??so w_a_ sa maybe bumping into langlands duality ?? .... os... sa "encoding dynkin diagram as extra dot coxeter diagram" .... sa the extra dot as relating to sa... affine reflection taking zero "root" to some other root .... ???os??? ..... asf os.... ????.... "affine weyl gp" .... ???os??? .... asf os... ??w_a_ the clay-and-toothpick model on my desk??????? asf os.... ?????..... the color scheme .... ?????asf os????...... ????sa... ????long root os???...... ???????sa..... ???????root ct weight or root ct coxeter vertex... ???os????? ....... asf os.... ???????.....

hmmm..... ??? sa extra coxeter vertex "at middle"????? os?????...... asf os... ????.....

???wa sa whether "langlands dual" lives at level including gl(1) (??? or something... sa "abelian" langlands reciprocity.... ???asf os??...) and/or even "the additive group" ??? os??? .... asf os... ???


??finding interesting stuff in old paper notebook #287 from october of 2002...
stuff about g2 and octonions and rolling ball (something about on projective plane...), and getting nice basis for tangent space of partial flag manifold by something about "hemispheres that the figure (?= coxeter vertex?) lies in", and so forth... more relevant to point of moment though, is... ??some stuff about affine weyl groups and so forth.... ??idea that "the blacks are the roots" ...???or something like that?? ...that seems pretty helpful/crucial or something,,,

??some "grand overlay" stuff here??? .... ???...

hmm, there really _is_ a pretty grand overlay picture here... in margin there's a quincunx and a crown ... don't see explicit "roll, spin, skid" yet, though...

??hmm, so w_a_ sa "langlands duality" and "root ct weight" and sa kac-moody?? ... os, asf os... ???... ??transpose of cartan matrix?? ... asf .... ??? ???hmm, so w_a_ sa "co-root" os, asf os??...

hmmm... i found this:

"???so... ??is it maybe the case that ... ??the extended coxeter complex "for" (?...) a loop group g-tilde has black dots forming a lattice which looks like the root lattice for the _dynkin dual_ of g?... ?...o_s_? ??..."

...with a "??hopefully not?..." in the margin... ???also a "still not quite clear" ... ????....

??...also sa:

"??so wa sa a prescription for adding the extra dot (os...) that goes something like this:

?? "find the (...) parabolic that centralizes a short (??hmm...) root, and attach the extra dot so that it's touching just the dots out of that parabolic...." ... ?os..."

???...

??hmm, so wa langlands duality and short/long reversal here???? os... asf os... ?????....



???hmmmm.... ????sa voronoi cells of root lattice os??? as... ??what???....
??some nice thing with kaleidoscope symmetry, but _what_?? and is there some "langlands duality" here, os??? ...asf os...

???maybe sa "coxeter supertile associated to extra dot" ???? os???....

wait a minute, i still think that there's some voronoi cell thing going on here, but maybe it's not quite the root lattice... staring at some of these pictures in #287... ??maybe in the _good_ picture it _is_ the root lattice?? os??? ... g2 case ... os ... ???...

hmm, perhaps more salient than "voronoi cell" here is ... ??well, some sort of fundamental domain for the "affine kaleidoscope" group acting on the root lattice space... ??something about some canonical way of getting a nice fundamental domain or something??


????wa sa "systematically discretizable polytope" vs "tile polytope" here???? os???.... asf os... ???...


hmmm... reference to pressley & segal p16-17 ... ??...

???sa "co-root lattice and weight lattice as dual" ???? os???....

??hmm, sa qa326 h83, p2... manin... "moduli space of curves of arbitrary genus : virasoro gp :: partial flag variety : semi-simple lie group" ... ????os??? ...
and me asking what in the world that's supposed to mean ... ??...

??so... the coadjoint partial flag variety tends to be "very partial", so the corresponding parabolic subalgebra tends to be close to maximal... ??which maybe fits with the idea that the extra dot tends to touch not that many other dots?? ... or something...

??so _is_ it clear whether "langlands duality" is really showing up here (?..) ??
??maybe it's _not_ ?? because... ??... hmmm... ??seems like it shouldn't be that hard to check ... ??? ...unless the killing form maybe causes confusion here... ??...

??hmm, so wa sa semi-direct product of kaleidoscope group acting on "heisenberg lattice" of root system here??? or something...