A general Weyl-type Integration Formula for Isometric Group Actions

TL;DR

This paper introduces a unified Weyl-type integration formula for isometric group actions, reducing complex integrals over manifolds to simpler integrals over sections and homogeneous spaces, generalizing classical results and connecting to random matrix theory.

Contribution

It presents a general integration formula for isometric group actions, extending classical Weyl formulas and relating to polar actions and random matrix ensembles.

Findings

Derived a reduction of integrals over G-manifolds to sections and homogeneous spaces.

Connected the formula to classical Weyl integration and polar actions.

Established a reductive decomposition of Killing fields.

Abstract

We show that integration over a $G$-manifold $M$ can be reduced to integration over a minimal section $\Sigma$ with respect to an induced weighted measure and integration over a homogeneous space $G/N$. We relate our formula to integration formulae for polar actions and calculate some weight functions. In case of a compact Lie group acting on itself via conjugation, we obtain a classical result of Hermann Weyl. Our formula allows to view almost arbitrary isometric group actions as generalized random matrix ensembles. We also establish a reductive decomposition of Killing fields with respect to a minimal section.