Lie Group Representations on Polynomial Rings

0.Introduction. 1.Let G be a group of linear transformations on a finite dimensional real or complex vector space X.Assume X is completely reducible as a G-module.Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let JC5 be the subring of all G-invariant polynomials on X.Now let J + be the set of all ƒ £ J having zero constant term and let HQS be any graded subspace such that S=J + S+H is a G-module direct sum.It is then easy to see that