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Maximal stable lattices in representations over discretely valued fields

Amit Ophir, Ariel Weiss

Abstract

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 $W_j$.

Maximal stable lattices in representations over discretely valued fields

Abstract

Let be an continuous irreducible representation of a compact group over a complete discretely valued field . Let be two irreducible subrepresentations of , the semisimplification of the residual representation. We study the structure of the -stable lattices with a view to understanding the question of when realises a non-split extension of by . In particular, we introduce the notion of a maximal -stable lattice and prove that any non-split extension of by that can be realised by 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 . On the other hand, we also show that, if the representations occur with multiplicity one in , then can realise at most one non-split extension of by .
Paper Structure (16 sections, 19 theorems, 27 equations)

This paper contains 16 sections, 19 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Maximal stable lattices
  3. Generalisations of Ribet's Lemma
  4. Uniqueness of extensions realised by $\rho$
  5. The geometry of maximal lattices
  6. Preliminaries
  7. The Bruhat--Tits building of $\mathop{\mathrm{PGL}}\nolimits(V)$
  8. The invariant complex of a representation
  9. Maximal lattices
  10. Existence of maximal vertices
  11. Sharp subquotients
  12. Sharp subquotients and maximal vertices
  13. Applications
  14. Uniqueness of extensions realised by modular representations
  15. On Bellaïche's generalisation of Ribet's Lemma
  16. ...and 1 more sections

Key Result

Theorem 1.3

For each $i = 1, \ldots r$, there exists a maximal lattice $\Lambda_i$ with $\mathop{\mathrm{soc}}\nolimits(\Lambda_i/\pi\Lambda_i)\simeq W_i$. For each $j = 1, \ldots r$, suppose that $W_j$ is in the $m$-th level of the socle filtration of $\Lambda_{i}/\pi\Lambda_{i}$. Then there is a directed path

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2: Bellaïche's graph of extensions
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Bruhat--Tits building of $\mathop{\mathrm{PGL}}\nolimits(V)$
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Bellaiche-apropos*Prop. 3.1.3
  • Proposition 2.5
  • proof
  • ...and 45 more