so in exploring the idea that invariant distributions on a homogenous space g/h (for example the flag variety of a nice simple algebraic group g) serve as "infinitesimalizations" of "geometric orientations between points of g/h" (aka "(h,h) double cosets"), i got somewhat confused by the case g = pgl(2) and h = gl(1), where there seems to be an excess of geometric orientations and a shortage of invariant distributions to serve as their infinitesimalizations...

the tautological action of pgl(2) on the projective line is "sharply triply transitive" in the sense that a "frame" for the geometry is equivalent to a non-degenerate ordered triple ("infinity", "0", "1") of points on the line:

first, nailing down "the point at infinity" increases the structure from that of a projective line to that of an affine line (with adjoined point at infinity); or in other words reduces (or "breaks") the symmetry group from pgl(2) down to the group "al(1)" of affine-linear transformations of the affine line.

second, nailing down "the origin" further increases the structure from affine line to vector line; or, breaks the symmetry from al(1) down to gl(1).

finally, nailing down "1" increases the structure from vector line to the completely rigid structure of a "scaled" vector line; or, breaks the symmetry from gl(1) down to nothing.

in other words, there's a natural 4-input function f on any projective line, valued in the standard projective line, with x |-> f("infinity", "0", "1", x) being the isomorphism that takes "infinity" to infinity, "1" to 1, and "0" to 0. f is a semi-famous function known as "cross-ratio" (probably because there's an algebraic formula for it involving ratios... maybe in some sort of cross-like arrangement?). f in fact encodes the structure of being a projective line (over the unspoken base field...).

since 2+2 = 4, you can use cross-ratio to help classify the geometric orientations between nondegenerate 2-tuples of points of the projective line. so there's a lot of such orientations, in particular a continuous family of them because cross-ratio is continuous-valued.

on the other hand, there's not a lot of invariant distributions on the homogeneous space of nondegenerate 2-tuples on the projective line. there's no "non-trivial" ones. so what happened in this particular case to the general idea of studying geometric orientations by studying their corresponding "infinitesimal geometric orientations" (aka invariant distributions), that seemed so useful in the case of flag varieties?

well, i thought that i understood this pretty well at one point, back here, and that the explanation was mostly common sense but involved a certain amount of staring at pictures and/or using some mild "geometric inutition", and that the upshot was cautionary but not overly cautionary; that there are situations where the microscopic geometric orientations tell you a lot about the macroscopic ones and situations where they don't, and that it's probably a good idea to try to understand why flag varieties are one of the cases where they tell you a lot.

i think that the basic explanation of "what goes wrong" in the case of nondegenerate 2-tuples on the projective line is that there are a lot of geometric orientations to a 2-tuple p that don't allow you to get anywhere near p, or in other words whose corresponding infinitesimal geometric orientation is trivial (namely the zero invariant distribution). i think that the picture that i wound up staring at was of the level curves of the function g(x,y)=xy, and how most of them are hyperbolas that don't get anywhere near the central point (0,0). but i should try to work this out again more carefully sometime to see if i really understand it.

i have a vague feeling that so-called "geometric invariant theory" might say something interesting about the different flavors of cases here. but so might the study of orbit stacks of algebraic groups acting on tangent bundles of homogeneous varieties, which (coincidentally...?) i just started thinking about. i think that i was annoyed at first that "geometric invariant theory" stole that name, but perhaps the potentially more interesting subjects from which i was imagining it got stolen would better be called "geometric covariant theory" or something...

meanwhile, even in the case of flag varieties, where the relationship between macroscopic and corresponding microscopic geometric orientations seems perhaps the nicest, there's a lot of interesting peculiarities to the correspondence that i don't understand yet. a lot of this is presumably to be explained in terms of schubert varieties being singular or tangent to each other at the basepoint, but some of the peculiarities still seem pretty peculiar to me so far. (is either or both of "singularity" and "tangency" a special case of each other??)