Title: Steinberg representations and harmonic cochains for split adjoint quasi-simple groups,
Abstract: Let $G$ be an adjoint quasi-simple group defined and split over a
non-archimedean local field $K$. We prove that the dual of the Steinberg
representation of $G$ is isomorphic to a certain space of harmonic cochains on
the Bruhat-Tits building of $G$. The Steinberg representation is considered
with coefficients in any commutative ring.