Geometric realizations for some series of representations of the quantum group SU2,2

Анотація

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 2x2 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).