Harmonic Analysis on Quantum Complex Hyperbolic Spaces

TL;DR

This paper develops harmonic analysis on quantum complex hyperbolic spaces by introducing a quantum Laplace-Beltrami operator, analyzing its spectral properties, and establishing a Plancherel theorem, thus extending classical harmonic analysis to a quantum setting.

Contribution

It introduces a quantum analog of the Laplace-Beltrami operator and analyzes its spectral properties, including eigenfunctions related to Al-Salam-Chihara polynomials.

Findings

Eigenfunctions related to Al-Salam-Chihara polynomials

Second order q-difference operator identified

Plancherel theorem established for the quantum setting

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.