# Semistability of principal bundle vs vector bundle

Ramanathan has defined the semistability of a principal $G-$bundle $E$ over a curve $X$ as follows:

$E$ is semistable iff for any parabolic subgroup $P\subset G$, for any reduction of the structure group of $E$ to $P$: $\sigma:X\rightarrow E(G/P)$, and for any dominant caracter $\chi:P\rightarrow \mathbb C^*$, one has $$\sigma^* E(\chi)\le 0$$ where $E(\chi)$ is the line bundle associated to the $P$ bundle $E\rightarrow E(G/P)$ by $\chi$

Question: Taking $G=GL_r$, How could one proof that this definition is the same as the usual one (due to Mumford)?

Thanks

The following paper contains a proof (Corollary 1) of the identification of Ramanathan-semistability for principal $GL_n$-bundles and Mumford-semistability for the associated vector bundles: