On automorphisms of semistable G-bundles with decorations

Abstract We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of G-bundles on a smooth projective curve for a reductive algebraic group G. For example, our result applies to the stack of semistable G-bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolic G-bundles. Similar arguments apply to Gieseker semistable G-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic 0 every stack of semistable decorated G-bundles admitting a quasiprojective good moduli space can be written naturally as a G-linearized global quotient Y/G, so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphic G-Higgs bundles on a family of curves is smooth over any base in characteristic 0.