Plancherel formula for the quantum matrix ball - I

TL;DR

This paper develops a q-analog of the Plancherel formula for the spherical transform on the quantum matrix ball, providing an explicit radial measure and involving q-Jacobi polynomials as spherical functions.

Contribution

It introduces a novel q-analog of the classical Plancherel formula specifically for the quantum matrix ball, with explicit formulas and spherical functions.

Findings

Explicit formula for the radial part of the Plancherel measure

Identification of q-Jacobi polynomials as spherical functions

Advancement in harmonic analysis on quantum symmetric spaces

Abstract

Plancherel formula is one of the celebrated result of harmonic analysis on semisimple Lie groups and their homogeneous spaces. The main goal of this work is to find a q-analog of the Plancherel formula for spherical transform the unit matrix ball. Here we present an explicit formula for the radial part of the Plancherel measure. q-Jacobi polynomials as spherical functions naturally arise on the way.