  Let G = SL(n,R) (or, more generally, let G be a connected, noncompact, simple
Lie group). For any compact Lie group K, it is easy to find a compact manifold
M, such that there is a volume-preserving, connection-preserving, ergodic
action of G on some smooth, principal K-bundle P over M. Can M can be chosen
independent of K? We show that if M = H/L is a homogeneous space, and the
action of G on M is by translations, then P must also be a homogeneous space
H'/L'. Consequently, there is a strong restriction on the groups K that can
arise over this particular M.
