Maximal stable lattices in representations over discretely valued fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916096870711296 |
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| author | Ophir, Amit Weiss, Ariel |
| author_facet | Ophir, Amit Weiss, Ariel |
| contents | Let $ρ\colon G\to \mathrm{GL}_n(K)$ be an continuous irreducible representation of a compact group over a complete discretely valued field $K$. Let $W_i,W_j$ be two irreducible subrepresentations of $\overlineρ^{ss}$, the semisimplification of the residual representation. We study the structure of the $G$-stable lattices $Λ\subseteq K^n$ with a view to understanding the question of when $ρ$ realises a non-split extension of $W_i$ by $W_j$.
In particular, we introduce the notion of a maximal $G$-stable lattice and prove that any non-split extension of $W_i$ by $W_j$ that can be realised by $ρ$ can also be realised by a maximal lattice. As applications, we give a new proof and a strengthening of Bellaïche's generalisation of Ribet's Lemma, which assures the abundancy of non-split extensions that can be realised by $ρ$. On the other hand, we also show that, if the representations $W_i, W_j$ occur with multiplicity one in $\overlineρ^{ss}$, then $ρ$ can realise at most one non-split extension of $W_i$ by $W_j$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal stable lattices in representations over discretely valued fields Ophir, Amit Weiss, Ariel Number Theory Group Theory Representation Theory 20G25, (11F70 20G05, 51E24) Let $ρ\colon G\to \mathrm{GL}_n(K)$ be an continuous irreducible representation of a compact group over a complete discretely valued field $K$. Let $W_i,W_j$ be two irreducible subrepresentations of $\overlineρ^{ss}$, the semisimplification of the residual representation. We study the structure of the $G$-stable lattices $Λ\subseteq K^n$ with a view to understanding the question of when $ρ$ realises a non-split extension of $W_i$ by $W_j$. In particular, we introduce the notion of a maximal $G$-stable lattice and prove that any non-split extension of $W_i$ by $W_j$ that can be realised by $ρ$ can also be realised by a maximal lattice. As applications, we give a new proof and a strengthening of Bellaïche's generalisation of Ribet's Lemma, which assures the abundancy of non-split extensions that can be realised by $ρ$. On the other hand, we also show that, if the representations $W_i, W_j$ occur with multiplicity one in $\overlineρ^{ss}$, then $ρ$ can realise at most one non-split extension of $W_i$ by $W_j$. |
| title | Maximal stable lattices in representations over discretely valued fields |
| topic | Number Theory Group Theory Representation Theory 20G25, (11F70 20G05, 51E24) |
| url | https://arxiv.org/abs/2401.10183 |