so given a partial flag x, let's consider how the stabilizer subgroup of x in the automorphism group g of the flag building (or perhaps in some related group g'??...) intersects the conjugacy classes of g (??or of g'?? ...or something like that...we should also consider coadjoint orbits and so forth here)...

i think that there's some tendency towards... the stabilizer subgroup intersecting just those conjugacy classes that are shaped like the partial flag variety in which x lives, and intersecting each such class in precisely one element, so that the stabilizer subgroup is actually a classification of those such classes... except that this is almost certainly wildly over-(not to mention mis-)stating the case... (for example consider the conjugacy class of the identity element...)... the vague intuition that's leading me in roughly this direction has to do with the idea of "a structure canonically inducing an automorphism of itself" that i learned about from david joyce's papers on quandles... and with the special case of classical grassmanians as hermitian symmetric spaces (which idea we've been running into again lately as a very degenerate example of the theory of invariant distributions on partial flag varieties... ??maybe that's again related to what we're trying to get at here??...)

(??some of what we're thinking about here could be explored in a very austere "pure group-theoretic" context... starting with an arbitrary coset and seeing how it intersects the conjugacy classes of the group... not sure how this might go... again might relate to joyce's (and/or freyd's and/or yetter's...) ideas...??)

(there's also another (??) whole complex of vaguely remembered ideas that this is reminding me of...???... from several years ago.... the idea of "classes of conjugacy classes" and so forth... and how this relates to stuff like "geometric quantization vs [borel-weil theorem and/or verma modules and/or certain other ideas that i can't quite articulate right at the moment]" (??resp "coadjoint orbits vs partial flag varieties" and so forth...) and "k-a-n decomposition" (and/or stuff like that) and hecke operators and young diagrams and so forth... ??vaguely remembered conversation with allen knutson... hmmm.... i think that part of it had to do with partial flag varieties that correspond to different subsets of the dot-set of the dynkin diagram, but which are "equivalent via hecke operator" (or something like that... (??i guess that this also relates to stuff like understanding the kernel of the homomorphism from the burnside ri(n)g to the "green ri(n)g", though i don't think that it was stored in my memory that way just now...) ...of course all of this is tied in (in my memory at least, but probably in more than just that) with weird stuff about hecke operators and general sorts of "braids" (for example in an artin-brieskorn-coxeter sense...hmmm, this is leading me towards yet other whole huge complexes of ideas... "cardinal braid" ... (??for example "kleinian"??) singularity... milnor... "multi-homogeneous..." ... hyperplane quandle... schubert calculus... ?????...... ??????....)

anyway, to try to get back to the vague humble idea that i started with here... motivated in part by trying to visualize elements of g2 as explicit automorphisms of the geometry of a fermion rolling on a projective plane of 3 times the radius... let's consider some configuration x of this rolling fermion... perhaps we should identify this with the octonion "(j,j)"... and let's consider the stabilizer of x... so this is a 9-dimensional parabolic subgroup... or something like that... ??so am i really suggesting that there's a 9-dimensional space of conjugacy classes that are shaped like the configuration space in which x lives?? ...i think that that is what i was suggesting, and that it's not probably not completely wrong-headed, but screwed up in certain ways... if we considered a total flag instead of a partial flag here, then it's stabilizer would be an 8-dimensional borel subgroup, and... sorry, i think that i'm just trying to remind myself here of some basic numerology concerning dimension of the space of coadjoint orbits...

while i'm trying to straighten that out, let me also mention another source of confusion here... i'm being guided by certain intuitions concerning how partial flag varieties of the _complex_ form of a simple lie group relate to conjugacy classes of the _compact_ form... which for some (mysterious to me at the moment...) reason gives a very simple version of one of the games that i'm trying to play here... not sure how much more complicated the version that i'm trying to play is... ??...

ok, sorry, there's a lot of mistakes here, and/or obvious omissions that i forgot about... i'll try to fix some of this...