Noncommutative Harmonic Analysis on Quantum Hyperbolic Spaces. The Laplace-Beltrami Operator

In this paper we study the Laplace-Beltrami operator on quantum complex hyperbolic spaces. We describe its action in terms of certain $q$-difference operators of second order and prove spectral theorems for these operators. The corresponding eigenfunctions are related to Al-Salam-Chihara polynomials. The obtained Plancherel measure is related to a quantum analog for the Harish-Chandra c-function.