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In these notes we reformulate the classical Hilbert-Mumford criterion for GITstability in terms of algebraic stacks, this was independently done byHalpern-Leinster. We also give a geometric condition that guarantees theexistence of separated coarse moduli spaces for the substack of stable objects.This is then applied to construct coarse moduli spaces for torsors underparahoric group schemes over curves.

Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves