Construction and non-vanishing of a family of vector-valued Siegel Poincaré series
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916236213878784 |
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| author | Žunar, Sonja |
| author_facet | Žunar, Sonja |
| contents | Using Poincaré series of $ K $-finite matrix coefficients of integrable antiholomorphic discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_ρ(Γ) $ of Siegel cusp forms of weight $ ρ$ for $ Γ$, where $ ρ$ is an irreducible polynomial representation of $ \mathrm{GL}_n(\mathbb C) $ of highest weight $ ω\in\mathbb Z^n $ with $ ω_1\geq\ldots\geqω_n>2n $, and $ Γ$ is a discrete subgroup of $ \mathrm{Sp}_{2n}(\mathbb R) $ commensurable with $ \mathrm{Sp}_{2n}(\mathbb Z) $. Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction and non-vanishing of a family of vector-valued Siegel Poincaré series Žunar, Sonja Number Theory Representation Theory 11F03, 11F46 Using Poincaré series of $ K $-finite matrix coefficients of integrable antiholomorphic discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_ρ(Γ) $ of Siegel cusp forms of weight $ ρ$ for $ Γ$, where $ ρ$ is an irreducible polynomial representation of $ \mathrm{GL}_n(\mathbb C) $ of highest weight $ ω\in\mathbb Z^n $ with $ ω_1\geq\ldots\geqω_n>2n $, and $ Γ$ is a discrete subgroup of $ \mathrm{Sp}_{2n}(\mathbb R) $ commensurable with $ \mathrm{Sp}_{2n}(\mathbb Z) $. Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series. |
| title | Construction and non-vanishing of a family of vector-valued Siegel Poincaré series |
| topic | Number Theory Representation Theory 11F03, 11F46 |
| url | https://arxiv.org/abs/2405.02966 |