Real polarizable hodge structures arising from foliations

We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kahlerin analogue of the Weil conjectures carries over as well. Generalizing a construction of Looijenga and Lunts one obtains possibly infinite-dimensional Lie algebras attached to Kahler-Riemann foliations. Finally using (frak g, K)-cohomology we discuss a class of examples obtained by dividing a product of symmetric spaces by a cocompact lattice and considering the foliations coming from the factors.