Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 1

Semidirect products and invariant connections

Pages: 1 – 20

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

Author

Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Bombay, India)

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

Published 1 September 2015