Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 078, 19 pages      arXiv:1108.3357      https://doi.org/10.3842/SIGMA.2011.078
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Harmonic Analysis on Quantum Complex Hyperbolic Spaces

Olga Bershtein and Yevgen Kolisnyk
Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61103, Kharkov, Ukraine

Received April 30, 2011, in final form August 10, 2011; Published online August 18, 2011

Abstract
In this paper we obtain some results of harmonic analysis on quantum complex hyperbolic spaces. We introduce a quantum analog for the Laplace-Beltrami operator and its radial part. The latter appear to be second order q-difference operator, whose eigenfunctions are related to the Al-Salam-Chihara polynomials. We prove a Plancherel type theorem for it.

Key words: quantum groups, harmonic analysis on quantum symmetric spaces; q-difference operators; Al-Salam-Chihara polynomials; Plancherel formula.

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