Proper actions on reductive homogeneous spaces. (Actions propres sur les espaces homogènes réductifs.) (French) Zbl 0868.22013

Benoist, Yves

Ann. Math. (2) 144, No. 2, 315-347 (1996).

Let \(G\) be a linear semisimple real Lie group and let \(H\) be a reductive subgroup. This paper gives necessary and sufficient conditions for the existence of a nonabelian free subgroup \(\Gamma\) of \(G\) that acts properly on \(G/H\). In particular, the author shows that such a group \(\Gamma\) does not exist for \(SL(2n+1,R)/SL(2n,R)\). When \(\Gamma\) does not exist, it follows that \(G/H\) does not have a compact quotient (i.e., there is no discrete subgroup \(\Gamma\) acting properly on \(G/H\) such that \(\Gamma \backslash G/H\) is compact). It follows that \(SL(2n+1,R)/SL(2n,R)\) does not have a compact quotient, which has also been shown by Margulis. Moreover, the case of complex symmetric spaces is studied extensively. We state the main theorem more precisely. Let \(A_H\) be a maximal split abelian subgroup of \(H\) and let \(A\) be a maximal split abelian subgroup of \(G\) containing \(A_H.\) Let \(A^+\) be a Weyl chamber and let \(B=\{ x\in A^+ |w_0(x)=x^{-1} \}\), where \(w_0\) is the long element of the Weyl group of \(G.\) The main theorem asserts that there is a discrete subgroup \(\Gamma\) which does not contain an abelian subgroup of finite index acting properly on \(G/H\) if and only if \(wA_H\) does not contain \(B\) for any \(w\) in the Weyl group. Moreover, one may assume \(\Gamma\) acts freely on \(G/H\) and is Zariski dense in \(G.\) The condition is easy to check in examples. It is known that \(G/H\) contains an infinite discrete subgroup acting properly on \(G/H\) if and only if \(A_H\not= A,\) so we may as well assume \(A_H\not= A.\) Note that \(A^+=B\) whenever \(w_0(x)=x^{-1},\) which happens except for \(A_n, n\geq 2, D_{2n+1}, \) and \(E_6.\) When \(A^+=B\) and \(A_H\not= A,\) then of course, \(wA_H \) does not contain \(B\) for any \(w\) by reasons of dimension. The main technique of the proof is to study the Cartan projection of a discrete group onto the dominant Weyl chamber. To show that no \(\Gamma\) action exists in the main theorem, the author produces infinitely many elements in a subset of the positive Weyl chamber. The author deduces the results about compact quotients by showing that a quotient by a nilpotent discrete group cannot be compact via a reduction to the rank one case. The main results hold for groups over any local field of characteristic zero.

Reviewer: S.Evens (Tucson)

MSC:

22E15	General properties and structure of real Lie groups
43A85	Harmonic analysis on homogeneous spaces
53C30	Differential geometry of homogeneous manifolds
22E40	Discrete subgroups of Lie groups

Keywords:

proper action; semisimple real Lie group; complex symmetric spaces; Weyl chamber; Weyl group; Cartan projection