Let p be a parabolic subalgebra of 𝔰⁢𝔩⁢(V) of maximal dimension and let n⊂p be the corresponding nilradical. In this paper, we classify the set of sl(V)-modules whose restriction to U(n) is free of rank 1. It turns out that isomorphism classes of such modules are parametrized by polynomials in dimV−1 variables. We determine the submodule structure for these modules and we show that they generically are simple.