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