  Let G be a complex Lie group, G_R a real form of G and X a G_R-stable domain
of holomorphy in a complex G-manifold. If there is a G_R-invariant strictly
plurisubharmonic function on X which has certain exhaustion properties, then we
show that the extended domain G.X is also a domain of holomorphy. As an
application we give a proof of the extended future tube conjecture. This is the
assertion that G.X is a domain of holomorphy in the case where X is the N-fold
product of the tube domain in C^4 over the positive light cone in R^4 and G is
the connected complex Lorentz group acting diagonally.
