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. 



Polynomial invariants for simple algebraic groups
Generally, one has $\dim V//G=\dim V-\dim G$ but there are exceptions. For simple $G$ the exceptions have been classified in


Èlašvili, A. G. Canonical form and stationary subalgebras of points in general position for simple linear Lie groups ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=304554) [English article](https://link.springer.com/article/10.1007/BF01075509)). Funkcional. Anal. i Priložen. 6 (1972), no. 1, 51–62


Popov, A. M. Stationary subgroups in general position for certain actions of simple Lie groups ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=425013) [English article](https://link.springer.com/article/10.1007/BF01075537)). Funkcional. Anal. i Priložen. 10 (1976), no. 3, 88–90.


Generally, the quotient $V//G$ is singular. For simple $G$, the representations with $V//G$ being smooth (i.e. isomorphic to an affine space) were classified in


Schwarz, Gerald W. Representations of simple Lie groups with regular rings of invariants ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=511189) [article](https://link.springer.com/article/10.1007%2FBF01403085)).
Invent. Math. 49 (1978), no. 2, 167–191


Concerning your example $G=Sp(2n)$ and $V=L(\omega\_1)⊕L(\omega\_1)⊕L(\omega\_2)$: Here $V//G$ is smooth of dimension $2n-1=\dim V-\dim G$ (see the paper of Schwarz above). The ring of invariants is freely generated by polynomials of multidegrees
$(0,0,d)$, $d=2,\ldots, n$ and $(1,1,d)$, $d=0,\ldots,n-1$. The quotient map $V\to V//G$ is equidimensional (by another paper of Schwarz).


