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

Abstract

For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra \(\mathcal {A}\) 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.