Levi-flat CR structures on compact Lie groups

Abstract Pittie (Proc Indian Acad Sci Math Sci 98:117-152, 1988) proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We generalized Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. The main tools we used was a version of the Leray–Hirsch theorem for CR principal bundles and the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups (Charbonnel and Khalgui in J Lie Theory 14:165-198, 2004) .