Hermitian Symmetric Spaces Research Articles

Isoparametric surfaces and the tt*-equations

In this paper we provide a link between tt*-bundles, which are solutions of a general version of the equations of topological–antitopological fusion considered by Cecotti–Vafa, Dubrovin and Hertling and isoparametric spherical surfaces, i.e. spherical surfaces with constant principal curvatures. More precisely, we construct a tt*-bundle from a pluriharmonic map into the hermitian symmetric space S O ( l + 2 ) / S O ( 2 ) × S O ( l ) which can be seen as the Grassmannian G r ( 2 , n + 2 ) of oriented two-planes in R n + 2 . Applying the above construction to the Gauss map of an isoparametric spherical surface we associate to a given isoparametric spherical surface a new solution to the t t ∗ -equations.

Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials

Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: orthogonality, generating and determinantal formulae, difference equations. For proving these properties we use the composition of the following transformations: Cayley transform, Laplace transform, and spherical Fourier transform associated to Hermitian symmetric spaces of tube type. In particular the difference equation for the multivariate Meixner-Pollaczek polynomials is obtained from an Euler type equation on a bounded symmetric domain.

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $\sigma$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $\sigma$-minuscule elements in the affine Weyl group of $\mathfrak g$ and that of strongly orthogonal roots in Hermitian symmetric spaces.

A Note on Vertex-transitive K\"ahler graphs

In this paper we construct finite vertex-transitive K\ahler graphs, which may be considered as discrete models of Hermitian symmetric spaces admitting K\ahler magnetic fields. We give a condition on cardinality of the set of vertices and the principal and the auxiliary degrees for a vertex-transitive K\ahler graphs. Also we give some examples of K\ahler graphs corresponding typical vertex-transitive ordinary graphs.

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

Abstract We first introduced a linear stationary equation with a quadratic operator in ∂ x and ∂ y , then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

Geometric generators for braid-like groups

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group $P{\Gamma}$. The classical example is the braid group, but there are many similar "braid-like" groups that arise in topology and algebraic geometry. Our main result is that if $P{\Gamma}$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then G is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for G in a particular intricate example in complex hyperbolic space of dimension 13. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group M, that gives geometric meaning to the generators and relations in the Conway-Simons presentation of $(M \times M):2$.

Hermitian symmetric space, flat bundle and holomorphicity criterion

Let K\\G be an irreducible Hermitian symmetric space of noncompact type and Γ⊂G a closed torsionfree discrete subgroup. Let X be a compact Kähler manifold and ρ:π1(X,x0)⟶Γ a homomorphism such that the adjoint action of ρ(π1(X,x0)) on the Lie algebra Lie(G) is completely reducible. A theorem of Corlette associates to ρ a harmonic map H:X⟶K\\G/Γ. We give a criterion for this harmonic map H to be holomorphic. We also give a criterion for it to be anti-holomorphic.

Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m + n)/S(U(m) × U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schrödinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat’s dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.

Cotangent bundles over all the Hermitian symmetric spaces

We construct the N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. In order to construct them we use the projective superspace formalism which is an N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the N =1 superfields, once the Kähler potentials of the base manifolds are obtained. Starting with N = 1 supersymmetric Kähler nonlinear sigma models on the Hermitian symmetric spaces, we extend them into the N = 2 supersymmetric models by using the projective superspace formalism and derive the general formula for the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. We apply to the formula for the non-compact Hermitian symmetric space E7/E6 × U(1)1.

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space

In this paper we investigate invariant domains in $\, \Xi^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,\Xi^+$, recently introduced by Kr\otz and Opdam, contains the crown domain $\,\Xi\,$ and it is maximal with respect to properness of the $\,G$-action. In the tube case, it also contains $\,S^+$, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of $\,\Xi$. We prove that the envelope of holomorphy of an invariant domain in $\,\Xi^+$, which is contained neither in $\,\Xi\,$ nor in $\,S^+$, is univalent and coincides with $\,\Xi^+$. This fact, together with known results concerning $\,\Xi\,$ and $\,S^+$, proves the univalence of the envelope of holomorphy of an arbitrary invariant domain in $\,\Xi^+\,$ and completes the classification of invariant Stein domains therein.

PROJECTIVELY FLAT IMMERSIONS OF HERMITIAN SYMMETRIC SPACES OF COMPACT TYPE

In this article, we define a projectively flat map of a compact Kähler manifold into the complex Grassmannian manifold, which is a kind of extension of a holomorphic map into the complex projective space. Then we show a rigidity theorem of a holomorphic isometric projectively flat immersion of Hermitian symmetric spaces of compact type. As an application, we classify holomorphic isometric projectively flat immersions of compact Kähler manifolds with parallel second fundamental form.

Complete integrability from Poisson–Nijenhuis structures on compact hermitian symmetric spaces

We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis tensor are a choice of action variables. Our proof relies on an explicit formula for the contravariant connection defined on vector bundles that are Poisson with respect to the Bruhat-Poisson structure.

The index of symmetry of a flag manifold

We study the index of symmetry of a compact generalized flag manifold M = G/H endowed with an invariant Kähler structure. When the group G is simple we show that the leaves of symmetry are irreducible Hermitian symmetric spaces which depend only on the invariant complex structure and we estimate their dimension

Differential symmetry breaking operators: I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.

A multivariate version of the disk convolution

We present an explicit product formula for the spherical functions of the compact Gelfand pairs (G,K1)=(SU(p+q),SU(p)×SU(q)) with p≥2q, which can be considered as the elementary spherical functions of one-dimensional K-type for the Hermitian symmetric spaces G/K with K=S(U(p)×U(q)). Due to results of Heckman, they can be expressed in terms of Heckman–Opdam Jacobi polynomials of type BCq with specific half-integer multiplicities. By analytic continuation with respect to the multiplicity parameters we obtain positive product formulas for the extensions of these spherical functions as well as associated compact and commutative hypergroup structures parametrized by real p∈]2q−1,∞[. We also obtain explicit product formulas for the involved continuous two-parameter family of Heckman–Opdam Jacobi polynomials with regular, but not necessarily positive multiplicities. The results of this paper extend well known results for the disk convolutions for q=1 to higher rank.

Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, we introduce the notion of Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. By using a new method of simultaneous diagonalizations, we give a complete classification for real hypersurfaces in complex hyperbolic two‐plane Grassmannians with the Reeb parallel Ricci tensor.

CLASSIFICATION OF SMOOTH SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIANS

Abstract. A Schubert variety in a rational homogeneous variety G/P isdefined by the closure of an orbit of a Borel subgroup Bof G. In general,Schubert varieties are singular, and it is an old problem to determinewhich Schubert varieties are smooth. In this paper, we classify all smoothSchubert varieties in the symplectic Grassmannians. 1. IntroductionA rational homogeneous manifold S= G/Pis a projective manifold, wherea connected complex semisimple group Gacts transitively. Under the actionof a Borel subgroup Bof G, Shas finitely many orbits. The closure of a B-orbit in Sis called a Schubert variety of S. In general, Schubert varieties aresingular, and it is an old problem to determine which Schubert varieties aresmooth. Lakshmibai-Weyman and Brion-Polo have studied the singular lociof Schubert varieties of S, when Sis a compact Hermitian symmetric space([9] and [2]). In particular, they showed that in this case any smooth Schubertvariety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S. For example, a Schubert variety of theGrassmannian Gr(k,V) of k-subspaces in a vector space V is smooth if andonly if it is a linearly embedded sub-Grassmannian.More generally, when Sis associated to a long simple root, we have:Theorem 1.1 (Proposition 3.7 of Hong-Mok [4]). Let S= G/P be a rationalhomogeneous manifold associated to a long simple root. Then any smooth Schu-bert variety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S.On the other hand, when Sis associated to a short simple root, there is asmooth Schubert variety that is not homogeneous. Let V be a vector spacewith a symplectic form ω, i.e., a nondegenerate skew-symmetric bilinear form.

Supersymmetry and cotangent bundle over non-compact exceptional Hermitian symmetric space

Abstract We construct $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the non-compact exceptional Hermitian symmetric spaces $$ \mathrm{\mathcal{M}}={E}_{6\left(-14\right)}/\mathrm{SO}(10)\times \mathrm{U}(1) $$ ℳ = E 6 − 14 / SO 10 × U 1 and E 7(−25) /E 6 × U(1). In order to construct them we use the projective superspace formalism which is an $$ \mathcal{N}=2 $$ N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the $$ \mathcal{N}=1 $$ N = 1 superfields, once the Kähler potentials of the base manifolds are obtained. We derive the $$ \mathcal{N}=1 $$ N = 1 supersymmetric nonlinear sigma models on the Kähler manifolds $$ \mathrm{\mathcal{M}} $$ ℳ . Then we extend them into the $$ \mathcal{N}=2 $$ N = 2 supersymmetric models with the use of the result in arXiv:1211.1537 developed in the projective superspace formalism. The resultant models are the $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the Hermitian symmetric spaces $$ \mathrm{\mathcal{M}} $$ ℳ . In this work we complete constructing the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces.

Correction to: “The intersection of two real forms in Hermitian symmetric spaces of compact type”

We correct the proof of Theorem 1.1 in our previous paper specially in the non-irreducible case.

ON DIFFERENTIAL INVARIANTS OF HYPERPLANE SYSTEMS ON NONDEGENERATE EQUIVARIANT EMBEDDINGS OF HOMOGENEOUS SPACES

Given a complex submanifold M of the projective space P(T), the hyperplane system R on M characterizes the projective embedding of M into P(T) in the following sense: for any two nondegenerate com- plex submanifolds MP(T) and M ' � P(T ' ), there is a projective linear transformation that sends an open subset of M onto an open subset of M ' if and only if (M,R) is locally equivalent to (M ' ,R ' ). Se-ashi developed a theory for the differential invariants of these types of systems of linear differential equations. In particular, the theory applies to systems of lin- ear differential equations that have symbols equivalent to the hyperplane systems on nondegenerate equivariant embeddings of compact Hermitian symmetric spaces. In this paper, we extend this result to hyperplane sys- tems on nondegenerate equivariant embeddings of homogeneous spaces of the first kind.