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 $\mathfrak{sl}_2$?
The question subsumes all of 19th Century invariant theory, so I don't think there is much chance of a really explicit answer.
For example, take $\mathfrak{g} = \mathfrak{sl}_d(\mathbb{C})$ and let $V$ be the $\mathfrak{sl}_d(\mathbb{C})$-module obtained from the polynomial representation $\nabla^\lambda(\mathbb{C}^d)$ of $\mathrm{GL}_d(\mathbb{C})$ canonically labelled by the partition $\lambda$ of $n$ with at most $n-1$ parts. (So $\lambda$ is a dominant integral weight.) The invariants for $\mathfrak{sl}_d(\mathbb{C})$ in the degree $r$ component of $S(V)$ lift to elements of $\mathrm{Sym}^r \nabla^\lambda(\mathbb{C}^d)$ on which $\mathrm{GL}_d(\mathbb{C})$ acts as a power of the determinant: since det has polynomial degree $d$, the relevant power is $m = r|\lambda|/d$. These elements span a subspace of $\mathrm{Sym}^r \nabla^\lambda(\mathbb{C}^d)$ isomorphic to $\nabla^{(m^d)}(\mathbb{C}^d)$. Thus finding the dimension of the invariant spaces (as $d$ varies) is equivalent to finding the multiplicities of polynomial representations labelled by rectangular partitions in $\mathrm{Sym}^r \nabla^\lambda(\mathbb{C}^d)$.
This can be restated in the language of Schur functions as asking for $\langle s_{(r)} \circ s_\lambda, s_{(m^d)} \rangle$, where $\circ$ is the plethystic product. Decomposing plethysms into Schur functions is a notorious open problem in algebraic combinatorics. As far as I know, it is not made easier by restricting to constituents labelled by rectangular partitions.
This paper https://arxiv.org/pdf/0807.0430.pdf has a formula for the dimensions in the special case when $\lambda = (n)$ has one part, i.e. it finds the dimension of the invariant spaces of $\mathfrak{sl}_d(\mathbb{C})$ on the space $\mathrm{Sym}^n \mathbb{C}^d$ of $n$-ary forms in $d$ variables. The formula looks somewhat intractable to me.
For $\mathfrak{sl}_2(\mathbb{C})$ the simple modules are the symmetric powers of the natural module $E \cong \mathbb{C}^2$ and much more is known. Thinking of $\mathrm{Sym}^2 E$ as the space of quadratic forms, the invariant algebra $S(\mathrm{Sym}^2 E)^{\mathfrak{sl}_2(\mathbb{C})}$ is generated by the discriminant in degree $2$. Hence for $\mathrm{Sym}^2 E$ there is a unique invariant (up to scalars) in each even degree.
Still for $\mathfrak{sl}_2(\mathbb{C})$, but for all $r$ and $\ell$, the dimension of the degree $r$ component of $S(\mathrm{Sym}^\ell \!E)^{\mathfrak{sl}_2(\mathbb{C})}$ is the number of partitions of $\ell r/2$ that fit in an $\ell \times r$ box, minus the number of partitions of $\ell r/2-1$ in the same box. This is the Cayley–Sylvester formula: see e.g. Lecture XVII in Hilbert Theory of algebraic invariants. The dimensions are easily computed but do not grow in a regular way when $r \ge 3$. The symmetry swapping $r$ and $\ell$ is known as Hermite reciprocity. There is some more recent work on generalizations of the Cayley–Sylvester formula, see e.g. papers by King, Manivel and (please excuse the self-publicity) Paget and Wildon.
Please, check the work of Stephen Yau published in memoirs of AMS, i believe you will find a partial answer to your questions. best Jorge
