Transformation Groups | 2019
THE CATEGORY OF WEIGHT MODULES FOR SYMPLECTIC OSCILLATOR LIE ALGEBRAS
Abstract
The rank n symplectic oscillator Lie algebra 𝔤 n is the semidirect product of the symplectic Lie algebra 𝔰𝔭 2 n and the Heisenberg algebra H n . In this paper, we first study weight modules with finite-dimensional weight spaces over 𝔤 n . When the central charge z ̇ $$ \\dot{z} $$ ≠ 0, it is shown that there is an equivalence between the full subcategory 𝒪𝔤 n z ̇ $$ \\left[\\dot{z}\\right] $$ of the BGG category 𝒪𝔤 n for 𝔤 n and the BGG category 𝒪𝔰𝔭 2 n for 𝔰𝔭 2 n . Then using the technique of localization and the structure of generalized highest weight modules, we give the classification of simple weight modules over 𝔤 n with finite-dimensional weight spaces. As a byproduct we also determine all simple 𝔤 n -modules (not necessarily weight modules) that have a simple H n -submodule.