  Looijenga--Lunts and Verbitsky showed that the cohomology of a compact
hyper-K\"ahler manifold $X$ admits a natural action by the Lie algebra
$\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition
for compact K\"ahler manifolds. In this paper, we determine the
Looijenga--Lunts--Verbitsky (LLV) decomposition for all known examples of
compact hyper-K\"ahler manifolds, and propose a general conjecture on the
weights occurring in the LLV decomposition, which in particular determines
strong bounds on the second Betti number $b_2(X)$ of hyper-K\"ahler manifolds.
  Specifically, in the $K3^{[n]}$ and $\mathrm{Kum}_n$ cases, we give
generating series for the formal characters of the associated LLV
representations, which generalize the well-known G\"ottsche formulas for the
Euler numbers, Betti numbers, and Hodge numbers for these series of
hyper-K\"ahler manifolds. For the two exceptional cases of O'Grady we refine
the known results on their cohomology. In particular, we note that the LLV
decomposition leads to a simple proof for the Hodge numbers of hyper-K\"ahler
manifolds of O'Grady 10 type. In a different direction, for all known examples
of hyper-K\"ahler manifolds, we establish the so-called Nagai's conjecture on
the monodromy of degenerations of hyper-K\"ahler manifolds. More
consequentially, we note that Nagai's conjecture is a first step towards a more
general and more natural conjecture, that we state here. Finally, we prove that
this new conjecture is satisfied by the known types of hyper-K\"ahler
manifolds.
