Hilbert measures on orbit spaces of coregular $\operatorname{O}_m$-modules

TL;DR

This paper introduces Hilbert measures on orbit spaces of coregular orthogonal group representations, highlighting their singularities related to the number of defining representation copies.

Contribution

It constructs canonical Hilbert measures on orbit spaces of classical coregular O_m-modules and analyzes their singularity structure.

Findings

Hilbert measures are canonical on orbit spaces of coregular representations.

Singularities occur along non-principal strata when the number of defining representations equals m.

The measures' singularity behavior is characterized explicitly.

Abstract

We construct canonical measures, referred to as Hilbert measures, on orbit spaces of classical coregular representations of the orthogonal groups $\operatorname{O}_m$. We observe that the measures have singularities along non-principal strata of the orbit space if and only if the number of copies of the defining representation of $\operatorname{O}_m$ is equal to $m$.