Real polarizable Hodge structures arising from foliations

We construct real polarizable Hodge structures on the reduced leafwise cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's K\"ahlerian 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 K\"ahler-Riemann foliations. Finally using $(\mathfrak{g},K)$-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.