Cocompact Subgroups of Semisimple Lie Groups

Lattices and parabolic subgroups are the obvious examples of cocompact subgroups of a connected, semisimple Lie group with finite center. We use an argument of C. C. Moore to show that every cocompact subgroup is, roughly speaking, a combination of these. We study a cocompact subgroup H of a connected, semisimple Lie group G with finite center. The case where H is discrete is very important and has been widely studied [6, 91, though it is not yet fully understood. Apparently, C. C. Moore [5] made the first (and, up to now, the only) assault on the case where H is not discrete: he treated the case where the identity component of H is nilpotent. In this paper, we modify Moore's argument to eliminate the assumption of nilpotence. Given G, we will explicitly describe what subgroups can arise as the identity component of a cocompact subgroup H. This essentially reduces the problem of finding all the cocompact subgroups to the problem of finding discrete cocompact subgroups. The identity component of H plays a key role in the proof of Moore's theorem [ 5 ] . Perhaps the only major difference between Moore's proof and ours is that we have selected a different subgroup-the unipotent radical of H-to play this key role. For the statement of the main theorem, it will be convenient to construct a refinement of the Langlands decomposition of a parabolic subgroup. Notation. For any Lie group X, we let X O be the identity component of X. Definition 1.1. Suppose P is a parabolic subgroup of a connected, semisimple Lie group G with finite center. Recall that P has a Langlands decomposition P = MAN [8, p. 811. Let L be the product of all the noncompact simple factors of MO, and let E be the maximal compact factor of MO. Then Po = LEAN; we call this the refined Langlands decomposition of Po . Main Theorem 1.2. Let G be a connected, semisimple Lie group with finite center, let P be any parabolic subgroup of G, and let Po = LEAN be the refined Langlands decomposition of Po . For any connected, normal subgroup X of L, and any connected, closed subgroup Y of EA, there is a closed, cocompact subgroup H of G such that (a) H is contained in P, and (b) HO = XYN. Conversely, given any closed, cocompact subgroup of H of G, there is a parabolic subgroup P and corresponding subgroups X and Y satisfying (a) and (b). 1980 Mathematics Subject Classification (1985 Revision). Primary 22E46. This paper is in final form and no version of it will be submitted for publication elsewhere. @ 1990 American Mathematical Society 0271-4132/90 $1.00 + $.25 per page