Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a Riemann surface and constructed a projective moduli space of such objects. We generalize Ramanathan’s notion and construction to higher dimension, allowing also objects which we call semistable principal G-sheaves, in order to obtain a projective moduli space: a principal G-sheaf on a projective variety X is a triple (P,E,ψ), where E is a torsion free sheaf on X, P is a principal G-bundle on the open set U where E is locally free and ψ is an isomorphism between E|U and the vector bundle associated to P by the adjoint representation.
We say it is (semi)stable if all filtrations E∙ of E as sheaf of (Killing) orthogonal algebras, i.e. filtrations with E⊥i=E−i−1 and [Ei,Ej]⊂E∨∨i+j, have
where PEi is the Hilbert polynomial of Ei. After fixing the Chern classes of E and of the line bundles associated to the principal bundle P and characters of G, we obtain a projective moduli space of semistable principal G-sheaves. We prove that, in case dimX=1, our notion of (semi)stability is equivalent to Ramanathan’s notion.