Adam Korányi

Function spaces and reproducing kernels on bounded symmetric domains

There are several natural Hilbert spaces of holomorphic functions on a bounded symmetric domain. Such are the Bergman-type spaces, on which the holomorphic discrete series operates, and the Hardy-type spaces which are related to the analytic continuation of the holomorphic discrete series. Also closely related is the Bergmann space of entire functions on the ambient @” which arises as the closure of the polynomials with respect to a natural inner product. The space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group K of the domain. The main facts about this decomposition were proved by Schmid [21]; for another proof see [22]. Each irreducible subspace contains a unique normalized L-invariant (“spherical”) polynomial, where L is the isotropy group of the Shilov boundary in K. Our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the Hilbert spaces considered. For the domains of classical type they were considered by Hua [9] and for some of the Hilbert spaces this was done before by Upmeier [24] using different methods; for certain others there are partial results in [22]. We are able to do this in a fairly simple unified way by making strong use of Gindikin’s generalized Gamma function [S]. Next we obtain a description of the reproducing kernels of the K-irreducible subspaces in each of our Hilbert spaces, and an expansion in terms of these for

H-type groups and Iwasawa decompositions☆

Since their introduction by A. Kaplan [Kpl] some ten years ago, generalised Heisenberg groups, also known as groups of Heisenberg type or H-type groups, have provided a framework in which to construct interesting examples in geometry and analysis (see, for instance, [C2], [Kp2], [Kp3], [KpR], [K2], [Rl], [R2], [TL], [TV]). The Iwasawa N-groups associated to all the real rank one simple Lie groups are H-type, so one has a convenient vehicle for studying these in a unified way: many problems on these simple Lie groups can be reduced to a problem on H-type groups, via the so-called noncompact picture, and often problems on H-type groups can be solved on all the groups of the family in one fell swoop (as in, for example, [CH], [CK], [DR]). Out of this approach to studying simple Lie groups several problems arise, such as why only some H-type groups correspond to simple Lie groups of real rank one. In this paper, we discuss various features of Iwasawa N-groups which distinguish them in the class of all H-type groups. We shall show that all H-type groups which possess certain geometric properties, clearly possessed by Iwasawa N-groups, satisfy a Lie-algebraic condition (implicit in the work of B. Kostant [Kt2]) that we shall call the J’-condition. We shall also use elementary Clifford algebra to classify the

Analysis and geometry on complex homogeneous domains

Part 1 Function spaces on complex semi-groups, Jacques Faraut: Hilbert spaces of holomorphic functions invariant cones and complex semi-groups positive unitary representations Hilbert function spaces on complex semi-groups Hilbert function spaces on SL(2,C) Hilbert function spaces on a complex semi-simple Lie group. Part 2 Graded Lie algebras and pseudo-hermitian symmetric spaces, Soji Kaneyuki: semi-simple graded Lie algebras symmetric R-spaces pseudo-hermitian symmetric spaces. Part 3 Function spaces on bounded symmetric domains, Adam Koranyi: Bergman kernel and Bergman metric symmetric domains and symmetric spaces construction of the hermitian symmetric spaces structure of symmetric domains the weighted Bergman spaces differential operators function spaces. Part 4 The heat kernels of non-compact symmetric spaces, Qi-keng Lu: introduction the Laplace-Beltrami operator in various co-ordinates the integral transformations the heat kernel of the hyperball Rr(m,n) the harmonic forms on the complex Grassmann manifold the horo-hypercircle coordinate of a complex hyperball the heat kernel of R11(m) the matrix representation of NIRGSS. Part 5 Jordan triple systems, Guy Ross: polynomial identities Jordan algebras the quasi-inverse the generic minimal polynomial tripotents and Pierce decomposition hermitian positive JTS further results and open problems. References.

An approach to symmetric spaces of rank one via groups of Heisenberg type

We give an elementary unified approach to rank one symmetric spaces of the noncompact type, including proofs of their basic properties and of their classification, with the development of a formalism to facilitate future computations.Our approach is based on the theory of Lie groups of H-type. An algebraic condition of H-type algebras, called J2,is crucial in the description of the symmetric spaces. The classification of H-type algebras satisfying J2 leads to a very simple description of the rank one symmetric spaces of the noncompact type.We also prove Kostant’s double transitive theorem; we describe explicitly the Riemannian metric of the space and the standard decompositions of its isometry group.Examples of the use of our theory include the description of the Poisson kernel and the admissible domains for convergence of Poisson integrals to the boundary.

New constructions of homogeneous operators

Abstract New examples of homogeneous operators involving infinitely many parameters are constructed. They are realized on Hilbert spaces of holomorphic functions with reproducing kernels which are computed explicitly. All the examples are irreducible and belong to the Cowen–Douglas class. Even though the construction is completely explicit, it is based on certain facts about Hermitian holomorphic homogeneous vector bundles. These facts also make possible a description of all homogeneous Cowen–Douglas operators, in a somewhat less explicit way. To cite this article: A. Koranyi, G. Misra, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous

Equivariant first order differential operators for parabolic geometries

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Admissible limit sets of discrete groups on symmetric spaces of rank one

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Poisson Transforms for Line Bundles from the Shilov Boundary to Bounded Symmetric Domains

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Some Applications of Gelfand Pairs in Classical Akalysis

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