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For a projective $2n$-dimensional irreducible holomorphic symplectic manifold$Y$ of generalized Kummer deformation type and $j>1$ the smallest divisor of$n+1$, we prove the Lefschetz standard conjectures in degree $<2(n+1)(j-1)/j$.We show that the restriction homomorphism from the cohomology of a projectivedeformation of a moduli space of Gieseker-stable sheaves on an abelian surfaceto the cohomology of $Y$ is surjective in these degrees. A corollary is thatthe Lefschetz standard conjectures hold for $Y$ when $n+1$ is prime. The proofsrely on Markman's description of the monodromy of generalized Kummer varietiesand construction of a universal family of moduli spaces of sheaves, Verbitsky'stheory of hyperholomorphic sheaves, and the decomposition theorem.

The Lefschetz standard conjectures for IHSMs of generalized Kummer deformation type in certain degrees