Berezin kernels and analysis on Makarevich spaces

Abstract

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels.We consider the conformal group Conf(V ) of a simple real Jordan algebra V . The maximal degenerate representations π s (s ∈ C) we shall study are induced by a character of a maximal parabolic subgroupP of Conf(V ). These representations π s can be realized on a space I s of smooth functions on V . There is an invariant bilinear form B s on the space I s . The problem we consider is to diagonalize this bilinear form B s , with respect to the action of a symmetric subgroup G of the conformal group Conf(V ). This bilinear form can be written as an integral involving the Berezin kernel B ν , an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of B ν . From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity :where D(ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I −s to I s . Furthermore we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V ).2000 Mathematics Subject Classification. 43A35, 22E46.

Covariant quantization: spectral analysis versus deformation theory

“…[38][39][40]55] Reproducing kernel theory, which has its roots in classical complex analysis, was systematized by Aronszajn [11] and Bergman [19] in the context of Hilbert subspaces of functions and by L. Schwartz [145] in the more general setting of topological locally convex quasi-complete Hausdorff vector spaces. Once applied to the theory of unitary representations, these ideas allowed the treatment of spectral problems by means of spherical function-theoretic methods.…”

“…For example, one may see it as an integral transformation on a bounded symmetric domain with the kernel generalizing the cross-ratio [52,55], or as an invariant bilinear form on the representation space of the principal series of a conformal group [35,36,55,63]. This latter approach turns out to be efficient in solving the branching problem for representations of conformal groups.…”

“…In [38,55] we have explicitly calculated the spectrum of decomposition of canonical representations that turned out to be continuous. More precisely, the 262 M. Pevzner representations T m decompose multiplicity free in a direct integral of irreducible spherical unitary representations of H and the spectral measure of this decomposition is given by the following theorem.…”

“…An interesting phenomenon arises when one considers the same problem for the real compact form U of the conformal group G. It was already mentioned by Berezin, developed by Neretin [109] and Zhang [170] and generalized in [55] for every compact symmetric space dual of a Makarevich space of tube type , a space that is not Hermitian in general. More precisely, the spherical Fourier coefficients of the Berezin kernel on are given by…”

“…In [55] one shows that for κ ∈ N, the "compact" Berezin form B c s is positive-definite. Thus, the restriction of π m to the compact group U decomposes into a direct sum of principal spherical series representations Π m of U:…”