On very stablity of principal $G-$bundles

Abstract

Let $X$ be a smooth irreducible projective curve. Recently, Pauly and Pe\'on-Nieto shows that a vector bundle over $X$ is very stable if and only if the Hitchin map on the vector space of Higgs field on that vector bundle is proper. In this notes, we generalize this result to principal $G-$bundles for any semisimple linear algebraic group $G$. We also study the relation between very stability and other stability conditions in the case of $\text{SL}_2-$bundles.