so let's try understanding all the coadjoint orbits and/or conjugacy classes in sl(2,reals); seems like it should be pretty easy in terms of stuff that i've already learned about...

actually i mean to focus on projective coadjoint orbits...
also i probably meant to try something like sl(3,r) rather than sl(2,r), though sl(2,r) seems like a good idea too... also sl(2,c) and sl(3,c)... i mean the complex projective coadjoint orbits in those cases, i think...

consider for example the projective coadjoint orbit of the element

0 -1
1 0

of sl(2,r).

i guess that i should think of the groups here as groups rather than as lie algebras for purposes of trying to make precise exactly what i mean by "coadjoint orbit"...

a b
c -a

a^2+bc = -1

??

??a generic coadjoint orbit in sl(2,c) is equivalent as an affine variety to the "apartment" variety?? ??or something??... hmm...this sounds like something that we might have known at some point, or perhaps should have known... because it vaguely reminds me somehow of how an element in the compact form gives not just a partial flag but also a "complementary" partial flag... or something...
hmm, some confusion here...

??the projective space of the adjoint representation can be thought of as the space of "1-parameter subgroups" of the group.... ????.....

so what about the "downward homogeneization" of an ideal in a graded commutative algebra?? hmm...

??apartment variety as open subvariety of (flag variety)^2 ...??

ok, so i think that part of what i was getting confused about is that there's a lot of "projective coadjoint orbits" that are just quasi-projective varieties rather than projective... or something like that...

but let's look at the projective coadjoint orbit of

0 0 0
1 0 0
0 1 0

in sl(3,c) ... or something...

clearly we get a flag from this, as the kernels or images of the powers. but we also get isomorphisms between the three associated grades of the flag, i think. so the coadjoint orbit seems to be 5 dimensional, with stabilizer the nilpotent part of the borel. so the projective coadjoint orbit is i guess a 4-dimensional quasi-projective variety... or something...

well, all of this is tending to make me suspect that in fact the coadjoint partial flag variety is generally going to be the only partial flag variety that gets an invariant contact distribution from being a projective coadjoint orbit... which seems good because it fits with our naive guesses coming from staring at the root systems...