Theorem 6

Let $g$ be a Lie algebra, $V$ a semi-simple $g$-module of finite dimension over $K$, $S$ the symmetric algebra of $V$ and $x_S$ the derivation of $S$ which extends $x_V$ (so that $x \mapsto x_S$ is a representation of $g$ on $S$).

(a) The algebra $S_0$ of invariants of $S$ is generated by a finite number of elements.

(b) For every class $\delta$ of simple representations of $g$ of finite dimension over $K$, let $S_\delta$ be the isotypical component of $S$ of species $\delta$. Then $S_\delta$ is a finitely generated $S_0$-module.

Let $\overline{S} \subset S$ be the ideal of elements of $S$ with no constant term. Let $I$ be the ideal of $S$ generated by $S_0 \cap \overline{S}$ and let $(s_1, s_2, \ldots, s_p)$ be a finite system of generators of the ideal I (Commutative Algebra, Chapter III, § 3). It can be assumed that the s_i belong to S_0 ∩ S̄ and are homogeneous (the x_S preserve degrees and hence each S_δ is a graded submodule). Let S_1 be the subalgebra of S generated by 1 and the s_i. Then S_1 ⊂ S_0. We show that S_1 = S_0. For this we prove that every homogeneous element s of S_0 is in S_1, arguing by induction on the degree n of s. If n = 0, our assertion is obvious. Suppose therefore n > 0 and that our assertion has been proved when the degree of s is < n. As s ∈ I, s = $\sum_{i=1}^{p} s_i s'_i$, where the s'_i are elements of S which can be assumed to be homogeneous, with deg(s'_i) = deg(s) − deg(s_i) < n. Lemma 5 can be applied, as the g-module S is semi-simple (no. 5, Corollary 2 to Theorem 4); in the notation of this lemma,

The {s'_i}^{\natural} are elements of S_0 which are homogeneous and of degree < n (since each S_δ is a graded submodule). Hence they are in S_1 by the induction hypothesis. Hence s ∈ S_1, which proves (a).

We now consider a simple representation of g of class δ on a finite-dimensional space M. Let L = $\mathscr{L}_K(M, S)$. For all s ∈ S and all f ∈ L, let sf be the element of L defined by (sf)(m) = s.f(m) (m ∈ M); an S-module structure is thus defined on L; as M is finite-dimensional over K, clearly L is a finitely generated S-module and hence a Noetherian S-module since the ring S is Noetherian. On the other hand, L has a canonical g-module structure. For every integer n ≥ 0 let S^n be the set of homogeneous elements of S of degree n; then the g-module $\mathscr{L}_K(M, S^n)$ is semi-simple (no. 5, Corollary 3 to Theorem 4) and hence the g-module L is semi-simple. Moreover, for s ∈ S, f ∈ L, x ∈ g and m ∈ M,

and hence $x_L(sf) = (x_S s)f + s(x_L f)$. We can therefore apply Lemma 5.

The subset L_0 of invariant elements of L is just the set of homomorphisms of the g-module M into the g-module S. Hence, if φ denotes the canonical homomorphism of M ⊗_K L onto S, $\phi(M \otimes_K L_0) = S_\delta$. As φ is obviously an S-module homomorphism, it suffices to show that L_0 is a finitely generated S_0-module. Let J be the sub-S-module of L generated by L_0. Since L is a Noetherian S-module, there exists a finite sequence (f_1, ..., f_q) of elements of L_0 generating the S-module J. Let L_1 be the S_0-module generated by f_1, \ldots, f_q. Then L_1 \subset L_0. On the other hand, if f \in L_0, then $f = \sum_{i=1}^q s_i f_i$ with $s_i \in S$ for all $i$ and hence by Lemma 5 whose notation we adopt:

Hence $L_0 = L_1$, so that $L_0$ is a finite generated $S_0$-module.