Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's "Classical Groups"

Question

First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but didn't get any answer or comment yet, maybe due to the vague formulation.

In the "Lectures on the Fourteenth Problem of Hilbert" by Nagata, he mentioned that if $G$ is a complex connected semisimple Lie group, then the answer to the following problem is "yes". This is supposed to be proven in Weyl's "Classical Groups".

Let $G\subseteq\operatorname{GL}_n(\mathbb{C})$ act on $\mathbb{C}[x_1,\dots,x_n]$ via $x\mapsto A\cdot x$ for $A\in G$. Is the invariant ring $\mathbb{C}[x_1,\dots,x_n]^G$ a finitely generated $\mathbb{C}$-algebra?

I have no familiarity at all with Lie groups, and just wish to reference the result for the sake of completeness. But I can't seem to find it in the above book. Does anyone of you maybe know which result exactly is meant? Also, is there a nice classification of the subgroups of $\operatorname{GL}_n$ which are connected semisimple Lie groups? Or rather, what are "known" examples of such groups?

Thank you very much in advance!

Answer 1

Your basic question about a reference does go back to Weyl's complete reducibility theorem (I'd have to check his book on classical groups, but it isn't just a result about classical Lie groups). For a standard modern proof in the wider context of semisimple (or more generally, reductive) algebraic groups over an algebraically closed field of characteristic 0, see for instance Section 14.3 in my 1975 Springer graduate text Linear Algebraic Groups. In any case, complete reducibility of finite dimensional representations in the appropriate category for Lie groups or algebraic groups is the essence of the matter.

Your later questions are much more open-ended and difficult to answer in any detail, even if you limit yourself to classical type Lie groups. Subgroups of general linear groups which are connected semisimple groups are determined by the faithful linear representations, especially the irreducible ones. The highest weight classification for each $n$ would tell you a lot, but there is no list for arbitrary $n$.

P.S. I should add that the ideas of Hilbert and later Weyl developed out of classical invariant theory, where the "first fundamental theorem" deals with the finite generation of certain rings of invariants. Modern treatments, probably more readable than Weyl's, are given in recent texts by Goodman-Wallach and Procesi. But there's no need to discuss the finite generation only for finite or classical groups, though historically they were the natural examples to start with.