S. Cupit-Foutou
A two-orbit variety is a normal complete complex algebraic variety on which a reductive complex algebraic group acts with exactly two orbits. The aim of this paper is to give the classification of all two-orbit varieties and to prove Luna's conjecture, namely that two-orbit varieties are spherical.
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Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties.
PGL orbits in tree varieties
该研究定义树簇并分析其上的 PGL 作用，刻画有限轨道情形，给出稠密轨道判据与反例，并确定 $F(k_1,k_2;n)^3$ 的稠密轨道充要条件。
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.
In this article, we investigate the geometry of reductive group actions on algebraic varieties. Given a connected reductive group $G$, we elaborate on a geometric and combinatorial approach based on Luna-Vust theory to describe every normal $G$-variety with spherical orbits. This description encompasses the classical case of spherical varieties and the theory of $\mathbb{T}$-varieties recently introduced by Altmann, Hausen, and Süss.
Toric subvarieties of projective space are classified up to projective automorphisms.
This article gives a description of the diagonal $GL_3$-orbits on the quadruple projective variety $(\mathbb P^2)^4$. We give explicit representatives of orbits, and describe the closure relations of orbits. A distinguished feature of our setting is that it is the simplest case where $\mathrm{diag}(GL_n)$ has infinitely many orbits but has an open orbit in the multiple projective space $(\mathbb P^{n-1})^m$.
For a connected, simply-connected complex simple algebraic group $G$, we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we compute the Poincaré polynomials and irreducible components of these varieties in Lie type $A$. Furthermore, we show these Hessenberg varieties to be GKM with respect to the action of a maximal torus $T\subseteq G$. The corresponding GKM graphs are then explicitly determined. Finally, we present the ordinary and $T$-equivariant cohomology rings of our varieties as quotients of those of the flag variety.
Let $N$ be a point of an orbit closure $\bar{O_M}$ in a module variety such that its orbit $O_N$ has codimension two in $\bar{O_M}$. We show that under some additional conditions the pointed variety $(\bar{O_M},N)$ is smoothly equivalent to a cone over a rational normal curve.
We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety $G/P$ are homogeneously embedded cominuscule $G'/P'$, and are classified by subdiagrams of a Dynkin diagram. This generalizes the classification of smooth Schubert varieties in cominuscule $G/P$.
In the setting of strict wonderful varieties we answer positively to Luna's conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits or model spaces. To make the paper self-contained as much as possible, we shall gather some known results on these families and more generally on wonderful varieties.
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。
In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.
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。
Let M_d(k) denote the space of dxd-matrices with coefficients in an algebraically closed field k. Let X be an orbit closure in the product [M_d(k)]^t equipped with the action of the general linear group GL_d(k) by simultaneous conjugation. We show that X is regular at any its point y such that the orbit of y has codimension one in X. The proof uses mainly the representation theory of associative algebras.
On the Classification of Schubert Varieties in Partial Flag Varieties
本文在部分旗流形中推广了Schubert流形的分类，通过根系折叠得到等价对，并确定了Schubert三重流形等价类的上界。
We determine which nilpotent orbits in $E_6$ have normal closure and which do not. We also verify a conjecture about small representations in rings of functions on nilpotent orbit covers for type $E_6$.
With the help of Lusztig's canonical basis, we study local intersection cohomology of the Zariski closures of orbits of representations of a quiver of type A, D or E. In particular, we characterize the rationally smooth orbits and prove that orbit closures are smooth if and only if they are rationally smooth. This provides an analogue of theorems of V. Deodhar, and J. Carrell and D. Peterson on Schubert varieties.
Singularities of orbit closures in loop spaces of symmetric varieties II: twisted cases
本研究将轨道闭包奇点理论推广至扭曲情形，证明IC复形的逐点纯性与奇偶消失性，并应用于扭曲仿射Lusztig--Vogan模及Langlands对偶。
The invariant Hilbert schemes considered in \cite{BC1} were proved to be affine spaces. The proof relied on the classification of strict wonderful varieties. We obtain in the present article a classification-free proof of the affinity of these invariant Hilbert scheme by means of deformation theoretical arguments. As a consequence we recover in a shorter way the classification of strict wonderful varieties. This provides an alternative and new approach to answer Luna's conjecture.
We unify aspects of the equivariant geometry of type $D$ quiver representation varieties, double Grassmannians, and symmetric varieties $GL(a+b)/GL(a)\times GL(b)$; in particular we translate results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant $K$-theory between these three families. These results are all obtained from our generalization of a construction of Zelevinsky for type $A$ quivers to the type $D$ setting. More precisely, we give explicit embeddings with nice properties of homogeneous fiber bundles over type $D$ quiver repres…
Let X be the direct product of two Grassmann varieties of k- and l-planes in a finite-dimensional vector space V, and let B be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog to Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott--Samelson desingularizations for Schubert varieties.
For the classical groups, Kraft and Procesi have resolved the question of which nilpotent orbits have closures which are normal and which are not, with the exception of the very even orbits in $D_{2l}$ which have partition of the form $(a^{2k}, b^2)$ for $a, b$ distinct even natural numbers with $a k + b = 2 l$. In this article, we show that these orbits do have normal closure.
Singularities of orbit closures in loop spaces of symmetric varieties
本文研究对称簇环路空间中Iwahori轨道闭包的奇点，推广Lusztig-Vogan工作至仿射情形，证明IC-复形的纯性与奇偶消失，并应用于仿射Lusztig-Vogan模及相对Langlands对偶，回答Lusztig的问题。
This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric varieties of classical types. Here, we determine the generating series the numbers of Borel orbits in $\mathbf{SO}_{2n+1}/\mathbf{S(O}_{2p}\times \mathbf{O}_{2q+1}\mathbf{)}$ (type $BI$) and in $\mathbf{Sp}_n/\mathbf{Sp}_p\times \mathbf{Sp}_q$ (type $CII$). In addition, we explore relations to lattice path enumeration.
Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--orbits in flag varieties $G/P$. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths--Yukawa coupling to the variety of lines on $G/P$ (under a minimal homogeneous embedding), construct a large class of polarized $G_{\mathbb{R}}$--orbits in $G/P$, and compute the associated Hodge--theoretic boundary components. An emphasis is placed through…
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 …
Let F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with finitely many orbits, and let V be an H-orbit closure in F. Expanding the cohomology class of V in the basis of Schubert classes defines a union V_0 of Schubert varieties in F with positive multiplicities. If G is simply-laced, we show that these multiplicites are equal to the same power of 2. For arbitrary G, we show that V_0 is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational, and we construct a flat degeneration of…
Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most $3$. In each fixed dimension $d$ up to isomorphism there are only finitely many $d$-dimensional toric quiver varieties. A procedure for their classification is outlined.
Let $G:= (C^*)^k\times SL_2(C)$ act linearly on a vector space or its projectivisation. We obtain an effective criterion to detect whether a number of orbits in an orbit-closure is finite or not.
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.
For a connected semisimple algebraic group $G$, we consider some special infinite series of tensor products of simple $G$-modules whose $G$-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to the existence of open $G$-orbits in multiple flag varieties and address the problem of classifying such series.
Let g be a semisimple complex Lie algebra. Let O be a nilpotent orbit in g. Fix a triangular decomposition g=n+h+n^-. An irreducible component of the intersection of O and n is called an orbital variety associated to O. It is a Lagrangian subvariety of O. In this note we discuss the closure of an orbital variety as a union of varieties. We show that if g contains factors not of type A_n then there are orbital varieties whose closure contains components which are not Lagrangian. We show that the argument does not work if all the factors are of type A_n and provide the facts supporting the conje…
We try to write (generic) elements of classical simple Lie algebras as sums of as few elements of the orbit C of long root vectors as possible. If the Lie algebra in question is sl_n or sp_n, then C consists of all rank 1 matrices in the Lie algebra, and an element of rank k is the sum of k elements of C. If, on the other hand, the Lie algebra is o_n, then C consists of all rank 2 matrices with square zero, and the situation is much more complicated. We explicitly describe, in this case as well, the closure of the set of all sums of k elements of C, also known as the (k-1)st secant variety of …
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.
We define the Barbasch-Evens-Magyar varieties. We show they are isomorphic to the smooth varieties defined in [D.~Barbasch-S.~Evens '94] that map generically finitely to symmetric orbit closures, thereby giving resolutions of singularities in certain cases. Our definition parallels [P.~Magyar '98]'s construction of the Bott-Samelson varieties [H.~C.~Hansen '73, M.~Demazure '74]. From this alternative viewpoint, one deduces a graphical description in type $A$, stratification into closed subvarieties of the same kind, and determination of the torus-fixed points. Moreover, we explain how these ma…
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no reference to root systems. Our proofs use properties of root systems, but not their classification.
Secant varieties of generalised Grassmannians
该研究利用余极小簇的抛物轨道性质，给出其直线割线簇可识别与奇异轨迹的完整统一描述，提出第二类强Terracini轨迹并确定其形式，还对比了非余极小的迷向Grassmann簇的差异。
Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.
Let $H$ be a connected spherical subgroup of a semisimple algebraic group $G$. In this paper, we give a criterion for $H$-orbit closures in the flag variety of $G$ to have nice geometric and cohomological properties. Our main tool is the method of Frobenius splitting and of global F-regularity.
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.
We describe noncommutative desingularizations of determinantal varieties, determinantal varieties defined by minors of generic symmetric matrices, and pfaffian varieties defined by pfaffians of generic anti-symmetric matrices. For maximal minors of square matrices and symmetric matrices, this gives a non-commutative crepant resolution. Along the way, we describe a method to calculate the quiver with relations for any non-commutative desingularizations coming from exceptional collections over partial flag varieties.
Equivariant geometry of symmetric quiver orbit closures
本文将有限型对称箭图表示簇的等变几何问题与对称簇GL(n)/K的相应问题统一，通过构造显式嵌入转换了奇点、组合学及等变上同调与K理论的结果。
Using the notion of a Lagrangian covering, W. Graham and D. Vogan proposed a method of constructing representations from the coadjoint orbits for a complex semisimple Lie group $G$. When the coadjoint orbit $\calO$ is nilpotent, a representation of $G$ is attached to each orbital variety of $\calO$ in this way. In the setting of classical groups, we show that whenever it is possible to carry out the Graham-Vogan construction for an orbital variety of a spherical $\mathcal{O}$, its infinitesimal character lies in a set of characters attached to $\mathcal{O}$ by W. M. McGovern. Furthermore, we s…
The HHMP decomposition of the permutohedron and degenerations of torus orbits in flag varieties
本文在全旗流形中构造了一般环面轨道闭包到Richardson簇的显式环面退化，使其矩映射像与HHMP分解一致，从而为Anderson-Tymoczko公式提供了新证明。
Let $ G $ be a connected reductive algebraic group over $ \C $. We denote by $ K = (G^θ)_{0} $ the identity component of the fixed points of an involutive automorphism $ θ$ of $ G $. The pair $ (G, K) $ is called a symmetric pair. Let $Q$ be a parabolic subgroup of $K$. We want to find a pair of parabolic subgroups $P_{1}$, $P_{2}$ of $G$ such that (i) $P_{1} \cap P_{2} = Q$ and (ii) $P_{1} P_{2}$ is dense in $G$. The main result of this article states that, for a simple group $G$, we can find such a pair if and only if $(G, K)$ is a Hermitian symmetric pair. The conditions (i) and (ii) yield …
In the first half of this article, we review the Steinberg theory for double flag varieties for symmetric pairs. For a special case of the symmetric space of type AIII, we will consider $ X = GL_{2n}/P_{(n,n)} \times GL_n / B_n^+ \times GL_n / B_n^- $ on which $ K = GL_n \times GL_n $ acts diagonally. We give a classification of $ K $-orbits in $ X $, and explicit combinatorial description of the Steinberg maps. In the latter half, we develop the theory of embedding of a double flag variety into a larger one. This embedding is a powerful tool to study different types of double flag varieties i…
Classification of Levi-spherical Schubert varieties
本研究针对完全旗流形中Levi-球面舒伯特簇的分类问题，给出组合分类结果，证实相关猜想并提出标准Coxeter元的新表述，证明采用关键多项式理论。
Let $L$ be a Levi subgroup of $GL_N$ which acts by left multiplication on a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. We say that $X(w)$ is a spherical Schubert variety if $X(w)$ is a spherical variety for the action of $L$. In earlier work we provide a combinatorial description of the decomposition of the homogeneous coordinate ring of $X(w)$ into irreducible $L$-modules for the induced action of $L$. In this work we classify those decompositions into irreducible $L$-modules that are multiplicity-free. This is then applied towards giving a complete classification of the spherical…
Singularities of orthogonal and symplectic determinantal varieties
本文研究正交与辛行列式簇的奇点，证明辛情形下轨道闭包正规且具有有理奇点，并精确刻画正交情形下的例外情况。
Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-…
We gave a classification of P and Q with a finite number of K-orbits of a double flag variety G/P*K/Q for a symmetric pair (G, K) when G=GL_{m+n} and K=GL_{m}*GL_{n}, and a description of K-orbits when the number of K-orbits of G/P*K/Q is finite. We solved the problem by providing a correspondence between the K-orbits and the quiver representations.
The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $\mathfrak g$, we obtain several results on completeness of homogeneous Poisson-commutative subalgebras of ${\mathcal S}(\mathfrak g)$ on coadjoint orbits. This concerns, in particular, Mishchenko-Fomenko and Gelfand-Tsetlin subalgebras.
We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We describe parametrizations of each orbit set and the combinatorics of its weak order, allowing us to compute formulas for the equivariant classes of all remaining orbit closures using divided difference operators. In each of the cases in type A, we realize the $K$-orbit closures as universal degeneracy loci of a certain type, involving a vector bundle $V$ over …
Let G be a reductive algebraic group over the complex number filed, and K = G^θ be the fixed points of an involutive automorphism θof G so that (G, K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal K-action. The double flag variety G/P \times K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P^1 \times G/P^2 \times G/P^3 is a special case of our double flag varieties, and there are many interesting works on the triple flag …
Let g be a semisimple Lie algebra with h a Cartan subalgebra. The orbit method attempts to assign representations of g to orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits which should lead to highest weight representations of g. It is known that all unitary highest weight representations can be obtained in this fashion. A hypersurface orbital variety is one which is of codimension 1 in the nilradical of a parabolic. Their classification for g = sl(n) obtains from general results. Recently Benlolo and Sanderson conjectured the form of the (non-linear) elemen…
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).
Enumerating Finite Braid Group Orbits on $SL_2(\C)$-Character Varieties
研究辫群在 $n$ 次穿孔球面的 $SL_2(\C)$ -特征簇上的有限轨道，基于中间卷积和有限复反射群的结果，运用卡茨中间卷积分类有限轨道，给出非本原群相关理论结果、本原群计算结果及相关计算机代码。
Orbit structures on real double flag varieties for the Siegel parabolic subgroups
本文分类了实李群 U(n,n) 和 Sp_{2n}(R) 在双旗簇上的对称子群轨道，证明它们有相同参数化，并揭示了与松木对偶和松木-大岛氏族的联系。
For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In sl(n) orbital varieties are described by Young tableaux. In this paper we consider so called Richardson orbital varieties in sl(n). A Richardson orbital variety is an orbital variety whose closure is a standard nilradical. We show that in sl(n) a Richardson orbital variety closure is a union of orbital varieties. We give a complete combinatorial description …
Chimney retractions in affine buildings encode orbits in affine flag varieties
本文通过引入带标签的折叠画廊并刻画烟囱收缩的原像，在任意仿射建筑中建立了收缩几何与折叠画廊组合学之间的统一联系，从而推广了仿射旗簇中双陪集相交的研究。
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