Left-right regular representation of an affine algebraic group
= Left-right regular representation of an affine algebraic group
The group $G\times G$ acts on $k[G]$ by
$$
((a,b)f)(x)=f(a^{-1}xb).
$$
For $G=\mathbb G_m$, this gives $k[G]=\bigoplus_{r\in\mathbb Z}k_{(-r,r)}$. For $G=\mathbb G_a$, the action factors through $(a,b)\mapsto b-a$ and the degree filtration of $k[t]$ has trivial one-dimensional successive quotients.