Irreducible modules over finite simple Lie pseudoalgebras IV. Non-primitive pseudoalgebras

Let d ⇢ d0 be finite-dimensional Lie algebras, H D U.d/; H0 D U.d0 / the corresponding universal enveloping algebras endowed with the canonical cocommutative Hopf algebra structure. We show that if L is a primitive Lie pseudoalgebra over H then all finite irreducible L0 D CurH0 H L-modules are of the form CurH0 H V , where V is an irreducible L-module, with a single class of exceptions. Indeed, when L ' H.d; ;!/, we introduce non-current L0-modules VH ;!;t;d0.R/ that are obtained by modifying the current pseudoaction with an extra term depending on an element t 2 d0 n d, which must satisfy some technical conditions. This, along with results from [2–4], completes the classification of finite irreducible modules of finite simple Lie pseudoalgebras over the universal enveloping algebra of a finite-dimensional Lie algebra.