Cohomology of $p$-adic Chevalley groups
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909694693474304 |
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| author | Dotto, Andrea Hung, Bao V. Le |
| author_facet | Dotto, Andrea Hung, Bao V. Le |
| contents | Let $G$ be a split connected reductive group over the ring of integers of a finite unramified extension $K$ of $\mathbf{Q}_p$. Under a standard assumption on the Coxeter number of $G$, we compute the cohomology algebra of $G(\mathcal{O}_K)$ and its Iwahori subgroups, with coefficients in the residue field of $K$. Our methods involve a new presentation of some graded Lie algebras appearing in Lazard's theory of saturated $p$-valued groups, and a reduction to coherent cohomology of the flag variety in positive characteristic. We also consider the case of those inner forms of $\mathrm{GL}_n(K)$ that give rise to the Morava stabilizer groups in stable homotopy theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomology of $p$-adic Chevalley groups Dotto, Andrea Hung, Bao V. Le Number Theory Representation Theory Let $G$ be a split connected reductive group over the ring of integers of a finite unramified extension $K$ of $\mathbf{Q}_p$. Under a standard assumption on the Coxeter number of $G$, we compute the cohomology algebra of $G(\mathcal{O}_K)$ and its Iwahori subgroups, with coefficients in the residue field of $K$. Our methods involve a new presentation of some graded Lie algebras appearing in Lazard's theory of saturated $p$-valued groups, and a reduction to coherent cohomology of the flag variety in positive characteristic. We also consider the case of those inner forms of $\mathrm{GL}_n(K)$ that give rise to the Morava stabilizer groups in stable homotopy theory. |
| title | Cohomology of $p$-adic Chevalley groups |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2507.13500 |