so let's consider for example the g2 root system:

===a===
b=c=d=e
=f=g=h=
j=k=m=n
===p===

with borel subalgebra abcdefg; the invariant distributions on the flag variety are the submodules wrt this subalgebra of the complementary hjkmnp, which it seems reasonable to guess are essentially the down-sets in the poset p>n>m>k>h,j; namely [],[h],[j],[hj],[khj],[mkhj],[nmkhj],[pnmkhj].

now let's consider those of these submodules that contain h, or those that contain j. there are six of each... ??...

for comparison purposes let me try b2 here as well...

abc
def
ghj

borel abcde, down-sets in j>h>f,g = [],[f],[g],[fg],[hfg],[jhfg]

containing f = [f],[fg],[hfg],[jhfg]
containing g = [g],[fg],[hfg],[jhfg]

hmm... ???....

might as well do a2 as well...

ab=
cde
=fg

borel abcd, down-sets in g>e,f = [],[e],[f],[ef],[efg]

containing e = [e],[ef],[efg]
containing f = [f],[ef],[efg]


i'm somewhat confused at the moment about how the b2 calculation above relates to our previous alleged calculations about the singularity of the 2d schubert variety on one of the b2 grassmanians... ??...