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

Shibananda Biswas, Prahllad Deb, Somnath Hazra, Dinesh Kumar Keshari, Gadadhar Misra

TL;DR

The paper develops a curvature- and kernel-based framework to classify $G$-homogeneous analytic Hilbert modules when the group action is not transitive, showing that unitary invariants can be recovered from values on a fundamental set $\Lambda$. It combines kernel quasi-invariance and curvature transformation with Wallach-set based constructions to produce new Aut$(\mathbb{G}_2)$-homogeneous modules beyond weighted Bergman spaces, and proves that none of the weighted Bergman metrics on the symmetrized bidisc $\mathbb{G}_2$ are Kähler-Einstein. The main contributions include a practical curvature criterion for homogeneity, explicit families of homogeneous modules on $\mathbb{G}_2$ (such as $\mathbb{A}^{(\lambda,\nu)}$ and $\mathbb{H}^{(\lambda,\nu)}$), and rigorous inequivalence results across and within these families. These results significantly broaden the landscape of homogeneous analytic Hilbert modules and provide concrete tools for distinguishing unitary classes via curvature determinants and fundamental-set data.

Abstract

This work investigates analytic Hilbert modules $\mathcal{H}$, over the polynomial ring, consisting of holomorphic functions on a $G$-space $Ω\subset \mathbb{C}^m$ 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 $Λ$ 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 Kähler-Einstein.

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

TL;DR

The paper develops a curvature- and kernel-based framework to classify -homogeneous analytic Hilbert modules when the group action is not transitive, showing that unitary invariants can be recovered from values on a fundamental set . It combines kernel quasi-invariance and curvature transformation with Wallach-set based constructions to produce new Aut-homogeneous modules beyond weighted Bergman spaces, and proves that none of the weighted Bergman metrics on the symmetrized bidisc are Kähler-Einstein. The main contributions include a practical curvature criterion for homogeneity, explicit families of homogeneous modules on (such as and ), and rigorous inequivalence results across and within these families. These results significantly broaden the landscape of homogeneous analytic Hilbert modules and provide concrete tools for distinguishing unitary classes via curvature determinants and fundamental-set data.

Abstract

This work investigates analytic Hilbert modules , over the polynomial ring, consisting of holomorphic functions on a -space that are homogeneous under the natural action of the group . 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 of the group action. Next, utilizing these techniques, we examine the analytic Hilbert modules associated with the symmetrized bi-disc and its homogeneity under the automorphism group of . It follows from one of our main theorems that none of the weighted Bergman metrics on the symmetrized bi-disc is Kähler-Einstein.

Paper Structure

This paper contains 8 sections, 30 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Open Questions
  3. Curvature criterion for homogeneous modules
  4. Homogeneous analytic Hilbert modules over symmetrized bidisc
  5. Weighted Bergman modules LG
  6. Hilbert modules TT over AA
  7. The Hilbert modules AB over BB
  8. Inequivalence

Key Result

Theorem 2.1

Let $\Omega\subseteq \mathbb{C}^m$ be a $G$-space, where $G$ is a subgroup of the bi-holomorphic automorphism group of $\Omega$. Let $(\mathcal{H}, K)$ be an analytic Hilbert module over the polynomial ring $\mathbb{C}[\boldsymbol{z}]$ consisting of holomorphic functions defined on $\Omega$. The com for all $g\in G$.

Theorems & Definitions (64)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6: Cowen-Douglas
  • ...and 54 more