Mukai-Sakai bound for equivariant principal bundles (Q2368531)

Let \(X\) be a smooth genus \(g\) curve. Let \(E\) be a rank \(r\) vector bundle on \(X\) and \(s\) an integer such that \(1 \leq s < r\). \textit{S. Mukai} and \textit{F. Sakai} [Manuscr. Math. 52, 251--256 (1985; Zbl 0572.14008)] proved the existence of a rank \(s\) subbundle \(S\) of \(E\) such that \(\deg \Hom (S,E/S) \leq s(r-s)g\). This means that \(E\) cannot be too stable; related results were proved by H. Lange and A. Hirschowitz. This inequality was extended to the case of principal bundles in [\textit{Y. I. Holla} and \textit{M. S. Narasimhan}, Compos. Math. 127, 321--332 (2001; Zbl 1047.14018)]. Here the authors extend this result to the case of equivariant principal bundles. Their proof also simplifies the known proof for principal bundles.