It is shown that the cycle space of an arbitrary orbit of a non-Hermitian real form G in a flag manifold $Z=G^\mathbb C/Q$ of its complexification is naturally equivalent to a certain universal domain which depends only on G. This makes use of complex geometric methods which were recently developed for the purpose of handling the case of open orbits together with a better understanding of the connection to Schubert varieties and the related complex slices along lower-dimensional Gorbits.
This is a survey of history, methods and developments in the theory of cycle spaces of flag domains, and new results on double fibration transforms and their applications.
In the study of complex flag manifolds, flag domains and their cycle spaces, a key point is the fact that the cycle space $\mathcal M_D$ of a flag domain $D$ is a Stein manifold. That fact has a long history. The earliest approach relied on construction of a strictly plurisubharmonic function on $\mathcal M_D$, starting with a $q$--convex exhaustion function on $D$, where $q$ is the dimension of a particular maximal compact subvariety of $D$ (we use the normalization that 0--convex means Stein). Construction of that exhaustion function on $D$ required that $D$ be measurable. In that case the e…
Using Ram's theory of alcove walks, we give a proof of the Bernstein presentation of the affine Hecke algebra. The method works also in the case of unequal parameters. We also discuss how these results help in studying sheaves of nearby cycles on affine flag manifolds.
We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson-Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an earlier paper, which encode the data of the Jordan-Hoelder series. We prove the multiplicity functions are polynomials in q, and we give a sharp bound for their degrees. Our results apply as well to the nearby cycles in the p-adic deformation of Laumon-Haines-Ngo, and also to Wakimoto sheaves.
Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexification. A special class of flag domains constitute the classifying spaces for variations of Hodge structure, namely period domains or more generally Mumford-Tate domains. In this paper we classify all proper, equivariant embeddings of a flag domain of Hodge type, required to satisfy a certain transversality condition, in a period domain and show that any such flag domain must be a Hermitian symmetric space of non-compact type of classical type. In particular, the classification describes also the pr…
We study the Jordan-Hoelder series for nearby cycles on certain Shimura varieties and Rapoport-Zink local models, and on finite-dimensional pieces of Beilinson's deformation of the affine Grassmannian to the affine flag variety (and their p-adic analogues). We give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups, and we also provide an algorithm to compute multiplicities, in terms of the affine Hecke algebra.
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 …
We classify the GL_p x GL_q-orbits in the flag variety for GL_{p+q} with rationally smooth closure, showing that they are all either already closed or are pullbacks from orbits with smooth closure in a partial flag variety.
Let $G_\mathbb{R}$ be a Lie group and $G$ its complexification. An open $G_\mathbb{R}$-orbit in a $G$-flag manifold is measurable whenever it carries a $G_\mathbb{R}$-invariant volume element. In this paper the notion of measurability is generalized to the supersymmetric setting and a classification of measurable open orbits in flag supermanifolds is given.
The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D = G_0(z) \subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous holomorphic vector bundle $\mathbb E \to D$. The topic here is the double fibration transform ${\cal P}: H^q(D;{\cal O}(\mathbb E)) \to H^0({\cal M}_D;{\cal O}(\mathbb E'))$ where $q$ is given by the geometry of $D$, ${\cal M}_D$ is the cycle space of $D$, and $\mathbb E' \to {\cal M}_D$ is a certain naturally derived holomorphic vector bundle. Schubert intersecti…
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 connected real semi-simple Lie group and H a closed connected subgroup. Let P be a minimal parabolic subgroup of G. It is shown that H has an open orbit on the flag manifold G/P if and only if it has finitely many orbits on G/P. This confirms a conjecture by T. Matsuki.
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.
We characterize the Sp_{2n} orbits in the flag variety for SL_{2n} with rationally smooth closure via a pattern avoidance criterion, also showing that the singular and rationally singular loci of such orbit closures coincide.
We show that each orbit of a Borel subgroup $B_{n-1}$ of GL(n-1) (respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is a bundle over a $B_{n-1}$-orbit on a generalized flag variety of GL(n-1) (respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous subgroup on a smaller flag variety. As a consequence, we develop an inductive procedure to classify $B_{n-1}$-orbits on the flag variety. Our method is essentially uniform in the two cases. As further consequences, in the sequel to this paper we give an explicit combinatorial classification of orbits and det…
We characterize the O_{2n} orbits in the flag variety for GL_{2n} with rationally smooth closure via a graph-theoretic criterion. We also give a necessary pattern avoidance criterion for rational smoothness and conjecture its sufficiency.
It is known that the closure of an arbitrary K_c-orbit on a flag manifold is expressed as a product of a closed K_c-orbit and a Schubert cell ([M2], [Sp]). We already applied this fact to the duality of orbits on flag manifolds ([GM]). We refine here this result and and give its new applications to the study of domains arising from the duality.
We study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic, we show that there is a geometrically defined bijection between the sets of adjoint and coadjoint orbits of such groups. In addition, we show that the corresponding orbits, although different, are homotopy equivalent. We also provide a geometric description of the adjoint and coadjoint orbits of the Euclidean and orthogonal groups as a special class of flag manifold which we call a Hermitian flag manifold. These man…
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 …
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.
In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of $\mathrm{PGL}(4,\mathbb{C})$.
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 give a pattern avoidance criterion to classify the orbits of Sp(p,C) x Sp(q,C) (resp. GL(n,C)) on the flag variety of type C_{p+q} (resp. D_n) with rationally smooth closure. We show that all such orbit closures fiber (with smooth fiber) over a smaller flag variety, and hence are in fact smooth. In addition we prove that the classification is insensitive to isogeny.
In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of $n\times n$ complex matrices $\fgl(n,\C)$ introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of $K_{n}=GL(n-1,\C)\times GL(1,\C)$-orbits on the flag variety $\mathcal{B}_{n}$ of $GL(n,\C)$ to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory o…
Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the irreducible *-representations of the corresponding quantized flag manifolds are also parametrized by their Schubert cells. An important step is the determination of suitable algebraic generators of the quantized flag manifolds. These algebraic generators can be naturally expressed in terms of quantum Plucker coordinates. This note complements the paper of the…
We calculate Bernstein-Sato b-functions for $f_{G^3}^λ$, a $SL_n$-invariant section of a line bundle on $SL_n/B \times SL_n/B \times \mathbb{P}^{n - 1}$ whose zero-set is the complement of the open $G$-diagonal orbit. The proof uses a similar calculation by Kashiwara of the b-function for $f^λ$, a $B^-$-semiinvariant section of a line bundle on $SL_n/B$ whose zero-set is the complement of the big Bruhat cell.
We prove that generic higher Deligne-Lusztig representations over truncated formal power series are non-nilpotent, when the parameters are non-trivial on the biggest reduction kernel of the centre; we also establish a relation between the orbits of higher Deligne-Lusztig representations of $\mathrm{SL}_n$ and of $\mathrm{GL}_n$. Then we introduce a combinatorial analogue of Deligne-Lusztig construction for general and special linear groups over local rings; this construction generalises the higher Deligne--Lusztig representations and affords all the nilpotent orbit representations, and for $\m…
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.
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…
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 …
Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup H of B acting with finitely many orbits on the flag variety G/B, and we classify the H-orbits in G/B in terms of suitable combinatorial invariants. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G/B, and we give a combinatorial model for this action in terms of weight polytopes.
We give a necessary and sufficient condition in terms of pattern avoidance for the conjugates of the bottom vertex in the Bruhat graph attached to an O_n orbit O in the flag variety for GL_n to have degree equal to the rank of this graph as a poset, showing that this condition is equivalent to the rational smoothness of the closure of O. We also give a necessary and sufficient condition in terms of pattern avoidance for the closure of O to be smooth.
Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. Certain group theoretically defined compact complex submanifolds, which are regarded as cycles, are of basic importance for their complex geometric and representation theoretic properties. It is known that there are optimal Schubert varieties which intersect the cycles transversally in finitely many points and in particular determine them in homology. Here we give a precise description of these Schubert varieties in terms of certain subsets of the Weyl group and compute their total number f…
In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity will be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ1,WZ2,FH,M6] and for open S in [M6]. In this paper, we prove the conjecture for all the other orbits when G_R is of non-Hermitian type.
The subgroup K=GL_p x GL_q of GL_{p+q} acts on the (complex) flag variety GL_{p+q}/B with finitely many orbits. We introduce a family of polynomials that specializes to representatives for cohomology classes of the orbit closures in the Borel model. We define and study K-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the H-polynomials and the Kazhdan-Lusztig-Vogan polynomials.
In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type by computing many examples. In this paper, we first prove (in Theorem 1.3 and Corollary 1.4) this conjecture for the open K_C-orbit S on an ``arbitrary'' flag manifold generalizing the result of Barchini. This conjecture for closed S was solved in [WZ1], [WZ2] (Hermitian cases) and [FH] (non-Hermitian cases). We also deduce an alternative p…
We continue our study of the inclusion posets of diagonal $SL(n)$-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets that we determine explicitly. Furthermore, our computations show that the number of diagonal $SL(n)$-orbits in any of these posets is at most 10 for any positive integer $n$. This is in contrast with the complexity 0 case, where, in some cases, the resulting posets attain arbitrary heights.
In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ2,WZ3,FH,M4] and for open S in [M4]. It was proved for the other orbits in [M5] when G_R is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed K_C-orbit when G_R is of Hermitian type. Thus the conjecture is completely solved affirmatively.
A real form $G_0$ of a complex semisimple Lie group $G$ has only finitely many orbits in any given compact $G$-homogeneous projective algebraic manifold $Z=G/Q$. A maximal compact subgroup $K_0$ of $G_0$ has special orbits $C$ which are complex sub-manifolds in the open orbits of $G_0$. These are referred to as cycles. The cycles intersect Shubert varieties $S$ transversely in finitely many points. In particular, determining these points of intersection yields a description of the topological class of the given cycle. This was carried out for all real forms of $SL(n,\mathbb{C})$ in the work of…
Let G be the split special orthogonal group of degree 2n+1 over a field F of char F \ne 2. Then we describe G-orbits on the triple flag varieties G/P\times G/P\times G/P and G/P\times G/P\times G/B with respect to the diagonal action of G where P is a maximal parabolic subgroup of G of the shape (n,1,n) and B is a Borel subgroup. As by-products, we also describe GL_n-orbits on G/B, Q_{2n}-orbits on the full flag variety of GL_{2n} where Q_{2n} is the fixed-point subgroup in Sp_{2n} of a nonzero vector in F^{2n} and 1\times Sp_{2n}-orbits on the full flag variety of GL_{2n+1}. In the same way, …
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.
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 …
We give positive combinatorial descriptions of Schubert structure constants $c_{u,v}^w$ for the full flag variety in type $A_{n-1}$ when $u$ and $v$ form what we refer to as a "$(p,q)$-pair" ($p+q=n$). The key observation is that a certain subset of the $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on the flag variety (those satisfying an easily stated pattern avoidance condition) are Richardson varieties. The result on structure constants follows when one combines this observation with a theorem of Brion concerning intersection numbers of spherical subgroup orbit closures and Schu…
Let $G$ be a connected reductive algebraic group over an algebraically closed field ${\bf k}$ of characteristic not equal to 2, let $\B$ be the variety of all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing a closed $K$-orbit in $\B$, we associate to every $K$-orbit on $\B$ some subsets of the Weyl group of $G$, and we study them as invariants of the $K$-orbits. When ${\bf k} = {\mathbb C}$, these invariants are used to determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup of $G$ have non-empty intersection in $\B$. We also characterize the i…
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…
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formu…
In [Wyser-Yong '13] we introduced polynomial representatives of cohomology classes of orbit closures in the flag variety, for the symmetric pair $(GL_{p+q}, GL_p \times GL_q)$. We present analogous results for the remaining symmetric pairs of the form $(GL_n,K)$, i.e., $(GL_n,O_n)$ and $(GL_{2n},Sp_{2n})$. We establish the well-definedness of certain representatives from [Wyser '13]. It is also shown that the representatives have the combinatorial properties of nonnegativity and stability. Moreover, we give some extensions to equivariant $K$-theory.
Flag domains are open orbits of real forms $G_\mathbb{R}$ of complex reductive Lie supergroups $G$ in $G$-flag supermanifolds $Z = G/P$. This thesis discusses three topics from the theory of these flag domains: 1. Measurability(i.e. existence of $G_\mathbb{R}$-invariant Berezinian densities) 2. Global holomorphic superfunctions 3. Cycle spaces and the Double Fibration Transform A revision of the respective classical results is included in the second chapter. This thesis provides a classification of all measurable flag domains and of the global holomorphic functions on all flag domains. Moreove…
This paper is the sequel to ``$B_{n-1}$-bundles on the flag variety, I". We continue our study of the orbits of a Borel subgroup $B_{n-1}$ of $G_{n-1}=GL(n-1)$ (resp. $SO(n-1)$) acting on the flag variety $\mathcal{B}_{n}$ of $G=GL(n)$ (resp. $SO(n)$). We begin by using the results of the first paper to obtain a complete combinatorial model of the $B_{n-1}$-orbits on $\mathcal{B}_{n}$ in terms of partitions into lists. The model allows us to obtain explicit formulas for the number of orbits as well as the exponential generating functions for the sequences $\{|B_{n-1}\backslash \mathcal{B}_{n}|…
In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of $K_{i}=GL(i…
Let $\mathfrak{g}$ be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline{D}$ on the coordinate ring $\mathbb{C}[N]$ related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when $\mathfrak{g}$ is of type $A_n$ or $…
For a complex semi-simple group G and its real form G0 we define a Poisson structure on the flag variety of G such that all the Bruhat cells (for a suitable choice of a Borel subgroup of G) as well as all the G0-orbits are Poisson submanifolds. We show that each intersection of a G0-orbit with a Bruhat cell is a regular Poisson submanifold and we compute the dimensions of its symplectic leaves.
