We study a category of representations of the Lie algebras of
vector fields on an affine algebraic variety X that admit a compatible
action of the algebra of polynomial functions on X. We investigate two
classes of simple modules in this category: gauge modules and Rudakov
modules, and establish a covariant pairing between modules of these two
types.
We show that every module in this category, which is finitely generated
over the algebra of functions, is projective, and state a conjecture that
gauge modules exhaust all such modules. We give a proof of this conjecture
when X is the affine space.
This is a joint work with Slava Futorny, Jonathan Nilsson, Andre Zaidan,
Colin Ingalls and Amir Nasr.