Symplectic Dirac operators on Hermitian symmetric spaces

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.     

Isometries and Hermitian operators on complex symmetric sequence spaces

We consider a complex symmetric sequence space E that possesses the Fatou property and is different from l₂. We prove that, for every surjective linear isometry V on E, there exist λ _n ∈ ? with |λ _n | = 1 and a bijective mapping π on the set ? of natural numbers such that

$$V\left( {\left\{ {\xi _n } \right\}_{n \in \mathbb{N}} } \right) = \left\{ {\lambda _n \xi _{\pi (n)} } \right\}_{n \in \mathbb{N}}$$

for every {ξ _n {_n∈? ∈ E.

     

Hardy-type spaces for eigenfunctions of invariant differential operators on homogeneous line bundles over Hermitian symmetric spaces

We prove a Fatou-type theorem on a homogeneous line bundle over a Hermitian symmetric space, and characterize the range of the Poisson transform of L_p -functions on the maximal boundary as a Hardy-type space.     

Shimura invariant differential operators and their eigenvalues

Let be the covariant Cauchy-Riemann operator and the covariant holomorphic differential operator on a line bundle over a Hermitian symmetric space . We study the Shimura invariant differential operators defined via and . We find the eigenvalues of a family of the Shimura operators and of the generators. Received October 2, 1999 / Published online October 30, 2000     

Rotations and Hermitian symmetric spaces

On an almost Hermitian manifold (M, g, J) one considers the naturally defined field of local diffeomorphismsj _m =exp _m J _m exp _m ^–1 ,mM, and in particular, one studies isometric, harmonic, holomorphic and symplecticj _m . This leads to some characterizations of special classes of almost Hermitian manifolds, including the class of Hermitian symmetric spaces. In addition, one treats some intrinsic and extrinsic geometrical properties of geodesic spheres relating to these local diffeomorphisms.Supported by grant 203.01.50 of the C.N.R., Italy.     

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].     

Projective ranks of Hermitian symmetric spaces

This paper is a written version of the NAM William Claytor Lecture delivered at the 1991 Annual Meeting of the AMS-MAA-NAM, January 1991 in San Frandsco, CA.     

Equivariant embeddings of Hermitian symmetric spaces

We prove that equivariant, holomorphic embeddings of Hermitian symmetric spaces are totally geodesic (when the image is not of exceptional type).     

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. Received: 10 September 1996 / Revised version: 10 September 1997     

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 . We show that all such systems can be derived from a decomposition . Here gives the Laplace-Beltrami operator and 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 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.