Semisimple Lie algebras of differential operators

Starting from the commutation relations in a complex semisimple Lie algebra g, one may obtain a space (g) over cap of vector fields on Euclidean space such that g and (g) over cap are isomorphic when (g) over cap is equipped with the usual Lie bracket between vector fields and the isotropy subalgebr a of (g) over cap is a Borel subalgebra b. Furthermore, one may adjoin to t he vector fields in (g) over cap multiplication operators to obtain an h*-p arameter family of distinct presentations of g as spaces of differential op erators, where h* is the dual of a Cartan subalgebra. Some of these present ations will preserve a space of polynomials on Euclidean space, and, in fac t, all the finite-dimensional representations of g can be presented in this way. All of this is carried out explicitly for arbitrary g. In doing so, o ne discovers there is a Lie group of diffeomorphisms of the unipotent subgr oup N complementary to B which acts on these presentations and preserves a certain notion of weight.