Non-vanishing on the first cohomology

— It is shown here that, for any finitely generated lattice r in certain semi simple groups over local fields of positive characteristics, H (r. Ad) is nonvanishing; this is in sharp contrast with the situation in characteristic zero. Let Kbe a local field (i. e. a non-discrete locally compact field), and let G be a connected semi simple algebraic group defined over K. Let G = G (K), and let r = j^—rank G. The topology on K induces a locally compact Hausdorff topology on G; in the sequel, we assume G endowed with this topology. G is then a ^-analytic group. Let F be a lattice in G i.e., a discrete subgroup of G such that Gyr carries a finite Cr-invariant Borel measure. We assume that F is irreducible, i.e. no subgroup of r of finite index is a direct product of two infinite normal subgroups. In case K = R and G is not locally isomorphic to either SL (2, R) or SL (2, C), it is known that H 1 (T, Ad) = 0; where, as usual, Ad denotes the adjoint representation of G on its Lie algebra (see WEIL [9], [10] for uniform lattices; for non-uniform lattices in groups of R-rank > 1, this vanishing theorem follows from the results of RAGHUNATHAN [8], combined with the results of MARGULIS [4] on arithmeticity; for non-uniform lattices in groups of R-rank 1, it is contained in GARLAND-RAGHUNATHAN [2]). It is also known, in view of a recent result of MARGULIS ([5], theorem 8), that in case Kis non-archimedean but of characteristic zero, H (T, Ad) = 0 when r > 1. BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE