Representations of the Lie algebra of vector fields on a sphere

Abstract: For an affine algebraic variety $X$ we study a category of modules that admit compatible actions of both the algebra of functions on $X$ and the Lie algebra of vector fields on $X$. In particular, for the case when $X$ is the sphere $\mathbb{S}^2$, we construct a set of simple modules that are finitely generated over $A$. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational $\mathrm{GL}_2$-modules.