The role of Reynolds operator in GIT has always been a little mystery to me. Actually I see that in some proofs it gets used in an efficient way, but what I cannot grasp is the general philosophy. I see of course it is a projector onte the invariant subspace of a G-representation. But, as I asked, when do I want to use the R.O. and what for? I guess that one of the most important features is the Reynolds identity...