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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.

Gauge Modules for the Lie Algebras of Vector Fields on Affine Varieties