On the First Cohomology of Discrete Subgroups of Semi-Simple Lie Groups

Abstract

Introduction. Let G be a connected semi-simple Lie group and r a discrete subgroup such that the quotient G/r is compact. Let po be a finite dimensional representation of G. Our aim in this paper, is to show that for a large class of representations po, the first cohomology group of r with coefficients in the representation pr (the restriction of po to r) is zero (for a precise statement, see Theorem 1). Our results say in particular that if po does not contain the trivial representation of G and if no simple component of G is compact or locally isomorphic to SO, (n, 1) or SU (n, 1), then this first cohomology group vanishes. Even if G has components locally isomorphic to SOo (n, 1) or SU (n, 1) we give a sufficient condition for the vanishing of the cohomology in terms of the highest weights of the irreducible components of the complexification p of po. The importance of these cohomology groups arises from the role they play in deformation theory; for instance, when poO is the adjoint representation, these cohomology groups are intimately connected with the theory of deformations of discrete subgroups of Lie groups [6]. It has been proved essentially by A. Weil [8] (see also [5] and [6]) that when po is the adjoint representation, this cohomology group vanishes if G has no compact or three dimensional components. This result is a special case of our theorem (see Corollary 1 to Theorem 1). The case of trivial representations has been treated by Y. Matsushima in [4].