Flat vector bundles over parallelizable manifolds (Q2753166)

Suppose \(G\) is a connected complex Lie group and \(\Gamma\) is a lattice in \(G\). The author continues his study of holomorphic vector bundles on \(G/\Gamma\) begun in his Habilitationsschrift. To this end he indicates some connections between flatness and homogeneity of bundles and considers bounded representations. Next he introduces the notion of an essentially antiholomorphic representation. Given complex Lie groups \(G\) and \(H\) and \(\Gamma\subset G\) a discrete subgroup an essentially antiholomorphic representation is a group homomorphism \(\rho: \Gamma\to H\) such that \(\rho(\gamma) = \zeta(\gamma)\cdot \xi(\gamma)\) for all \(\gamma\in\Gamma\), where \(\xi: \Gamma \to H\) is a map with relatively compact image (a bounded part that turns out to be a homomorphism!) and \(\zeta: G \to H\) is an antiholomorphic Lie group homomorphism. NEWLINENEWLINENEWLINEThe author then uses essentially antiholomorphic representations to prove several results about holomorphic vector bundles on \(G/\Gamma\). He provides a classification of such bundles over \(S/\Gamma\), where \(S\) is a simply connected semisimple complex Lie group that does not have any \(SL(2,{\mathbb C})\) factor having an orbit with finite volume. He shows that if \(E \to X = G/\Gamma\) is a homogeneous vector bundle, then \(\Gamma(X,E)\) is finite dimensional even though \(X\) need not be compact. Sections of such bundles come from a vector subbundle that is parallel with respect to the flat connection. The final section contains a result about vector subbundles of a flat vector bundle given by an essentially antiholomorphic representation.