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

TL;DR

This paper derives an explicit formula for the natural measure on the space of conjugacy classes of a unitary group, expressed through matrix-valued rational functions, linking Haar measures to probability distributions.

Contribution

It provides a novel explicit expression for the measure on conjugacy classes of unitary groups in terms of Livshits characteristic functions.

Findings

Explicit formula for the measure on conjugacy classes

Connection between Haar measure and probability distributions

Representation of measures via rational matrix functions

Abstract

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.