Let G be a simple algebraic group over the complex numbers. Let N be the cone of nilpotent elements in the Lie algebra of G. Let K_{G x C^*}(N) denote the Grothendieck group of the category of G x C^*-equivariant coherent sheaves on N. In this note we construct a Kazhdan-Lusztig type canonical basis of K_{G x C^*}(N) over representation ring of C^*. This basis is parametrized by the set of dominant weights for G. On the other hand we conjecture that this basis is close to the basis consisting of irreducible G-equivariant bundles on nilpotent orbits. This would give us a natural construction of Lusztig's bijection between two sets: \{dominant weights for G\} and \{pairs consisting of a nilpotent orbit O and irreducible G-equivariant bundle on O.
Let $G$ be a simple complex algebraic group, $P$ a parabolic subgroup of $G$ and $N$ the unipotent radical of $P.$ The so-called equivariant compactification of $N$ by $G/P$ is given by an action of $N$ on $G/P$ with a dense open orbit isomorphic to $N$. In this article, we investigate how many such equivariant compactifications there exist. Our result says that there is a unique equivariant compactification of $N$ by $G/P$, up to isomorphism, except $¶^n$.
We consider aspects of the geometry and topology of nilpotent orbits in finite-dimensional complex simple Lie algebras. In particular, we give the equivariant cohomologies of the regular and minimal nilpotent orbits with respect to the action of a maximal compact torus of the overall group in question.
We exhibit a new proof, relying on bivariant theory, that the nilpotent cone is rationally smooth. Our approach enables us to prove a slightly more general statement.
In this article, we investigate an analogue of the global nilpotent cone in the case of the cotangent stack of the moduli stack of parahoric torsors on curves.
We exhibit the irreducible components of the nilpotent cone of the tadpole quiver of arbitrarily high codimension and introduce an approach that could allow to describe all of them.
We study the nilpotent cone in the Mukai system for rank two and genus two. We compute the degrees and multiplicities of its irreducible components and describe their cohomology classes.
We study mirror symmetry of complete intersection Calabi-Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.
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…
A short note to show that the elements of the (open) cone underlying a nilpotent orbit on a period domain are pairwise congruent under the symmetry group of the period domain.
In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig …
We define and study an exotic t-structure on the bounded derived category of equivariant coherent sheaves on partial resolutions of the nilpotent cone.
The $\mathrm{GL}(V)$-orbits in the enhanced nilpotent cone $V\times\mathcal{N}(V)$ are (essentially) in bijection with the orbits of a certain parabolic $P\subseteq\mathrm{GL}(V)$ (the mirabolic subgroup) in the nilpotent cone $\mathcal{N}(V)$. We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different pa…
We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural $\mathbb{C}^*$-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle.
In characteristic zero, Bezrukavnikov has shown that the category of perverse coherent sheaves on the nilpotent cone of a simply connected semisimple algebraic group is quasi-hereditary, and that it is derived-equivalent to the category of (ordinary) coherent sheaves. We prove that graded versions of these results also hold in good positive characteristic.
We continue the study of the closures of $GL(V)$-orbits in the enhanced nilpotent cone $V\times\cN$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably-defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.
This paper gives a combinatorial description of the set of irreducible components of the semistable locus of the global nilpotent cone, in genus $\ge2$. The first main result of this paper states that the set of irreducible components of the global nilpotent cone is given by the very natural decomposition in twisted Jordan strata, which are smooth. Then we move on to the semistable locus and obtain purely combinatorial conditions on the twisted Jordan type to be semistable. The proof uses an analogous result obtained in the context of moduli stacks of chains, and shows that semistability can b…
Let $G$ be a real reductive Lie group, and $H^{\mathbb{C}}$ the complexification of its maximal compact subgroup $H\subset G$. We consider classes of semistable $G$-Higgs bundles over a Riemann surface $X$ of genus $g\geq2$ whose underlying $H^{\mathbb{C}}$-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of $G$-Higgs bundles over $X$ to the moduli space of $H^{\mathbb{C}}$-bundles over $X$, in contrast with the situation when $g=1$, and to show reducibility of the nilpotent cone of the moduli space of $G$-Higgs bundles, for $G$ c…
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.
In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar ``Langlands dual'' description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik.
We compute the equivariant Hirzebruch class of the quadric cone in C^n degenerating it to an intersection to hyperplanes. The difference of the Hirzebruch classes (measured by the Milnor class) turns out to be the Hirzebruch class of the complement of the quadratic cone of smaller dimension.
We define and study several equivariant stratifications of the isotropy and coisotropy representations of a parabolic subgroup in a complex reductive group.
We describe two approaches to classifying the possible monodromy cones C arising from nilpotent orbits in Hodge theory. The first is based upon the observation that C is contained in the open orbit of any interior point N in C under an associated Levi subgroup determined by the limit mixed Hodge structure. The possible relations between the interior of C its faces are described in terms of signed Young diagrams. The second approach is to understand the Tannakian category of nilpotent orbits via a category D introduced by Deligne in a letter to Cattani and Kaplan. In analogy with Hodge theory, …
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of complex nilpotent matrices. We obtain generic normal forms of the orbits and describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the $U$-invariant ring. The latter is described in more detail in terms of algebraic quotients by a special toric variety closely related. The study of a GIT-quotient for the Borel-action is initiated.
Let $n$ be a fixed positive integer and $h: \{1,2,...,n\} \rightarrow \{1,2,...,n\}$ a Hessenberg function. The main result of this manuscript is to give a systematic method for producing an explicit presentation by generators and relations of the equivariant and ordinary cohomology rings (with $\mathbb{Q}$ coefficients) of any regular nilpotent Hessenberg variety $\mathrm{Hess}(h)$ in type A. Specifically, we give an explicit algorithm, depending only on the Hessenberg function $h$, which produces the $n$ defining relations $\{f_{h(j),j}\}_{j=1}^n$ in the equivariant cohomology ring. Our resu…
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…
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $\mathbb{C}P^2\# N\overline{\mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of…
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 …
For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are…
In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada and Tymoczko also prove that in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of a successful outcome of Betti poset pinball for …
We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun's Conjecture which says that the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kahler-Einstein Fano manifolds only depends on the algebraic structure of the singularity. The second result says that for any lo…
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…
In the approach to Gromov-Witten theory developed by Givental, genus-zero Gromov-Witten invariants of a manifold X are encoded by a Lagrangian cone in a certain infinite-dimensional symplectic vector space. We give a construction of this cone, in the spirit of S^1-equivariant Floer theory, in terms of S^1-equivariant Gromov-Witten theory of the product X \times P^1. This gives a conceptual understanding of the "dilaton shift": a change-of-variables which plays an essential role in Givental's theory.
It is shown that projectivized irreducible components of nilpotent cones of complex symmetric spaces are projective self-dual algebraic varieties. Other properties equivalent to their projective self-duality are found.
In Part VII, we proved that the range of the big J-function in permutation-equivariant genus-0 quantum K-theory is an overruled cone, and gave its adelic characterization. Here we show that the ruling spaces are D_q-modules in Novikov's variables, and moreover, that the whole cone is invariant under a large group of symmetries defined in terms of q-difference operators. We employ this for the explicit reconstruction of the cone from one point on it, and apply the result to toric target spaces, when such a point is given by the q-hypergeometric function.
The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the Geometric Langlands program, due to Beilinson-Drinfeld, since it insures that the D-modules on the moduli space of G-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic, hence, e.g. have finite length.
Mukai's space, parametrizing simple sheaves on a K3 surface S whose numerical invariants are those of a line bundle on a curve C in S, is interpreted as a deformation of Hitchin's system on C. This is used to show that the nilpotent cone in Mukai space is Lagrangian. In rank 2, components of this nilpotent cone are described as affine bundles over symmetric products of the curve. The underlying vector bundles give the corresponding components of the Hitchin nilpotent cone.
Institut Camille Jordan, UMR 5208 CNRS, Univ Lyon, Université Jean Monnet（让·莫内大学）；Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université Bourgogne Franche-Comté（勃艮第弗朗什-孔泰大学）
The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a formal slice. Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C*-equivariant version of the Darboux-Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqu…
This is a continuation of math.AG/0408274, where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution. The purpose of this paper is, to describe the remainder. We shall first construct a deformation of the nilpotent orbit closure in a canonical manner according to Brieskorn and Slodowy, and next describe all its crepant simultaneous resolutions. This construction enables us to d…
For d > 1, we consider the Veronese map of degree d on a complex vector space W , Ver_d : W -> Sym^d W , w -> w^d , and denote its image by Z. We describe the characters of the simple GL(W)-equivariant holonomic D-modules supported on Z. In the case when d is 2, we obtain a counterexample to a conjecture of Levasseur by exhibiting a GL(W)-equivariant D-module on the Capelli type representation Sym^2 W which contains no SL(W)-invariant sections. We also study the local cohomology modules H_Z^j(S), where S is the ring of polynomial functions on the vector space Sym^d W. We recover a result of Og…
We describe a geometric-combinatorial algorithm that allows one, using solely the system of weights and roots, to determine the Hesselink strata of the null-cone of a linear representation of a reductive algebraic group and calculate their dimensions. In particular, it provides a constructive approach to calculating the dimension of the null-cone and determining all its irreducible components of maximal dimension. In the case of the adjoint representation (and, more generally, a $θ$-representation), the algorithm turns into the classification algorithm for the conjugacy classes of nilpotent el…
The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decompositio…
