The weighted Bergman spaces and complex reflection groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866909690578862080 |
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| author | Ghosh, Gargi |
| author_facet | Ghosh, Gargi |
| contents | We consider a bounded domain $Ω\subseteq \mathbb C^d$ which is a $G$-space for a finite complex reflection group $G$. For each one-dimensional representation of the group $G,$ the relative invariant subspace of the weighted Bergman space on $Ω$ is isometrically isomorphic to a weighted Bergman space on the quotient domain $Ω/G.$ Consequently, formulae involving the weighted Bergman kernels and projections of $Ω$ and $Ω/G$ are established. As a result, a transformation rule for the weighted Bergman kernels under a proper holomorphic mapping with $G$ as its group of deck transformations is obtained in terms of the character of the sign representation of $G$. Explicit expressions for the weighted Bergman kernels of several quotient domains (of the form $Ω/ G$) have been deduced to demonstrate the merit of the described formulae. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_14162 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The weighted Bergman spaces and complex reflection groups Ghosh, Gargi Functional Analysis Complex Variables 32A25, 32A36, 32H35 We consider a bounded domain $Ω\subseteq \mathbb C^d$ which is a $G$-space for a finite complex reflection group $G$. For each one-dimensional representation of the group $G,$ the relative invariant subspace of the weighted Bergman space on $Ω$ is isometrically isomorphic to a weighted Bergman space on the quotient domain $Ω/G.$ Consequently, formulae involving the weighted Bergman kernels and projections of $Ω$ and $Ω/G$ are established. As a result, a transformation rule for the weighted Bergman kernels under a proper holomorphic mapping with $G$ as its group of deck transformations is obtained in terms of the character of the sign representation of $G$. Explicit expressions for the weighted Bergman kernels of several quotient domains (of the form $Ω/ G$) have been deduced to demonstrate the merit of the described formulae. |
| title | The weighted Bergman spaces and complex reflection groups |
| topic | Functional Analysis Complex Variables 32A25, 32A36, 32H35 |
| url | https://arxiv.org/abs/2104.14162 |