Hermitian Symmetric Spaces Research Articles

Minimality and Hamiltonian Stability of Lagrangian Submanifolds in Adjoint Orbits

In this article, we improve and generalize the result of [O]; the one is the necessary and sufficient condition of the minimality of Lagrangian submanifolds in adjoint orbits, which are Hermitian symmetric spaces, and the other is the necessary and sufficient condition of the Hamiltonian stability of minimal Lagrangian submanifolds in adjoint orbits, which are not necessarily Hermitian symmetric spaces.

The Maslov triple index on the Shilov boundary of a classical domain

Let D be an irreducible Hermitian symmetric space of tube-type, S its Shilov boundary, G its group of holomorphic diffeomorphisms. For a generic triple of points ( σ 1, σ 2, σ 3)∈ S× S× S, a characteristic G-invariant ι( σ 1, σ 2, σ 3), called the Maslov index was introduced in [Transform. Groups 6 (2001) 303]. For D of classical type (i.e. for all cases except for the exceptional domain associated to Albert’s algebra), the definition of the Maslov index is extended to all triples, by using a holomorphic embedding of D into a Siegel disc, which corresponds to an embedding of S into a Lagrangian manifold. When D is the Lie ball, the extension of the definition is obtained through a realization of S in the Lagrangian manifold of a spinor space.

Construction of supersymmetric nonlinear sigma models on noncompact Calabi-Yau manifolds with isometry

We propose a class of N = 2 supersymmetric nonlinear sigma models on the noncompact Ricci-flat Kähler manifolds, interpreted as the complex line bundles over the hermitian symmetric spaces. Kähler potentials and Ricci-flat metrics for these manifolds with isometries are explicitly constructed by using the techniques of supersymmetric gauge theories. Each of the metrics contains a resolution parameter which controls the size of these base manifolds, and the conical singularity appears when the parameter vanishes.

Schrödinger Flows on Compact Hermitian Symmetric Spaces and Related Problems

In this note, we prove that the Schrodinger flow of maps from a closed Rieman surface into a compact irreducible Hermitian symmetric space admits a global weak solution. Also, we show the existence of weak solutions to the initial value problem of Heisenberg model wit Lie algebra values, which is closely related to the Schrodinger flow on compact Hermitian symmetric spaces.

Surface group representations with maximal Toledo invariant

We study representations of compact surface groups on Hermitian symmetric spaces and characterize those with maximal Toledo invariant. To cite this article: M. Burger et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Coisotropic actions on compact homogeneous K�hler manifolds

Abstract. The main result of the paper is that a compact homogeneous Kahler manifold admitting an isometric and coisotropic action with a fixed point is isometric to a Hermitian symmetric space.

The Gromov norm of the Kaehler class and the Maslov index

Let D be a Hermitian symmetric space of the non-compact type, ! its Kaehler form. Fora geodesic triangle in D, we compute explicitly the integral R � !, generalizing previous results (see (D-T)). As a consequence, if X is a manifold which admits D as universal cover, we calculate the Gromov norm of (!) 2 H2(X, R). The formula for R � ! is extended to ideal triangles. Precise estimates are given and triangles for which the bound is achieved are studied. For tube-type domains we show the connection of these integrals with the Maslov index we introduced in a previous paper (see (C-O)). 0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral com- ponent of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form !. This real differential form of degree 2 is closed and hence can be integrated along any 2-cycle, in particular geodesic triangles (to mean triangles the sides of which are geodesic segments). When M is of type I,II or III (in E. Cartan's classification), the integral R � ! (the symplectic area of the geodesic triangle �) was computed in (D-T). By using their techniques, we give the result in the general case. It turns out that these quantities have an upper bound, and with the appropriate normalization, the bound depends only on the rank r of M. We extend these computations to ideal triangles, and we prove (new) sharp estimates for the areas. In particular, we determine precisely the triangles for which the upper bound is achieved. This turns out be of great geometric significance, as the summits of such an extremal triangle are contained in the image of a tight holomorphic totally geodesic imbedding of the complex unit disc into M (Theorem 4.7). Generally speaking, our study of the integrals R � ! requires the fine structure of Hermitian symmetric spaces : special role played by the tube-type case, behaviour of geodesics at infinity and structure of G-orbits in the boundary, use of partial Cayley transforms. This study is also related to a previous work (see (C-O)) where we extended the notion of Maslov index to the Shilov boundary S of a Hermitian symmetric space of tube-type. The Maslov index is (up to a factor �) nothing but the symplectic area of ideal triangles with summits in S, and in some sense the present work can be understood as a continuation of (C-O). The computation of the integrals was used in (D-T) to calculate the Gromov norm of the Kaehler class of a compact Hermitian locally symmetric manifold X = \M, where M is of type I and a discrete, torsion-free, co-compact subgroup of the group G. They observed that it has a nice topological corollary. Let S be a Riemann surface of genus g > 1 and f : S → X a continuous map. Then

Berezin–Toeplitz Quantization and Invariant Symbolic Calculi

It is well known that one can often construct an invariant star-product by expanding the product of two Toeplitz operators asymptotically into a series of another Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin–Toeplitzquantization. We show that on bounded symmetric domains (Hermitian symmetric spaces of noncompact type), one can in fact obtain in a similar way any invariant star-productwhich is G-equivalent to the Berezin–Toeplitz star-product, by using, instead of Toeplitz operators, other suitable assignments f↦Qf from compactly supported C∞ functions f to bounded linear operators Qf on the corresponding Hilbert spaces. (This procedureis referred to as prime quantization by some authors.) Along the way, we establish two technical results which are of interest in their own right, namely a controlled-growth parameter generalization of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, and the fact that any invariant bi-differential operator (Hochschild two-cochain) on a bounded symmetric domain automatically maps the Schwartz space into itself.

The pure state space of quantum mechanics as Hermitian symmetric space

The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kähler manifold. The classical principles of quantum mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non-linear geometrical context.

Volumes of complex analytic subvarieties of Hermitian symmetric spaces

We give lower bounds of volumes of k -dimensional complex analytic subvarieties of certain naturally defined domains in n -dimensional complex space forms of constant (positive, zero, or negative) holomorphic sectional curvature. For each 1 ≤ k ≤ n , the lower bounds are sharp in the sense that these bounds are attained by k -dimensional complete totally geodesic complex submanifolds. Such lower bounds are obtained by constructing singular potential functions corresponding to blow-ups of the Kähler metrics involved. Similar lower bounds are also obtained in the case of Hermitian symmetric spaces of noncompact type. In this case, the lower bounds are sharp for those values of k at which the Hermitian symmetric space contains k -dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature.

Higher dimensional geometries related to fuzzy odd-dimensional spheres

We study $SO(m)$ covariant Matrix realizations of $ \sum_{i=1}^{m} X_i^2 = 1 $ for even $m$ as candidate fuzzy odd spheres following hep-th/0101001. As for the fuzzy four sphere, these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset. The fuzzy $S^{2k-1} $ is related to a higher dimensional coset $ {SO(2k) \over U(1) \times U(k-1)}$. These cosets are bundles where base and fibre are hermitian symmetric spaces. The detailed form of the generators and relations for the Matrix algebras related to the fuzzy three-spheres suggests Matrix actions which admit the fuzzy spheres as solutions. These Matrix actions are compared with the BFSS, IKKT and BMN Matrix models as well as some others. The geometry and combinatorics of fuzzy odd spheres lead to some remarks on the transverse five-brane problem of Matrix theories and the exotic scaling of the entropy of 5-branes with the brane number.

The Hodge star operator on Schubert forms

Let X= G/ P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds \(N\subset M,\) having positive holomorphic bisectional curvature with the induced Kahler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, \({\rm Pr} \left( M\right),\) the usual rank, \(rk\left( M\right),\) and the 2-number \(\#_2\left( M\right)\) (which is known to be equal to the Euler-Poincare characteristic in these spaces).

Perverse Sheaves on Grassmannians

AbstractWe compute the category of perverse sheaves on Hermitian symmetric spaces in types A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety Λ.

Bruhat Poisson structure on CPn and integrable systems

In this note we present a simple formula for the Bruhat Poisson structure on complex projective spaces in terms of the momentum coordinates. We also give a simple description of a family of functions in involution on compact Hermitian symmetric spaces obtained via the bi-Hamiltonian approach using the Bruhat Poisson structure and an invariant symplectic structure. We compute these functions explicitly on CPn and relate them to the Gelfand–Tsetlin coordinates. We also show how the Lenard scheme can be applied.

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

Gauge Theoretical Construction of Non-compact Calabi–Yau Manifolds

We construct the non-compact Calabi–Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CPN−1. Imposing an F-term constraint on the line bundle over CPN−1, we obtain the line bundle over the complex quadric surface QN−2. On the other hand, when we promote the U(1) gauge symmetry in CPN−1 to the non-abelian gauge group U(M), the line bundle over the Grassmann manifold is obtained. We construct the non-compact Calabi–Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. Each of these manifolds contains the resolution parameter which controls the size of the base manifold, and the conical singularity appears when the parameter vanishes.

Plancherel Formula for Berezin Deformation of L2 on Riemannian Symmetric Space

Consider natural representations of the pseudounitary group U(p, q) in the space of holomorphic functions on the Cartan domain (Hermitian symmetric space) U(p, q)/(U(p)×U(q)). Berezin representations of O(p, q) are the restrictions of such representations to the subgroup O(p, q). We obtain the explicit Plancherel formula for the Berezin representations. The support of the Plancherel measure is a union of many series of representations. The density of the Plancherel measure on each piece of the support is an explicit product of Γ-functions. We also show that the Berezin representations give an interpolation between L2 on noncompact symmetric space O(p, q)/O(p)×O(q) and L2 on compact symmetric space O(p+q)/O(p)×O(q).

Symmetric spaces and star representations: II. Causal symmetric spaces

We construct and identify star representations canonically associated with holonomy-reducible simple symplectic symmetric spaces. This leads a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley-type and Hermitian symmetric spaces of tube-type.

Canonical and Boundary Representations on the Lobachevsky Plane

Canonical representations on Hermitian symmetric spaces G/K were introduced by Vershik, Gelfand and Graev and Berezin. They are unitary. We study canonical representations in a wider sense. In this paper we restrict ourselves to a crucial example – the Lobachevsky plane: G=SU(1,1), K=U(1). Canonical representations are labelled by the complex parameter λ (Vershik–Gelfand–Graev's representations correspond to −3/2<λ<0). We decompose the canonical representations into irreducible components. The decomposition includes boundary representations generated by the canonical representations. So we study these boundary representations themselves. The decomposition of boundary representations is closely connected with the meromorphic structure of Poisson and Fourier transforms associated with canonical representations. In particular, second-order poles give second-order Jordan blocks. Finally, we give a full decomposition of the Berezin transform using ‘generalized powers’ (Pochhammer symbols) instead of usual powers of λ.