Let $G=SL(2,\mathbb{C})$ and let $H$ be the set of unipotent matrices $ \left\{ \left[ \begin{array}{cc} 1 & b \\ 0 & 1 \\ \end{array} \right] : b\in \mathbb{C}\right\}. $
I am trying to work out the details to show that $G/H$ can be identified with $\mathbb{C}^2\setminus \{ 0\}$ via the transitive action of $G$ on $\mathbb{C}^2\setminus \{ 0\}$.
This action is then supposed to extend to a linear action on its projective completion $\mathbb{P}^2=\overline{G/H}$ where we take the point $[1:1:0]$ to represent the identity coset $H$.
Any help/suggestion is greatly appreciated. Thank you.
Added: note that $\dim SL(2,\mathbb{C})/H$ is clearly 2, but I am not certain of how $G/H$ and $\mathbb{C}^2\setminus \{ 0\}$ can be identified (i.e., construct an explicit map between the two).
Just continuing to think about the above question, for $SL(3,\mathbb{C})/H$ where $ H =\left\{ \left[ \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 &1 \\ \end{array} \right] : a,b,c\in\mathbb{C} \right\}, $ can we conclude that $SL(3,\mathbb{C})/H$ also acts transitively on some subset $S$ of $\mathbb{C}^5$, and identify $SL(3,\mathbb{C})/H$ with $S$? Would then the projective closure $\overline{SL(3,\mathbb{C})/H}$ equal $\mathbb{P}^5$?