Berezin transform on the quantum unit ball

We introduce and study, in the framework of a theory of quantum Cartan domains, a q-analogue of the Berezin transform on the unit ball. We construct q-analogues of weighted Bergman spaces, Toeplitz operators and covariant symbol calculus. In studying the analytical properties of the Berezin transform we introduce also the q-analogue of the SU(n,1)-invariant Laplace operator (the Laplace-Beltrami operator) and present related results on harmonic analysis on the quantum ball. These are applied to obtain an analogue of one result by A.Unterberger and H.Upmeier. An explicit asymptotic formula expressing the q-Berezin transform via the q-Laplace-Beltrami operator is also derived. At the end of the paper, we give an application of our results to basic hypergeometric q-orthogonal polynomials.Comment: 38 pages, accepted by Journal of Mathematical Physic

Covariant q-differential operators and unitary highest weight representations for U_q su(n,n)

We investigate a one-parameter family of quantum Harish-Chandra modules of U_q sl(2n). This family is an analog of the holomorphic discrete series of representations of the group SU(n,n) for the quantum group U_q su(n, n). We introduce a q-analog of "the wave" operator (a determinant-type differential operator) and prove certain covariance property of its powers. This result is applied to the study of some quotients of the above-mentioned quantum Harish-Chandra modules. We also prove an analog of a known result by J.Faraut and A.Koranyi on the expansion of reproducing kernels which determines the analytic continuation of the holomorphic discrete series.Comment: 26 page