  For a projective $2n$-dimensional irreducible holomorphic symplectic manifold
$Y$ of generalized Kummer deformation type and $j$ the smallest prime number
dividing $n+1$, we prove the Lefschetz standard conjectures in degrees
$<2(n+1)(j-1)/j$. We show that the restriction homomorphism from the cohomology
of a projective deformation of a moduli space of Gieseker-stable sheaves on an
Abelian surface to the cohomology of $Y$ is surjective in these degrees. A
corollary is that the Lefschetz standard conjectures hold for $Y$ when $n+1$ is
prime. The proofs rely on Markman's description of the monodromy of generalized
Kummer varieties and construction of a universal family of moduli spaces of
sheaves, Verbitsky's theory of hyperholomorphic sheaves, and the decomposition
theorem.
