Actions of compact groups on coherent sheaves

We consider actions of compact real Lie Groups K on complex spaces X such t hat the associated reduced K-space admits a semistable quotient, e.g. X is a Stein space. We show that there is a complex space X-C endowed with a hol omorphic action of the universal complexification G of K that contains X as an open K-stable subset. As our main result, we prove that every coherent K-sheaf on X extends uniquely to a holomorphic G-sheaf on X-C.