Finiteness properties of generalized Montréal functors with applications to mod $p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$
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| Format: | Preprint |
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2025
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| _version_ | 1866917918423384064 |
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| author | Jakovác, Gergely Zábrádi, Gergely |
| author_facet | Jakovác, Gergely Zábrádi, Gergely |
| contents | The second named author previously constructed a functor $\mathbb{V}^\vee\circ D^\vee_Δ$ from the category of smooth $p$-power torsion representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ to the category of inductive limits of continuous representations on finite $p$-primary abelian groups of the direct product $G_{\mathbb{Q}_p,Δ}\times \mathbb{Q}_p^\times$ of $(n-1)$ copies of the absolute Galois group of $\mathbb{Q}_p$ and one copy of the multiplicative group $\mathbb{Q}_p^\times$. In the present work we show that this functor attaches finite dimensional representations on the Galois side to smooth $p$-power torsion representations of finite length on the automorphic side. This has some implications on the finiteness properties of Breuil's functor, too. Moreover, $\mathbb{V}^\vee\circ D^\vee_Δ$ produces irreducible representations of $G_{\mathbb{Q}_p,Δ}\times \mathbb{Q}_p^\times$ when applied to irreducible objects on the automorphic side and detects isomorphisms unless it vanishes. Further, we determine the kernel of $D^\vee_Δ$ when restricted to successive extensions of subquotients of principal series. We use this to characterize representations that are parabolically induced from the product of a torus and $\mathrm{GL}_2(\mathbb{Q}_p)$. Finally, we formulate a conjecture and prove partial results on the essential image. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness properties of generalized Montréal functors with applications to mod $p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ Jakovác, Gergely Zábrádi, Gergely Number Theory Representation Theory 11S37, 11F70, 11F80, 11F85, 22E50 The second named author previously constructed a functor $\mathbb{V}^\vee\circ D^\vee_Δ$ from the category of smooth $p$-power torsion representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ to the category of inductive limits of continuous representations on finite $p$-primary abelian groups of the direct product $G_{\mathbb{Q}_p,Δ}\times \mathbb{Q}_p^\times$ of $(n-1)$ copies of the absolute Galois group of $\mathbb{Q}_p$ and one copy of the multiplicative group $\mathbb{Q}_p^\times$. In the present work we show that this functor attaches finite dimensional representations on the Galois side to smooth $p$-power torsion representations of finite length on the automorphic side. This has some implications on the finiteness properties of Breuil's functor, too. Moreover, $\mathbb{V}^\vee\circ D^\vee_Δ$ produces irreducible representations of $G_{\mathbb{Q}_p,Δ}\times \mathbb{Q}_p^\times$ when applied to irreducible objects on the automorphic side and detects isomorphisms unless it vanishes. Further, we determine the kernel of $D^\vee_Δ$ when restricted to successive extensions of subquotients of principal series. We use this to characterize representations that are parabolically induced from the product of a torus and $\mathrm{GL}_2(\mathbb{Q}_p)$. Finally, we formulate a conjecture and prove partial results on the essential image. |
| title | Finiteness properties of generalized Montréal functors with applications to mod $p$ representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ |
| topic | Number Theory Representation Theory 11S37, 11F70, 11F80, 11F85, 22E50 |
| url | https://arxiv.org/abs/2501.18396 |