Link Groups – Homogeneous Space for Intersecting Subgroups

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 with representations of $$P_1$$ and $$P_2$$. For example, let's take representations $$rho_1$$ and $$rho_2$$ respective $$P_i$$ and imagine that we have studied the associated packages and perhaps found 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 of course a representation of $$P_1 cap P_2$$, and so there is an associated package on $$M_ {12}$$. In addition, there are natural subspaces of $$T M_ {12}$$ correspond to $$mathfrak {p} _i / ( mathfrak {p} _1 cap mathfrak {p} _2)$$. In the case where $$P_1$$ is conjugated to $$P_2$$we can interpret $$M_ {12}$$ like a $$G$$-orbit in the configuration space of the point pairs in $$M_1$$. If there is only one orbit in this configuration space, the subspaces $$mathfrak {p} _i / ( mathfrak {p} _1 cap mathfrak {p} _2)$$ span $$TM_ {12}$$.

I am working on a problem in this general configuration (in my case, I have three $$P$$and they are conjugated parabolic subgroups), but I imagine that it's a fairly standard construct. My question is if this is the case and if so, what is a good reference or keywords to look for? Thank you in advance.