Cusp forms and parabolic cohomology classes for symmetric spaces of rank one
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918481269620736 |
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| author | Bruggeman, Roelof Choie, YoungJu Miatello, Roberto Pohl, Anke |
| author_facet | Bruggeman, Roelof Choie, YoungJu Miatello, Roberto Pohl, Anke |
| contents | For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $Γ$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the cusp forms of $Γ$. To that end, we identify certain $Γ$-submodules of smooth semi-analytic vectors in the spherical principal series representation with spectral parameter $ν$ as well as certain subspaces of parabolic cohomology spaces of $Γ$ of degree dim S-1 with these $Γ$-submodules. We provide explicit isomorphisms between the spaces of cusp forms of spectral parameter $ν$ and these specific cohomology subspaces. The isomorphisms from cusp forms to cohomology are given by an integral transform, and the explicit form of the inverse isomorphism takes advantage of a certain reproducing property of the integral transform. The result is uniform for all these symmetric spaces and does not rely on their classification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02599 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cusp forms and parabolic cohomology classes for symmetric spaces of rank one Bruggeman, Roelof Choie, YoungJu Miatello, Roberto Pohl, Anke Number Theory Primary: 11F75, 32N15, Secondary: 22E30, 22E40 For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $Γ$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the cusp forms of $Γ$. To that end, we identify certain $Γ$-submodules of smooth semi-analytic vectors in the spherical principal series representation with spectral parameter $ν$ as well as certain subspaces of parabolic cohomology spaces of $Γ$ of degree dim S-1 with these $Γ$-submodules. We provide explicit isomorphisms between the spaces of cusp forms of spectral parameter $ν$ and these specific cohomology subspaces. The isomorphisms from cusp forms to cohomology are given by an integral transform, and the explicit form of the inverse isomorphism takes advantage of a certain reproducing property of the integral transform. The result is uniform for all these symmetric spaces and does not rely on their classification. |
| title | Cusp forms and parabolic cohomology classes for symmetric spaces of rank one |
| topic | Number Theory Primary: 11F75, 32N15, Secondary: 22E30, 22E40 |
| url | https://arxiv.org/abs/2605.02599 |