E₈ · Neutral gradings
70 neutral gradings · KL, Fourier, Arthur packets
| Ambient orbit | KL block sizes | Arthur parameters | Parameters in packets | Graded orbits | Local systems |
|---|---|---|---|---|---|
| 1 | 1 × 1 | 1 | 1 | 1 | 1 |
| A₁ | 2 × 2 | 2 | 2 | 2 | 2 |
| 2A₁ | 3 × 3 | 3 | 3 | 3 | 3 |
| 3A₁ | 4 × 4 | 4 | 4 | 4 | 4 |
| A₂ | 6 × 6 | 5 | 6 | 5 | 6 |
| 4A₁ | 5 × 5 | 5 | 5 | 5 | 5 |
| A₂ + A₁ | 12 × 12 | 10 | 12 | 10 | 12 |
| A₂ + 2A₁ | 12 × 12 | 10 | 12 | 10 | 12 |
| A₃ | 6 × 6 | 2 | 2 | 6 | 6 |
| A₂ + 3A₁ | 12 × 12 | 10 | 12 | 10 | 12 |
| 2A₂ | 13 × 13 | 10 | 13 | 10 | 13 |
| 2A₂ + A₁ | 20 × 20 | 10 | 12 | 18 | 20 |
| A₃ + A₁ | 13 × 13 | 6 | 6 | 13 | 13 |
| D₄(a₁) | 18 × 18 | 11 | 14 | 15 | 18 |
| D₄ | 8 × 8 | 2 | 2 | 8 | 8 |
| 2A₂ + 2A₁ | 19 × 19 | 11 | 14 | 16 | 19 |
| A₃ + 2A₁ | 20 × 20 | 10 | 10 | 20 | 20 |
| D₄(a₁) + A₁ | 36 × 36 | 22 | 28 | 30 | 36 |
| A₃ + A₂ | 38 × 38 | 21 | 28 | 31 | 38 |
| A₄ | 24 × 24 | 3 | 6 | 21 | 24 |
| A₃ + A₂ + A₁ | 31 × 31 | 20 | 26 | 25 | 31 |
| D₄ + A₁ | 16 × 16 | 4 | 4 | 16 | 16 |
| D₄(a₁) + A₂ | 32 × 32 | 21 | 30 | 23 | 32 |
| A₄ + A₁ | 54 × 54 | 6 | 12 | 48 | 54 |
| 2A₃ | 20 × 20 | 3 | 3 | 20 | 20 |
| D₅(a₁) | 36 × 36 | 6 | 8 | 32 | 36 |
| A₄ + 2A₁ | 67 × 67 | 15 | 27 | 55 | 67 |
| A₄ + A₂ | 40 × 40 | 12 | 20 | 32 | 40 |
| A₅ | 20 × 20 | 2 | 2 | 20 | 20 |
| A₄ + A₂ + A₁ | 56 × 56 | 8 | 12 | 48 | 56 |
| D₅(a₁) + A₁ | 40 × 40 | 8 | 10 | 35 | 40 |
| E₆(a₃) | 39 × 39 | 6 | 8 | 35 | 39 |
| D₄ + A₂ | 47 × 47 | 10 | 14 | 38 | 47 |
| D₅ | 26 × 26 | 2 | 2 | 26 | 26 |
| A₄ + A₃ | 68 × 68 | 4 | 6 | 60 | 68 |
| A₅ + A₁ | 44 × 44 | 4 | 4 | 44 | 44 |
| D₅(a₁) + A₂ | 78 × 78 | 8 | 12 | 70 | 78 |
| E₆(a₃) + A₁ | 78 × 78 | 12 | 16 | 70 | 78 |
| D₆(a₂) | 65 × 65 | 6 | 10 | 58 | 65 |
| E₇(a₅) | 74 × 74 | 20 | 28 | 64 | 74 |
| E₈(a₇) | 87 × 87, 1 × 1 | 34 | 58 | 63 | 88 |
| D₅ + A₁ | 52 × 52 | 4 | 4 | 52 | 52 |
| D₆(a₁) | 72 × 72 | 6 | 8 | 64 | 72 |
| A₆ | 55 × 55 | 3 | 3 | 48 | 55 |
| E₇(a₄) | 92 × 92 | 12 | 14 | 84 | 92 |
| A₆ + A₁ | 70 × 70 | 4 | 4 | 64 | 70 |
| E₆(a₁) | 71 × 71 | 2 | 4 | 65 | 71 |
| D₅ + A₂ | 106 × 106 | 18 | 26 | 86 | 106 |
| E₆ | 40 × 40 | 2 | 2 | 40 | 40 |
| D₆ | 48 × 48 | 2 | 2 | 48 | 48 |
| D₇(a₂) | 144 × 144 | 4 | 8 | 130 | 144 |
| E₆(a₁) + A₁ | 146 × 146 | 4 | 8 | 134 | 146 |
| A₇ | 70 × 70 | 2 | 2 | 70 | 70 |
| E₇(a₃) | 106 × 106 | 4 | 6 | 98 | 106 |
| E₈(b₆) | 123 × 123 | 8 | 16 | 105 | 123 |
| E₆ + A₁ | 80 × 80 | 4 | 4 | 80 | 80 |
| D₇(a₁) | 136 × 136 | 6 | 10 | 118 | 136 |
| E₇(a₂) | 110 × 110 | 4 | 4 | 110 | 110 |
| E₈(a₆) | 127 × 127 | 4 | 10 | 110 | 127 |
| E₈(b₅) | 148 × 148 | 8 | 12 | 128 | 148 |
| D₇ | 96 × 96 | 2 | 2 | 96 | 96 |
| E₇(a₁) | 120 × 120 | 2 | 2 | 120 | 120 |
| E₈(a₅) | 159 × 159 | 6 | 8 | 145 | 159 |
| E₈(b₄) | 184 × 184 | 4 | 6 | 168 | 184 |
| E₇ | 128 × 128 | 2 | 2 | 128 | 128 |
| E₈(a₄) | 192 × 192 | 2 | 4 | 178 | 192 |
| E₈(a₃) | 212 × 212 | 4 | 6 | 196 | 212 |
| E₈(a₂) | 220 × 220 | 2 | 2 | 220 | 220 |
| E₈(a₁) | 240 × 240 | 2 | 2 | 240 | 240 |
| E₈ | 256 × 256 | 2 | 2 | 256 | 256 |
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.