Quantum Spheres for OSp_q(1/2)

Using the corepresentation of the quantum supergroup OSp_q(1/2) a general method for constructing noncommutative spaces covariant under its coaction is developed. In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed. It is observed that embedding of the supersphere in the OSp_q(1/2) algebra is possible. This realization admits infinitesimal characterization a la Koornwinder. A covariant oscillator realization of the supersphere is also presented.Comment: 30pages, no figure. to be published in J. Math. Phy

On a q-analog of the Wallach-Okounkov formula

We obtain a $q$-analog of the well known Wallach-Okounkov result on a joint spectrum of invariant differential operators with polynomial coefficients on a prehomogeneous vector space of complex $n \times n$-matrices. We are motivated by applications to the problems of harmonic analysis in the quantum matrix ball: our main theorem can be used while proving the Plancherel formula (to be published). This paper is dedicated to our friend and colleague Dmitry Shklyarov who celebrates his 30-th birthday on April 8, 2006.Comment: 10 pages, corrected minor misprint

Spherical principal series of quantum Harish-Chandra modules

The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmertic domains.Comment: 14 page

Intertwining Operators And Quantum Homogeneous Spaces

In the present paper the algebras of functions on quantum homogeneous spaces are studied. The author introduces the algebras of kernels of intertwining integral operators and constructs quantum analogues of the Poisson and Radon transforms for some quantum homogeneous spaces. Some applications and the relation to $q$-special functions are discussed.Comment: 20 pages. The general subject is quantum groups. The paper is written in LaTe