Modules induced from large subalgebras of the Lie algebra of differential operators of degree at most one

Abstract

We study the representation theory of the Lie algebra $D$ of differential operators of degree at most one on the algebra of complex Laurent polynomials. We define a natural family of subalgebras of $D$, which we call polynomial subalgebras. After classifying the one-dimensional modules for polynomial subalgebras, we use these one-dimensional modules to construct corresponding induced modules for the full Lie algebra $D$. These induced modules are frequently simple and generalize a family of recently discovered simple modules. In order to understand these induced modules in certain complicated cases, we take advantage of some general results on tensor products of modules.