Very little has been known about representation theory of Lie
algebras of polynomial vector fields on affine algebraic varieties beyond
the cases of affine space and a torus. We study a category of
representations of the Lie algebras of vector fields on a smooth algebraic
variety X that admit a compatible action of the algebra of polynomial
functions on X. We construct simple modules in this category and state a
conjecture on the general structure of such modules.
This is a joint work with Slava Futorny and Jonathan Nilsson.