INVARIANT DIFFERENTIAL OPERATORS ON A SEMISIMPLE LIE ALGEBRA

Let R and C be the fields of real and complex numbers, respectively, and Eo a vector space over R of finite dimension. Then, E being the complexification of Eo, we consider the symmetric algebra S(E) and the algebra Q(E) of polynomial functions on E. We regard Eo as a differentiable manifold and, for any differential operator D and indefinitely differentiable function f, denote by f(X; D) the value of Df at X Eo. Corresponding to any X in Eo we define the differential operator 6(X) by