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

We construct canonical measures, referred to as Hilbert measures, on orbit spaces of classical coregular representations of the orthogonal groups ${\mathrm{O}}_m$. 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 ${\mathrm{O}}_m$ is equal to $m$.