Non-connected Lie groups, twisted equivariant bundles and coverings

Abstract

AbstractLet $$\Gamma $$ Γ be a finite group acting on a Lie group G. We consider a class of group extensions $$1 \rightarrow G \rightarrow \hat{G} \rightarrow \Gamma \rightarrow 1$$ 1 → G → G ^ → Γ → 1 defined by this action and a 2-cocycle of $$\Gamma $$ Γ with values in the centre of G. We establish and study a correspondence between $$\hat{G}$$ G ^ -bundles on a manifold and twisted $$\Gamma $$ Γ -equivariant bundles with structure group G on a suitable Galois $$\Gamma $$ Γ -covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group $$\hat{G}$$ G ^ , since such a group is always isomorphic to an extension as above, where G is the connected component of the identity and $$\Gamma $$ Γ is the group of connected components of $$\hat{G}$$ G ^ .