An exotic nilpotent cone for symplectic group

辛李代数中幂零切片的等变形变理论

Equivariant deformation theory for nilpotent slices in symplectic Lie algebras

针对辛李代数中双块若尔当形幂零轨道的Slodowy切片非通用泊松形变问题，证明其幂零Slodowy簇与D型同构，C型切片为$\u2124_2$-等变通用泊松形变且有非交换对应。

Symplectic Resolutions for Coverings of Nilpotent Orbits

Let $\0$ be a nilpotent orbit in a semisimple complex Lie algebra $\g$. Denote by $G$ the simply connected Lie group with Lie algebra $\g$. For a $G$-homogeneous covering $M \to \0$, let $X$ be the normalization of $\bar{\0}$ in the function field of $M$. In this note, we study the existence of symplectic resolutions for such coverings $X$.

Birational geometry of symplectic resolutions of nilpotent orbits II

In this paper we shall study symplectic resolutions of a nilpotent orbit closure of a complex simple Lie algebra \g. We shall introduce an equivalence relation in the set of parabolic subgroups of $G$ in terms of marked Dynkin diagrams. We start with a nilpotent orbit closure which admits a Springer resolution with a parabolic subgroup $P_0$ of $G$. Then we prove that all symplectic resolution of the nilpotent closure are Springer resolutions with $P$ which are equivalent to $P_0$. Here all symplectic resolutions are connected by Mukai flops. We need three types of Mukai flops (types A, D and …

Equivariant coherent sheaves on the exotic nilpotent cone

Let $G=Sp_{2n}(\mathbb{C})$, and $\mathfrak{N}$ be Kato's exotic nilpotent cone. Following techniques used by Bezrukavnikov in [5] to establish a bijection between $Λ^+$, the dominant weights for a simple algebraic group $H$, and $\textbf{O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $\widetilde{\mathfrak{N}}$ have vanishing higher cohomology, and compute their g…

sheets，Jordan类和诱导轨道在exotic和enhanced模块中

Sheets, Jordan classes and induced orbits in the exotic and enhanced modules

本文研究了exotic和enhanced模块中的sheets和Jordan类，通过引入Jordan分层和诱导轨道理论，对adjoint表示和对称空间进行了推广，并基于Borho定理对这些模块中的sheets进行了分类。

General Linear and Symplectic Nilpotent Orbit Varieties

The condition of nilpotency is studied in the general linear Lie algebra $\mathfrak{gl}_{n}(\mathbb{K})$ and the symplectic Lie algebra $\mathfrak{sp}_{2m}(\mathbb{K})$ over an algebraically closed field of characteristic 0. In particular, the conjugacy class of nilpotent matrices is described through nilpotent orbit varieties $\mathcal{O}_λ$ and an algorithm is provided for computing the closure $\overline{\mathcal{O}_λ} \cong \text{Spec}\left(\mathbb{K}[X]\big/J_λ\right).$ We provide new generators for the ideal $J_λ$ defining the affine variety $\overline{\mathcal{O}_λ}$ which show that the…

Deformations of nilpotent cones and Springer correspondences

Let G = Sp (2n) be the symplectic group over Z. We present a certain kind of deformation of the nilpotent cone of G with G-action. This enables us to make direct links between the Springer correspondence of sp_{2n} over C, that over characteristic two, and our exotic Springer correspondence. As a by-product, we obtain a complete description of our exotic Springer correspondence.

An exotic Springer correspondence for symplectic groups

This paper is a sequel to math.RT/0601155. Let G be a complex symplectic group. In math.RT/0601155, we constructed a certain G-variety N = N_1, which we call the (1-) exotic nilpotent cone. In this paper, we study the set of G-orbits of the variety N. It turns out that the variety N gives a variant of the Springer correspondence for the Weyl group of type C, but shares a similar flavor with that of type A case. (I.e. there appears no non-trivial local system and the correspondence is bijective.) As an application, we present one sufficient condition for the bijectivity of our exotic Deligne-La…

Birational Geometry of symplectic resolutions of nilpotent orbits

In this paper, we shall prove that any two (projective) symplectic resolutions of a nilpotent orbit closure in a classical simple Lie algebra are connected by a finite sequence of diagrams which are locally trivial families of Mukai flops of type A or of type D. A Mukai flop of type A is the same as a stratified Mukai flop in our previous paper "Mukai flops and derived categories II". On the other hand, a Mukai flop of type D comes from a nilpotent orbit in so(2k). Let G^+ and G^- be two connected components of the orthogonal Grassmannian G_{iso}(k,2k). Then the closure of this orbit has two S…

Simple examples of symplectic fourfolds with exotic properties

We construct examples of simply connected nonalgebraic symplectic fourfolds with a prescribed number of nonintersecting symplectic curves with positive self-intersections.

Four-dimensional conical symplectic hypersurfaces

We show that every indecomposable conical symplectic hypersurface of dimension four is isomorphic to the known one, namely, the Slodowy slice $X_n$ which is transversal to the nilpotent orbit of Jordan type $[2n-2, 1, 1]$ in the nilpotent cone of $\mathfrak{sp}_{2n}$ for some $n\ge 2$. In the appendix written by Yoshinori Namikawa, conical symplectic varieties of dimension two are classified by using contact Fano orbifolds.

Irreducible components of exotic Springer fibres

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the …

Admissible modules and normality of classical nilpotent orbits I

In the case of complex symplectic and orthogonal groups, we find $(\mathfrak{g}, K)-$modules with the property that their $K-$structure matches the structure of regular functions on the closures of nilpotent orbits. This establishes a version of the Orbit Method of Kirrilov-Kostant-Souriau as proposed by Vogan. In the process we give another proof of the classification of nilpotent orbits with normal closure in the Lie algebra of a classical group first established by Kraft-Procesi.

Springer theory for symplectic Galois groups

A classical and beautiful story in geometric representation theory is the construction by Springer of an action of the Weyl group on the cohomology of the fibres of the Springer resolution of the nilpotent cone. We establish a natural extension of Springer's theory to arbitrary symplectic resolutions of conical symplectic singularities. We analyse features of the action in the case of affine quiver varieties, constructing Weyl group actions on the cohomology of $ADE$ quiver varieties, and also consider "symplectically dual" examples arising from slices in the affine Grassmannian. Along the way…

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。

A new family of isolated symplectic singularities with trivial local fundamental group

We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of $\mathbb{C}^4$ by the dihedral group of order $2d$, (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order $2d$ at equal parameters, (3) as singularities of a certain Slodowy slice in the $d$-fold cover of the nilpotent cone in ${\mathfrak{sl}}_d$.

Symplectic geometry of $p$-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles

The aim of the present paper is to provide a new aspect of the $p$-adic Teichmüller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichmüller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in…

Deformation of Singular Fibers of Genus $2$ Fibrations and Small Exotic Symplectic $4$-Manifolds

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to ${\mathbb{CP}}^{2}\#6({\overline{\mathbb{CP}}^{2}})$, ${\mathbb{CP}}^{2}\#7({\overline{\mathbb{CP}}^{2}})$, and $3{\mathbb{CP}}^{2}\#k({\overline{\mathbb{CP}}^{2}})$ for $k=16, 17, 18, 19$ using combinations of such deformations, symplectic blowups, and (…

Symplectic complexity of reductive group actions

Let a complex algebraic reductive group $\bf G$ act on a complex algebraic manifold $\bf X$. For a $\bf G$-invariant subvariety $Ξ$ of the nilpotent cone $\mathcal{N}(\mathfrak{g}^*)\subset \mathfrak{g}^*$ we define a notion of $Ξ$-symplectic complexity of $\bf X$. This notion generalizes the notion of complexity defined in [Vin86]. We prove several properties of this notion, and relate it to the notion of $Ξ$-complexity defined in [AG] motivated by its relation with representation theory.

Gurtas Lefschetz 纤维化、有理爆破与奇异辛四流形

Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds

本文通过Gurtas Lefschetz纤维化的例外截面构造负球面，利用有理爆破在奇偶性条件下生成单连通的奇异辛四流形。

Kaehler spaces, nilpotent orbits, and singular reduction

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and pre-homogeneous spaces; in particular, in the world of singular Poisson geometry, the closures of principal holom…

Normality of closure of orthogonal nilpotent symmetric orbits

We study closures of conjugacy classes in the symmetric matrices of the orthogonal group and we determine which one are normal varieties. In contrast to the result for the symplectic group where all classes have normal closure, there is only a relatively small portion of classes with normal closure. We perform a combinatorial computation on top of the same methods used by Kraft-Procesi and Ohta.

假想 exotic $CP^2$ 的辛对称性：潜在构造或不存在性证明

Symplectic symmetries of a hypothetical exotic $CP^2$: a potential construction or a nonexistence proof

本文研究同胚但不同胚于 CP^2 的辛 4-流形及其有限阶辛同胚的存在性，将其归结为 12 个接触管道流形的辛填充问题，并确定了伪自由辛 Z_p-作用的阶与局部表示，提出潜在构造或不存在性证明。

Steinberg理论的推广与一个异动量映射

A generalization of Steinberg theory and an exotic moment map

本研究将Steinberg关于约化群的理论推广到对称对情形，构造了广义Steinberg映射与异动量映射，聚焦特定一般线性群对称对，给出对应几何背景与组合算法，拓展了Robinson-Schensted对应。

Poisson-de Rham homology of hypertoric varieties and nilpotent cones

We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson-de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson-de Rham-Poincare polynomial, and prove …

New Exotic 4-Manifolds via Luttinger Surgery on Lefschetz Fibrations

In [2], the first author constructed the first known examples of exotic minimal symplectic $\CP#5\CPb$ and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to $3\CP#7\CPb$. The construction in [2] uses Y. Matsumoto's genus two Lefschetz fibrations on $M = \mathbb{T}^{2}\times \mathbb{S}^{2} #4\CPb$ over $\mathbb{S}^2$ along with the fake symplectic $\mathbb{S}^{2} \times \mathbb{S}^{2}$ construction given in [1]. The main goal in this paper is to generalize the construction in [2] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz an…