Vanishing Cohomology of Dominant Line Bundles for Real Groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915552426983424 |
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| author | Cook, Jack A. |
| author_facet | Cook, Jack A. |
| contents | In \cite{Broer1993}, it was shown that certain line bundles on $\widetilde{\mathcal{N}}=T^*G/B$ have vanishing higher cohomology. We prove a generalization of this theorem for real reductive algebraic groups. More specifically, if $\mathcal{N}_θ$ denotes the cone of nilpotent elements in a Cartan subspace $\mathfrak{p},$ we have a similar construction of a resolution of singularities $\widetilde{\mathcal{N}_θ}.$ We prove that for a certain cone of weights $H^i(\widetilde{\mathcal{N}_θ},\mathcal{O}_{\widetilde{\mathcal{N}_θ}}(λ))=0$ for $i> 0.$ This follows by combining a simple calculation of the canonical bundle for $\widetilde{\mathcal{N}_θ}$ with Grauert-Riemenschneider vanishing. Restricting to the structure sheaf, we get a characterization of the singularities of the normalization of $\mathcal{N}_θ.$ We use this to show that for groups of QCT (Definition 2), $\mathbb{C}[\mathcal{N}_θ]$ is equivalent as a $K$-representation to a certain cohomologically induced module giving a new proof of a result in \cite{KostantRallis1971}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13473 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vanishing Cohomology of Dominant Line Bundles for Real Groups Cook, Jack A. Representation Theory Algebraic Geometry In \cite{Broer1993}, it was shown that certain line bundles on $\widetilde{\mathcal{N}}=T^*G/B$ have vanishing higher cohomology. We prove a generalization of this theorem for real reductive algebraic groups. More specifically, if $\mathcal{N}_θ$ denotes the cone of nilpotent elements in a Cartan subspace $\mathfrak{p},$ we have a similar construction of a resolution of singularities $\widetilde{\mathcal{N}_θ}.$ We prove that for a certain cone of weights $H^i(\widetilde{\mathcal{N}_θ},\mathcal{O}_{\widetilde{\mathcal{N}_θ}}(λ))=0$ for $i> 0.$ This follows by combining a simple calculation of the canonical bundle for $\widetilde{\mathcal{N}_θ}$ with Grauert-Riemenschneider vanishing. Restricting to the structure sheaf, we get a characterization of the singularities of the normalization of $\mathcal{N}_θ.$ We use this to show that for groups of QCT (Definition 2), $\mathbb{C}[\mathcal{N}_θ]$ is equivalent as a $K$-representation to a certain cohomologically induced module giving a new proof of a result in \cite{KostantRallis1971}. |
| title | Vanishing Cohomology of Dominant Line Bundles for Real Groups |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2509.13473 |