Hermitian Symmetric Spaces Research Articles

Geometry of domains with the uniform squeezing property

We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmüller spaces and Hermitian symmetric spaces of non-compact type. The properties supported by such manifolds include pseudoconvexity, hyperconvexity, Kähler-hyperbolicity, vanishing of cohomology groups and quasi-isometry of various invariant metrics. It also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of Serre about Stein properties of holomorphic fiber bundles.

Rankin–Cohen brackets, transvectants and covariant differential operators

We give a geometric construction of the transvectant on a Hermitian symmetric space (which in the case of the unit disk is also called the Rankin–Cohen bracket) in terms of the covariant differentiation.

On the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits

In the finite-dimensional setting, every Hermitian-symmetric space of compact type is a coadjoint orbit of a finite-dimensional Lie group. It is natural to ask whether every infinite-dimensional Hermitian-symmetric space of compact type, which is a particular example of an Hilbert manifold, is transitively acted upon by a Hilbert Lie group of isometries. In this paper we give the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits of connected simple L *-groups of compact type using the notion of simple roots of non-compact type. The key step is, given an infinite-dimensional symmetric pair (, ), where is a simple L *-algebra of compact type and a subalgebra of , to construct an increasing sequence of finite-dimensional subalgebras n of together with an increasing sequence of finite-dimensional subalgebras n of such that , , and such that the pairs ( n , n ) are symmetric. Comparing with the classification of Hermitian-symmetric spaces given by W. Kaup, it follows that any Hermitian-symmetric space of compact or non-compact type is an affine-coadjoint orbit of an Hilbert Lie group.

Chiral formulation for hyperKähler sigma-models on cotangent bundles of symmetric spaces

Starting with the projective-superspace off-shell formulation for four-dimensional = 2 supersymmetric sigma-models on cotangent bundles of arbitrary Hermitian symmetric spaces, their on-shell description in terms of = 1 chiral superfields is developed. In particular, we derive a universal representation for the hyperkahler potential in terms of the curvature of the symmetric base space. Within the tangent-bundle formulation for such sigma-models, completed recently in arXiv:0709.2633 and realized in terms of = 1 chiral and complex linear superfields, we give a new universal formula for the superspace Lagrangian. A closed form expression is also derived for the Kahler potential of an arbitrary Hermitian symmetric space in Kahler normal coordinates.

On the unitarity of gauged non-compact world-sheet supersymmetric WZNW models

In this paper we generalize our investigation of the unitarity of non-compact WZNW models connected to Hermitian symmetric spaces to the N = 1 world-sheet supersymmetric extension of these models. We will prove that these models have a unitary spectrum in a BRST approach for antidominant highest weight representations if the level and weights of the gauged subalgebra are integers. We will find new critical string theories in 7 and 9 space–time dimensions.

Twistors, 4-symmetric spaces and integrable systems

An order four automorphism of a Lie algebra gives rise to an integrable system introduced by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. As applications, we find that surfaces with holomorphic mean curvature in 4-dimensional real or complex space forms constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in 4-dimensional Hermitian symmetric spaces (this last providing a conceptual explanation of a result of Hélein-Romon).

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, \({{\rm Sp}(2,\mathbb {R})}\) or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.

Coupled Vortex Equations on Complete Kähler Manifolds

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

Jordan triples and Riemannian symmetric spaces

We introduce a class of real Jordan triple systems, called JH-triples, and show, via the Tits–Kantor–Koecher construction of Lie algebras, that they correspond to a class of Riemannian symmetric spaces including the Hermitian symmetric spaces and the symmetric R-spaces.

Twistor fibrations over Hermitian symmetric spaces and harmonic maps

Given a twistor space over a Hermitian symmetric space of compact type we construct a map onto a twistor space over another inner symmetric space of compact type. This map is holomorphic and preserves the superhorizontal distributions. We describe an application to harmonic maps.

Corwin–Greenleaf multiplicity functions for Hermitian symmetric spaces

Kobayashi’s multiplicity-free theorem asserts that irreducible unitary highest weight representations $\pi$ are multiplicity-free when restricted to any symmetric pairs if $\pi$ is of scalar type. The Hua–Kostant–Schmid–Kobayashi branching laws embody this abstract theorem with explicit irreducible decomposition formulas of holomorphic discrete series representations with respect to symmetric pairs. In this paper, we study the ‘classical limit’ (geometry of coadjoint orbits) of a special case of these representation theoretic theorems in the spirit of the Kirillov–Kostant–Duflo orbit method. \\ First, we consider the Corwin–Greenleaf multiplicity function $n (\mathcal{O}^{G},\,\mathcal{O}^{K})$ for Hermitian symmetric spaces $G/K$. The first main theorem is that $n(\mathcal{O}^{G},\,\mathcal{O}^{K}) \le 1$ for any $G$-coadjoint orbit $\mathcal{O}^{G}$ and any $K$-coadjoint orbit $\mathcal{O}^{K}$ if $\mathcal{O}^{G} \cap \sqrt{-1} ([\mathfrak{k}, \mathfrak{k}] + \mathfrak{p})^{\bot} \ne \emptyset$. Here, $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ is the Cartan decomposition of the Lie algebra $\mathfrak{g}$ of $G$. The second main theorem is a necessary and sufficient condition for $n (\mathcal{O}^{G},\,\mathcal{O}^{K}) \ne 0$ by means of strongly orthogonal roots.

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(= 1, 2, 3) distinct principal curvatures.

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space.

Homogeneous bundles and the first eigenvalue of symmetric spaces

In this note we prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kähler metric on a compact Hermitian symmetric spaces of ABCD–type.

Symplectic duality of symmetric spaces

Let M ⊂ C n be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ω hyp . Denote by ( M ∗ , ω FS ) the compact dual of ( M , ω hyp ) , where ω FS is the Fubini–Study form on M ∗ . Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism Ψ M : M → R 2 n = C n ⊂ M ∗ satisfying Ψ M ∗ ω 0 = ω hyp and Ψ M ∗ ω FS = ω 0 for the pull-back of Ψ M , where ω 0 is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map Ψ M , we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of C n . As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin.

On the unitarity of gauged non-compact WZNW strings

In this paper we investigate the unitarity of gauged non-compact WZNW strings, i.e., string theories formulated as G / H ′ WZNW models, where G is a non-compact group. These models represent string theories on non-trivial curved space–times with one time-like component. We will prove that for the class of models connected to Hermitian symmetric spaces, and a natural set of discrete highest weight representations, the BRST formulation, in which the gauging is defined through a BRST condition, yields unitarity. Unitarity requires the level times the Dynkin index to be an integer, as well as integer valued highest weights w.r.t. the compact subalgebra. We will also show that the BRST formulation is not equivalent to the conventional GKO coset formulation, defined by imposing a highest weight condition w.r.t. H ′ . The latter leads to non-unitary physical string states. This is, to our knowledge, the first example of a fundamental difference between the two formulations.

Polar supermultiplets, hermitian symmetric spaces and hyperkähler metrics

We address the construction of four-dimensional N=2 supersymmetric nonlinear sigma models on tangent bundles of arbitrary Hermitian symmetric spaces starting from projective superspace. Using a systematic way of solving the (infinite number of) auxiliary field equations along with the requirement of supersymmetry, we are able to derive a closed form for the Lagrangian on the tangent bundle and to dualize it to give the hyperkahler potential on the cotangent bundle. As an application, the case of the exceptional symmetric space E_6/SO(10) \times U(1) is explicitly worked out for the first time.

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $\sigma$ such that $\sigma |_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the action of any symmetric subgroup $H$ is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold $S$. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite and infinite dimensional cases, and for both discrete and continuous spectra.

Polar actions on Hermitian and quaternion-Kähler symmetric spaces

We analyze polar actions on Hermitian and quaternion-Kahler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric action on a compact homogeneous simply connected Kahler manifold.

On tangential varieties of rational homogeneous varieties

The sphericality of tangential varieties of rational homogeneous varieties is determined. The homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS) are determined. Bounds on the degrees of generators of the ideals of tangential varieties of CHSS are given, and more explicit information is obtained about the ideals in certain cases.