Hermitian Symmetric Spaces Research Articles

Composition series for analytic continuations of holomorphic discrete series representations of 𝑆𝑈(𝑛,𝑛)

We study a certain family of holomorphic discrete series representations of the semisimple Lie group G = S U ( n , n ) G\, = \,SU(n,\,n) and the corresponding analytic continuation in the inducing parameter λ \lambda . At the values of λ \lambda where the representations become reducible, we compute the composition series in terms of a Peter-Weyl basis on the Shilov boundary of the Hermitian symmetric space for G.

On the hypersurfaces of Hermitian symmetric spaces of compact type

On the hypersurfaces of Hermitian symmetric spaces of compact type

On the hypersurfaces of Hermitian symmetric spaces of compact type. II

On the hypersurfaces of Hermitian symmetric spaces of compact type. II

On the nullities of Kahler C-spaces in $P_{N}(C)$

Let Mbe a Kahler C-space which Is holomorphically and isometricallyimbedded in an N-dimensional complex projectivespace Pjv(C). Then M is a minimal submanifold of Pn(C). Let na(M) be theanalytic nullityof M which was definedin [2]. We know that the nullityn{M) of M is equal tona{M) if M is a Hermitian symmetric space (Kimura [2]). In this note we prove that w(M)-=na(M) forany Kahler C-space M. By a theorem of Simons [5], the nullity of a Kahler submanifold coincides with the real dimension of the space of holomorphic sectionsof a normal bundle of the submanifold. Put M=G/U where G is a complex semi-simple Lie group and U is a parabolic subgroup of G. By a resultof Nakagawa and Takagi [4],we know that every imbedding of M in Pn(C) is induced by a holomorphic linear representationof G. From thisresultwe see that the normal bundle N(M) over M is a homogeneous vector bundle. We prove Theorem 1 which generalizes the generalized Borel-Weil theorem of Bott [1]. Applying the theorem to calculatethe dimension of the space of holomorphic sectionsof N(M) and prove that n(M) =na(M). The auther proved the above result before Proffesor Takeuchi gave another proofof it. His proof does not use Theorem 1 and is more simple than our proof (c.f.Takeuchi [6]). §1. The generalization of Bott'sresultLet G be a simply connected compact semi-simple Lie group with Lie algebra 8. Take a Cartan subalgebra I)of 0. Denoto by A therootsystem ofgwith respect to fy We fixa linear order on the realvector space spaned by theelements ereA. Let A+ (resp.A~)be the setof allpositive(resp.negative)roots. LetIT={au･･･,ai) be the fundamental root system, where /is therank of g and IL.be a subsystem ofII. We put

Reflective submanifolds. IV. Classification of real forms of Hermitian symmetric spaces

Reflective submanifolds. IV. Classification of real forms of Hermitian symmetric spaces

A hypersurface of the irreducible Hermitian symmetric space of type E III

A hypersurface of the irreducible Hermitian symmetric space of type E III

Complex submanifolds of some Hermitian symmetric spaces

Complex submanifolds of some Hermitian symmetric spaces

Exceptional Lie algebras and the structure of Hermitian symmetric spaces

Exceptional Lie algebras and the structure of Hermitian symmetric spaces

On cut loci and first conjugate loci of the irreducible symmetric $R$-spaces and the irreducible compact hermitian symmetric spaces

On cut loci and first conjugate loci of the irreducible symmetric $R$-spaces and the irreducible compact hermitian symmetric spaces

The characters of the holomorphic discrete series

This paper summarizes the derivation of an explicit and global formula for the character of any holomorphic discrete series representation of a reductive Lie group G which satisfies certain conditions. The only very restrictive condition is that G/K be a Hermitian symmetric space. (Here K is the maximal compact subgroup of G.).

Real forms of hermitian symmetric spaces

Real forms of hermitian symmetric spaces

Coherent states and bounded homogeneous domains

The system of coherent states for complex bounded homogeneous domains is constructed and the properties of coherent states are investigated. The question of selecting complete subsystems connected with discrete subgroups of the motion groups of Hermitian symmetric spaces is also studied.

Orbit structure of the exceptional Hermitian symmetric spaces. I

Orbit structure of the exceptional Hermitian symmetric spaces. I

Holomorphic vector bundles on homogeneous spaces

LET G be a connected, compact, semisimple, Lie group, and V a Lie subgroup of maximal rank. We shall assume that the coset space G/V is provided with a holomorphic structure which is invariant under the action of G. Any representation p of I’ gives rise to a vector bundle on G/V, viz. the fibre bundle associated to the fibration G + G/V. Moreover, this bundle can be provided with a natural holomorphic structure. (There are, in fact, several ways of doing this, but if p is irreducible, this procedure is unique.) We shall denote this holomorphic vector bundle by E,, and refer to this as a homogeneous vector bundle. Using the results of Bott on homogeneous vector bundles, we prove: Assume that V does not contain any simple component of G. If p is irreducible, then any holomorphic endomorphism of Ee is a scalar multiple of the identity. Moreover, tfp, and pz are two irreducible representations of V, then E,,, and EPI are holomorphically equivalent tfand only zfp, and pz are equivalent. This in particular implies that EP is indecomposable. The latter was proved by Griffiths [7, theorem 81. However, the fact that the only endomorphism of EP are scalars, seems likely to be useful in the study of deformation of such bundles. The equivalence problem was first raised by Ise [9, $141. We then study the homogeneous vector bundles over an irreducible Hermitian symmetric space. In this case, the second Betti number of G/V is 1 and since G/V is an algebraic manifold, there is a canonical generator for H’(G/V, Z). This enables one to talk of positive and negative elements of H2(G/V, Z). Following the definition of Mumford [13] of stable vector bundles over projective algebraic curves, we define a stable vector bundle E over G/V to be a vector bundle such that any proper subbundle F has the property

A Schwarz lemma for bounded symmetric domains

The purpose of this note is to generalize Pick's invariant formulation of the classical Schwarz lemma. For the four classical types of bounded symmetric domains such a generalization was given by K. H. Look [3]; the present treatment will be independent of classification theory and will also include the exceptional Cartan domains. The results are also independent of the particular realization of the domain in Cn, they depend only on its structure as a hermitian manifold. The results will therefore be formulated for hermitian symmetric spaces of noncompact type; these are known to be in one-to-one correspondence with the holomorphic equivalence classes of bounded symmetric domains. We shall make use of Harish-Chandra's canonical realization of the hermitian symmetric spaces as bounded domains; this could perhaps be avoided, but it makes the proofs considerably simpler. In the following M=G/K will be a hermitian symmetric space of noncompact type; the identity coset will be denoted by po, 9 and 1 will denote the Lie algebras of G and K, respectively, and Ha, Ea, ■ ■ • will be a Weyl basis of q with respect to a Cartan subalgebra of Q contained in f. By a result of Harish-Chandra there exists a set A of strongly orthogonal roots of Q such that ct= ^„eA P(P«4-P-a) is a Cartan subalgebra of the symmetric pair (g, f). So every point pEM can be represented in the form p = kexp(^2aeAta(Ea-\-E-a)) -p0 with kEK, ta^0. For any p, qEM we denote by d(p, q) the distance of p and q in the metric induced by the hermitian structure of M. In any realization of M as a complex domain this is the Bergman metric. We denote by d*(p, q) the Caratheodory distance, which is defined by

Realization of Hermitian Symmetric Spaces as Generalized Half-planes

Realization of Hermitian Symmetric Spaces as Generalized Half-planes

Transformation groups on compact symmetric spaces

The symmetric spaces constitute the most important class of Riemannian manifolds; some of them have been the standard spaces in various branches of geometry, and many authors threw light upon their deep properties. Still there seems to be no thorough study of general transformation groups L (other than the isometry groups H, but containing G) of compact(2) symmetric spaces M, which we call geometric transformation groups of M in this introduction. Its need will be patent if it will reveal interrelations of symmetric spaces, and if L will be the automorphism group of some geometric structure of M, more or less closely related with the Riemannian structure of M, by which M is geometrically distinguished from the other symmetric spaces. Let us observe a few examples. Let M be the sphere as a symmetric space. M has the projective [respectively, conformal] structure; it can be thought of as the set of all geodesics [respectively, the function which gives the angles between two tangent vectors at the same points] of M. This can be defined, of course, for any Riemannian manifold, but the automorphism group L, or the projective [respectively, conformal] transformation group, differs from the isometry group G for the sphere M, and by this fact, M is distinguished from all other symmetric spaces (except the real projective space which is locally isometric with M), as asserted by E. Cartan [Oeuvres completes, Partie I, Vol. II, Gauthier-Villars, Paris, 1952, p. 659]. (See [7], [8] for the proof.) M is the standard space in the projective [respectively, conformal] differential geometry. And, 'a la F. Klein, this group L on M gives rise to the (real) projective [respectively, conformal or Moebius] geometry. Next, to observe another example, we select a compact hermitian symmetric space for M. The structure is the complex structure connected with the Riemannian metric in a certain way. The automorphism group L is the holomorphic transformation group. L is a complex Lie group whose complex structure essentially determines that of M.

On certain cohomology groups attached to Hermitian symmetric spaces. II

On certain cohomology groups attached to Hermitian symmetric spaces. II

On Kaehlerian Homogeneous Spaces of Unimodular Lie Groups

Recently the results of E. Cartan on Hermitian symmetric space have been extended to the case of Kaehlerian homogeneous spaces by several authors. In particular the structure of compact Kaehlerian homogeneous spaces has been fully exploited. As for the non-conmpact case, there are few known results except in the case where the groups are semi-simple or reductive. A. Borel [1] and J. L. Koszul [6] have studied the structure of Kaehlerian homogeneous spaces of semi-simple Lie groups and have shown, independently of each other, an interesting result that a bounded domain in a unitary space admitting a transitive semi-simple Lie group of complex analytic homeomorphisms is symmetric. This result gives a partial answer to the problem of E. Cartan. In his imiportant paper in 1935 [2], E. Cartan raised the question whether a bounded homogeneous domain is always a symmetric bounded domain. Y. Matsushima [8] has st-udied the structure of Kaehlerian homogeneous spaces of reductive Lie groups and has shown that these spaces are the direct produict of a locally flat Kaehlerian space and a Kaehlerian homogeneous space of a semi-simple Lie group. The purpose of the present paper is to study the structure of EKaehlerian homogeneous spaces of a more general class of Lie groups, that is, those of unimodular Lie groups. Specifically, we shall deal with the following two cases. First we shall consider the case where the isotropy group is semisimple and obtain the following two theorems:

Manifolds with nonnegative isotropic curvature

We prove that if (M-n, g), n >= 4, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature,then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature. (ii) (M, g) is isometric to a locally symmetric space. (iii) (M, g) is Kahler and biholomorphic to CPn/2. (iv) (M, g) is quaternionic-Kahler. This is implied by the following result: Let (M-2n, g) be a compact, locally irreducible Kahler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CPn or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of Brendle and Schoen on the Ricci flow.