Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free O_X-modules of rank n and vector bundles of rank n. So, equivalently, principal GL(n,C)-bundles are given by locally free sheaves of rank n.

So...what about other groups? I guess that SL(n,C) bundles are then locally free sheaves of rank n with top exterior power trivial, but can we phrase everything in terms of the properties of a sheaf and a group?

My guess is that in this context, if we can do it, we'll end up with something that's not quite locally free sheaves of rank n for GL(n,C), but which will be equivalent.

Note: I'm aware that we could just say something like "the sheaf of local sections of a G-bundle" but I'm looking for something intrinsic, a set of properties of the sheaf without reference to the geometric bundle, which can be reconstructed from the sheaf description.