  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 $\mathfrak{gl}_N$-module is irreducible
as a module over the Lie algebra of vector fields unless it appears in the de
Rham complex.
