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This paper contains a proof of a conjecture of Breuil and Schneider on the existence of an invariant norm on any locally algebraic representation of GL(n), with integral central character, whose smooth part is given by a generalized Steinberg representation. In fact, we prove the analogue for any connected reductive group G. This is done by passing to a global setting, using the trace formula for an R-anisotropic model of G. The ultimate norm comes from classical p-adic modular forms.

A proof of the Breuil-Schneider conjecture in the indecomposable case