Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ a finite dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$.

Is there a description of the invariants $S(V)^\mathfrak{g}$?

If $V$ is the standard module of the classical algebras, then this reduces to the fundamental theorems of invariant theory. Is there something in the literature of this more general kind? Is this known, at least for $\mathfrak{sl}_2$?

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$.

Is there a description of the invariants $S(V)^\mathfrak{g}$?

If $V$ is the standard module of the classical algebras, then this reduces to the fundamental theorems of invariant theory. Is there something in the literature of this more general kind? Is this known, at least for $\mathfrak{sl}_2$?