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

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 bidisc ${\mathbb{G}}_2$ 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 bidisc is Kähler-Einstein.

Proposition 3.10 in the previous version was incorrect. Theorem 3.9 provides the corrected formulation of that proposition in this new version. Furthermore, Proposition 3.10 is new