Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic components of $S(\mathfrak{g})$ finite-dimensional?

This seems to be true for $\mathfrak{sl}_n$, since this is the subject of "plethysm" and there are explicit formulae for these multiplicities. See e.g. this mathoverflow question and the reference therein.