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| Natura: | Preprint |
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2022
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| Accesso online: | https://arxiv.org/abs/2210.01381 |
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| _version_ | 1866915701895200768 |
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| author | Qian, Zicheng |
| author_facet | Qian, Zicheng |
| contents | Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute the higher $\mathrm{Ext}$-groups between locally analytic generalized Steinberg representations (LAGS for short) of $\mathrm{GL}_n(K)$ via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each $\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their $\mathrm{Ext}$-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an $\mathrm{Ext}$-group, and then describe the cup product maps between such $\mathrm{Ext}$-groups using these bases. As an application, we generalize Breuil's $\mathscr{L}$-invariants for $\mathrm{GL}_2(\mathbb{Q}_p)$ and Schraen's higher $\mathscr{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ to $\mathrm{GL}_n(K)$. Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for $\mathrm{GL}_3(\mathbb{Q}_p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_01381 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On generalization of Breuil--Schraen's $\mathscr{L}$-invariants to $\mathrm{GL}_n$ Qian, Zicheng Number Theory Representation Theory Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute the higher $\mathrm{Ext}$-groups between locally analytic generalized Steinberg representations (LAGS for short) of $\mathrm{GL}_n(K)$ via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each $\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their $\mathrm{Ext}$-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an $\mathrm{Ext}$-group, and then describe the cup product maps between such $\mathrm{Ext}$-groups using these bases. As an application, we generalize Breuil's $\mathscr{L}$-invariants for $\mathrm{GL}_2(\mathbb{Q}_p)$ and Schraen's higher $\mathscr{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ to $\mathrm{GL}_n(K)$. Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for $\mathrm{GL}_3(\mathbb{Q}_p)$. |
| title | On generalization of Breuil--Schraen's $\mathscr{L}$-invariants to $\mathrm{GL}_n$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2210.01381 |