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

We consider the space $\mathcal C$ of conjugacy classes of the unitary group $\mathrm U(n+m)$ with respect to a smaller unitary group $\mathrm U(m)$. It is known that to every element of $\mathcal C$ 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 $\mathcal C$ obtained as the push-forward of the Haar measure of $\mathrm U(n+m)$ in terms of characteristic functions.