Non-connected Lie groups, twisted equivariant bundles and coverings

Abstract

Let Gamma 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 defined by this action and a 2-cocycle of Gamma with values in the centre of G. We establish and study a correspondence between hat{G}-bundles on a manifold and twisted Gamma -equivariant bundles with structure group G on a suitable Galois Gamma -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}, since such a group is always isomorphic to an extension as above, where G is the connected component of the identity and Gamma is the group of connected components of hat{G}.