Looijenga-Lunts and Verbitsky (LLV) have shown that the cohomology of a compact hyper-Kaehler manifold admits the action of a big Lie algebra g, generalizing the usual sl(2) Hard Lefschetz action (in the presence of a Kaehler class). We compute the LLV decomposition of the cohomology for the known classes of hyper-Kaehler manifolds (i.e. K3^n, Kim_n, OG6, and OG10). As an application, we easily recover the Hodge numbers of the exceptional example OG10. In a different direction, we establish the so-called Nagai’s conjecture (on the nilpotency index for higher degree monodromy operators) for the known cases. More interestingly, based on the known examples, we conjecture a new restriction on the cohomology of compact hyper-Kaehler manifolds, which in particular implies the vanishing of the odd cohomology as soon as the second Betti number is large enough relative to the dimension.

This is joint work with M. Green, Y. Kim, and C. Robles.