Exercise 34, § 9

Let $G$ be a finite-dimensional connected real or complex Lie group and $\rho$ an analytic linear representation of $G$ on a finite-dimensional complex vector space $V$. Suppose that $\rho$ is semi-simple. The group $G$ operates by automorphisms on the algebra $S(V^*)$ of polynomial functions on $V$. (a) Let $S(V^*)^G$ be the set of $G$-invariant elements of $S(V^*)$. There exists a projector $p$ of $S(V^*)$ onto $S(V^*)^G$ which commutes with the operations of $G$ and which leaves stable every $G$-stable vector subspace of $S(V^*)$.

(b) Let $I, J$ be ideals of $S(V^*)$ which are stable under $G$. Let $A$ (resp. $B$) be the set of zeros of $I$ (resp. $J$) in $V$. Suppose that $A \cap B = \varnothing$. There then exists $u \in I$ such that $u$ is $G$-invariant and that $u = 1$ in $B$. (By Commutative Algebra, Chapter 7, § 3, no. 3, Proposition 2, there exists $v \in I, w \in J$ such that $v + w = 1$ Let $u = pv$. Show that $u$ has the required properties.)