Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group $SO_0(2,m)$
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| Format: | Preprint |
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2025
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| _version_ | 1866909618776571904 |
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| author | Pal, Ankita Paul, Pampa |
| author_facet | Pal, Ankita Paul, Pampa |
| contents | Let $G = SO_0(2,m),$ the connected component of the Lie group $SO(2,m);\ K = SO(2) \times SO(m),$ a maximal compact subgroup of $G;$ and $θ$ be the associated Cartan involution of $G.$ Let $X = G/K,\ \frak{g}_0$ be the Lie algebra of $G$ and $\frak{g} = \frak{g}_0^\mathbb{C}.$ In this article, we have considered the special cycles associated with all possible involutions of $G$ commuting with $θ.$ We have determined the special cycles which give non-zero cohomology classes in $H^*(Γ\backslash X; \mathbb{C})$ for some $θ$-stable torsion-free arithmetic uniform lattice $Γ$ in $G,$ by a result of Millson and Raghunathan. For each cohomologically induced representation $A_\frak{q}$ with trivial infinitesimal character, we have determined the special cycles for which the non-zero cohomology class has no $A_\frak{q}$-component, via Matsushima's isomorphism. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group $SO_0(2,m)$ Pal, Ankita Paul, Pampa Representation Theory 22E40, 22E46, 17B20, 17B40, 57S15 Let $G = SO_0(2,m),$ the connected component of the Lie group $SO(2,m);\ K = SO(2) \times SO(m),$ a maximal compact subgroup of $G;$ and $θ$ be the associated Cartan involution of $G.$ Let $X = G/K,\ \frak{g}_0$ be the Lie algebra of $G$ and $\frak{g} = \frak{g}_0^\mathbb{C}.$ In this article, we have considered the special cycles associated with all possible involutions of $G$ commuting with $θ.$ We have determined the special cycles which give non-zero cohomology classes in $H^*(Γ\backslash X; \mathbb{C})$ for some $θ$-stable torsion-free arithmetic uniform lattice $Γ$ in $G,$ by a result of Millson and Raghunathan. For each cohomologically induced representation $A_\frak{q}$ with trivial infinitesimal character, we have determined the special cycles for which the non-zero cohomology class has no $A_\frak{q}$-component, via Matsushima's isomorphism. |
| title | Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group $SO_0(2,m)$ |
| topic | Representation Theory 22E40, 22E46, 17B20, 17B40, 57S15 |
| url | https://arxiv.org/abs/2505.15583 |