Suppose we have a Lie group $G$ and two subgroups $P_1$ and $P_2$. We can then study the homogeneous spaces $M_1=G/P_1$ and $M_2=G/P_2$, and bundles on these spaces associated to representations of $P_1$ and $P_2$. For example, let us take representations $\rho_1$ and $\rho_2$ of the respective $P_i$ and imagine that we studied the associated bundles and maybe found some interesting structures.
I now want to consider the homogeneous space $M_{12}=G/(P_1\cap P_2)$. Now $\rho_1\otimes \rho_2$ is naturally a representation of $P_1\cap P_2$, and therefore there is an associated bundle on $M_{12}$. Furthermore, there are natural subspaces of $T M_{12}$ corresponding to $\mathfrak{p}_i/(\mathfrak{p}_1\cap \mathfrak{p}_2)$. In the case when $P_1$ is conjugate to $P_2$, one can interpret $M_{12}$ as a $G$-orbit in the configuration space of pairs of points in $M_1$. If there is only one orbit in this configuration space, then the subspaces $\mathfrak{p}_i/(\mathfrak{p}_1\cap \mathfrak{p}_2)$ span $TM_{12}$.
I am working on a problem in this general setup (in my case I have three $P$'s and they are conjugate parabolic subgroups), but I imagine that this is some fairly standard construction. My question is whether it is the case, and if yes, what is a good reference or some keywords to look for? Many thanks in advance.
