Irreducible Hermitian Symmetric Space Research Articles

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.

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.

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.

Formula omitted]-types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series

Let G0 be a simply connected non-compact real simple Lie group and let K0 be a maximal compact subgroup of G0. Suppose that K0 is semisimple and that rank(K0)=rank(G0). Let Δ+ be a Borel–de Siebenthal positive root system and let πλ be the Borel–de Siebenthal discrete series of G0 with Harish-Chandra parameter λ. One has a certain subgroup L0⊂K0 so that K0/L0 is an irreducible Hermitian symmetric space. Also, there is a holomorphic discrete series πλ′ of K0⁎, the non-compact dual of K0, with Harish-Chandra parameter λ′:=λ−(1/2)∑α, where the sum is over non-compact roots in Δ+. We prove that there are infinitely many L0-types common to πλ and πλ′ under certain hypotheses.

The intersection of two real forms in Hermitian symmetric spaces of compact type

We show that the intersections of two real forms, certain totally geodesic Lagrangian submanifolds, in Hermitian symmetric spaces of compact type are antipodal sets. The intersection number of two real forms is invariant under the replacement of the two real forms by congruent ones. If two real forms are congruent, then their intersection is a great antipodal set of them. It implies that any real form in Hermitian symmetric spaces of compact type is a globally tight Lagrangian submanifold. Moreover we describe the intersection of two real forms in the irreducible Hermitian symmetric spaces of compact type.

Witt groups of complex cellular varieties

We show that the Grothendieck-Witt and Witt groups of smooth complex cellular varieties can be identified with their topological KO-groups. As an application, we deduce the values of the Witt groups of all irreducible hermitian symmetric spaces, including smooth complex quadrics, spinor varieties and symplectic Grassmannians.

A Berger type normal holonomy theorem for complex submanifolds

We prove a Berger type theorem for the normal holonomy \({\Phi^\perp}\) (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \({\mathbb{C} P^n}\). Namely, if \({\Phi^\perp}\) does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \({\mathbb{C}^n}\) the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \({\mathbb{C} P^n}\) case) and basic facts of complex submanifolds.

Homogeneous isoparametric hypersurfaces in spheres with four distinct principal curvatures and moment maps

We study relations between moment maps of Hamiltonian actions and isoparametric hypersurfaces in spheres with four distinct principal curvatures. In this paper, we deal with the isoparametric hypersurfaces given by the isotropy representations of compact irreducible Hermitian symmetric spaces of classical type and of rank two. We show that such isoparametric hypersurfaces can be obtained by moment maps. More precisely, certain squared-norms of moment maps coincide with Cartan-Münzner polynomials, which are defining-equations, of above isoparametric hypersurfaces.

A characterization of non-tube type Hermitian symmetric spaces by visible actions

Let G/K be an irreducible Hermitian symmetric space of non-compact type, and \({G_{\mathbb{C}}/K_{\mathbb{C}}}\) its complexification by forgetting the original complex structure. Then, \({D :=G_{\mathbb{C}}/[K_{\mathbb{C}}, K_{\mathbb{C}}]}\) is a non-symmetric Stein manifold. We prove that a maximal compact subgroup of \({G_{\mathbb{C}}}\) acts on D in a strongly visible fashion in the sense of Kobayashi (Publ Res Inst Math Sci 41:497–549, 2005) if and only if G/K is of non-tube type. Our proof uses the theory of multiplicity-free representations and a construction of a slice and an anti-holomorphic involution on D.

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.

A Higgs bundle on a Hermitian symmetric space

Let M : = Γ\G/K be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice $${\Gamma \subset G}$$ . Let V be a complex irreducible representation of G. We give a Hodge decomposition of the cohomology of the Γ-module V in terms of the cohomologies of automorphic vector bundles on M associated to the Lie algebra cohomologies $${H*({\mathfrak p}^{+} ,V)}$$ .

Hermitian symmetric spaces and Kahler rigidity

We characterize irreducible Hermitian symmetric spaces which are not of tube type, both in terms of the topology of the space of triples of pairwise transverse points in the Shilov boundary, and of two invariants which we introduce, the Hermitian triple product and its complexification. We apply these results and the techniques introduced in [6] to characterize conjugacy classes of Zariski dense representations of a locally compact group into the connected component G of the isometry group of an irreducible Hermitian symmetric space which is not of tube type, in terms of the pullback of the bounded Kahler class via the representation. We conclude also that if the second bounded cohomology of a finitely generated group Γ is finite dimensional, then there are only finitely many conjugacy classes of representations of Γ into G with Zariski dense image. This generalizes results of [6].

Hamiltonian actions and homogeneous Lagrangian submanifolds

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired by recent results, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is given. Also, we describe under which condition a Lagrangian orbit is isolated. If $M$ is a compact Kähler manifold, we give a necessary and sufficient condition for an isometric action to admit a Lagrangian orbit. Then we investigate homogeneous Lagrangian submanifolds on the symplectic cut and on the symplectic reduction. As an application of our results, we exhibit new examples of homogeneous Lagrangian submanifolds on the blow-up at one point of the complex projective space and on the weighted projective spaces. Finally, applying our result which may be regarded as Lagrangian slice theorem for a Hamiltonian group action with a fixed point, we give new examples of homogeneous Lagrangian submanifolds on irreducible Hermitian symmetric spaces of compact or noncompact type.

Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

We obtain the full classification of coisotropic and polar isometric actions of compact Lie groups on irreducible Hermitian symmetric spaces.

The Normal Holonomy Group of Kähler Submanifolds

We study the (restricted) holonomy group Hol(∇⊥) of the normal connection ∇⊥ (shortened to normal holonomy group) of a Kähler submanifold of a complex space form. We prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible Hermitian symmetric space. In particular, it is a compact group and the complex structure J belongs to its Lie algebra.We prove that the normal holonomy group acts irreducibly if the submanifold is full (that is, it is not contained in a totally geodesic proper Kähler submanifold) and the second fundamental form at some point has no kernel. For example, a Kähler–Einstein submanifold of CPn has this property.We define a new invariant μ of a Kähler submanifold of a complex space form. For non‐full submanifolds, the invariant μ measures the deviation of J from belonging to the normal holonomy algebra. For a Kähler–Einstein submanifold, the invariant μ is a rational function of the Einstein constant. By using the invariant μ, we prove that the normal holonomy group of a not necessarily full Kähler–Einstein submanifold of CPn is compact, and we give a list of possible holonomy groups.The approach is based on a definition of the holonomy algebra hol(P) of an arbitrary curvature tensor field P on a vector bundle with a connection and on a De Rham type decomposition theorem for hol(P). 2000 Mathematics Subject Classification 53C40 (primary), 53B25 (secondary).

Differential systems of type (1,1) on Hermitian symmetric spaces and their solutions

This paper concerns G-invariant systems of second-order differential operators on irreducible Hermitian symmetric spaces G/ K. The systems of type (1,1) are obtained from K-invariant subspaces of p +⊗ p − . We show that all such systems can be derived from a decomposition p +⊗ p −= H′⊕ L⊕ H c . Here L gives the Laplace–Beltrami operator and H= H′⊕ L is the celebrated Hua system, which has been extensively studied elsewhere. Our main result asserts that for G/ K of rank at least two, a bounded real-valued function is annihilated by the system L⊕ H c if and only if it is the real part of a holomorphic function. In view of previous work, one obtains a complete characterization of the bounded functions that are solutions for any system of type (1,1) which contains the Laplace–Beltrami operator.

A Pieri rule for Hermitian symmetric pairs I

Let X be an irreducible Hermitian symmetric space of non-compact type and rank r. Let p ∈ X and let K be the isotropy group of p in the group of biholomorphic transformations. Let S denote the symmetric algebra in the holomorphic tangent space to X at p. Then S is multiplicity free as a representation of K and the irreducible constituents are parametrized by r-tuples, (m 1 ...,m r ) with m 1 >... > m r > 0. That is, the same parameters as the irreducible polynomial representations of GL(r). Let S[m 1 ,...,m r ] be the corresponding isotypic component. In this article we show that the product in S, S[m 1 ,...,m r ]S[k,0,0,...,0] is a direct sum of constituents following precisely the classical Pieri rule.

Cohomology operations in the space of loops on the exceptional Lie group $E_6$

Let $E_{6}$ be the compact 1-connected exceptional Lie group of rank 6. In [9] we determined the Hopf algebra structure of $H_{*}(\Omega E_{6}; \mathbb{Z})$ by the generating variety approach of R. Bott [1]. In this case, as a generating variety we can take $EIII$, the irreducible Hermitian symmetric space of exceptional type. Then as Bott pointed out in [1], §6, we can determine the action of the mod $p$ Steenrod algebra $\mathcal{A}_{p}$ on $H^{*}(\Omega E_{6}; \mathbb{Z}_{p})$ from that on $H^{*}(EIII; \mathbb{Z}_{p})$ for all primes $p$. In this paper, for ease of algebraic description, we compute the action of $\mathcal{A}_{p_{*}}$, the dual of $\mathcal{A}_{p}$, on $H^{*}(\Omega E_{6}; \mathbb{Z}_{p})$ for $p = 2, 3$ (For larger primes the description is easy). In the course of computation we also determine the action of $\mathcal{A}_{3}$ on $H^{*}(E_{6}/T; \mathbb{Z}_{3})$, where T is a maximal torus of $E_{6}$. The paper is constructed as follows: In Section 2 we recall some results concerning the cohomology of some homogeneous spaces of $E_{6}$. In Section 3 by considering the action of the Weyl group on $E_{6}/T$, we determine the cohomology operations in $EIII$. Using the results obtained, in Section 4 we shall determine the cohomology operations in $\Omega E_{6}$. Throughout this paper $\sigma _{i}(x_{1},\ldots , x_{n})$ denotes the $i$-th elementary symmetric function in the variables $x_{1},\ldots , x_{n}$.

Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces

A compact minimal Lagrangian submanifold immersed in a Kahler manifold is called Hamiltonian stable if the second variation of its volume is nonnegative under all Hamiltonian deformations. We study compact Hamiltonian stable minimal Lagrangian submanifolds with parallel second fundamental form embedded in complex projective spaces. Moreover, we completely determine Hamiltonian stability of all real forms in compact irreducible Hermitian symmetric spaces, which were classified previously by M. Takeuchi.

Invariant hyperkaehler structures on the cotangent bundles of Hermitian symmetric spaces

Let G/K be an irreducible Hermitian symmetric space of compact type with standard homogeneous complex structure. Then the real symplectic manifold (T*(G/K),{omega}) has the natural complex structure J{sup -}. All G-invariant Kehler structures (J,{omega}) on G-invariant subdomains of T*(G/K) anticommuting with J{sup -} are constructed. Each hypercomplex structure of this kind, equipped with a suitable metric, defines a hyperkehler structure. As an application, a new proof of the theorem of Harish-Chandra and Moore for Hermitian symmetric spaces is obtained.