Equivariant holomorphic Hermitian principal bundles over a Euclidean space (Q2852538)

Let \(V\) be a complex vector space of finite dimension. Fix a Hermitian inner product \(h\) on \(V\). Let \(G\) be the group of all affine automorphisms of \(V\) of the form \(v\mapsto v_0+T(v),\) where \(v_o\in V\) and \(T \in GL(V)\) preserves \(h.\) NEWLINENEWLINENEWLINELet \(H\) be a connected reductive affine algebraic group defined over \(\mathbb C.\) A Hermitian structure on a \(C^{\infty}\) principal \(H\)-bundle \(E_H\) over \(V\) is a \(C^{\infty}\) reduction of the structure group of \(E_H\) to a fixed maximal compact subgroup \(K.\) A holomorphic Hermitian principal \(H\)-bundle on \(V\) is a pair of the form \((E_H, E_k),\) where \(E_H\) is a holomorphic principal \(H\)-bundle on \(V\) and \(E_K\subset E_H\) is a Hermitian structure. The holomorphic Hermitian principal \(H\)-bundle on \(V\) equipped with a lift of the tautological action \(G\) on \(V\) is considered. NEWLINENEWLINENEWLINEThe present paper provides all the isomorphisms classes of \(G\)-invariant holomorphic Hermitian principal \(H\)-bundle over \(V.\)