Automorphism group of principal bundles, Levi reduction and invariant connections

Abstract

Let M be a compact connected complex manifold and G a connected reductive complex affine algebraic group. Let $$E_G$$ be a holomorphic principal G–bundle over M and $$T\, \subset \, G$$ a torus containing the connected component of the center of G. Let N (respectively, C) be the normalizer (respectively, centralizer) of T in G, and let W be the Weyl group N / C for T. We prove that there is a natural bijective correspondence between the following two: The composition of maps $$E'_C\, {\mathop {\longrightarrow }\limits ^{\psi }}\, E_W \, {\mathop {\longrightarrow }\limits ^{\phi }}\, M$$ defines a principal N–bundle on M. This principal N–bundle $$E_N$$ is a reduction in structure group of $$E_G$$ to N. Given a complex connection $$\nabla $$ on $$E_G$$ , we give a necessary and sufficient condition for $$\nabla $$ to be induced by a connection on $$E_N$$ . This criterion relates Hermitian–Einstein connections on $$E_G$$ and $$E'_C$$ in a very precise manner.