Principal bundles on the projective line.
Let \(G\) be a reductive group defined over any field \(k\) of characteristic different from \(2\) and \(3\). Let \(E\) denote a principal \(G\)-bundle on the projective line \({\mathbb P}^1_k\) which is trivial at the origin i.e., the restriction of \(E\) to the \(k\)-rational point \(0\in {\mathbb P}^1_k\) is a trivial principal homogeneous space over Spec \(k\). The authors show that \(E\) is associated to the line bundle \({\mathcal O}(1)\) on \({\mathbb P}^1_k\) (regarded as a \(G_m\)-bundle) for some one parameter subgroup \(G_m \to G\) defined over \(k\). In particular, the structure group of \(E\) has reduction to a torus. The result is proved using Harder-Narasimhan filtrations. The result was proved by \textit{P. Gille} using Bruhat-Tits theory [Transform. Groups 7, No. 3, 231--245 (2002; Zbl 1062.14061)]. In particular cases, the result is due to \textit{A. Grothendieck} when \(k={\mathbb C}\) [Am. J. Math. 79, 121--138 (1957; Zbl 0079.17001)] and to G. Harder when \(G\) is split over \(k\) and the principal \(G\)-bundle is Zariski locally trivial.
scientific article; zbMATH DE number 1870282