维，证明它们可被光滑Hessenberg簇上的迭代仿射空间丛的完美化铺砌。
We state a conjecture on how to construct affine pavings for cohomologically pure projective algebraic varieties, which admit an action of torus such that the fixed points and $1$-dimensional orbits are finite. Experiments on the affine grassmannian for $\mathrm{GL}_{3}$ under the guideline of this conjecture, together with the work of Berenstein-Fomin-Zelevinsky and Kamnitzer, have led to the conjecture that the truncated affine grassmannians for $\mathrm{GL}_{r+1}$ admit affine pavings. For the group $\mathrm{GL}_{3}$, we construct affine pavings for the truncated affine grassmannians, and w…
On the dependence of the affine Springer fibers on their root valuation datum
本研究针对群$\mathrm{GL}_{d}$证实了Goresky、Kottwitz与MacPherson的猜想，即仿射斯普林格纤维的上同调仅依赖于定义元素的根赋值数据，证明借助Ngô支撑定理中间扩张的微局部分析及平面曲线奇点万有形变相关层的Whitney正则性。
Purity of generalized affine Springer fibers from generic planar curve singularities
本文通过构造显式仿射铺砌，证明了广义不可约平面曲线奇点上点状Hilbert概型的上同调纯性，由此确立了广义$GL_N$-仿射Springer纤维纯性猜想的一个新情形，并推广了关于紧化Jacobian的组合学结果。
We describe Springer fibers corresponding to the minimal and minimal special nilpotent orbits of simple Lie algebras. As a result, we give an answer to the conjecture of Humphreys regarding some graphs attached to Springer fibers.
We decompose the fibers of the Springer resolution for the odd nilcone of the Lie superalgebra $\osp(2n+1,2n)$ into locally closed subsets. We use this decomposition to prove that almost all fibers are connected. However, in contrast with the classical Springer fibers, we prove that the fibers can be disconnected and non equidimensional.
For G a semisimple algebraic group, we revisit the description of the components of the affne Springer fiber given by ts, with s a regular semisimple element. We then compute the fixed points of each component of a particular affne Springer fiber for Type A
Let ${\mathcal B}_u$ be the Springer fiber over a nilpotent endomorphism $u\in End(\mathbb{C}^n)$. Let $J(u)$ be the Jordan form of $u$ regarded as a partition of $n$. The irreducible components of ${\mathcal B}_u$ are all of the same dimension. They are labelled by Young tableaux of shape $J(u)$. We study the question of singularity of the components of ${\mathcal B}_u$ and show that all the components of ${\mathcal B}_u$ are nonsingular if and only if $J(u)\in\{(λ,1,1,...), (λ_1,λ_2), (λ_1,λ_2,1), (2,2,2)\}$.
Perverse filtration on the cohomology of affine Springer fibers
本文通过构造仿射Springer纤维上同调纯部分上的“反常过滤”，证明了任何仿射Springer纤维的顶部上同调作为Weyl群表示包含某些Springer纤维总上同调的大部分。
Affine Springer Fibers and Generalized Haiman Ideals
本研究在等变形式假设下计算GL_n非分歧仿射Springer纤维的Borel-Moore同调，关联Haiman理想，针对n=3给出理想显式刻画并验证相关猜想。
A decomposition theorem for the affine Springer fibers
该研究构造仿射Springer纤维对应紧致化雅可比有限阿贝尔覆叠的等变形变，分解其导出直像复形并取极限得到仿射Springer纤维同调的分解，可约化至Λ⁰-不变子空间，还给出Goresky等人纯度假设的层论重述并提出相关猜想。
We determine a formula for the dimension of a family of affine Springer fibers associated to a symmetric space arising from the block diagonal embedding $\mathrm{GL}_n\times\mathrm{GL}_n\hookrightarrow\mathrm{GL}_{2n}$ . As an application, we determine the dimension of affine Springer fibers attached to certain unitary symmetric spaces.
Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for $GL_n(\mathbb{C})$ using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equ…
Parabolic Multiplicative Affine Springer Fibers
该研究引入抛物型乘法仿射Springer纤维及其全局对应物抛物型乘法Hitchin纤维，通过相关纤维化证明前者等维并确定其维数，为相关代数几何研究提供新对象与结论。
For a Weyl group $W$ of classical type, we present a formula to calculate the restriction of (graded) Springer representations of $W$ to a maximal parabolic subgroup $W'$ where the types of $W$ and $W'$ are in the same series. As a result, we obtain recursive formulas for the Betti numbers of Springer fibers for classical types.
These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of Springer fibers, affine Springer fibers and Hitchin fibers; representations of (affine) Weyl groups arising from these objects; relation between affine Springer fibers and orbital integrals.
We give a topological description of the two-row Springer fiber over the real numbers. We show its cohomology ring coincides with the oddification of the cohomology ring of the complex Springer fiber introduced by Lauda-Russell. We also realize Ozsváth-Rasmussen-Szabó odd TQFT from pullbacks and exceptional pushforwards along inclusion and projection maps between hypertori. Using these results, we construct the odd arc algebra as a convolution algebra over components of the real Springer fiber, giving an odd analogue of a construction of Stroppel-Webster.
This article has two parallel perspectives: to demonstrate an inductive structure of Shalika germs, and to show an analogous inductive structure for affine Springer fibers. More precisely, we give an algorithm to compute arbitrary Shalika germs (resp. affine Springer fibers up to stratification) in terms of three ingredients: Shalika germs (resp. affine Springer fibers) for twisted Levi subgroups, a finite list of combinatorial objects, and the numbers of rational points on varieties over the residue field (resp. varieties themselves) among an explicit finite list of such. We also discuss some…
We calculate the Borel-Moore homology of affine Springer fibers of type $A$ associated to some regular semisimple nil elliptic elements. As a result, we obtain bigraded $\mf{S}_{n}$-modules whose bigraded Frobenius series are generalization of the symmetric functions introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov.
We construct an explicit topological model (similar to the topological Springer fibers appearing in work of Khovanov and Russell) for every two-row Springer fiber associated with the even orthogonal group and prove that the respective topological model is homeomorphic to its corresponding Springer fiber. This confirms a conjecture by Ehrig and Stroppel concerning the topology of the equal-row Springer fiber for the even orthogonal group. Moreover, we show that every two-row Springer fiber for the symplectic group is homeomorphic (even isomorphic as an algebraic variety) to a connected componen…
Close fields, affine Springer fibers and fundamental lemmas
研究近场中仿射弹簧纤维的几何局部恒定性，证明稳定轨道积分在这些家族中局部恒定，并转移基础引理到任意正特征。
Affine Springer fiber and the small quantum group
本文通过仿射斯普林纤维的微sheaves子类，提出量子群在根单位处主块的几何诠释，并证明与几何朗兰兹理论相关的等价性。
In this note we give a geometric realization of the cohomology of Springer fibers in type A. More precisely, we describe the cohomology by the coordinate ring of a scheme theoretic intersection of a Cartan subalgebra with a certain union of cones of nilpotent elements inside the Lie algebra of the group SL_n. As such, the main result of this note is similar to an earlier description given by de Concini and Procesi.
This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.
Starting from a homogeneous affine Springer fiber $Fl_ψ$, we construct three moduli spaces that correspond to the Dolbeault, de Rham and Betti aspects of a hypothetical Simpson correspondence with wild ramifications. We show that $Fl_ψ$ is homeomorphic to the central Lagrangian fiber in the Dolbeault space, prove that the Dolbeaut and de Rham spaces both have the same cohomology as $Fl_ψ$, and construct a map from the de Rham space to the Betti space which we conjecture to be an analytic isomorphism.
Witt vector affine Springer fibers
本文在混合特征局部域上，对本质驯分歧且剩余特征非坏的重约群，建立了Witt向量仿射Springer纤维的维数公式，涉及判别式赋值、Artin导子和Kottwitz不变量。
Ideals defining components of two-row Springer fibers
本文针对两行Springer纤维，通过非交叉匹配定义多项式理想，证明这些理想定义了Springer纤维的对应分支，并给出分支上同调类的两个猜想公式。
We consider the Springer fiber ${\mathcal B}_u$ corresponding to a nilpotent endomorphism $u$ of nilpotent order 2. As a first result, we give a description of the elements of a given component of ${\mathcal B}_u$ which are fixed by the action of the standard torus relative to some Jordan basis of $u$. By using this result, we establish a necessary and sufficient condition of singularity for the components of ${\mathcal B}_u$.
We give an explicit description of the irreducible components of two-row Springer fibers in type A as closed subvarieties in certain Nakajima quiver varieties in terms of quiver representations. By taking invariants under a variety automorphism, we obtain an explicit algebraic description of the irreducible components of two-row Springer fibers of classical type. As a consequence, we discover relations on isotropic flags that describe the irreducible components.
Springer correspondence and mirror symmetries for parabolic Hitchin systems
本文通过构造幂零轨道闭包相关的一般Hitchin纤维并分析其与抛物型Hitchin系统一般纤维的关系，证明了B、C型抛物型Hitchin系统的SYZ与拓扑镜像对称，并给出Springer对偶和Lusztig典范商的新几何解释。
Let $\mathbf{G}$ be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $γ_0\in(\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $γ=tγ_0$. Using methods from $p$-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $γ$ modulo $Z_{\mathbf{G}((t))}(γ)$ is equal to the order of the Weyl group.
Cell closures for two-row Springer fibers via noncrossing matchings
本文研究两行 Springer 纤维的胞腔闭包，通过定义“切割弧”操作显式描述 Springer Schubert 胞腔的闭包关系，证明其边界由所有切割弧方式对应的仿射子集构成。
We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.
We study basic geometric properties of some group analogue of affine Springer fibers and compare with the classical Lie algebra affine Springer fibers. The main purpose is to formulate a conjecture that relates the number of irreducible components of such varieties for a reductive group $G$ to certain weight multiplicities defined by the Langlands dual group $\hat{G}$. We prove our conjecture in the case of unramified conjugacy class.
Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{t}$ its Cartan subalgebra and $W$ the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus \mathfrak{t}^*)/W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimo…
Geometric and combinatorial properties of extended Springer fibers
本文研究扩展 Springer 解消的纤维，证明其几何受 tableau 组合学支配，并给出仿射铺砌、Betti 数及 Lusztig 层茎维数公式。
The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky-Kottwitz-MacPherson and Bezrukavnikov-Varshavsky. We first prove this result for the global Springer action, and…
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…
For $G$ a connected reductive group, $γ\in \kg(F)$ semisimple regular integral, we introduce a fundamental domain $F_γ$ for the affine Springer fibers $\xx_γ$. There is a beautiful way to reduce the purity conjecture of $\xx_γ$ to that of $F_γ$, we call it the Arthur-Kottwitz reduction. When restricted to the unramified case, it turns out that these fundamental domains behave well in family. We formulate a rationality conjecture about a generating series of their Poincaré polynomials. We then study them in detail for the group $\gl_{3}$. In particular, we pave them in affine spaces and we prov…
We identify, up to homeomorphisms, the affine Springer fibers for GL(n) on a local field of equal characteristics with some coverings of compactified jacobians of singular projective curves. This allows us to prove an irreducibility property of these affine Springer fibers and to deform them.
Torus Equivariant Cohomology for the $Δ$-Springer Fiber
本文定义了一个环面U作用于Δ-斯普林纤维Y_{n,λ,s}，并给出其等变上同调环的博雷尔式呈现，方法基于轨道谐波变形技术及Chou-Matsumura-Rhoades和Abe-Horiguchi的方法。
Purity of the anisotropic affine Springer fibers for $\mathbf{GL}_{n}$
对于 $\mathrm{GL}_{n}$ 群和各向异性元素，证明了 Goresky-Kottwitz-MacPherson 的纯度猜想，即仿射 Springer 纤维在 Grothendieck-Deligne 意义上是上同调纯的。
Irreducible components of two-column $Δ$-Springer fibers
本文证明两列$Δ$-Springer纤维$Y_{n,n-1}$的所有不可约分支均光滑，其任意分支交为光滑Hessenberg簇，并给出分支奇异上同调环表示及分支交并的Poincaré多项式组合公式。
For any connected reductive group $G$ over $\mathbb{C}$, we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibers $\mathrm{Sp}_γ\subset \mathrm{Gr}_G$, where $γ=at^d$, and $a$ is a regular semisimple element in the Lie algebra of $G$. In the case $G = GL_n$, we relate the equivariant cohomology of $\mathrm{Sp}_γ$ to Haiman's work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of $(n, dn)$-torus links, and formulate a conjecture describing the homology of the Hilb…
Richardson tableaux and components of Springer fibers equal to Richardson varieties
本文定义了标准表的新子集Richardson表，从组合与几何层面进行刻画，证明其对应Springer纤维分量为Richardson簇且光滑，并发现其计数与Motzkin数相关。
Webs and smooth components of two column Springer fibers
本文通过编织与双列矩形斯宾纳纤维的对应关系，揭示了光滑成分的几何结构，并证明了其Poincaré多项式在编织的二面体作用下保持不变。
Following Goresky, Kottwitz and MacPherson, we compute the homology of truncated affine Springer fibers in the unramified case but under a purity assumption. We prove this assumption in the equivalued case. The truncation parameter is viewed as a divisor on an l-adic toric variety. For each fiber, we introduce a graded quasi-coherent sheaf on the toric variety whose space of global sections is precisely the l-adic homology of the truncated affine Springer fiber. Moreover, for some families of endoscopic groups, these sheaves show up in an exact sequence. As a consequence we prove Arthur's weig…
The affine Springer fiber-sheaf correspondence
本文提出仿射Springer纤维-层对应，利用仿射Springer理论在Langlands对偶群的三角交换簇部分解消上构造拟凝聚层，体现3d镜像对称，并探讨了$GL_n$情形及齐次元的量子化。
This paper is a subsequent paper of math.RT/0607673. Here we consider the irreducible components of Springer fibres (or orbital varieties) for two-column case in GL}_n. We describe the intersection of two irreducible components, and specially give the necessary and sufficient condition for this intersection to be of codimension one. Since an orbital variety in two-column case is a finite union of the Borel orbits, we solve the initial question by determining orbits of codimension one in the closure of a given orbit. We show that they are parameterized by a specific set of involutions called de…
Let $u\in\mathrm{End}(\mathbb{C}^n)$ be nilpotent. The variety of $u$-stable complete flags is called the Springer fiber over $u$. Its irreducible components are parameterized by a set of standard Young tableaux. The Richardson (resp. Bala-Carter) components of Springer fibers correspond to the Richardson (resp. Bala-Carter) elements of the symmetric group, through Robinson-Schensted correspondence. Every Richardson component is isomorphic to a product of standard flag varieties. On the contrary, the Bala-Carter components are very susceptible to be singular. First, we characterize the singula…
The $Δ$-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo, and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall-Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $Δ$-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb{F}_q$ and partitioning the $Δ$-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geome…
On the local behavior of weighted orbital integrals and the affine Springer fibers
本文研究加权轨道积分局部行为与仿射Springer纤维对根赋值数据的依赖性，证明加权轨道积分Arthur-Shalika芽展开的齐次性及抛物下降，验证基本域点数生成级数为有理分式并界定分母。
We study the exotic t-structure on the derived category of coherent sheaves on two-block Springer fibre (i.e. for a nilpotent matrix of type (m+n,n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories of coherent sheaves on different two-block Springer fibres (i.e. for different values of n). After checking some exactness properties of these functors, w…
Springer fibers and the Delta Conjecture at $t=0$
本研究引入推广A型Springer纤维的Δ-Springer簇，给出其上同调环的显式表示与对称群作用，为t=0时的Delta猜想提供几何实现，并推广了对角幂零矩阵概形的相关结果。
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