The moduli stack of principal $\rho$-sheaves and Gieseker-Harder-Narasimhan filtrations

Let X be a smooth projective variety and let G be a connected reductive group, both defined over a field of characteristic 0. Given a faithful representation $\rho$ of G into a product of general linear groups, we define a moduli stack of principal $\rho$-sheaves that compactifies the stack of G-bundles on X. We apply the theory developed by Alper, Halpern-Leistner and Heinloth to construct a moduli space of Gieseker semistable principal $\rho$-sheaves. This provides an intrinsic stack-theoretic construction of the moduli space of semistable singular principal bundles as constructed by Schmitt and G\'omez-Langer-Schmitt-Sols. Our second main result is the definition of a schematic Gieseker-Harder-Narasimhan filtration for $\rho$-sheaves, which induces a stratification of the stack by locally closed substacks. This filtration for a general reductive group G is a refinement of the canonical slope parabolic reductions previously considered at the level of points by Anchouche-Azad-Biswas and as a stratification of the stack by Gurjar-Nitsure. In an appendix, we apply the same techniques to define Gieseker-Harder-Narasimhan filtrations in arbitrary characteristic and show that they induce a stratification of the stack by radicial morphisms.