Question about the inverse operator on PSL(2,R) with respect to Liouville measure

Question

In GTM 259 chapter 9 and Katok Hasselbaltt Introduction to Modern Theory of Dynamical System chapter 5 (using the Iwasawa KAN decomposition) we see the Unit Tangent bundle of Hyperbolic half plane is identified as Lie group $$PSL(2,R) \cong TH^1:=\{(z=x+yi,v)|z\in H^2,y>0,v=y(\cos(\theta)+i\sin(\theta)) \}$$

The correspondence given as group action $$\left( \begin{array}{ccc} a &b\\ c& d \end{array} \right)\circ(z,v) \rightarrow \left(\frac{az+b}{cz+d},\frac{v}{(cz+d)^2}\right) $$ The geodesic flow can identified as right muliplication in $PSL(2,R)$ $$g_t:PSL(2,R)\rightarrow PSL(2,R)\\g_t(A)=A\left(\begin{array}{ccc} e^{\frac{t}{2}} &0\\0&e^\frac{{-t}}{2}\end{array}\right)$$ I read some paper and know there is some Jacobian way to check such geodesic flow and also the horocycle flows are invariant with the Liouville measure $\frac{1}{y^2}dx\wedge dy\wedge d\theta$

My question is how do we verify the inverse operator $$\tau: PSL(2,R)\rightarrow PSL(2,R)\\\tau(A)=A^{-1}$$ this operator preserves Liouville measure too?

I think the left mulitiplication preserving this measure can be seen from KAN decomposition. But for those flows, they are right multiplications. If we proved the inverse operator preserves measure, we can avoid the jacobian verfications.

Thank you for opinions on my question.

Answer 1

The keywords here are "Haar measure" and "unimodularity". The Liouville measure is actually translation invariant with respect to the right action of the group on itself, i.e., is a Haar measure on $PSL(2,\mathbb R)$. The latter group is unimodular, which means precisely that its Haar measure is preserved by the inversion.