i've been taking a pretty unsystematic approach (as usual) to the relationship between "geometric orientations" and invariant distributions, in part because in the case of flag varieties (as usual) an unsystematic approach seems to pay off, due to the humanly accessible rich-but-not-too-rich combinatorial flavor of the subject.

a more systematic approach though might include exploring the different possibilities more thoroughly. for example consider the case of the homogeneous space g/h of what i call "frames", where h is the trivial subgroup. then the atomic invariant distributions are the points of the projective space of the lie algebra, while the geometric orientations are essentially the elements of g ...

perhaps one of the distincive aspects of flag varieties is their "almost doubly transitive" nature... i just tried googling on "almost doubly transitive" and there were some hits but i couldn't tell right away how relevant they were...