I was reading a paper by Labourie https://arxiv.org/pdf/math/0401230.pdf and I'm having trouble understanding a proof. First, here's the setup.

$G = PSL(n,\mathbb{R})$ and $U$ is the subgroup of diagonal matrices.

$P^+, P^-$ are the opposite Borel subgroups (i.e. the upper and lower triangular matrices).

Let $M = G/U$ with $m_0$ the class of identity.

and we have

$H=PSL(2,\mathbb{R})$ and $A$ is the subgroups of diagonal matrices (in $PSL(2,\mathbb{R}$)).

Notice there is a flow on $H$ (the geodesic flow) defined by right multiplication by elements $\begin{pmatrix} e^t & 0\\ 0 &e^{-t} \end{pmatrix}$ in $A$.

We have an action of $PSL(2,\mathbb{R})$ on $\mathbb{R}^{n} = \mathrm{Symm}^{n-1}\mathbb{R}^2$. This gives us an embedding of $PSL(2,\mathbb{R})$ into $PSL(n,\mathbb{R})$.

now we can define a map

$F:PSL(2,\mathbb{R}) \to M$ with $g\mapsto gU$ (i.e. the left multiplication by the image of the embedding from above).

Define $E =F^* TM$ the pullback bundle.

We can equip $E$ with an obvious choice of left-invariant metric

First let $q_{m_0}(\cdot,\cdot)$ be an arbitrary metric on the fiber over $m_0\in M = G/U$.

For any $gm_0$, define $q_{gm_0}(u,u) = (g^*q_{m_0})(u,u) = q_{m_0}(g_*^{-1}(u), g_*^{-1}(u))$ where $g_∗$ the linear map from $E_{m_0}$ to $E_{gm_0}$ associated to the action of an element g. This metric is left-invariant by construction.

So far I'm OK with the proof. My problem is with the following paragraph:

We finally have a right action of $A$ on $M$ commuting with the left $H$ action, this action of $A$ preserves globally the orbit $F(H)$; the corresponding action of $A$ on $H$ is the geodesic flow, ... . We therefore obtain a right action of $A$ on $E$. If $a$ is an element of $A$ and $q$ a left $H$ invariant metric, $\tilde{q} = a^* q$ is also a $H$-invariant metric completely determined by $q$. By construction, we have $\tilde{q}_{m_0} = Ad(a)q_{m_0}$...

So the questions are

- How is this right action by $A$ defined? If it's simply right multiplication by $A$, the image of $A$ under the embedding mentioned above is entirely contained in $U$ and the action on $M=G/U$ will be trivial. I guess it can be a different action but Labourie clearly states that it is the one that corresponds to the geodesic flow on $H=PSL(2,\mathbb{R})$ (which is by right multiplication).
- When constructing $\tilde{q}$ is $a^*$ defined in the same manner with $g^*$ in the definition of original $q$ because in that case, I'm not quite sure how he's getting $Ad(a)$ at the end. How is this $\tilde{q}$ constructed exactly?