I am reading the lecture notes. On page 19, let $G=SL_2$. It is said that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures on the upper half plane $\mathcal{H}$.
How to show that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures? Thank you very much.
Every element of $SL_2(\mathbb{R})$ is the product of elementary matrices $$\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad\text{and}\quad \begin{pmatrix}1&a\\0&1\end{pmatrix} $$ so it suffices to check invariance under these. In terms of complex maps $(az+b)/(cz+d)$, these are $z\mapsto -1/z$ and $z\mapsto z+a$. The invariance under translation $z\mapsto z+a$ is obvious, since $y$ stays the same. You just have to work a bit on the inversion, using the fact that $\operatorname{Im} (-1/z) = (\operatorname{Im}z)/|z|^2$, where the denominator is conveniently the modulus of the derivative of the map. Plug $z=1/\zeta$ into the differential form, and simplify.
