Bershtein O. Principal series of representations of noncompact quantum groups

Quantum bounded symmetric domains. Quantum analogs of main problems of real reductive Lie groups representation theory. The methods include methods of geometrical representation theory, theory of operators in Hilbert spaces and theory of holomorphic vector bundles. A principal degenerate series of quantum Harish-Chandra modules related to the Shilov boundary of the unit ball in the space of complex nxn-matrices is introduced and studied. A quantum analog of the unit ball in the space of symmetric complex nxn-matrices is studied. Joint spectrum of quantum invariant differential operators in the space of complex nxn-matrices is found. A scalar spherical principal non-degenerate series of quantum Harish-Chandra modules is obtained. The results are important for further investigations in quantum group theory, noncommutative geometry and in studying of exactly solvable models of quantum physics