Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $G$-stable.



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> Is there any "nice" description of the invariant ring $\mathbb{K}[Y]^G$ in terms of the invariant ring $\mathbb{K}[X]^G$? Maybe something like- it's a quotient or a subring of $\mathbb{K}[X]^G$.
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For my situation, $X$ is an affine space.



Invariant ring of the subvariety
As we discussed in the comments, by linear reductivity of $G$, there is a $G$-module splitting of the surjection $\mathbb K[X] \to \mathbb K[Y]$; so the restriction map $\mathbb K[X]^G \to \mathbb K[Y]^G$ is a surjection. Thus $\mathbb K[Y]^G$ is a quotient of $\mathbb K[X]^G$, as both a $G$-module and an algebra; and is a $G$-submodule of $\mathbb K[X]^G$, but (even if $G$ is trivial) need not be a subalgebra.


