The first question might be too much in general.
The cases I'd like to understand in practice are quotients (as algebraic varieties) of GL(n,C) (or SL(n,C) if you prefer) by finite subgroups. Is there anything I can say about the quotient (set of cosets)?
The babiest case would be the standard rep of the dihedral/symmetric group $S_3$. What is $GL(2,C)/S_3$? Or $SL(2,C)/S_3$ if it makes things easier?
Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what the quotient is.
Any ideas or references?