Hermitian Symmetric Spaces Research Articles

Boundary Value Problems for Various Boundaries of Hermitian Symmetric Spaces

The image of the Poisson transform on a principal series representations on a boundary component of a Hermitian symmetric space is considered. We prove that the image is characterized by a covariant differential operator on a homogeneous line bundle on the symmetric space.

Hermitian Symmetric Spaces, Cycle Spaces, and the Barlet–Koziarz Intersection Method for Construction of Holomorphic Functions

Under certain conditions, a recent method of Barlet and Koziarz (2) constructs enough holomorphic functions to give a direct proof of the Stein con- dition for a cycle space. Here we verify those conditions for open G0-orbits on X, where G0 is the group of a bounded symmetric domain and X is its compact dual viewed as a flag quotient manifold of the complexification G of G0 . This Stein result was known for a few years (9), and in fact a somewhat more precise result is known (12) for the flag domains to which we apply the Barlet-Koziarz method, but the proof here is much more direct and holds the possibility of greater generality. Also, some of the tools developed here apply directly to open orbits that need not be measurable, avoiding separate arguments of reduction to the measurable case.

Quantized Dirac Operators

We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained.

On Congruent Hypersurfaces of Hermitian Symmetric Spaces

On Congruent Hypersurfaces of Hermitian Symmetric Spaces

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kahler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kahler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.

A BGG type resolution of holomorphic Verma modules

For a Hermitian symmetric space $X = G/K$ of non-compact type let $\theta$ denote the Cartan involution of the semisimple Lie group $G$ with respect to the maximal compact subgroup $K$ of $G$, and let $q$ denote a $\theta$-stable parabolic subalgebra of the complexified Lie algebra $g$ of $G$ with corresponding Levi subgroup $L$ of $G$. Given a finite-dimensional irreducible $L$ module $F_{L}$ we find Bernstein-Gelfand-Gelfand type resolutions of the induced $(g, L \cap K)$ module $U(g) \otimes_{U(q)}F_{L}$ and its Hermitian dual, the produced module $\mathrm{Hom}_{U(\bar{q})}(U(g),F_{L})_{L \cap K-\mathrm{finite}}$, where $U(\cdot)$ is the universal enveloping algebra functor and $\bar{q}$ is the complex conjugate of $q$. The results coupled with a Grothendick spectral sequence provide for application to certain $(g,K)$ modules obtained by cohomological parabolic induction, and they extend results obtained initially by Stanke.

Orthogonal complex structures on certain Riemannian 6-manifolds

It is shown that the Hermitian-symmetric space CP 1× CP 1× CP 1 and the flag manifold F 1,2 endowed with any left invariant metric admit no compatible integrable almost complex structures (even locally) different from the invariant ones. As an application it is proved that any stable harmonic immersion from F 1,2 equipped with an invariant metric into an irreducible Hermitian symmetric space of compact type is equivariant. It is also shown that CP 1× CP 1× CP 1 and F 1,2 with its invariant Kähler–Einstein structures are the only compact Kähler–Einstein spin 6-manifolds of non-negative, non-identically vanishing holomorphic sectional curvature that admit another orthogonal complex structure of Kähler type. A necessary and sufficient condition on a compact oriented 6-manifold to admit three mutually commuting almost complex structures is given; it is used to characterize CP 1× CP 1× CP 1 and F 1,2 as Fano 3-folds admitting three mutually commuting complex structures which satisfy certain compatibility conditions.

Integrable coupling of optical waves in higher-order nonlinear Schrödinger equations

Integrable higher-order generalizations of the nonlinear Schrödinger equation that describes the propagation of multi-mode optical pulses in a fiber are presented. We construct the coupled higher-order nonlinear Schrödinger equation (CHONSE) in association with each Hermitian symmetric spaces and demonstrate its integrability by deriving the Lax pair. We show that two distinct types of higher-order generalizations are possible, which we call as the `type-I' and the `type-II' CHONSE. The type-I and the type-II CHONSE generalize the Hirota and the Sasa–Satsuma equations respectively and it is shown that the type-II CHONSE can be obtained via a consistent reduction of the type-I CHONSE based on the AIII symmetric spaces.

Gauged nonlinear sigma model and boundary diffeomorphism algebra

We consider Chern-Simons gauged nonlinear sigma model with boundary which has a manifest bulk diffeomorphism invariance. We find that the Gauss's law can be solved explicitly when the nonlinear sigma model is defined on the Hermitian symmetric space, and the original bulk theory completely reduces to a boundary nonlinear sigma model with the target space of Hermitian symmetric space. We also study the symplectic structure, compute the diffeomorphism algebra on the boundary, and find an (enlarged) Virasoro algebra with classical central term.

Canonical representations associated to hyperbolic spaces II

In this paper we determine explicit models for the unitary representations which occur discretely in the decomposition of canonical representations for hyperbolic spaces. This generalizes work of Gelfand, Graev and Vershik for the case SU(1,1). In a previous paper [1] we have introduced canonical representations π λ for hyperbolic spaces of Riemannian type, generalizing Berezin's definition for the case of Hermitian symmetric spaces. These unitary representations have a rich internal structure and prove that not only quasi-regular representations are important in harmonic analysis. They have an interesting decomposition into irreducible constituents: for large λ only unitary principal series representations occur, for small λ however a finite number of complementary series representations has to be added. A detailed description of the latter, as subrepresentations of π λ , was started in [1]. The present paper contains much more precise results. The classical notions of Fourier and Poisson transform play a dominant role and are treated in Section 4. In Section 5 the actual projection of a given function onto the complementary series is then given. Since the proof makes use of some results on maximal degenerate series representations of SL( n + 1), these representations are studied in Section 1 in some detail. As a by-product we obtain an alternative introduction of the canonical representations, see Sections 2 and 5. This paper has a considerable, but unevitable optical overlap with our joint paper [2] with V.F. Molchanov. The method described here has been introduced in a different setting by Molchanov, but the observation that it works also for hyperbolic spaces is new and surprising. To leave out parts of the proof would make this paper almost unreadable. We thank S.C. Hille and V.F. Molchanov for several discussions on these matters and for their encouragement to publish the results obtained in this paper.

Classification of Irreducible Holonomies of Torsion-Free Affine Connections

The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poisson geometry. The latter theory is especially important for the construction and classification of those torsion-free connections whose holonomy falls into one of the so-called `exotic' cases, i.e., those that were not included in Berger's original lists. Some remarks involving an interpretation of some of the examples in terms of supersymmetric constructions are also included.

Bogomol'nyi solitons and Hermitian symmetric spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N − 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.

On congruent holomorphic mappings into a Hermitian symmetric space

It is one of basic problems in differential geometry to find geometric conditions that determine the congruence class of a submanifold of a given manifold. For example, a general hypersurface of n-dimensional Euclidean space R n is determined up to congruence by its first and second fundamental forms. It is another classical result that for a certain generic hypersurfaces of R n (n > 4), the first fundamental form alone is sufficient to determine the congruence class. On the other hand, we know some remarkable facts in the complex-analytic case. First, any complex submanifold of a complex space form has the metric rigidity, i.e., it is determined up to congruence by its first fundamental form alone (the rigidity theorem of Calabi [1]). Secondly, Green [2] asserts even if the ambient space 5 is a general Kahler manifold with real-analytic Kahler metric, a certain generic holomorphic mapping (a nondegenerate mapping in his terminology) into it has the local metric rigidity: There is actually shown the existence of such a finite-dimensional family of local real-analytic hypersurfaces of S that a holomorphic mapping into S has the local metric rigidity unless its image lies in any of them. Only one cannot find any geometric conditions that assure the nondegeneracy for a mapping. In this article, we shall give local differential-geometric conditions that determine the congruence class of a holomorphic mapping into a Hermitian symmetric space under considerably general settings. For a technical reason, we have to restrict ourselves to the case where mappings are infinίtesimally in our sense. In particular we cannot deal with totally geodesic complex submanifolds. But our results extend Calabi's rigidity theorem to a direction different from [2], since our infinitesimally condition is equivalent to the full condition used in [1] in the case of complex space forms.

Berezin transform on¶compact Hermitian symmetric spaces

Let X=G * be a compact Hermitian symmetric space. We study the Berezin transform on L 2(X) and calculate its spectrum under the decomposition of L 2(X) into the irreducible representations of G *. As applications we find the expansion of powers of the canonical polynomial (Bergman reproducing kernel for the canonical line bundle) in terms of the spherical polynomials on X, and we find the irreducible decomposition of tensor products of Bergman spaces on X.

Unitarity of strings and non-compact Hermitian symmetric spaces

If G is a simple non-compact Lie Group, with K its maximal compact subgroup, such that K contains a one-dimensional center C, then the coset space G/ K is an Hermitian symmetric non-compact space. SL(2, R)/U(1) is the simplest example of such a space. It is only when G/ K is an Hermitian symmetric space that there exists unitary discrete representations of G. We will here study string theories defined as G/ K′, K′= K/ C, WZNW models. We will establish unitarity for such string theories for certain discrete representations. This proof generalizes earlier results on SL(2, R) , which is the simplest example of this class of theories. We will also prove unitarity of G/ K conformal field theories generalizing results for SL(2, R)/U(1) . We will show that the physical space of states lie in a subspace of the G/ K state space.

Compressions and contractions of Hermitian symmetric spaces

Compressions and contractions of Hermitian symmetric spaces

Integrable extension of nonlinear sigma model

We propose an integrable extension of nonlinear sigma model on the target space of Hermitian symmetric space (HSS). Starting from a discussion of soliton solutions of O(3) model and an integrally extended version of it, we construct general theory defined on arbitrary HSS by using the coadjoint orbit method. It is based on the exploitation of a covariantized canonical structure on HSS. This term results in an additional first-order derivative term in the equation of motion, which accommodates the zero-curvature representation. Infinite conservation laws of nonlocal charges in this model are derived.

A vanishing theorem for modular symbols on locally symmetric spaces

A modular symbol is the fundamental class of a totally geodesic submanifold 0 nG 0 =K 0 embedded in a locally Riemannian symmetric space nG=K, which is dened by a subsymmetric space G 0 =K 0 ,! G=K. In this paper, we consider the modular symbol dened by a semisimple symmetric pair (G;G 0 ), and prove a vanishing theorem with respect to the -component (2b) in the Matsushima-Murakami formula based on the discretely decomposable theorem of the re- striction jG0 . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV.

Quantized Flag Manifolds and Irreducible *-Representations

We study irreducible *-representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold. All irreducible *-representations are classified for a subclass of flag manifolds containing in particular the irreducible compact Hermitian symmetric spaces. For this subclass it is shown that the irreducible *-representations are parametrized by the symplectic leaves of the underlying Poisson bracket. We also discuss the relation between the quantized flag manifolds studied in this paper and the quantum flag manifolds studied by Soibelman, Lakshimibai and Reshetikhin, Jurco and Stovicek, and Korogodsky.

Rigidity of irreducible Hermitian symmetric spaces of the compact type under K�hler deformation

Rigidity of irreducible Hermitian symmetric spaces of the compact type under K�hler deformation