What is the space $\text{SL}(n, \mathbb{C})/\text{SL}(n,\mathbb{R})$?

Question

Is a description of that space other than as that quotient known? I'm interested in the case $n = 2$ in particular.

Answer 1

You can identify $SL(n,\mathbb C)/SL(n,\mathbb R)$ with the set of real structures on the complex vector space $\mathbb C^n$. Here a real structure on a complex vector space $V$ is defined as a map $\sigma:V\to V$ such that $\sigma^2=id_V$ and such that $\sigma(\lambda v)=\overline{\lambda}\sigma(v)$ for all $v\in V$ and $\lambda\in\mathbb C$. The standard real structure on $\mathbb C^n$ is given by complex conjugation of the coordinates of a vector.

Now you define an action of $SL(n,\mathbb C)$ on the set of real structures on $\mathbb C^n$ by letting $A\in SL(n,\mathbb C)$ act on $\sigma$ via $A\cdot\sigma:=A\circ\sigma\circ A^{-1}$. One directly shows that this action is transitiv (i.e. that any real structure is given by conjugating the coordinates with respect to some complex basis of $\mathbb C^n$). Moreover, $A$ satbilizes the standard structure if and only if $A\bar v=\overline{Av}$ for all $v\in\mathbb C^n$ and looking at the elements of the standard basis, you conclude that this is equivalent to all entries of $A$ being real. Hence the stabilizer of the standard real structure is $SL(n,\mathbb R)\subset SL(n,\mathbb C)$ which completes the argument.

Alternatively, you can identify real structures with totally real subspaces in $\mathbb C^n$ of real dimension $n$, i.e. subspaces $W$ such that $W\cap i\cdot W=\{0\}$. These can be identified with real structures via the fixed point set.

Answer 2

For $n=2$ your space is (unnaturally) homeomorphic to ${\mathbb R}\times S^2$. More naturally, it is the 3-dimensional anti-de Sitter space $adS_3$, aka the 1-sheeted hyperboloid in ${\mathbb R}^{1,3}$. To see this, consider the natural action of $SL(2, {\mathbb C})$ on $adS_3$ and note that point stabilizers are conjugates of $SL(2, {\mathbb R})$.