Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V_1\oplus \cdots \oplus V_n$ where $V_i$'s are irreducible representations. Let $I:=\mathbb{C}[V]^G$ denote the space of invariant polynomials. We know that $I$ is a finitely generated $\mathbb{C}$-algebra. Let $V/\!/G:=\mathrm{Spec}(I)$. Then $V/\!/G$ is an affine variety.

Question: What do we know about $V/\!/G$? For instance, for which pairs $(G,V)$ do we know the dimension of $V/\!/G$? For which pairs $(G,V)$ is $V/\!/G$ isomorphic to an affine space?

The most familiar case is when $V$ is the adjoint representation, in which case, $V/\!/G$ is an affine space of dimension equal to the rank of $G$. Vinberg's invariant theory for finite gradings of Lie algebras provides a generalisation. I'm looking for more examples, or a general theory if there is one.

Of particular interest to me is $G=\mathrm{Sp}_{2n}$ and $V=L(\omega_1)\oplus L(\omega_1)\oplus L(\omega_2)$ where $\omega_i$ is the $i$th fundamental weight.