Identifying $SL(2,\mathbb{C})/H$ with $\mathbb{C}^2\setminus \{ 0\}$

Question

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 $\mathbb{P}^5$?

Answer 1

Define a map $SL_2(\mathbb{C}) \to \mathbb{C}^2 - 0$ by sending $[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}] \to [\begin{smallmatrix} a \\ c \end{smallmatrix}]$.

You can check that $g$ and $g.h$ map to the same element for all $g \in SL_2(\mathbb{C})$ and $h \in H$.

Answer 2

Let $G$ be a Lie group and let $X$ be a smooth manifold. An action of $G$ on $X$ is a smooth map $\left(\cdot,\cdot\right):G\times X\to X$ such that $(g,(h,x))=(gh,x)$ for all $g,h\in G$ and $x\in X$ and $(1_{G},x)=x$ for all $x\in X$ where $1_{G}\in G$ is the multiplicative identity.

Exercise 1: Prove that the "natural action" of the Lie group $\text{SL}(2,\mathbb{R})$ on the smooth manifold $\mathbb{C}\setminus \{0\}$ is indeed an action according to the definition above (i.e., it is smooth).

Exercise 2: Let $G$ be a Lie group and let $X$ be a smooth manifold. Let there be a transitive action of $G$ on $X$. If $x\in X$ and if $H\subseteq G$ is the stablizer of $x\in X$, i.e., $H=\{g\in G:gx=x\}$, then prove that $H$ is a subgroup of $G$. We refer to the quotient $G/H$ as a homogeneous space.

(a) Prove that the natural map $G/H\to X$ given by $gH\to (g,x)$ (where we recall that $\left(\cdot,\cdot\right):G\times X\to X$ denotes the action) is a bijection.

(b) Prove that there exists a unique structure of a smooth manifold on $G/H$ such that the bijection of (a) is a diffeomorphism.

Exercise 3: Prove that the stablizer of $1\in \mathbb{C}\setminus \{0\}$ with respect to the action of Exercise 1 is the subgroup of $G$ consisting of all unipotent matrices.

We conclude that the quotient $\text{SL}(2,\mathbb{R})/H$ is a homogeneous space and this homogeneous space can be identified with $\mathbb{C}\setminus \{0\}$ by Exercise 2(a) and Exercise 3 via the action of Exercise 1. We also know according to Exercise 2(b) that there exists a unique smooth structure on $\text{SL}(2,\mathbb{R})/H$ such that this identification defines a diffeomorphism $\text{SL}(2,\mathbb{R})/H\cong \mathbb{C}\setminus \{0\}$.

I hope this helps!