Maximal stable lattices in representations over discretely valued fields

Abstract

Let $\rho :G\to {\mathrm{GL}}_n(K)$ 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 ${\overline{\rho }}^{ss}$, the semisimplification of the residual representation. We study the structure of the $G$-stable lattices $\Lambda \subseteq K^n$ with a view to understanding the question of when $\rho$ 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 $\rho$ 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 $\rho$. On the other hand, we also show that, if the representations $W_i,W_j$ occur with multiplicity one in ${\overline{\rho }}^{ss}$, then $\rho$ can realise at most one non-split extension of $W_i$ by $W_j$.

13 pages, comments welcome!