# Grothendieck's paper on principal bundles, reduction to a torus step

In Grothendieck's paper "Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann", there is a step I don't understand in section 4, where he proves reduction to a torus. He states (lemma 4.1) that if $P$ is a principal $G$ bundle on a compact space and the associated bundle $P\times ^G ad$ has a section which is regular, then $P$ has a reduction to the normalizer of a Cartan, with reduction to a Cartan in case the base space is simply connected. I do not understand why the simply connected assumption is necessary. Can someone explain the problem with the following argument (if it makes a difference, I am interested in the case where $G$ is semisimple): suppose $s$ is a regular section of $P\times ^G ad$. We have that the regular locus of the adjoint representation, $ad^r$, is the disjoint union of orbits of the form $G/H$. So $P\times ^G ad^r$ is really a disjoint union of bundles of the form $P\times ^G G/H$. The section $s$ actually has to land in exactly one of these, because we have a map $\phi: P\times ^G ad\rightarrow ad/G$ such that each of the bundles $P\times G/H$ are the fiber of a point under this map, and the composition $\phi \circ s$ is constant, since it is a map from a compact space to an affine space. But then $s$ is really a section of $P\times^G G/H$, and hence $P$ has a reduction of structure to a Cartan.

Your map $\phi$ is $W$-equivariant, with $W$ being the Weyl group. So each fiber $\phi^{-1}(G.x)$, for $G.x \in ad/G$, is "closed" under the Weyl group. So you have shown that there is a holomorphic section of $P \times^G G/N$, where $N$ is the normalizer of $T$, and not that there is a holomorphic section of $P \times^G G/T$, as you thought. I believe this is where your mistake was.