  Let G be a connected complex Lie group. We show that any flat principal
G-bundle over any finite CW-complex pulls back to a trivial bundle over some
finite covering space of the base space if and only if each real characteristic
class of positive degree of G vanishes. A third equivalent condition is that
the derived group of the radical of G is simply connected. As a corollary, the
same conditions are equivalent if G is a connected amenable Lie group. In
particular, if G is a connected compact Lie group then any flat principal
G-bundle over any finite CW-complex pulls back to a trivial bundle over some
finite covering space of the base space.
