Kolisnyk Y. Harmonic analysis on quantum symmetric spaces

The thesis concerns quantum bounded symmetric domains and quantum hyperbolic spaces. Problems of noncommutative harmonic analysis are solved in the thesis. 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 nxn-matrices is obtained. Also an explicit formula for the radial part of the Plancherel measure is obtained, and the the formula for expansion in spherical functions of an invariant function on the quantum matrix ball, the Fourier transform for the invariant functions are provided. $q$-Jacobi polynomials as spherical functions naturally arise on the way. An explicit formula for the radial part of the Laplace-Beltrami operator on the quantum hyperbolic space is obtained. The corresponding eigenfunctions can be expressed in terms of Al-Salam-Chihara polynomials.