Hermitian Symmetric Spaces Research Articles

Remarks on quadratic bundles related to Hermitian symmetric spaces

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)×U(n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.

Phases of Lagrangian-invariant objects in the derived category of an abelian variety

We continue the study of Lagrangian-invariant objects (LI-objects for short) in the derived category $D^b(A)$ of coherent sheaves on an abelian variety, initiated in arXiv:1109.0527. For every element of the complexified ample cone $D_A$ we construct a natural phase function on the set of LI-objects, which in the case $\dim A=2$ gives the phases with respect to the corresponding Bridgeland stability (see math.AG/0307164). The construction is based on the relation between endofunctors of $D^b(A)$ and a certain natural central extension of groups, associated with $D_A$ viewed as a hermitian symmetric space. In the case when $A$ is a power of an elliptic curve, we show that our phase function has a natural interpretation in terms of the Fukaya category of the mirror dual abelian variety. As a byproduct of our study of LI-objects we show that the Bridgeland's component of the stability space of an abelian surface contains all full stabilities.

Quotients of non-classical flag domains are not algebraic

Aag domain D = G=V forG a simple real non-compact groupG with compact Cartan subgroup is non-classical if it does not ber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that for an innite, nitely generated discrete subgroup of G, the analytic space nD does not have an algebraic structure. We also give another proof of the theorem of Huckleberry that any two points in a non-classical domain D can be joined by a nite chain of compact subvarieties of D.

The classification of real forms of simple irreducible pseudo-Hermitian symmetric spaces

The main purpose of this paper is to classify the real forms $M$ of simple irreducible pseudo-Hermitian symmetric spaces $G/R$ with $G$ non-compact. That provides an extension of Jaffee's results (Bull. Amer. Math. Soc. '75; J. Differential Geom. '78), Leung's result (J. Differential Geom. '79) and Takeuchi's result (Tohoku Math. J. '84) concerning the classification of real forms of irreducible Hermitian symmetric spaces of the non-compact type. Moreover, that enables us to classify the pairs of simple para-Hermitian symmetric Lie algebras and their para-holomorphic involutions, which includes Kaneyuki-Kozai's result (Tokyo J. Math. '85) of the classification of simple para-Hermitian symmetric Lie algebras.

Dispersive geometric curve flows

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flow is integrable if the evolution equation of the local differential invariants of a solution of the curve flow is a soliton equation. For example, the Hodge star mean curvature flow on $R^3$ and on $R^{2,1}$, the geometric Airy flow on $R^n$, the Schrodingier flow on compact Hermitian symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a compact Lie group are integrable. In this paper, we give a survey of these results, describe a systematic method to construct integrable curve flows from Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the Cauchy problem of these curve flows.

A Geometric Proof of a Result of Takeuchi

In 1984 Masaru Takeuchi showed that every real form of a hermitian symmetric space of compact type is a symmetric $R$-space and vice-versa. In this note we present a geometric proof of this result.

The classification of homogeneous Einstein metrics on flag manifolds with [formula omitted

Consider a compact connected simple Lie group G. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler–Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been recently completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds E8/U(1)×SU(4)×SU(5) and E8/U(1)×SU(2)×SU(3)×SU(5). In order to determine explicitly the Ricci tensors of an E8-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold G/H with b2(M)=1 admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality in the case where $M$ is irreducible. As its application, we prove that the totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations.

Singular loci of cominuscule Schubert varieties

Let X=G/P be a cominuscule rational homogeneous variety. Equivalently, X admits the structure of a compact Hermitian symmetric space. I give a uniform description (that is, independent of type) of the irreducible components of the singular locus of a Schubert variety Y⊂X in terms of representation theoretic data. The result is based on a recent characterization of the Schubert varieties using an integer a≥0 and a marked Dynkin diagram. Corollaries include: (1) the variety is smooth if and only if a=0; (2) if G is of type ADE, then the singular locus occurs in codimension at least 3.

The volume entropy of local Hermitian symmetric space of noncompact type

We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant r, a, b.

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kahler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szego kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Englis (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.

Symplectic branching laws and Hermitian symmetric spaces

Let $G$ be a complex simple Lie group, and let $U \subseteq G$ be a maximal compact subgroup. Assume that $G$ admits a homogenous space $X=G/Q=U/K$ which is a compact Hermitian symmetric space. Let $\mathscr{L} \rightarrow X$ be the ample line bundle which generates the Picard group of $X$. In this paper we study the restrictions to $K$ of the family $(H^0(X, \mathscr{L}^k))_{k \in \N}$ of irreducible $G$-representations. We describe explicitly the moment polytopes for the moment maps $X \rightarrow \fk^*$ associated to positive integer multiples of the Kostant-Kirillov symplectic form on $X$, and we use these, together with an explicit characterization of the closed $K^\C$-orbits on $X$, to find the decompositions of the spaces $H^0(X,\mathscr{L}^k)$. We also construct a natural Okounkov body for $\mathscr{L}$ and the $K$-action, and identify it with the smallest of the moment polytopes above. In particular, the Okounkov body is a convex polytope. In fact, we even prove the stronger property that the semigroup defining the Okounkov body is finitely generated.

Nearly holomorphic sections on compact Hermitian symmetric spaces

Let X be a Kähler manifold, and E be a Hermitian vector bundle on X. We investigate the space N(X,E) of nearly holomorphic sections in E, which generalizes the notion of nearly holomorphic functions introduced by Shimura. If X=U/K is a compact Hermitian symmetric space, and E is U-homogeneous, it turns out that N(X,E) coincides with the space of U-finite vectors in C∞(X,E), and we obtain new results on the U-type decomposition of the Hilbert space of square integrable sections. As an application, we determine this decomposition for the holomorphic tangent bundle of X.

On a generalization of a theorem of McDuff

We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.

Classification of homogeneous holomorphic hermitian principal bundles over G/K

Abstract We give a complete classification of isomorphism classes of homogeneous holomorphic hermitian principal bundles over an irreducible hermitian symmetric space of noncompact type. This generalizes the results of our earlier work [Kyoto J. Math. 50 (2010), 325–363] where homogeneous holomorphic hermitian principal bundles over the upper half-plane were considered.

Cotangent bundle over hermitian symmetric space E 7 /E 6 × U(1) from projective superspace

We construct an $\mathcal{N}$ supersymmetric sigma model on the cotangent bundle over the Hermitian symmetric space $E_7/(E_6\times U(1))$ in the projective superspace formalism, which is a manifest $\mathcal{N}=2$ off-shell superfield formulation in four-dimensional spacetime. To obtain this model we elaborate on results developed in arXiv:0811.0218 and present a new closed formula for the cotangent bundle action, which is valid for all Hermitian symmetric spaces. We show that the structure of cotangent bundle action is intimately related to the analytic structure of the K\"ahler potential with respect to a uniform rescaling of coordinates.

Satake-Furstenberg compactifications, the moment map and λ 1

Let $G$ be a complex semisimple Lie group, $K$ a maximal compact subgroup and $\tau$ an irreducible representation of $K$ on $V$. Denote by $M$ the unique closed orbit of $G$ in $\Bbb{P}(V)$ and by $\cal{O}$ its image via the moment map. For any measure $\gamma$ on $M$ we construct a map $\Psi_\gamma$ from the Satake compactification of $G/K$ (associated to $V$) to the Lie algebra of $K$. If $\gamma$ is the $K$-invariant measure, then $\Psi_\gamma$ is a homeomorphism of the Satake compactification onto the convex envelope of $\cal{O}$. For a large class of measures the image of $\Psi_\gamma$ is the convex envelope. As an application we get sharp upper bounds for the first eigenvalue of the Laplacian on functions for an arbitrary K\"ahler metric on a Hermitian symmetric space.

Schur flexibility of cominuscule Schubert varieties

Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class [S] is Schur rigid if the only irreducible subvarieties Y of X with homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles and The identified a sufficient condition for a Schubert class to be Schur rigid. In this paper we show that the condition is also necessary.

Existence of nearly holomorphic sections on compact Hermitian symmetric spaces

Let be a compact Hermitian symmetric space, and let be a U -homogeneous Hermitian vector bundle on X . In a previous paper, we showed that the space of nearly holomorphic sections is well-adapted for harmonic analysis in provided that non-trivial nearly holomorphic sections do exist. Here we investigate the problem of extending local nearly holomorphic sections to global ones and prove the existence of non-trivial nearly holomorphic sections. This extends the results on the U -type decomposition of from our previous paper.

Canonical and Boundary Representations on Rank One Para-Hermitian Spaces

This work studies the canonical representations (Berezin representations) for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces G/H with G = SL(n,R),H = GL(n-1,R) . For Hermitian symmetric spaces G/K, canonical representations were introduced by Berezin and Vershik-Gelfand-Graev. They are unitary with respect to some invariant non-local inner product (the Berezin form). We consider canonical representations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distributions. For our spaces G/H, the canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. We decompose the canonical representations into irreducible constituents and decompose boundary representations.