Varieties of pairs of nilpotent matrices annihilating each other

幂零群特征簇上的混合霍奇结构

Mixed Hodge structures on character varieties of nilpotent groups

本文研究了幂零群到约化代数群的表示簇与特征簇，确定了其上的混合霍奇结构，并给出混合霍奇多项式的显式闭式与递归公式。

Toric varieties and a generalization of the Springer resolution

The Springer resolution of the nilpotent cone of a semisimple Lie algebra has played an important role in representation theory. The nilpotent cone is equal to Spec R, where R is the ring of regular functions on the nilpotent cone. This paper constructs and studies an analogue of the Springer resolution for the variety Spec S, where S is the ring of regular functions on the universal cover of the regular nilpotent orbit. The construction makes use of the theory of toric varieties. Using this construction, we provide new proofs of results of Broer and of Graham about Spec S. Finally, we show th…

On the number of points of nilpotent quiver varieties over finite fields

We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Finally we compute the number of points over a finite field of the various s…

Rational singular loci of nilpotent varieties

We present two methods for computing the rational singular locus of the closure of a nilpotent orbit in a complex semisimple Lie algebra and give a number of interesting examples.

Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones

We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…

Reducibility of nilpotent commuting varieties

Let $\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\ge 2$, let $C_r(\N_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that $C_2(\N_n)$ is irreducible for every $n$. We study in this note the reducibility of $C_r(\N_n)$ for various values of $n$ and $r$. In particular it will be shown that the reducibility of $C_r(\mathfrak{gl}_n)$, the variety of commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r(\N_n)$ under certain condition. Then we prove that $C_r(\N_n)$ is reduc…

Varieties of elementary subalgebras of maximal dimension for modular Lie algebras

Motivated by questions in modular representation theory, Carlson, Friedlander, and the first author introduced the varieties E(r, g) of r-dimensional abelian p-nilpotent subalgebras of a p-restricted Lie algebra g. In this paper, we identify the varieties E(r, g) for a reductive restricted Lie algebra g and r the maximal dimension of an abelian p-nilpotent subalgebra of g.

Equations for some nilpotent varieties

Let $\mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ over $\mathbb C$, induced from a Levi subalgebra whose simple roots are orthogonal short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{ \mathcal{O}}$ in $S \mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broer's result for the subregular nilpotent orbit. Along the way we give a…

Nilpotent commuting varieties of reductive Lie algebras

We prove that the nilpotent commuting variety of a reductive Lie algebra over an algebraically closed field of good characteristic is equidimensional. In characteristic zero, this confirms a conjecture of Vladimir Baranovsky. As a by-product, we obtain tat the punctual (local) Hilbert scheme parametrising the ideals of colength $n$ in $k[[X,Y]]$ is irreducible over any algebraically closed field $k$.

Parametrizations of canonical bases and irreducible components of nilpotent varieties

It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc and Schröer has recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper we show that their parametrization coincides with Lusztig's parametrization of the canonical basis.

A binary operation on irreducible components of Lusztig's nilpotent varieties {I}: definition and properties

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.

正则幂零 Hessenberg 簇上同调环的一组加法基

An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties

本文通过延拓更小正则幂零 Hessenberg 子簇的 Poincaré 对偶，构造了正则幂零 Hessenberg 簇上同调环的加法基，并证明这些对偶线性无关。

Dynamics on nilpotent character varieties

Let R(N,G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected compact Lie group G, and let X(N,G) be the corresponding moduli space. We show that there exists a natural Out(N)-invariant measure on X(N,G) and that whenever Out(N) has at least one hyperbolic element, the action of Out(N) on X(N,G) is mixing with respect to this measure.

Commuting varieties for nilpotent radicals

Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin-Röhrle concerning the commuting variety of Lie(U) for char(k)=0 to fields, whose characteristic is good for G

Nilpotent commuting varieties of the Witt algebra

Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimens…

Cell decompositions of quiver flag varieties for nilpotent representations of the oriented cycle

Generalizing Schubert cells in type A and a cell decomposition if Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of an oriented cycle admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation and describe the T-equivariant cohomology using Goresky-Kottwitz-MacPherson-theory. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by …

Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This le…

On the existence of smooth orbital varieties in simple Lie algebras

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers …

Categories of modules given by varieties of p-nilpotent operators

For a finite group scheme G over an algebraically closed field k of characteristic p>0 we study G-modules M, which are defined in terms of properties of their pull-backs along p-points of G. We show that the corresponding subcategories strongly depend on the structure of G. The second part of the paper discusses recent work by Carlson-Friedlander-Suslin concerning the subcategory of equal images modules from the vantage point of Auslander-Reiten theory.

On nilpotent commuting varieties and cohomology of Frobenius kernels

The paper studies the dimensions of irreducible components of commuting varieties of (restricted) nilpotent $r$-tuples in a classical Lie algebra $\mathfrak{g}$ defined over an algebraically closed field $k$. As applications, we obtain some new results on the structure of the (even) cohomology ring of Frobenius kernels $G_r$ for each $r\ge 1$, where $G$ is the simply connected, simple algebraic group such that $\text{Lie}(G)=\mathfrak{g}$. Explicit calculations for rank two groups are also presented.

幂零轨道及其割线簇

Nilpotent orbits and their secant varieties

本文利用二重作用理论描述单代数群幂零轨道的割线簇仿射锥，计算其维数、复杂度和秩，并揭示其与矩映射闭包的关系。

与伪极化相关的Spaltenstein流形

Spaltenstein Varieties Associated with Pseudo-Polarizations

本文研究了经典李代数中B、C和D类型nilpotent轨道的最小Richardson轨道和伪极化，证明了Spaltenstein流形的光滑性和纯维性，并扩展了Richardson轨道中的seesaw性质和对偶性到所有特殊轨道。

Anti-commuting varieties

We study the anti-commuting variety which consists of pairs of anti-commuting $n\times n$ matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT quotient of the anti-commuting variety with respect to the conjugation action of $GL_n$ is shown to be of pure dimension $n$. We also show the semi-nilpotent anti-commuting variety (in which one matrix is required to be nilpotent) is of pure dimension $n^2$ and describe its irreducible components.

Conjugation on varieties of nilpotent matrices

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL_n(C) on the variety of x-nilpotent complex matrices. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of 2-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. Concerning the action on the nilpotent cone, we obtain a generic normal form of the orbits which yields a U-normal form as well…

Admissible modules and normality of classical nilpotent orbits II

In this paper, we compute the character formula of the Brylinski model for all classical nilpotent varieties $\overline{\mathcal{O}}$. As a consequence, one can compute the multiplicities of all $K-$types of the ring of regular functions $R(\overline{\mathcal{O}})$ for all classical nilpotent varieties.

可解代数群作用的贪心开轨道准则及其在Lusztig幂零簇中的应用

A greedy open-orbit criterion for solvable algebraic group actions, with applications to Lusztig's nilpotent varieties

该研究针对可解代数群作用提出贪心开轨道准则，应用于Dynkin箭图无重表示的Aut_Q(M)作用，简化Lusztig幂零簇刚性问题，为等定向A型提供显式算法。

The combinatorics of orbital varieties closures of nilpotent order 2 in sl(n)

We consider two partial orders on standard Young tableaux. The first one is induced from the weak right Bruhat order on symmetric group by Robinson-Schensted algorithm. The second one is induced from the order on Young diagrams by considering a Young tableau as a chain of Young diagrams. We show that these two orders of completely different nature coincide on the subset of Young tableaux with 2 columns or with 2 rows. This fact has very interesting geometric implications for orbital varieties of nilpotent order 2 in sl(n).

Poincare polynomials of character varieties, Macdonald polynomials and affine Springer fibers

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…

Extension-orthogonal components of nilpotent varieties

Let Q be a Dynkin quiver, and let P(Q) be the corresponding preprojective algebra. Let I be a set of pairwise different indecomposable irreducible components of varieties of P(Q)-modules such that generically there are no extensions between these components. We show that the number of elements in I is at most the number of positive roots of Q. Furthermore, we give a module theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.

Finite Parabolic Conjugation on Varieties of Nilpotent Matrices

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$ on the variety of $x$-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of $2$-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. We show that in all non-finite cases, the correspon…

Existence of affine pavings for varieties of partial flags associated to nilpotent elements

The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In th…

Nested Punctual Hilbert Schemes and Commuting Varieties of Parabolic Subalgebras

It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of parabolic subalgebras of $M\_n(\K)$ and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the numerical conditions under which they are irreducible. In some reducible cases, we describe the irreducible components and their dimension.

A型箭袋簇、环路格拉斯曼流形与幂零锥的比较

Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A

本文在A型情形下建立Nakajima带标架箭袋簇、矩阵共轭类与环路格拉斯曼流形之间的显式几何等价，并由此实现紧化、分解及$(GL(m),GL(n))$对偶的几何化。

$A$ 型 Lusztig 幂零簇的刚性分量与伪刚性分量的一个遗传定理

A hereditary theorem for rigid and pseudorigid components of Lusztig's nilpotent varieties in type $A$

本文在 $A$ 型 Lusztig 幂零簇中，用组合方式刻画了 $C_\sigma\oplus C_\sigma$ 为不可约分量的条件，通过伪光滑置换的递归描述并引入遗传性质给出完整刻画。

A binary operation on irreducible components of Lusztig's nilpotent varieties {II}: applications and conjectures for representations of $GL_n$ over a non-archimedean local field

In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schröer, that this operation is compatible with taking the socle of parabolic induction of representations of general linear groups over a local non-archimedean field, at least when one of the irreducible components is rigid. We verify this conjecture in special cases.

Poincare series of character varieties for nilpotent groups

For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $Γ$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety) $Rep(Γ,G)_1$, with coefficients in a field $F$ with ${char} (F)$ either 0 or relatively prime to the order of the Weyl group $W$. We give explicit formulas for the Poincaré series. In addition we study $G$-equivariant stable decompositions of subspaces $X(q,G)$ of the free monoid $J(G)$ generated by the Lie group $G$, obtained from finitely generated free nilpotent group rep…

Paving Hessenberg varietes by affines

Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types, generalizing results of de Concini-Lusztig-Procesi and Kostant. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving and their dimensions are identified by combinatorial conditions on roots. We use the paving t…

Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

Let $G$ be a simple algebraic group with $\mathfrak g=\mathsf{Lie} G$ and $\mathcal O_{\sf min}\subset\mathfrak g$ the minimal nilpotent orbit. For a $\mathbb Z_2$-grading $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$, let $G_0$ be a connected subgroup of $G$ with $\mathsf{Lie} G_0=\mathfrak g_0$. We study the $G_0$-equivariant projections $φ:\overline{\mathcal O_{\sf min}}\to \mathfrak g_0$ and $ψ:\overline{\mathcal O_{\sf min}}\to\mathfrak g_1$. It is shown that the properties of $\overline{φ(\mathcal O_{\sf min})}$ and $\overline{ψ(\mathcal O_{\sf min})}$ essentially depend on whether the …

Regular orbits of symmetric subgroups on partial flag varieties

The main result of this paper is a new parameterization of the orbits of a symmetric subgroup on a partial flag variety. The parameterization is in terms of certain Spaltenstein varieties, on one hand, and certain nilpotent orbits, on the other. One of the motivations is related to enumerating special unipotent representations of real reductive groups. Another motivation is understanding (a portion of) the closure order on the set of nilpotent coadjoint orbits.

奇异Spaltenstein簇

Exotic Spaltenstein varieties

本文定义奇异Spaltenstein簇以推广经典Spaltenstein簇，证明二阶自伴幂零自同态情形下最高维不可约分量与半标准Young双表一一对应，并给出最高维数的组合公式。

二阶幂零元中心化子在旗簇中的轨道闭包：A、B、D 型的正规性与消解

Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

本文研究 A、B、D 型约化群中二阶幂零元中心化子在旗簇内的轨道闭包，证明其正规性、特定条件下的 Cohen–Macaulay 性与有理奇点，并构造涉及 Schubert 簇的双有理有理消解。

Spaltenstein varieties of pure dimension

We show that Spaltenstein varieties of classical groups are pure dimensional when the Jordan type of the nilpotent element involved is an even or odd partition. We further show that they are Lagrangian in the partial resolutions of the associated nilpotent Slodowy slices, from which their dimensions are known to be one half of the dimension of the partial resolution minus the dimension of the nilpotent orbit. The results are then extended to the $σ$-quiver-variety setting.

Unipotent Representations Attached to the Principal Nilpotent Orbit

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations.

Methods for Parametrizing Varieties of Lie Algebras

We describe how to smoothly parametrize certain families of nilpotent Lie algebras.

关于幂零轨道与抛物 Slodowy 簇的 Hikita-Nakajima 猜想

Around Hikita-Nakajima conjecture for nilpotent orbits and parabolic Slodowy varieties

本文针对幂零 Slodowy 横截与特殊幂零轨道覆盖的辛对偶，修正原始 Hikita 猜想，并对抛物 Slodowy 簇证明其成立，同时给出李理论与组合应用。

On support varieties and the Humphreys conjecture in type $A$

Let $G$ be a reductive algebraic group scheme defined over $\mathbb{F}_p$ and let $G_1$ denote the Frobenius kernel of $G$. To each finite-dimensional $G$-module $M$, one can define the support variety $V_{G_1}(M)$, which can be regarded as a $G$-stable closed subvariety of the nilpotent cone. A $G$-module is called a tilting module if it has both good and Weyl filtrations. In 1997, it was conjectured by J.E. Humphreys that when $p\geq h$, the support varieties of the indecomposable tilting modules coincide with the nilpotent orbits given by the Lusztig bijection. In this paper, we shall verif…

Components of the Springer Fiber and Domino Tableaux

Consider a complex classical semi-simple Lie group along with the set of its nilpotent coadjoint orbits. When the group is of type A, the set of orbital varieties contained in a given nilpotent orbit is described a set of standard Young tableaux. We parameterize both, the orbital varieties and the irreducible components of unipotent varieties in the other classical groups by sets of standard domino tableaux. The main tools are Spaltenstein's results on signed domino tableaux together with Garfinkle's operations on standard domino tableaux.

Classification of irreducible $(\mathfrak{g},\mathfrak{k})$-modules associated to the ideals of minimal nilpotent orbits for simple Lie groups of type $A$

We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.

特征为2的交换变种

Commuting varieties in bad characteristic

本文研究了特征为2的sp_{2n}李代数的交换变种和nilpotent交换变种的不可约性和维度。

Commuting Varieties and Cohomological Complexity

In this paper we determine, for all $r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting $r$-tuples of nilpotent elements of $\mathfrak{gl}_n$. Our main result is that in characteristic $\neq 2,3$, this nilpotent commuting variety has dimension $(r+1)\lfloor \frac{n^2}{4}\rfloor$ for $n\geq 4$, $r\geq 7$. We use this to find the dimension of the (ordinary) $r$-th commuting varieties of $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ for the same range of values of $r$ and $n$. Our principal motivation is the connection between nilpotent commuting variet…

On the irreducibility of associated varieties of W-algebras

We investigate the irreducibility of the nilpotent Slodowy slices that appear as the associated variety of W-algebras. Furthermore, we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves.

极小幂零轨道与环面簇

Minimal Nilpotent Orbits and Toric Varieties

本文研究了 $\mathfrak{sl}_{n+1}(\mathbb C)$ 中极小幂零轨道的概形论交集，证明其坐标环可平坦退化至 $C_n$ 型环面簇的等变上同调环，并证明了该交集及其轨道簇的既约性与 Cohen-Macaulay 性质。

B-orbits of nilpotent order 2 and link patterns

In this paper we describe geometry of orbits of upper triangular matrices of nilpotent order 2 under conjugation by the group of upper triangular invertible matrices in terms of link patterns. Further we apply this description to the computations of the closures of orbital varieties of nilpotent order 2 and intersections of components of a Springer fiber of nilpotent order 2. In particular we connect our results to the combinatorics of meanders and Temperley-Lieb algebras.

On singularities in B-orbit closures of 2-nilpotent matrices

This paper deals with singularities of closures of $2$-nilpotent Borel conjugacy classes in either a $\text{GL}_n$-conjugacy class or in the nilpotent cone of $\text{GL}_n$. In the latter case we construct a resolution of singularities, in the former we show that singularities are rational by applying a result of M. Brion. We reason why this generalizes the result of N. Perrin and E. Smirnov on the rationality of singularities of Springer fiber components in the two-column case. In the case of Borel orbit closures being contained in orbital varieties, we give an alternative version of L. Fress…