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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 $\mathbb{P}^5$?

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    No its not a group for the same reason. But $GL_3/B$ is a projective variety. Its called full flag variety of $GL_3$.2012-07-21

2 Answers 2

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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 $h \in H$.

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    @math-visitor I have elaborated on solbap's answer below. The idea is actually very general and leads to the theory of homogeneous spaces. Your question regarding whether an analogous statement is valid in the case of three (or higher) dimensions is best tackled with the general notion of homogeneous space in mind. You need to ask, in this context, whether or not the subgroup of $\text{SL}(3,\mathbb{R})$ consisting of all unipotent matrices is the stablizer of an element of $\mathbb{R}^{3}\setminus \{0\}$ under the natural action of $\text{SL}(3,\mathbb{R})$ on $\mathbb{R}^3\setminus \{0\}$.2012-07-21
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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!

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    Thanks Amitesh! Both answers are definitely helpful!2012-07-21