Semidirect products and invariant connections

TL;DR

This paper classifies $G$-equivariant holomorphic and almost holomorphic hermitian principal bundles on complex Lie groups, using the structure of semidirect products and invariant connections, under specific group action assumptions.

Contribution

It provides explicit descriptions of isomorphism classes of $G$-equivariant principal bundles on complex Lie groups, extending understanding of invariant connections in this context.

Findings

Explicit classification of $G$-equivariant almost holomorphic hermitian principal bundles.

Explicit classification of $G$-equivariant holomorphic hermitian principal bundles under certain conditions.

Connection between group actions and bundle isomorphism classes.

Abstract

Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivariant almost holomorphic hermitian principal bundles on $N$. Under the assumption that there is a central subgroup $Z= \text{U}(1)$ of $K(S)$ that acts on $\text{Lie}(N)$ as multiplication through a single nontrivial character, we give an explicit description of the isomorphism classes of $G$-equivariant holomorphic hermitian principal bundles on $N$.