Plancherel measure for the quantum matrix ball-1

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