Complex and CR-structures on compact Lie groups associated to Abelian actions (Q2462629)

Each compact Lie group \(\mathbf{K}\) can be given a left invariant structure that is either complex, when the real dimension of \(\mathbf{K}\) is even, or CR of hypersurface type, when \(\dim_{\mathbb{R}}\mathbf{K}\) is odd. The classical result for the complex structure has indeed been extended recently to a classification of the maximal CR structures on compact Lie groups [see \textit{J.-Y. Charbonnel} and \textit{H. Ounaïes Khalgui}, J. Lie Theory 14, No. 1, 165--198 (2004; Zbl 1058.22016)]. In this paper the authors present a geometrical construction of such left invariant structures, admitting a transverse CR-action of \(\mathbb{R}\). Let \(\mathbf{G}\) be the universal complexification of \(\mathbf{K}\), that is also assumed to be semisimple. From the Iwasawa decomposition of \(\mathbf{G}\) one obtains an embedding of \(\mathbf{K}\) into the quasi-projective manifold \(\mathbf{G}/\mathbf{U}\), where \(\mathbf{U}\) is a maximal unipotent Lie subgroup of \(\mathbf{G}\). The left-invariant CR structures on \(\mathbf{K}\) are defined by the datum of a complex subgroup \(\mathbf{L}\) of \(\mathbf{G}\) with \(\mathbf{L}\cap\mathbf{K}=\{e\}\). When \(\roman{dim}_{\mathbb{R}}\mathbf{K}\) is even, all left-invariant complex structures are defined, modulo equivalence, by closed solvable \(\mathbf{L}\) containing \(\mathbf{U}\). When \(\roman{dim}_{\mathbb{R}}\mathbf{K}\) is odd, an \(\mathbf{L}\) that yields a maximal CR structure may not contain a maximal unipotent \(\mathbf{U}\). Those \(\mathbf{L}\) that do contain a maximal unipotent \(\mathbf{U}\) are characterized by the fact that the corresponding CR structure admits a transverse CR action of \(\mathbb{R}\), and such structures exist on arbitrary odd dimensional compact Lie groups. An interesting point of these geometrical constructions is the possibility of obtaining non invariant CR structures by deformation of this action. Some generalizations are also obtained in the last part of the paper by considering \(\mathbb{C}^{\ell}\) actions on \(\mathbf{G}/\mathbf{U}\).