Suppose given a variety $X$ over an algebraically closed field $K$, $\mathrm{char}K = 0$, equipped with an action of a reductive group $G$. Suppose also that $X$ admits a good quotient $p\colon X\to Y:= X//G$. How can we describe the algebra of differential forms on $Y$? A naive guess is that at least when $X$ and $Y$ are both non-singular the natural map

$$ \Omega^\bullet_Y \to (p_*\Omega^\bullet_X)^G $$

is an isomorphism. Is this guess true? A similar question for tangent sheaves: if we denote by $\mathcal G$ the subsheaf of $\mathcal TX$ generated by the image of the Lie algebra of $G$, is it true that

$$ \mathcal TY \simeq p_*(\mathcal TX/\mathcal G) $$