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.