so let's proceed to the alleged calculation of the modular lattice of invariant distributions on the b2 flag variety.

abc
def
ghj

take abcde as flag stabilizer borel. now we want (we think) the down-sets in the poset
where j dominates h which in turn dominates f and g, namely [],[f],[g],[fg],[fgh],[fghj].

hmm. i don't feel at the moment too much like trying to guess which schubert varieties these might be the zariski tangent spaces of. not yet.

let's try the case a3. we want the down-sets in a certain poset whose elements are 12,13,14,23,24,34. maybe it's just the "interval containment" order? it should be easy to actually see what the correct order is, but instead let's just use this guess for now. the down-sets are [],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13],[13 34],[24],[12 24],[13 24],[14]. (i'm just listing the "generators" of the down-set.) hmm, suspiciously i seem to have just listed 14 such down-sets, which of course was my main guess as to what would happen here, what with the catalan numbers showing up in so many places. but is there any obvious nice way to biject these down-sets with something else that we know of offhand that's counted by the catalan numbers?

perhaps given a "partially bracketed string", you can get a down-set in the interval poset of the string by taking the intervals which "don't violate the bracketing" in some hopefully obvious sense... or something like that... ?? hmm, i don't think that that actually makes sense yet.