[from 2010/12/5]

was talking to john huerta about g2 and related stuff this evening...

got to wondering about... hmm, this idea is changing as i'm trying to describe it... i guess that what i'm really saying is... consider a g2 "line" as a projective line...

i guess that what i'm really saying has something to do with blowing up the basepoint singularity of the 2d schubert variety at a point of the g2 "line" grassmanian... ??or something like that??... ??and it's getting blown up into its projective line of g2 points?? or something like that??...

so... since the "base of the crown" consists of four dots, that should mean that we're interested in the line bundle over the projective line bundle whose space of sections is four-dimensional... ??or something like that?? ??and we're interested in viewing that four-dimensional space of sections as a representation of the g2 line stabilizer... ??or something like that??...

??so what _is_ this general idea that we're exploring here?? ... in the context of something like blowing up the basepoint singularity of an arbitrary schubert variety ... ???or something like that???....

at first i didn't even realize that this curve that i was getting had to be the g2 line itself... ???.... i was imagining that it might be some other weird curve... ???.... for example an elliptic curve... and then i was contemplating perhaps watching the elliptic curve vary as the ratio of the radius of the rolling ball to that of the stationary ball varies... or something like that... ??seems like we should still try to get some sort of variation like that going here?? ...hmm, but maybe that's problematic... in ways that huerta was hinting at... that for arbitrary values of the ratio you don't even get the rolling trajectory to nicely close up... and he was also hinting at some sort of "quantization condition" being involved in getting thw rolling trajectory to in fact close up... and this seems very suggestive now... that there's some sort of continuous variation going on here, but that what's really crucial is some sort of discrete variation related to that in a "quantization" way... ??or something like that?? ... have to try to work this out...

is there something going on here about ... ??interpreting parabolic subgroups as automorphism groups (involving both base and fiber... morally like a wreath product or something???....) of certain natural vector bundles, or something like that??? ???something about... ???the mysterious or at least confusing "extra stuff" on which a parabolic subgroup acts?? ??or something like that??....
??something about ... "quasi-tautological bundle over partial flag variety" ... ???or something??? hmmm.... ???and so forth.... ???.....

so how does this (??...) fit into the whole "invariant distribution" game?? ... or something... ???...

one of the themes of the discussion was that john wanted to talk a lot about the 2d schubert variety (and its infinitesimalization at the basepoint) on the g2 point grassmanian while i wanted to talk a lot about the 2d schubert variety (and its infinitesimalization at the basepoint) on the g2 line grassmanian... and it took us a while to remember that these are dual to each other and to contemplate exploiting that duality in certain ways... so maybe i should consciously try applying that duality here now... so... ??what's going on here?? are we getting some alleged 3d rep of the point stabilizer?? ... and so forth... ??seems like we're getting some pretty vanilla line bundle over the "dual" projective line of a g2 point... the g2 projective lines through it... ??....