Let $G$ be an algebraic group and $T$ the maximal torus. Suppose that $T$ acts on $G$. Do we have a multigrading on $\mathbb{C}[G]$? How to define the multigrading corresponding to the $T$-action? Thank you very much.
Edit: $\mathbb{C}[G]$ is the coordinate ring of $G$.
Suppose that $T$ acts on $G$. Then we have an action $T \times G \to G$. This gives a coation $\varphi: \mathbb{C}[G] \to \mathbb{C}[T] \otimes \mathbb{C}[G]$. An element $f \in \mathbb{C}[G]$ is called homogeneous of degree $\lambda \in \mathbb{C}[T]$ if $\varphi(f) = \lambda \otimes f$. Therefore there is a multigrading on $\mathbb{C}[G]$ corresponding to the $T$-action.