Covariant q-differential operators and unitary highest weight representations for Uq????su n,n

We investigate a one-parameter family of quantum Harish-Chandra modules of Uqsl2n. This family is an analog of the holomorphic discrete series of representations of the group SU(n,n) for the quantum group Uqsun,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.