TITLE: What is the Haar measures on $SL(2, \mathbb R)$ And $SL(2,\mathbb R) / SL(2,\mathbb Z)$?
QUESTION [5 upvotes]: How does one parametrize those spaces in order to do integration over them?
What's a good reference for doing integral a with Haar measures over matrix groups?

REPLY [7 votes]: An element of $SL_2(\mathbb{R})$ can be written as ($NAK$)
$$
\left(\begin{array}{cc}
1&x\\
0&1\\
\end{array}
\right)
\left(\begin{array}{cc}
y^{1/2}&0\\
0&y^{-1/2}\\
\end{array}
\right)
\left(\begin{array}{cc}
\cos\theta&-\sin\theta\\
\sin\theta&\cos\theta\\
\end{array}
\right)
$$
with $z=x+iy$ in the upper half-plane.  In these coordinates, the invariant measure is
$$
d\theta \frac{dx dy}{y^2},
$$
the product of the invariant measures on the two subgroups, hyperbolic area (upper half-plane model) and arclength on the circle.  See Lang's book $SL_2(\mathbb{R})$ for more details.