so let's try more or less explicitly (but not too carefully) calculating the poset (perhaps a modular lattice or something like that?) of invariant distributions on the flag variety, in for example the a-series case to start with.

let's consider a2, with the root system labeled as in the following low-tech picture:

-a-b-
c-d-e
-f-g-

let's take abcd as the borel subalgebra stabilizing a flag, thus efg is the tangent space at that basepoint of the flag variety, and let's guess that the stabilizer-invariant subspaces of the tangent space are the down-sets in the poset "e<-g->f". there seem to be 5 such down-sets [],[e],[f],[ef],[efg] and it seems reasonable to guess that these correspond to the tangent spaces of the 6 schubert varieties based at the basepoint, with the two 2d ones doubling up here as we've already decided that they seem to be tangent to each other. on the other hand if this guess is correct then it seems that none of the schubert varieties are singular at the basepoint, as the dimensions of the tangent spaces (the cardinalities of the basis sets such as card([ef])=2) seem to match the naive dimensions of the schubert varieties.

(at some point we should also think about logical combinations of schubert varieties...)

then we should try to test these guesses some more, as well as generalize them.