THE CATEGORY OF WEIGHT MODULES FOR SYMPLECTIC OSCILLATOR LIE ALGEBRAS

Abstract

The rank n symplectic oscillator Lie algebra 𝔤n is the semidirect product of the symplectic Lie algebra 𝔰𝔭2n and the Heisenberg algebra Hn. In this paper, we first study weight modules with finite-dimensional weight spaces over 𝔤n. When the central charge \( \dot{z} \) ≠ 0, it is shown that there is an equivalence between the full subcategory 𝒪𝔤n \( \left[\dot{z}\right] \) of the BGG category 𝒪𝔤n for 𝔤n and the BGG category 𝒪𝔰𝔭2n for 𝔰𝔭2n. Then using the technique of localization and the structure of generalized highest weight modules, we give the classification of simple weight modules over 𝔤n with finite-dimensional weight spaces. As a byproduct we also determine all simple 𝔤n-modules (not necessarily weight modules) that have a simple Hn-submodule.