(Co)tangent sheaves to good quotients

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

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

• Consider $X=G$ with $G$ acting by translation, then $p\colon X\to Y=\mathrm{pt}$ is a good quotient. In this case, the map which you wanted to be an isomorphism is $0 \to \mathfrak{g}^*$ (the zero space mapping to the dual of the Lie algebra of $G$). Commented Oct 15, 2019 at 9:18

The statement I made in the question is obviously false as Piotr pointed. However, the following holds when both $$X$$ and $$Y$$ are non-singular:
$$\Omega^\bullet Y \simeq (p_*(Ann(\mathcal G)\subset \Omega^\bullet X))^G$$