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Kyo Nishiyama

Publications and source records attributed to Kyo Nishiyama.

At least 19 recordsLinked to original sources
arXiv2026

Double flag varieties of Levi type with a finite number of orbits

We consider a double flag variety of the form $L/Q\times G/P$ where $G$ is a connected reductive group and $L$ is a Levi subgroup of $G$. The Levi subgroup $L$ acts diagonally on this double flag variety and a basic problem is to classify pairs of parabolic subgroups $P\subset G$ and $Q\subset L$ such that there are finitely many orbits for the considered action. We solve this problem in the case where $G$ is the general linear group by using representations of quivers.

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arXiv2026

Orbit structures on real double flag varieties for the Siegel parabolic subgroups

Let $ G $ be a connected reductive algebraic group over $ \mathbb{R} $, and $ H $ its symmetric subgroup. For parabolic subgroups $ P_{G} \subset G $ and $ P_{H} \subset H $, the product of flag varieties $ \mathfrak{X} = H/P_H \times G/P_G $ is called a double flag variety, on which $ H $ acts diagonally. Now let $G$ be either $\mathrm{U}(n,n)$ or $\mathrm{Sp}_{2n}(\mathbb{R})$. We classify the $H$-orbits on $ \mathfrak{X} $ in both cases and show that they admit exactly the same parametrization. Concretely, each orbit corresponds to a signed partial involution, which can be encoded by simple combinatorial graphs. The orbit structure reduces to several families of smaller flag varieties, and we find an intimate relation of the orbit decomposition to Matsuki duality and Matsuki-Oshima's notion of clans. We also compute the Galois cohomology of each orbit, which exhibits another classification of the orbits by explicit matrix representatives.

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arXiv2024

Overview on the theory of double flag varieties for symmetric pairs

Let $ G $ be a connected reductive algebraic group and its symmetric subgroup $ K $. The variety $ \dblFV = K/Q \times G/P $ is called a double flag variety, where $ Q $ and $ P $ are parabolic subgroups of $ K $ and $ G $ respectively. In this article, we make a survey on the theory of double flag varieties for a symmetric pair $ (G, K) $ and report entirely new results and theorems on this theory. Most important topic is the finiteness of $ K $-orbits on $ \dblFV $. We summarize the classification of $ \dblFV $ of finite type, which are scattered in the literatures. In some respects such classifications are complete, and in some cases not. In particular, we get a classification of double flag varieties of finite type when a symmetric pair is of type AIII, using the theorems of Homma who describes ``indecomposable'' objects of such double flag varieties. Together with these classifications, newly developed embedding theory provides double flag varieties of finite type, which are new. Other ingredients in this article are Steinberg theory, generalization of Robinson-Schensted correspondence, and orbit classification via quiver representations. We hope this article is useful for those who want to study the theory of double flag varieties.

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arXiv2023

Clifford algebras, symmetric spaces and cohomology rings of Grassmannians

We study various kinds of Grassmannians or Lagrangian Grassmannians over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, all of which can be expressed as $\mathbb{G}/\mathbb{P}$ where $\mathbb{G}$ is a classical group and $\mathbb{P}$ is a parabolic subgroup of $\mathbb{G}$ with abelian unipotent radical. The same Grassmannians can also be realized as (classical) compact symmetric spaces $G/K$. We give explicit generators and relations for the de Rham cohomology rings of $\mathbb{G}/\mathbb{P}\cong G/K$. At the same time we describe certain filtered deformations of these rings, related to Clifford algebras and spin modules. While the cohomology rings are of our primary interest, the filtered setting of $K$-invariants in the Clifford algebra actually provides a more conceptual framework for the results we obtain.

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arXiv2022

Action of Hecke algebra on the double flag variety of type AIII

Consider a connected reductive algebraic group $ G $ and a symmetric subgroup $ K $. Let $ \mathfrak{X} = K/B_K \times G/P $ be a double flag variety of finite type, where $ B_K $ is a Borel subgroup of $ K $, and $ P $ a parabolic subgroup of $ G $. A general argument shows that the orbit space $ \mathbb{C}\,\mathfrak{X}/K $ inherits a natural action of the Hecke algebra $ \mathscr{H} = \mathscr{H}(K, B_K) $ of double cosets via convolutions. However, to find out the explicit structure of the Hecke module is a quite different problem. In this paper, we determine the explicit action of $ \mathscr{H} $ on $ \mathbb{C}\,\mathfrak{X}/K $ in a combinatorial way using graphs for the double flag variety of type AIII. As a by-product, we also get the description of the representation of the Weyl group on $ \mathbb{C}\,\mathfrak{X}/K $ as a direct sum of induced representations.

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arXiv2022

On generalized Steinberg theory for type AIII

Given a symmetric pair $(G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_{p}(\mathbb{C})\times \mathrm{GL}_{q}(\mathbb{C}))$ of type AIII, we consider the diagonal action of $K$ on the double flag variety $\mathfrak{X}=\mathrm{Grass}(\mathbb{C}^{p+q},r)\times K/B_K$ whose first factor is a Grassmann variety for $G$ and whose second factor is a full flag variety of $K$. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization, dimensions, closure relations, and cover relations. Specifically, the orbits are parametrized by certain pairs of partial permutations. Each orbit in $\mathfrak{X}$ gives rise to a conormal bundle. As in the references [5] and [6], by using the moment map associated to the action, we define a so-called symmetrized Steinberg map, respectively an exotic Steinberg map, which assigns to each such conormal bundle (thus to each orbit) a nilpotent orbit in the Lie algebra of $K$, respectively in the Cartan complement of that Lie algebra. Our main result is an explicit description of these Steinberg maps in terms of combinatorial algorithms on partial permutations, extending the classical Robinson--Schensted procedure on permutations. This is a thorough generalization of the results in [5], where we supposed $p=q=r$ and considered orbits of special forms.

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arXiv2020

Orbit embedding for double flag varieties and Steinberg map

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 in terms of the known ones. We prove an embedding theorem of orbits in full generality and give an example of type CI which is embedded into type AIII.

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arXiv2020

A generalization of Steinberg theory and an exotic moment map

For a reductive group $G$, Steinberg established a map from the Weyl group to the set of nilpotent $G$-orbits by using moment maps on double flag varieties. In particular, in the case of the general linear group, it provides a geometric interpretation of the Robinson-Schensted correspondence between permutations and pairs of standard tableaux of the same shape. We extend Steinberg's approach to the case of a symmetric pair $(G,K)$ to obtain two different maps, namely a \emph{generalized Steinberg map} and an \emph{exotic moment map}. Although the framework is general, in this paper we focus on the pair $(G,K) = (\mathrm{GL}_{2n}(\mathbb{C}), \mathrm{GL}_n(\mathbb{C}) \times \mathrm{GL}_n(\mathbb{C}))$. Then the generalized Steinberg map is a map from \emph{partial} permutations to the pairs of nilpotent orbits in $ \mathfrak{gl}_n(\mathbb{C}) $. It involves a generalization of the classical Robinson--Schensted correspondence to the case of partial permutations. The other map, the exotic moment map, establishes a combinatorial map from the set of partial permutations to that of signed Young diagrams, i.e., the set of nilpotent $K$-orbits in the Cartan space $(\mathrm{Lie}(G)/\mathrm{Lie}(K))^* $. We explain the geometric background of the theory and combinatorial algorithms which produce the above mentioned maps.

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arXiv2019

Enhanced zeta distributions and its functional equations

We consider an ``enhanced symmetric space'', which is a prehomogeneous vector space. This vector space is intimately related to a double flag variety studied in \cite{NO.2018}. On a distinguished open orbit called ``enhanced positive cone'', we consider a zeta integral with two complex variables, which is analytically continued to meromorphic family of tempered distributions. One of the main results of this paper is to establish a precise formula for the meromorphic continuation which clarifies the location of poles (and may be useful to obtain residues). We also compute the Fourier transform of the zeta distribution and obtain a functional equation with explicit gamma factors.

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arXiv2017

Enhanced adjoint action and their orbits for the general linear group

We studied an enhanced adjoint action of the general linear group on a product of its Lie algebra and a vector space consisting of several copies of defining representations and its duals. We determined regular semisimple orbits (i.e., closed orbits of maximal dimension) and the structure of enhanced null cone, including its irreducible components and their dimensions.

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arXiv2017

Real double flag varieties for the symplectic group

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group $G$ and the symmetric subgroup $L$, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups $Q$ of $L$ corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the $L$-orbits here, and find natural explicit integral transforms between degenerate principal series of $L$ and $G$.

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arXiv2016

On the exotic Grassmannian and its nilpotent variety

Given a decomposition of a vector space $V=V_1\oplus V_2$, the direct product $\mathfrak{X}$ of the projective space $\mathbb{P}(V_1)$ with a Grassmann variety $\mathrm{Gr}_k(V)$ can be viewed as a double flag variety for the symmetric pair $(G,K)=(\mathrm{GL}(V),\mathrm{GL}(V_1)\times\mathrm{GL}(V_2))$. Relying on the conormal variety for the action of $K$ on $\mathfrak{X}$, we show a geometric correspondence between the $K$-orbits of $\mathfrak{X}$ and the $K$-orbits of some appropriate exotic nilpotent cone. We also give a combinatorial interpretation of this correspondence in some special cases. Our construction is inspired by a classical result of Steinberg and by the recent work of Henderson and Trapa for the symmetric pair $(\mathrm{GL}(V),\mathrm{Sp}(V))$.

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arXiv2016

The totally nonnegative part of the finite Toda lattice via a reducible rational curve

A totally nonnegative matrix is a real-valued matrix whose minors are all nonnegative. In this paper, we concern with the totally nonnegative structure of the finite Toda lattice, a classical integrable system, which is expressed as a differential equation of square matrices. The Toda flow naturally translates into a (multiplicative) linear flow on the (generalized) Jacobi variety associated with some reducible rational curve $X$. This correspondence provides an algebro-geometric characterization of the totally positive part of the Toda equation. We prove that the totally nonnegative part of the finite Toda lattice is isomorphic to a connected component of $\mathrm{Jac}(X)_{\mathbb{R}}$, the real part of the generalized Jacobi variety $\mathrm{Jac}(X)$, as semi-algebraic varieties.

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arXiv2014

Enhanced orbit embedding

Let $ \tilde{G} $ be an algebraic group acting on a variety $ \tilde{L} $, and $ G \subset \tilde{G} $ a subgroup which leaves a subvariety $ L \subset \tilde{L} $ stable. For a $ G $-orbit $ O_G = G u (u \in L) $ in $ L $, we can associate an orbit $ O_{\tilde{G}} = \tilde{G} u $ of $ \tilde{G} $ so that we get a map $ L/G \to \tilde{L}/\tilde{G} $ between orbit spaces, though this map is usually not injective. In this note, when $ G $ is a symmetric subgroup arising from an involutive anti-automorphism, we give certain sufficient conditions for the map $ L/G \to \tilde{L}/\tilde{G} $ to be injective after the method of Ohta (2008). Our main concern here is to produce examples of enhanced Lie algebras (or enhanced $ θ$-representations). We also analyze an obstruction which prevents the orbit space inclusion.

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arXiv2014

Codimension one connectedness of the graph of associated varieties

Let $ π$ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ AV(π) $. If $ AV(π) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus we are interested in the codimension one adjacency of nilpotent orbits for a symmetric pair $ (G, K) $. We define the notion of orbit graph and associated graph for $ π$, and study its structure for classical symmetric pairs; number of vertices, edges, connected components, etc. As a result, we prove that the orbit graph is connecetd for even nilpotent orbits. Finally, for indefinite unitary group $ U(p, q) $, we prove that for each connected component of the orbit graph $ Γ_K(O_λ) $ thus defined, there is an irreducible Harish-Chandra module $ π$ whose associated graph is exactly equal to the connceted component.

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arXiv2013

On orbits in double flag varieties for symmetric pairs

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 $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $.

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