TITLE: Kostant's $G$-invariant part in the sym power ring of adjoint representation?
QUESTION [5 upvotes]: Let $g$ be a Lie algebra, say $sl_n(\mathbb C)$. It is considered as the adjoint representation of $G=SL_n(\mathbb C)$. 
A famous theorem of Kostant from "Lie Group Representations on Polynomial Rings" states that $$S=I\otimes H,$$
where $S( g)=\oplus_{m=0}^\infty \vee^m {g}$ is a graded algebra of the symmetric powers of  the adjoint representation $g$, 
$H$ is $G$-harmonic part, and $I$ is $G$-invariant part.
I would like to see a formula for $I$, or the (degree of) generators of $I$. Is it true $I=\prod_{i=2}^n (1-q^{i})^{-1}$?
$q$ is the symbol for recording the grading.

REPLY [2 votes]: Unless I misunderstood something, in the case of $sl_n$ the Lie algebra is that of $n\times n$ matrices $M$ with zero trace. The invariants are given by traces of powers of $M$. Here, the algebra of invariants is generated by ${\rm tr}(M^k)$ for $k=2,\ldots,n$. These generators are algebraically independent, so your formula for the Cayley-Sylvester-Hilbert series of $I$, is I believe correct.