Genkai Zhang

Berezin transform on real bounded symmetric domains

Let D be a bounded symmetric domain in a complex vector space VC with a real form V and D = D∩V = G/K be the real bounded symmetric domain in the real vector space V . We construct the Berezin kernel and consider the Berezin transform on the L2-space on D. The corresponding representation of G is then unitarily equivalent to the restriction to G of a scalar holomorphic discrete series of holomorphic functions on D and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the L2-space. Introduction The main purpose of the present paper is to calculate the spectral symbol of the Berezin transform on real bounded symmetric domains. To explain our results and motivations we let D be the unit disk in the complex plane with the Lebesgue measure dm(z). We consider the weighted Bergman space H (ν > 1) of holomorphic functions on D square integrable with respect to the measure (1 − |z|2)ν−2dm(z). It has up to some positive constant the reproducing kernel Kw(z) = K(z, w) = (1 − zw)−ν . Moreover the group Gc = SU(1, 1) of fractional transformations of D acts on the space H via f(z) 7→ f(gz)g′(z) ν2 and it forms an irreducible unitary (projective) representation. Consider the subgroup SO(1, 1) consisting of transformations of the form z 7→ az+b bz+a with a, b ∈ R and a − b = 1. Thus it is of interest to study the irreducible decomposition of the weighted Bergman space under the subgroup G = SO(1, 1). For that purpose we consider the unit interval D = D ∩ R = (−1, 1) as a trivial symmetric space G/K = SO(1, 1)/{±1} and the restriction of holomorphic functions in H to the interval D. More precisely, consider R : H → C∞(D), Rf(x) = f(x)(1 − x) ν2 , x ∈ D. Let L(D, dμ0) be the L space on D with the SO(1, 1)-invariant measure dμ0(x) = dx (1−x2) , whose decomposition under SO(1, 1) can be done via the Mellin transform (see below). The restriction R is a bounded operator from H into the space L(D, dμ0) with dense image, and intertwines the respective actions of SO(1, 1), Received by the editors January 16, 2000 and, in revised form, October 10, 2000. 2000 Mathematics Subject Classification. Primary 22E46, 43A85, 32M15, 53C35.

Laplace and Segal–Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let D=G/K be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let DR=J∩D⊂D be its real form in a formally real Euclidean Jordan algebra J⊂V; DR=H/L is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal–Bargmann transform from a unitary G-space of holomorphic functions on D to an L2-space on DR. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to D of the spherical functions on DR and find their expansion in terms of the L-spherical polynomials on D, which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L2-space, the measure being the Harish–Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on D. Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.

Homogeneous multiplication operators on bounded symmetric domains

Let D=G/K be an irreducible bounded symmetric domain of dimension d and let Hν(D) be the analytic continuation of the weighted Bergman spaces of holomorphic functions on D. We consider the d-tuple M=(M1,…,Md) of multiplication operators by coordinate functions and consider its spectral properties. We find those parameters ν for which the tuple M is subnormal and we answer some open questions of Bagchi and Misra. In particular, we prove that when D=Bd is the unit ball in Cd, then Bd is a k-spectral set of M if and only if Hν(Bd) is the Hardy space or a weighted Bergman space.

TENSOR PRODUCTS OF ANALYTIC CONTINUATIONS OF HOLOMORPHIC DISCRETE SERIES

We give the irreducible decomposition of the tensor product of an an- alytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate. 0. Introduction. The work of Segal (IES) and Mautner (M) established the abstract Plancherel theorem for type I groups. This meant that for an arbitrary unitary represen- tation, one could find its spectral decomposition into irreducibles and a corresponding spectral measure. To make this program explicit on L 2 -spaces on homogeneous spaces is one of the main subjects of harmonic analysis. Another interesting case is that of decom- posing a tensor product of irreducible representations; our aim in this paper is to consider this for certain holomorphic representations. The problem of finding the irreducible decomposition of tensor products of holomor- phic discrete series of the group SL(2, ) has been studied by Repka (Re1). The results there were used by Howe (How) to give the decomposition of the metaplectic represen- tation for certain dual pairs. See also (OZ). For a general semisimple Lie group G of Hermitian type a similar problem is studied in (Re2). It is shown that the tensor prod- uct of a scalar holomorphic discrete series with its conjugate is unitarily equivalent to the L 2 -space on the corresponding Hermitian symmetric space, L 2 (G K). Therefore we know its decomposition from the known theory of Harish-Chandra; namely L 2 (G K) W H ( ) C( ) 2 d

Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2, R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L2 (R+,tα dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character Ψ↦ e−i(2n+α+1)Ψ; under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.

Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions includes the Szego and the Bergman kernel on the symmetrized polydisc.

Tensor products of Minimal Holomorphic Representations

Let D = G/K be an irreducible bounded symmetric domain with genus p and Hν(D) the weighted Bergman spaces of holomorphic functions for ν > p − 1. The spaces Hν(D) form unitary (projective) representations of the group G and have analytic continuation in ν; they give also unitary representations when ν in the Wallach set, which consists of a continuous part and a discrete part of r points. The first non-trivial discrete point ν = a 2 gives the minimal highest weight representation of G. We give the irreducible decomposition of tensor product H a 2 ⊗ H a 2 . As a consequence we discover some new spherical unitary representations of G and find the expansion of the corresponding spherical functions in terms of the K-invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials. Introduction Let D = G/K be an irreducible bounded symmetric domain of rank r in a complex vector space V with Lebesgue measure dm(z). The Bergman reproducing kernel ofD is of the form h(z, w)−p, where p is the genus ofD. Let ν > p−1 and consider the weighted Bergman space H with the weighted measure h(z, z)ν−pdm(z). They give naturally unitary representations of the group G and have analytic continuation in the parameter ν. The set of those ν for which H still form unitary representations is called the Wallach set and has been determined by various methods ([25], [30] and [5]). It is a union of an open interval and a discrete set, the last point in the discrete set is ν = 0 and corresponds to the trivial representation. Suppose that the rank r of D is bigger than 1. The other points in the discrete Wallach set correspond to some singular representations of G; the K-types appearing in the representations form some lower dimensional lattices. The first discrete point ν = a2 above the trivial point ν = 0 gives the minimal representation and the lattice of K-types is one-dimensional. Minimal and singular representations are of considerable interest since they normally cannot be constructed by standard methods. One may well expect that the representations appearing in the tensor product decomposition are also some minimal (singular) representations, thus it is worthwhile to study. Indeed we discover some new irreducible unitary (minimal) representations that appear in the decomposition. We also find the annihilating Received by the editors May 23, 2000 and, in revised form, April 10, 2001. 2000 Mathematics Subject Classification. Primary 22E46, 47A70, 32M15, 33C52.

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE

We extend the spectral analysis of differential forms on the disk (viewed as the non-Euclidean plane) in recent work by J. Peetre L. Peng G. Zhang to the dual situation of the Riemann sphere S2. In particular, we determine a concrete orthogonal base in the relevant Hilbert space Lν,2(S2), where −ν2-is the degree of the form, a section of a certain holomorphic line bundle over the sphere S2. It turns out that the eigenvalue problem of the corresponding invariant Laplacean is equivalent to an infinite system of one dimensional Schrödinger operators. They correspond to the Morse potential in the case of the disk. In the course of the discussion many special functions (hypergeometric functions, orthogonal polynomials etc.) come up. We give also an application to “Ha-plitz” theory.

Radon transform on real, complex, and quaternionic Grassmannians

LetGn,k(K) be the Grassmannian manifold of k-dimensional K-subspaces in K where K = R,C,H is the field of real, complex or quaternionic numbers. For 1 ≤ k ≤ k ≤ n − 1 we define the Radon transform (Rf)(η), η ∈ Gn,k′(K), for functions f(ξ) on Gn,k(K) as an integration over all ξ ⊂ η. When k+ k ≤ n we give an inversion formula in terms of the Garding-Gindikin fractional integration and the Cayley type differential operator on the symmetric cone of positive k × k matrices over K. This generalizes the recent results of Grinberg-Rubin for real Grassmannians.

Shimura invariant differential operators and their eigenvalues

Abstract. Let