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Yes. But the invariant polynomials form a polynomial algebra in rank $\mathfrak{g}$ (that is the dimension of a Cartan) variables and are the isotypic component corresponding to the trivial representation. Or am I completely misunderstanding the question?
I think your reference is for a different problem and the statement you are asking about is false even for $\mathfrak{sl}_2$ and the trivial representation (and therefore for every f.d. irreducible representation).
