Gauge modules for the Lie algebras of vector fields on affine varieties

For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $A$ of functions and the Lie algebra $\mathcal{V}$ of vector fields on the variety. We prove that a gauge module corresponding to a simple ${\mathfrak{g}\mathfrak{l}}_N$-module is irreducible as a module over the Lie algebra of vector fields unless it appears in the de Rham complex.