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

TL;DR

This work constructs canonical Hilbert measures on orbit spaces $V_k/\,O_m$ for coregular representations, using a Hilbert embedding $u:V_k\to\mathbb{R}^{\binom{k+1}{2}}$ built from Gram invariants $u_{i,j}$ and the Gram matrix $G_k$. The authors derive an explicit invariant density $\lambda_{k,m}(u)$, showing it factors into a Gram-determinant weight $|G_k|^{(m-k-1)/2}$ and a product of sphere volumes, with singularities only at the non-principal strata when $k=m$. The proof hinges on a two-step change of variables: (i) selecting orbit representatives in a fundamental domain via generalized Euler-angle rotations, and (ii) transforming to invariants $u_{i,j}$, carefully accounting for all Jacobians. The paper also provides concrete two-particle examples in dimensions $3$ and $2$ that yield explicit constants and a description of the Hilbert image $X$, illustrating how the theory can be applied to compute invariant integrals in practical settings.

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 $m$.

Theorems & Definitions (12)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3: SchwarzProcesi
• proof
• Example 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• Lemma 3.1
• proof