What is the Haar measures on $SL(2, \mathbb R)$ And $SL(2,\mathbb R) / SL(2,\mathbb Z)$? 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?
 A: 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.
A: You can parameterize $PSL(2,\mathbb{R})$ by its action on the unit tangent bundle of the upper half-space model $\mathbb{H}$ of the hyperbolic plane. Under this parameterization, Haar measure is the Liouville measure.
You can then get the Haar measure on $SL(2,\mathbb{R})$ by pulling back under the double covering map $SL(2,\mathbb{R}) \mapsto PSL(2,\mathbb{R})$.
Added: To get the Haar measure on $SL(2,\mathbb{R}) / SL(2,\mathbb{Z})$, you use the orbit map 
$$f : SL(2,\mathbb{R}) \mapsto SL(2,\mathbb{R}) \, / \, SL(2,\mathbb{Z}) = Q 
$$
This map $f$ is a regular covering map with deck transformation group $SL(2,\mathbb{Z})$ acting by multiplication from one side (the notation should perhaps be $Q = SL(2,\mathbb{Z}) \, \backslash \, SL(2,\mathbb{R})$ because usually one thinks of the orbits as defined by left multiplication of elements of $SL(2,\mathbb{Z})$, and in that case one should be using the left invariant Haar measure). Therefore, $Q$ has a basis consisting of all open subsets $U \subset Q$ that are evenly covered by $f$. 
For each such $U$, one uses the definition of an even covering to write $f^{-1}(U)$ as a disjoint union
$$f^{-1}(U) = \cup V_i
$$
such that for each $i$ the restriction $f \bigm| V_i$ is a homeomorphism from the open subset $V_i \subset SL(2,\mathbb{R})$ onto $U$. Since the covering is a regular covering, and since Haar measure on $SL(2,\mathbb{R})$ is invariant under the deck transformation subgroup $SL(2,\mathbb{Z})$, all of the $V_i$'s have the same Haar measure, and one defines the Haar measure of $U$ to equal the Haar measure of any of the $V_i$'s. Then there's a formal process in measure theory that one goes through, to extend this to a Haar measure defined on all Borel subsets of $Q = SL(2,\mathbb{R}) \, / \, SL(2,\mathbb{Z})$.
