For a fixed positive integer $n$, the group $G=GL_n(\mathbb{C})$ acts on the field $K=\mathbb{C}(t_1,\ldots,t_n)$ by linear change of variables. I would like to know if there is something like a Galois correspondence for this action. Specifically:
Is there a description of which subfields of $K$ are of the form $K^H$ for some subgroup $H\leq G$?
Is there a description of which subgroups of $G$ are of the form $$ G_L:=\{g\in G:g|_L=\mathrm{Id}_L\} $$ for some subfield $L\subset K$?
A few observations: For the first question, any subfield of the form $K^H$ contains $\mathbb{C}$ and is invariant under the diagonal action of $\mathbb{C}^\times$. I wonder if these conditions are sufficient.
For the second question, every subgroup of the form $G_L$ is Zariski closed, but this is not sufficient. For instance, the subgroup of diagonal matrices cannot be the stabilizer of any subfield, since anything fixed by the diagonal is fixed by all of $G$.
