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.

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.