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)?



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:

  • D. Hyeon and D. Murphy. Note on the stability for principal bundles. Proc. Amer. Math. Soc. 132 (2004), 2205-2213.

The basic point is that there is a direct relation between flags on the vector bundles and reductions to parabolic subgroups. This allows to relate slope of the vector bundles to the degree conditions for parabolic bundles. (I will not rewrite the proof in the above paper, it is really nice and short.)

This does not completely answer the question, because the definition used in the paper I mentioned is different from the formulation of the question. The remaining identification of semistability conditions for the principal bundle is done in Ramanathan's original paper.

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