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

Abstract

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