Stable lattices in p-adic representations III. Indecomposable reduction and holonomy

Abstract

Let G be a group, V a finite dimensional vector space over a complete discrete valuation field K, and ρ:G⟶GLK(V) an irreducible representation with bounded image (in the metric topology). First, we show that there are many stable lattices with indecomposable reduction, in the sense that their simplicial convex hull in the Bruhat-Tits building of GLK(V) is the set S(ρ) of stable norms.Then we investigate the reductions more closely, in case K has finite residue field and the simplicial set S(ρ) has dimension 2. When the reduction type (defined below) is (2,1), we show that there exists a stable lattice with indecomposable reduction that has at most 2 composition series (uniserial or biserial). When the reduction type is (3), we show that the number of v∈S(ρ)0 with semisimple reduction is bounded above by a constant times |S(ρ)0|.To prove these last two statements, we prove and apply an index theorem (of Gauss-Bonnet type).