Towards the wall-crossing of locally $\mathbb{Q}_p$-analytic representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913307401650176 |
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| author | Ding, Yiwen |
| author_facet | Ding, Yiwen |
| contents | Let $K$ be a finite extension of $\mathbb{Q}_p$. We study the locally $\mathbb{Q}_p$-analytic representations $π$ of $\mathrm{GL}_n(K)$ of integral weights that appear in spaces of $p$-adic automorphic representations. We conjecture that the translation of $π$ to the singular block has an internal structure which is compatible with certain algebraic representations of $\mathrm{GL}_n$, analogously to the mod $p$ local-global compatibility conjecture of Breuil-Herzig-Hu-Morra-Schraen. We next make some conjectures and speculations on the wall-crossings of $π$. In particular, when $π$ is associated to a two dimensional de Rham Galois representation, we make conjectures and speculations on the relation between the Hodge filtrations of $ρ$ and the wall-crossings of $π$, which have a flavour of the Breuil-Strauch conjecture. We collect some results towards the conjectures and speculations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06315 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards the wall-crossing of locally $\mathbb{Q}_p$-analytic representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$ Ding, Yiwen Number Theory Representation Theory Let $K$ be a finite extension of $\mathbb{Q}_p$. We study the locally $\mathbb{Q}_p$-analytic representations $π$ of $\mathrm{GL}_n(K)$ of integral weights that appear in spaces of $p$-adic automorphic representations. We conjecture that the translation of $π$ to the singular block has an internal structure which is compatible with certain algebraic representations of $\mathrm{GL}_n$, analogously to the mod $p$ local-global compatibility conjecture of Breuil-Herzig-Hu-Morra-Schraen. We next make some conjectures and speculations on the wall-crossings of $π$. In particular, when $π$ is associated to a two dimensional de Rham Galois representation, we make conjectures and speculations on the relation between the Hodge filtrations of $ρ$ and the wall-crossings of $π$, which have a flavour of the Breuil-Strauch conjecture. We collect some results towards the conjectures and speculations. |
| title | Towards the wall-crossing of locally $\mathbb{Q}_p$-analytic representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2404.06315 |