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

Consider the space $C$ of conjugacy classes of a unitary group $U(n+m)$ with respect to a smaller unitary group $U(m)$. It is known that for any element of the space $C$ we can assign canonically a matrix-valued rational function on the Riemann sphere (a Livshits characteristic function). In the paper we write an explicit expression for the natural measure on $C$ obtained as the pushforward of the Haar measure of the group $U(n+m)$ in the terms of characteristic functions.