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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2502.03883 |
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| _version_ | 1866916600517492736 |
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| author | Biswas, Shibananda Deb, Prahllad Hazra, Somnath Keshari, Dinesh Kumar Misra, Gadadhar |
| author_facet | Biswas, Shibananda Deb, Prahllad Hazra, Somnath Keshari, Dinesh Kumar Misra, Gadadhar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03883 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogeneous analytic Hilbert modules -- the case of non-transitive action Biswas, Shibananda Deb, Prahllad Hazra, Somnath Keshari, Dinesh Kumar Misra, Gadadhar Functional Analysis 47B32, 47B13, 47A13 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. |
| title | Homogeneous analytic Hilbert modules -- the case of non-transitive action |
| topic | Functional Analysis 47B32, 47B13, 47A13 |
| url | https://arxiv.org/abs/2502.03883 |