Homogeneous analytic Hilbert modules -- the case of non-transitive action

TL;DR

This paper studies homogeneous analytic Hilbert modules over polynomial rings on non-transitive G-spaces, showing that key invariants can be derived from a fundamental set and applying these results to the symmetrized bi-disc.

Contribution

It introduces methods to analyze invariants of homogeneous Hilbert modules without assuming transitivity, and applies these to the symmetrized bi-disc to derive geometric properties.

Findings

Unitary invariants can be determined from values on a fundamental set.

Homogeneity under the automorphism group of the symmetrized bi-disc is established.

Weighted Bergman metrics on the symmetrized bi-disc are not Kähler-Einstein.

Abstract

This work investigates analytic Hilbert modules $\mathcal{H}$, over the polynomial ring, consisting of holomorphic functions on a $G$-space $\Omega \subset \mathbb{C}^m$ that are homogeneous under the natural action of the group $G$. In a departure from the past studies of such questions, here we don't assume transitivity of the group action. The primary finding reveals that unitary invariants such as curvature and the reproducing kernel of a homogeneous analytic Hilbert module can be deduced from their values on a fundamental set $\Lambda$ of the group action. Next, utilizing these techniques, we examine the analytic Hilbert modules associated with the symmetrized bi-disc $\mathbb{G}_2$ and its homogeneity under the automorphism group of $\mathbb{G}_2$. It follows from one of our main theorems that none of the weighted Bergman metrics on the symmetrized bi-disc is K\"{a}hler-Einstein.