2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76538One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This allows us first to explain some strange feature pointed out by Gargoubi in his classification if the module of differential operators on the line. This also allows us to provide a so called sl(m+1,R) invariant symbol for differential operators acting on densities of non critical weights.LateX file, 20 pagesDifferential Geometry58B99 17B56 47E05text