Let $V$ be a (not necessarily irreducible) representation of a semisimple Lie group $G$. Define an equivalence relation on $V$ using the group action in the following way: for $v_1,v_2\in V$, we say$v_1\sim v_2$ if there exists $g\in G$ such that $g.v_1=v_2$ (I'm using the module notation). I'm interested in the structure of the quotient set $V/\sim$. In particular, is there a general way to figure out how many real parameters do I need to parametrize the quotient set $V/\sim$?
Motivation for this question:
I want to numerically solve a system of algebraic equations $$F_a[\{x_i\}_{i=1}^m]=0, a=1,2,...,n,$$ and I've found out that the equations are invariant under Lie group transformations $x_i\to \psi(g)_{ij} x_j$, where $\psi$ is a certain representation of $G$, meaning that if $x$ is a solution, so is $g.x$. This symmetry should allow me to reduce the number of unknown variables, and I want to know how many unknowns are left after I quotient out this group symmetry, and if there is always a simple way to parametrize the quotient set.