Simple 𝔰𝔩( V )-modules which are free over an abelian subalgebra

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