EXISTENCE OF LATTICES IN KAC–MOODY GROUPS OVER FINITE FIELDS

Abstract

Let [Formula: see text] be a Kac–Moody Lie algebra. We give an interpretation of Tits' associated group functor using representation theory of [Formula: see text] and we construct a locally compact "Kac–Moody group" G over a finite field k. Using (twin) BN-pairs (G,B,N) and (G,B-,N) for G we show that if k is "sufficiently large", then the subgroup B- is a non-uniform lattice in G. We have also constructed an uncountably infinite family of both uniform and non-uniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of non-uniform lattices in rank 2 is a spherical Tits system for G which we also construct.