a?? so having gotten some of the numerology a bit straighter let's try thinking a bit more about gl(2) .... ??? .....

q=2

6 = (1^2)*1 + (1^2 + 2^2)

q=3

48 = (2^2)*3 + (1^2 + 3^2)*2 + (4^2)*1

q=4

180 = (3^2)*6 + (1^2 + 4^2)*3 + (5^2)*3

?? hmmm ... ?? seemingly obvious naive numerological guess about which part of middle term lives over pgl(2) .... ?? seems to not work ??? .....

?? pgl(2,f_3) .... ???? .... ??? acting faithfully and transitively on 4 points of projective line .... ???? just 4! ?? ....

4 ?? 1

31 ?? 3

22 ?? 2 ?????

211 ?? 3

1111 ?? 1


gl(2,f_2) 3 irreps ...

gl(2,f_3) 8 irreps ...

gl(2,f_4) 15 irreps ....

... ???? .....


?? in principle it's just the cuspidals here (?? gl(2,f_q) for generic q ...) that we don't "understand" so far ?? ....

??? tensoring with 1d reps here (...) ... ??? ... ?? whether this puts some order into the variety of cuspidals here ... ?? ....

?? hmmm, trying to understand these 1d reps better ... ?? hmm, by 2 approaches ....

1 abelianization ...

2 ?? "vector bundle over projective line assigning to 1d subspace x f(x)#g(v/x) where f and g are functors from _1d f_q-vsp_ to _1d c-vsp_" .... ??? .... ?? then understanding "q-braiding" here and modding out by it .... ??? .... "hecke operator / algebra ..." .... ????? .....

?? abelianization and determinants ???? ....

?? abelianization of gl(n,field) vs of pgl(n,field) ?? .... ??? .....

?? abelianization of gl(n,field) as gl(1,field) ??? ..... ????? ....

?? "algebraic k-theory" ... ???? ...... ?????? ..... ???? ....

?? is "projectivization" always finer than "abelianization" ??? .... ?? meaning ... ?? ... when you abelianize gl(n,field), does it always happen that the constants get killed off ?? ..... ???? .... ?? meaning, maybe (?? ...) that the determinant of a constant is ...... ???? ......

gl(2,f_3) .... ????

?? here it really is true that the constants are determinant 1 ??? ..... ???? ....

?? but that seems like an extremely special case .... ?? ....

?? did i ask the question anywhere close to correctly ?? ... ???? ...

?? maybe we had an original intuition here (?? ...) that was right-track ... ?? that _of course_ determinant wouldn't survive too well to projective level .... ??? .... ??? .....

?? for some reason i feel tempted to say "determinants of constants form an obstruction to determinant surviving to projective level" .... ???? does that make _any_ sense ??? .... ???? ....

?? take a look at 1d irreps of gl(3,f_q) as somewhat classified in recent post here ... ??? ... ??? see if fits with "determinant" idea ??? .... ???? ....

?? yes, does seem to fit fairly straightforwardly ?? ... only 1d irreps that show up seem to be "applying trivial 3-box young diagram to gl(1)-cuspidals" .... ???? .... ??? hmm .... ??? idea that .... ?? abelianization of gl(n) is gl(1) via determinant (?? which idea maybe i'm beginning to remember some subtleties to ... ?? maybe only in beyond-field case ?? ... anyway never mind that for a moment ... ?? ...) as embodied here as "process of applying trivial n-box young diagram to gl(1) irrep" ... ??? .... ???? .... ????? ... ?? suggestive somehow ?? ... ???? .....

?? maybe enough progress here to satisfy original goal of being ready to take a look at what bump says about this stuff ... ?? though not yet running into any obvious way in which kinds of things bump seemed to be about to say ("induced rep from maximal torus rep" ?? ... ??? ...) show up ..... ???? .....

?? size numerology of maximal toruses here ??? ...

?? field = f_q .... ??? but ... ??? trying to see where field = c (or r ....) fit in here .... ??? .....

?? "maximal torus" in gl vs pgl vs sl case .... ???? .....

?? diagonal torus of gl(n) ..... ???? .....

q=2

diagonal torus 1 ... gl(2) 6

?? so 6d induced rep ?? ... ?? but maybe breaks into smaller irreps ??? ...

?? ok, so when bump talks about "induced reps", maybe they're really talking about "corresponding" irreps .... ?? how much does that help to straighten stuff out here ??? ..... ????? .....

?? so one stupid guess that doesn't quite make sense yet is something like ... ?? ... ?? bump is setting aside the cuspidal reps (my first term above ... ?? ....) for the moment (?? ...), and associating my second term with "unsplit torus", and my third term with "split torus" .... ...... ?? really not seeing it yet .... ????? .....

q=3

diagonal torus 4 .... gl(2) 48

??? 12 ..... ???? .... 4*3 ???

q=4

180 / 9 = 20 = 5*4

15*12 = 5*3*4*3 ..... ????? .....

?? "generic flag pair" .... ????? ......

?? vector bundle over [flag variety]^2 ??? .....

??? alleged other sort of "maximal torus" ..... ????? .....

?? splitting of unsplit torus ..... ????? ......

?? trying to associate "vector bundle over [flag variety]^2" with my third term as ... ?? seeming rather iffy and relying on special coincidences ... gl(2) vs gl(n) for more general n .... ???? .... ???? ....

?? mckay correspondence for binary tetrahedral group ..... ???? .... ?? is that e6 ?? .... bipartite graph .... ???? ...

?? having some trouble trying to get mckay correspondence numerology to work here .... ????? ... (actually having trouble at the moment getting it to work anywhere, so don't hold it against here too much .... ?? ...)

?? elt of gl(2,f_q) that .... ?? arises from .... ??? putting structure on [f_q]^2 making it into an f_[q^2] ... ??? and then .... ??? .... ?? picking generator for cyclic group gl(1,f_[q^2]) ... ?? and considering group it generates ... ??? hmmm, i guess that that's pretty much like saying ... ?? consider gl(1,f_[q^2]) -> gl(2,f_q) arising in hopefully obvious way .... ???? ....

?? so one stupid guess is that that's what an "unsplit (?? maximal ... ?? ....) torus" is ... ??

?? "ramified torus" ???? .... ???? gl(1,f_q[t]/t^2) -> gl(2,f_q) .... ?????

?? gl(1,z/p^2) -> ????? ..... ?? analog for q in place of p ??? .... ????? .... ??? ...

?? split torus and (q-1)^2, vs unsplit torus and q^2-1 ??? ..... ?? anything like that show up in our classification yet ??? .... ???? ......

?? hmm, almost getting a numerological glimmer of ... ?? cuspidal : unsplit :: non-cuspidal : split .... ???? ... try testing it out a bit further ... ?? or maybe generically .... ??? .....

(q^2-1)*(q^2-q) = (q-1)^2*(q*(q-1)/2) + (1^2 + q^2)*(q-1) + (q+1)^2*((q-1)*(q-2)/2)

?? the glimmer argument is something like ... ?? ... dividing left-hand side by unsplit torus size gives q^2-q, and factors of that get squared in first term ... ?? more specifically, the factor q-1 does .... ??? whereas dividing left-hand side by split torus size gives q^2+q, and factors of that get squared in last two terms .... ??? .... ?? so _is_ this a good hint of what bump's talking about ??? .... ???? .....

?? well, bump does say this :

The representations parametrized by maximal split tori are induced representations, those parametrized by nonsplit tori must be constructed by some other method.

... so yeah, it seems like we're on the right track here .... though in retrospect maybe it should have been somewhat obvious ... ?? ...

?? counting characters here .... ??? also trying to develop correspondences with real and complex cases .... ???? ....

?? not quite getting the non-squared numbers to correspond nicely to numbers of characters yet ..... ??? ..... hmmm, but ... ?? might be possibilities .... ??? "kaleidoscope folding" ?? .... ??? ....... ??? ..... ?? or sub rather than quotient ?? .... ?? "... chamber ..." ... ???? .... ??? hmmm, try adding together the raw counts for the second and third terms .... ??? .... ?? does seem to have some relationship to "(q-1)^2", while raw count for first term seems to relate to "q^2-1" .... ???? ..... ???? .....

?? wait a minute ... ?? might be right track, but some confusion here ... ?? between "parameterization by" as given by inducement (? which sort of seemed to work numerologically ... ?? ...), vs bit about those parameterized by nonsplit toruses as "_not_ induced" .... ???? .....