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

TL;DR

This paper investigates the Laplace-Beltrami operator on quantum hyperbolic spaces, describing its spectral properties and eigenfunctions, and establishing connections to quantum analogs of classical harmonic analysis functions.

Contribution

It introduces a detailed analysis of the Laplace-Beltrami operator on quantum hyperbolic spaces, linking it to $q$-difference operators and quantum special functions.

Findings

Spectral theorems for the quantum Laplace-Beltrami operator

Eigenfunctions related to Al-Salam-Chihara polynomials

Quantum Plancherel measure connected to a quantum Harish-Chandra c-function

Abstract

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.