Complex and CR-structures on compact Lie groups associated to Abelian actions

Abstract

It was shown by Samelson [A class of complex-analytic manifolds. Portugaliae Math. 12, 129–132 (1953)] and Wang [Closed manifolds with homogeneous complex structure. Amer. J. Math. 76, 1–32 (1954)] that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui [Classification des structures CR invariantes pour les groupes de Lie compactes. J. Lie theory 14, 165–198 (2004)] who have also given a complete algebraic description of these structures. In this article, we present an alternative and more geometric construction of this type of invariant structures on a compact Lie group K when it is semisimple. We prove that each left-invariant complex structure, or each CR-structure of maximal dimension with a transverse CR-action by $\mathbb{R}$ , is induced by a holomorphic $\mathbb{C}^l$ -action on a quasi-projective manifold X naturally associated to K. We then show that X admits more general Abelian actions, also inducing complex or CR-structures on K which are generically non-invariant.