# How does one write a three dimensional de Sitter space as the quotient $SL(2, C)/SL(2, R)$?

In arXiv:hep-th/0110108 the $(2+1)-$dimensional de Sitter space is represented as a quotient space of $SL(2, C)/SL(2, \mathbb{R})$. I couldn't understand how, both mathematically and intuitively, is the $(2+1)-$dimensional de Sitter space is written in such a way. If anyone could tell me it would be very useful.

Let $\mathbb{R}^{1,3}$ be the four dimensional real vector space with Minkowsky metric (quadratic form) $Q(x) = x_0^2 - x_1^2 - x_2^2 - x_3^2$ for each vector $x = (x_0,x_1,x_2,x_3) \, \in \, \mathbb{R}^{1,3}$. Define an isomorphism between $\mathbb{R}^{1,3}$ and the following four dimensional vector space of hermitian matrices $$R({1,3}) = \left\{X = \begin{pmatrix} x_0+x_3 & x_1 + i\,x_2\\ x_1 - i \, x_2 & x_0 - x_3 \end{pmatrix} \,\, : \,\, (x_0,x_1,x_2,x_3) \, \in \, \mathbb{R}^{1,3}\right\}$$ together with the quadratic form $\det(X) = Q(x) = x_0^2 - x_1^2 - x_2^2 - x_3^2$. Then the usual action (as a matrix times a vector) of the Lorentz group $SO^{+}(1,3)$ on $\mathbb{R}^{1,3}$ is isomorphic to the adjoint action of $SL(2,\mathbb{C})$ $$X \, \mapsto \, T\,X\,T^{-1}$$ for any matrix $T \in SL(2,\mathbb{C})$. In this matrix picture, the de Sitter space is defined as $$\text{de}S = \{X \in R(1,3) \,\, : \,\, \det(X) = -1 \}$$ The latter space is a homogeneous space with respect to the adjoint action (conjugation action) of $SL(2,\mathbb{C})$ and the stabilizer of each point of $\text{de}S$ is isomorphic to $SL(2,\mathbb{R})$. More precisely, fix the matrix $X_0 = \begin{pmatrix} 1 &0\\ 0 & - 1 \end{pmatrix}$ from $\text{de}S$. Then we have the following principle bundle construction $$\pi_0 \, : \, SL(2,\mathbb{R}) \, \to \, \text{de}S$$ $$\pi_0(T) = T\,X_0 \,T^{-1}$$ The stabilizer of $X_0$ in $SL(2,\mathbb{C})$ is exactly $SL(2,\mathbb{R})$ meaning that for any $S \in SL(2,\mathbb{C})$, $\,\,S \, X_0 \, S^{-1} = X_0$ if and only if $S \in SL(2,\mathbb{R})$. The latter fact means that the right action $T \mapsto TS$ of $SL(2,\mathbb{R})$ on $SL(2,\mathbb{C})$ is the action of the structure group $SL(2,\mathbb{R})$ on the total bundle space $SL(2,\mathbb{C})$ and the bundle projection $\pi_0$ is invariant with respect to this action, i.e. $\pi_0(TS) = \pi_0(T)$. Therefore $$SL(2,\mathbb{C}) / SL(2,\mathbb{R}) \,\, \text{ is diffeomorphic to the base space } \, \text{de}S$$ which is the de Sitter space.