Let $X \rightarrow Y$ be a torsor for a linear algebraic group $G$ (i.e. $X$ is a principal $G$-bundle over $Y$). Assume also that both $X$, and $Y$ are affine. What can be said about $\mathbb{C}[X]$ considered as $G$-module?
Assume now that we are given a homomorphism $\mathbb{C}^\times \rightarrow Aut(G)$. We obtain an action of $\mathbb{C}^\times$ on $G$. Assume that the algebra $\mathbb{C}[G]$ has finite-dimensional graded components. We also assume that we have an action of $\mathbb{C}^\times$ on $X$ such that the algebra $\mathbb{C}[X]$ has finite-dimensional graded components. We assume that the actions of $\mathbb{C}^\times$, $G$ on $X$ are compatible (in the sense that they glue to the action of $\mathbb{C}^\times \ltimes G$ on $X$) (the typical example is $G$ being upper-triangular matrixes and $\mathbb{C}^\times$ acting on $G$ via conjugation by regular cocharacter of diagonal matrices).
Now we have the induced action of $\mathbb{C}^\times$ on $Y$. Since graded components of $\mathbb{C}[X]$, $\mathbb{C}[Y]$, $\mathbb{C}[G]$ are finite-dimensional we can talk about dimensions of $\mathbb{C}[X]$, $\mathbb{C}[Y]$, $\mathbb{C}[G]$.
Question: is it true that $dim \mathbb{C}[X]=dim \mathbb{C}[G] \cdot dim \mathbb{C}[Y]$.
The condition that the action of $G$ on $X$ is free is equivalent to the fact that the natural morphism $G \times X \rightarrow X \times_Y X$ is an isomorphism i.e. that we have an isomorphism of algebras $\mathbb{C}[G] \otimes \mathbb{C}[X] \simeq \mathbb{C}[X] \otimes_{\mathbb{C}[Y]} \mathbb{C}[X]$ but I do not see how it helps.