"stabilizer subalgebra of point of coadjoint orbit depends only on what wall of weyl chamber point lives on, whereas stabilizer subalgebra of corresponding point of conjugacy class depends on what wall of weyl alcove the original point lives on" .... ???or something like that ...

???here talking about in the compact form, for the moment?? ???perhaps the compact, connected, simply connected form??? ... or something ...

??so what about something about ... ???the homogeneous spaces in question ???

???the coadjoint orbit ones as just the partial flag varieties of the complex form ??? or something ??? .... ???but what about the conjugacy class ones??? are those still just certain partial flag varieties of the complex form ?????? .... and so forth ... ???...