Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions

Abstract

We consider the space of conjugacy classes of the unitary group with respect to a smaller unitary group . It is known that to every element of we can canonically assign a rational matrix-valued function (the Livshits characteristic function) on the Riemann sphere. We find an explicit expression for the natural measure on obtained as the push-forward of the Haar measure of in terms of characteristic functions.