# Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's “Classical Groups”

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!

-
This holds more generally when the Zariski closure of $G$ is reductive, so for example when $G$ is finite, $GL_n, SL_n, SO_n,\ldots$. I don't have access to Weyl, but there are also number of alternative references that I'm sure someone here can suggest. –  Donu Arapura Jul 3 '13 at 12:53
(a) You only asked less than a day ago on MSE. (b) For future reference you should ask for migration instead of cross posting. meta.math.stackexchange.com/q/10067/1543 –  Willie Wong Jul 3 '13 at 12:58
@WillieWong Sorry! I guess there was some sort of union of the two sites that I missed. –  InvisiblePanda Jul 3 '13 at 16:14

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 $n$.