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# Pure and Applied Mathematics Quarterly

## Volume 11 (2015)

### Number 1

### Semidirect products and invariant connections

Pages: 1 – 20

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n1.a1

#### Author

#### 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 $Z = \mathrm{U}(1)$ of $K(S)$ that acts on $\mathrm{Lie}(N)$ as multiplication through a single nontriv- ial character, we give an explicit description of the isomorphism classes of $G$-equivariant holomorphic hermitian principal bundles on $N$.

#### Keywords

semidirect product, holomorphic hermitian bundle, invariant connection, parabolic subgroup

#### 2010 Mathematics Subject Classification

32L05, 53B35