Parabolic reductions of principal bundles

In this paper we describe the structure of the space of parabolic reductions, and their compactifications, of principal $G$-bundles over a smooth projective curve over an algebraically closed field of arbitrary characteristic. We first prove estimates for the dimensions of moduli spaces of stable maps to the twisted flag varieties $E/P$ and Hilbert schemes of closed subschemes of $E/P$ with same Hilbert polynomials as that of a $P$-reductions of $E$. This generalizes the earlier results of Mihnea Popa and Mike Roth to connected reductive groups and the results of Y. I. Holla and M. S. Narasimhan to the case of non-minimal sections. We then prove irreducibility and generic smoothness of the space of reductions for large numerical constraints, using the above result and the methods of G. Harder. We also study these space in more detail for generically stable $G$-bundles. As a consequence we can generalize the lower bound results of H. Lange to $G$.