Homological vanishing theorems for locally analytic representations

Let G be the group of rational points of a split connected reductive group over a p-adic local field, and let Γ be a discrete and cocompact subgroup of G. Motivated by questions on the cohomology of p-adic symmetric spaces, we investigate the homology of Γ with coefficients in locally analytic principal series and related representations of G. The vanishing and finiteness results that we find partially rely on the compactness of certain Banach–Hecke operators. We also give a new construction of P. Schneider’s reduced Hodge–de Rham spectral sequence and show that the induced filtration is the Hodge–de Rham filtration. In a previously unknown case, our vanishing theorems then also imply two other of P. Schneider’s conjectures.