Semisimple Lie Algebras of Differential Operators

Abstract

Starting from the commutation relations in a complex semisimple Lie algebra $${\mathfrak{g}}$$ , one may obtain a space $${\hat {\mathfrak{g}}}$$ of vector fields on Euclidean space such that $${\mathfrak{g}}$$ and $${\hat {\mathfrak{g}}}$$ are isomorphic when $${\hat {\mathfrak{g}}}$$ is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of $${\hat {\mathfrak{g}}}$$ is a Borel subalgebra $${\mathfrak{b}}$$ . Furthermore, one may adjoin to the vector fields in $${\hat {\mathfrak{g}}}$$ multiplication operators to obtain an $$\mathfrak{h}*$$ -parameter family of distinct presentations of $$\mathfrak{g}$$ as spaces of differential operators, where $$\mathfrak{h}*$$ is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of $$\mathfrak{g}$$ can be presented in this way. All of this is carried out explicitly for arbitrary $$\mathfrak{g}$$ . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.