Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}\_g$ be the space of holomorphic 1-forms over genus $g$ surface. The famous $\mathrm{SL}\_2\mathbb{R}$ action on $\Omega\mathcal{T}\_g$ is defined by composing each coordinate chart of $(X,\omega)$ with matrices in $\mathrm{SL}\_2\mathbb{R}$. It is claimed that by descending to $\mathbb{H}$ and $\mathcal{T}\_g$ from their tangent/cotangent bundle, the $\mathrm{SL}\_2\mathbb{R}$ action embeds the hyperbolic plane $\mathbb{H}$ isometrically into Teichmuller space $\mathcal{T}\_g$ of genus $g$.


A few things I have learned so far:


* $\mathrm{PSL}\_2\mathbb{R}$ is identified with $T^1\mathbb{H}$ by choosing $(i,i)\in T^1\mathbb{H}$ for identity matrix. For any $g\in \mathrm{PSL}\_2\mathbb{R}$ identified with $(x,v)\in T^1\mathbb{H}$, $ga\_t$ traces the geodesic in $\mathbb{H}$ passing through $x$ with tangent vector $v$.
* Let $a\_t:=\begin{pmatrix}e^{t/2}&0\\ 0&e^{-t/2} \end{pmatrix}, t\in\mathbb{R}$. The action of $a\_t$ streches the horizontal direction and shrinks the vertical direction of $(X,\omega)$. By the Teichmuller theorem, $a\_t\cdot (X,\omega)$ traces the geodesic in $\mathcal{T\_g}$ passing through $X$ with marking $id\_X$ as $t$ varies.


So if the embedding is given by the identification of $T^1\mathbb{H}$ with $\mathrm{PSL}\_2\mathbb{R}\cdot (X\_0,\omega\_0)$ above, the curve $t\mapsto a\_tg$ in $T^1\mathbb{H}$ is sent to the geodesic $t\mapsto a\_t\cdot (X,\omega)$ in $\mathcal{T}\_g$ given that $(X,\omega)=g\cdot (X\_0,\omega\_0)$. However, the former curve $t\mapsto a\_tg$ is NOT the geodesic but a scaling of the complex number $x$ to which $g$ is identified with. The fact that a non-geodesic is sent to a geodesic contradicts the claim that the embedding of $\mathbb{H}$ is isometric. I think the issue here is that the geodesic flow on $\mathrm{PSL}\_2\mathbb{R}$ is a RIGHT multiplication by $a\_t$ while the $\mathrm{PSL\_2}\mathbb{R}$ action on $\Omega\mathcal{T}\_g$ is from left.


Where does my argument go wrong? Or do we define the identification of $\mathbb{H}$ with $\mathrm{SO(2)}\backslash \mathrm{SL}\_2\mathbb{R}\cdot (X\_0,\omega\_0)$ differently? 


I apologize in advance if this kind of questions don't belong here. The link to the same post on MSE <https://math.stackexchange.com/questions/2822006/teichmuller-disk-and-mathrmsl-2-mathbbr-action> 



Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action
I have found an answer to my question with which I am satisfied by studying the original paper of Veech: *Teichmuller curves in moduli space, Eisenstein series adn an application to triangular billiards*. He considered two actions on $\Omega \mathcal{T}\_g$. One is the $PSL\_2(\mathbb{R})$ action from left mentioned above, and the other is an action by $Aff^+(X,\omega)$, the group of orientation-preserving affine homeomorphisms from right. Although the Veech group $SL(X,\omega)$ could be defined either as stabilizer of $PSL\_2(\mathbb{R})$ action or the image of differential $D:Aff^+(X,\omega)\rightarrow PSL\_2(\mathbb{R})$, the geodesic flow on $\Omega\mathcal{T}\_g$ should be really defined as the right action of $\phi\_t\in Aff^+(X,\omega)$ whose derivative $D\phi\_t=a\_t$ instead of the left multiplication by $a\_t$. The former is exactly what Veech did in his paper. However, the survey I read defined geodesic flow in the latter way and I was confused by this technical point. 


