TITLE: What is the space $\text{SL}(n, \mathbb{C})/\text{SL}(n,\mathbb{R})$?
QUESTION [8 upvotes]: Is a description of that space other than as that quotient known? I'm interested in the case $n = 2$ in particular.

REPLY [2 votes]: 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 $SL(2, {\mathbb R})$.