D8 overview

D₈ · Neutral gradings

75 neutral gradings · KL, Fourier, Arthur packets

Ambient orbit	KL block sizes	Arthur parameters	Parameters in packets	Graded orbits	Local systems
1111111111111111	1 × 1	1	1	1	1
22111111111111	2 × 2	2	2	2	2
31111111111111	3 × 3, 1 × 1	3	4	3	4
222211111111	3 × 3	3	3	3	3
322111111111	6 × 6, 2 × 2	6	8	6	8
331111111111	7 × 7, 2 × 2	6	9	6	9
511111111111	6 × 6, 2 × 2	2	4	6	8
2222221111	4 × 4	4	4	4	4
22222222+	5 × 5, 1 × 1	5	6	5	6
22222222-	5 × 5, 1 × 1	5	6	5	6
3222211111	9 × 9, 3 × 3	9	12	9	12
32222221	16 × 16, 4 × 4, 2 × 2, 2 × 2	16	24	16	24
3322111111	14 × 14, 4 × 4	12	18	12	18
3331111111	12 × 12, 4 × 4	10	16	10	16
4411111111	6 × 6	2	2	6	6
5221111111	12 × 12, 4 × 4	4	8	12	16
33222211	24 × 24, 6 × 6, 2 × 2, 2 × 2	21	34	21	34
5311111111	18 × 18, 8 × 8	9	19	16	26
7111111111	12 × 12, 4 × 4	2	4	12	16
33322111	26 × 26, 8 × 8, 2 × 2, 2 × 2	22	38	22	38
33331111	20 × 20, 6 × 6, 2 × 2, 2 × 2	16	30	16	30
52222111	18 × 18, 6 × 6	6	12	18	24
333322	22 × 22, 6 × 6	10	16	20	28
44221111	13 × 13	6	6	13	13
333331	19 × 19, 6 × 6	11	19	16	25
442222+	20 × 20, 2 × 2	10	12	20	22
442222-	20 × 20, 2 × 2	10	12	20	22
44311111	18 × 18, 6 × 6	6	8	18	24
53221111	36 × 36, 16 × 16	18	38	32	52
532222	45 × 45, 12 × 12, 4 × 4, 4 × 4	24	44	45	65
53311111	38 × 38, 18 × 18	21	46	31	56
443221	52 × 52, 13 × 13, 4 × 4, 4 × 4	24	38	52	73
55111111	25 × 25, 10 × 10	3	8	22	35
72211111	24 × 24, 8 × 8	4	8	24	32
443311	48 × 48, 12 × 12, 4 × 4, 4 × 4	14	28	42	68
533221	68 × 68, 28 × 28, 5 × 5, 5 × 5	40	86	60	106
73111111	36 × 36, 16 × 16	6	16	32	52
4444+	20 × 20, 3 × 3	3	6	20	23
4444-	20 × 20, 3 × 3	3	6	20	23
533311	62 × 62, 28 × 28, 5 × 5, 5 × 5	37	87	50	100
722221	48 × 48, 12 × 12, 4 × 4, 4 × 4	6	16	48	68
91111111	24 × 24, 8 × 8	2	4	24	32
544111	36 × 36, 12 × 12	4	8	36	48
552211	56 × 56, 20 × 20	6	16	50	76
5443	90 × 90, 24 × 24, 8 × 8, 8 × 8	16	40	90	130
553111	67 × 67, 32 × 32	15	45	55	99
732211	80 × 80, 32 × 32, 5 × 5, 5 × 5	12	36	72	122
5533	80 × 80, 33 × 33, 7 × 7, 7 × 7	23	65	70	127
661111	20 × 20	2	2	20	20
733111	80 × 80, 36 × 36, 5 × 5, 5 × 5	12	40	66	126
5551	68 × 68, 28 × 28	4	12	60	96
6622+	44 × 44, 4 × 4	4	6	44	48
6622-	44 × 44, 4 × 4	4	6	44	48
7333	78 × 78, 32 × 32	8	24	70	110
751111	65 × 65, 30 × 30	6	18	58	95
922111	48 × 48, 16 × 16	4	8	48	64
6631	80 × 80, 20 × 20, 6 × 6, 6 × 6	8	18	80	112
7441	96 × 96, 24 × 24, 8 × 8, 8 × 8	4	16	96	136
7522	110 × 110, 30 × 30, 8 × 8, 8 × 8	16	36	110	156
931111	72 × 72, 32 × 32	6	16	64	104
(11,1,1,1,1,1)	48 × 48, 16 × 16	2	4	48	64
7531	159 × 159, 75 × 75, 1 × 1, 15 × 15, 15 × 15	30	106	140	265
9322	120 × 120, 32 × 32, 8 × 8, 8 × 8	8	20	120	168
7711	100 × 100, 40 × 40, 7 × 7, 7 × 7	3	8	90	154
9331	136 × 136, 56 × 56, 10 × 10, 10 × 10	10	32	120	212
(11,2,2,1)	128 × 128, 32 × 32, 8 × 8, 8 × 8	4	12	128	176
88+	70 × 70, 6 × 6	2	4	70	76
88-	70 × 70, 6 × 6	2	4	70	76
9511	144 × 144, 60 × 60	4	16	130	204
(11,3,1,1)	160 × 160, 64 × 64, 10 × 10, 10 × 10	6	20	144	244
97	200 × 200, 55 × 55, 16 × 16, 16 × 16	8	24	200	287
(11,5)	220 × 220, 60 × 60, 16 × 16, 16 × 16	4	16	220	312
(13,1,1,1)	96 × 96, 32 × 32	2	4	96	128
(13,3)	240 × 240, 64 × 64, 16 × 16, 16 × 16	4	12	240	336
(15,1)	256 × 256, 64 × 64, 16 × 16, 16 × 16	2	8	256	352

Counting conventions

Matrix dimensions are listed separately for each block. Parameters in packets counts distinct IC parameters, without multiplicities or repeated membership. Local systems counts orbit–local-system pairs across all exported series. “Not computed” means that the selected export does not contain that result, rather than a count of zero.