Geometric realizations for some series of representations of the quantum group $SU_{2,2}$

The paper solves the problem of analytic continuation for the holomorphic discrete series of representations for the quantum group $SU(2,2)$. In particular, a new realization of the ladder representation of this group is produced. Besides, $q$-analogues are constructed for the Shilov boundary of the unit ball in the space of complex $2\times 2$ matrices and the principal degenerate series representations of $SU(2,2)$ associated to that boundary. A possibility is discussed of transferring some well known geometric constructions of the representation theory to the quantum case: the Penrose transform, the Beilinson–Bernstein approach to the construction of Harish–Chandra modules (for the case of the principal nondegenerate series).