6 = 1 + 4 + 1

48 = 1 + 9 + 1 + 9 + ...

28 ... ???? ....

15*12 = (1 + 16)*3 + 129

?? 129 = 3 * 43 .... ????? ....

?? 2d f_3 vsp .... ???

?? 4 1d subspaces ...

3d irrep ...

?? 2 cuspidals of gl(1,f_2) ?? ....

?? green convolution of 2 1d cuspidals ..... ????? .....

?? "take 2d f_3 vsp v, take 1d subspace x, take a-structure on x and b-structure on v/x" .... ????? ....


0

[]

1

[1]

2

[2] [11]

3

[3] [21] [111]

4

[4] [31] [22] [211] [1111]

5

[5] [41] [32] [311] [221] [2111] [11111]



??? gl(2) ....

?? cuspidal ??

?? green convolution of 2 different gl(1) cuspidals ...

?? apply 2-box young diagram (?? either [2] or [11] ?? ...) to gl(1) cuspidal ... ???? ....

c2 ....

c1 X c1 - c1 ... ????

c1 X 2 ... ???? ....

??? hmmmm .... q=2 .... ???? cj = 1 so c1 X c1 - c1 = 0 ??? .... ??? ...

6 = 1^2 + 2^2 + 1^2 .....

48 = 1^2 + 3^2 + 1^2 + 3^2 + 4^2 + 12 .... ???? 12 = (2^2)*3 ??? ....

15*12 = (1^2 + 4^2)*3 + (5^2)*3 + 54 ..... ???? 54 = (3^2)*6 ??? ....

24*20 = (1^2 + 5^2)*4 + (6^2)*6 + 160 .... ???? 160 = (4^2)*10 ??? ....

48*42 = (1^2 + 7^2)*6 + (8^2)*15 + (6^2)*21 ?????? .....

2016 = 300 + 960 + 756

?? so naive guesses seem to be checking out here .... ??? ....

7*6*4 = 168

26*24*18 = 11232

63*60*48 = 181440

?? "monocuspidal" :

gl(3) cuspidal .... ??? ....

gl(1) cuspidal with 3-box young diagram applied ... ??? ...

?? "bicuspidal" :

gl(2) cuspidal green-convolve gl(1) cuspidal

(gl(1) cuspidal with 2-box young diagram applied) green-convolve gl(1) cuspidal


?? "tricuspidal" :

gl(1) cuspidal green-convolve gl(1) cuspidal green-convolve gl(1) cuspidal

q=2 ... ??? ...

168 = 1^2 + 1^2 + ??? ....

?? nothing, point, flag .... ???? .....

?? induced vs irrep ... categorified gram-schmidt .... ???? ....

?? nothing, point - nothing, flag - point*2 + nothing

1, q^2+q+1, (q^2+q+1)*(q+1)

1, q^2+q, (q^2+q+1)*(q-1)+1 = q^3 ??? ....

q=2 ...

1, 6, 8 ???? ....

168 = (1^2) + (1^2 + 6^2 + 8^2) + (7^2)

1 + (1 + 36 + 64) + 49

?? could it be that there are 2 3d gl(3) cuspidals here ???? ..... ????

168 = (3^2 + 3^2) + (1^2 + 6^2 + 8^2) + (7^2)

11232 = (?^2)*?? + (1^2 + 12^2 + 27^2)*2 + ((13*2)^2)*3*2 + (13^2 + 39^2)*1

11232 - (1748 + 4056 + 3380) = 3738 = 2*3*7*89 ?? .... must be arithmetic mistake somewhere .... ???? ....

?? hmmm, so maybe the mistake was ... to neglect that for the case where you use a distinct pair of gl(1) cuspidals, the pair are distinguishable ...... ??? (maybe figure out what i mean by difference between "distinct" and "distinguishable" here sometime ??? ...) ... so that 1690 should be doubled to 3380 ... ?? and now the arithmetic seems a bit more encouraging ??? .... 2048 left over ... = 2^11 .... ??? .....

??? so that could fit with 2 32d gl(3,f_3) cuspidals ?? ... ?? or 8 16d ones ??? ... ?? ... or 32 8d ..... ???? or 128 4d, or 512 2d, or 2048 1d .... ???? .... ?? which seems more plausible offhand ??? ....

2 1d gl(1,f_3) cuspidals ....

3 2d gl(2,f_3) cuspidals ...

8 16d gl(3,f_3) cuspidals ... ???? .....

...

1 1d gl(1,f_2) cuspidal ....

1 1d gl(2,f_2) cuspidal ....

2 3d gl(3,f_2) cuspidals .... ???? ....

...

q-1 1d gl(1,f_q) cuspidals ....

q(q-1)/2 [q-1]d gl(2,f_q) cuspidals .... ????? ...

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

181440 = (?^2)*?? + (1^2 + 20^2 + 64^2)*3 + ((21*3)^2)*6*3 + (21^2 + (21*4)^2)*6 + ((21*5)^2)*1

181440 - 13491 - 71442 - 44982 - 11025 = 40500 = 2^2 * 3^4 * 5^3

?? so .... ???? gl(3,f_4) cuspidals ... ??? maybe 5 90d ??? or 20 45d ?? .... ?? or 45 30d ?? .... or 125 18d ?? .... or 180 15d ?? .... or 405 10d ... or 500 9d ... or 1125 6d ... or 1620 5d .... or 4500 3d ... or 10125 2d ... or 40500 1d ... ???? ......

124*120*100 = (?^2)*?? + (1^2 + 30^2 + 125^2)*4 + ((31*4)^2)*10*4 + (31^2 + (31*5)^2)*12 + ((31*6)^2)*4

1488000 - 66104 - 615040 - 299832 - 138384 = 368640 = 2^13 * 3^2 * 5

342*336*294 - (1^2 + 56^2 + 343^2)*6 - ((57*6)^2)*21*6 - (57^2 + (57*7)^2)*30 - ((57*8)^2)*20 =

33784128 - 724716 - 14737464 - 4873500 - 4158720 = 9289728 = 2^14 * 3^4 * 7

?? some sort of categorified gram-schmidt here ?? .... ???? .... ?? try to get q-polynomials for various sectors here ... ??? ... ?? ...