# Iterated quotients in GIT

Suppose that $$G$$ is a reductive group that acts algebraically on an affine variety $$X$$ over an algebraically closed field $$k$$.

Suppose also that $$G$$ is equipped with a normal abelian subgroup $$N$$ such that the quotient group $$G/N$$ is finite abelian.

Example: $$G = \text{O}(2,k)$$ and $$N= \text{SO}(2,k)$$, and $$k = \Bbb C$$ is the complex numbers.

My question: Under what circumstances can I conclude that a categorical/good quotient $$X/\!\!/G$$ coincides with a categorical/good quotient for $$G/N$$ acting on $$X/\!\!/N$$?

That is, under what circumstances is there a birational equivalence between $$X/\!\!/G$$ and $$(X/\!\!/N)/\!\!/(G/N) \,\, ?$$ If there is such a result, can someone point me to a place in the literature where it can be found?

• I'm confused about your condition. You seem just to be assuming that $G$ is an extension of $T$ by a finite, commutative group, which is either automatically reductive if you don't assume that reductive groups are connected (and then do we know that $X//G$ exists?), or never reductive unless $G$ equals $T$. In either case, what's the point of invoking reductivity? \\ See also Supposed generalization of $X/(G \times H)\simeq (X/G)/H$ for GIT-quotients. Commented Oct 29, 2023 at 17:09
• I have edited the question. Thanks. Commented Oct 29, 2023 at 18:20
• Doesn't this follow simply from $(A^N)^{G/N}=A^G$ for invariants? Commented Oct 30, 2023 at 9:07