On very stablity of principal G-bundles

Abstract

Let X be a smooth irreducible projective curve. In this note, we generalize the main result of Pauly and Peon-Nieto (Geometriae Dedicata 1–6, 2018) to principal G-bundles for any reductive linear algebraic group G. After defining very stability of principal G-bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of $$\text {SL}_2$$-bundles.