The bi-invariant Haar measure on the quotient $\mathrm{SL}\_2(\mathbb{Z}) \backslash \mathrm{SL}\_2(\mathbb{R}) / \mathrm{SO}\_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. An element of the quotient can be written (using the Iwasawa decomposition) as:
\[\begin{pmatrix} y^{1/2} & xy^{-1/2} \\
0 & y^{-1/2} \end{pmatrix}\]
with $z=x+iy$ in a fundamental domain of the action of $\mathrm{SL}\_2(\mathbb{Z})$ over the upper-half plane, such as $D = \{(x,y) : x^2+y^2 \geq 1,|x| \leq 1/2,y > 0\}$.
In these coordinates, the invariant measure is a scale of the hyperbolic measure, namely
\[\frac{3}{\pi} \frac{dx \, dy}{y^2}.\]
Is there a similar nice parametrisation for the rank 3 case (namely explicating the bi-invariant Haar measure of $\mathrm{SL}\_3(\mathbb{Z}) \backslash \mathrm{SL}\_3(\mathbb{R}) / \mathrm{SO}\_3(\mathbb{R})$?



Haar measure on $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$
Sure, and you can even do this for $\mathrm{SL}\_n$; I believe this goes back to Siegel. A good hands-on reference is Chapter 1 of *Automorphic Forms and $L$-Functions for the group $\mathrm{GL}(n,\mathbb{R})$* by Dorian Goldfeld.


More precisely, we have the Iwasawa decomposition $z = xy$ for $z \in \mathrm{SL}\_n(\mathbb{R}) / \mathrm{SO}\_n(\mathbb{R})$, where
\[x = \begin{pmatrix} 1 & x\_{1,2} & x\_{1,3} & \cdots & x\_{1,n} \\
 & 1 & x\_{2,3} & \cdots & x\_{2,n} \\
 & & \ddots & & \vdots \\
 & & & 1 & x\_{n-1,n} \\
 & & & & 1
\end{pmatrix}, \qquad
y = \begin{pmatrix} y\_1 \cdots y\_{n - 1} t & & & \\
 & \ddots & & \\
 & & y\_1 t & \\
 & & & t
\end{pmatrix}
\]
where $x\_{j,k} \in \mathbb{R}$, $y\_{\ell} \in (0,\infty)$, and $t = \prod\_{\ell = 1}^{n - 1} y\_{\ell}^{\frac{\ell}{n} - 1}$ (so that $\det z = \det x \det y = 1$). Then the Haar measure is
\[dz = \prod\_{1 \leq j < k \leq n} dx\_{j,k} \prod\_{\ell = 1}^{n - 1} y\_{\ell}^{-\ell(n - \ell) - 1} \, dy\_{\ell},\]
which gives $\mathrm{SL}\_n(\mathbb{Z}) \backslash \mathrm{SL}\_n(\mathbb{R}) / \mathrm{SO}\_n(\mathbb{R})$ volume
\[n 2^{n - 1} \prod\_{m = 2}^{n} \frac{\zeta(m)}{\mathrm{vol}(S^{m - 1})},\]
where $\zeta(s)$ denotes the Riemann zeta function and $S^{n - 1}$ denotes the $n - 1$-sphere, which has volume $2\pi^{n/2} / \Gamma(n/2)$.


In particular, the volume of $\mathrm{SL}\_2(\mathbb{Z}) \backslash \mathrm{SL}\_2(\mathbb{R}) / \mathrm{SO}\_2(\mathbb{R})$ with respect to the Haar measure $dz = y^{-2} \, dx \, dy$ is
\[4 \frac{\zeta(2)}{\mathrm{vol}(S^1)} = \frac{\pi}{3},\]
as $\zeta(2) = \pi^2/6$, while the volume of $\mathrm{SL}\_3(\mathbb{Z}) \backslash \mathrm{SL}\_3(\mathbb{R}) / \mathrm{SO}\_3(\mathbb{R})$ is
\[\frac{\zeta(3)}{4},\]
since $\Gamma(3/2) = \sqrt{\pi}/2$; note that there is no nice closed-form expression for $\zeta(3)$.


