E₇ · Neutral gradings
45 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, 1 × 1 | 4 | 5 | 4 | 5 |
| 3A₁' | 4 × 4 | 4 | 4 | 4 | 4 |
| A₂ | 6 × 6 | 5 | 6 | 5 | 6 |
| 4A₁ | 8 × 8, 2 × 2 | 8 | 10 | 8 | 10 |
| A₂ + A₁ | 14 × 14, 2 × 2 | 12 | 16 | 12 | 16 |
| A₂ + 2A₁ | 13 × 13, 2 × 2 | 11 | 15 | 11 | 15 |
| 2A₂ | 10 × 10 | 5 | 6 | 9 | 10 |
| A₃ | 6 × 6 | 2 | 2 | 6 | 6 |
| A₂ + 3A₁ | 12 × 12, 3 × 3 | 10 | 15 | 10 | 15 |
| (A₃ + A₁)'' | 13 × 13, 2 × 2 | 6 | 8 | 13 | 15 |
| 2A₂ + A₁ | 22 × 22, 3 × 3 | 10 | 14 | 20 | 25 |
| (A₃ + A₁)' | 13 × 13 | 6 | 6 | 13 | 13 |
| D₄(a₁) | 18 × 18 | 11 | 14 | 15 | 18 |
| A₃ + 2A₁ | 26 × 26, 4 × 4 | 12 | 16 | 26 | 30 |
| D₄ | 8 × 8 | 2 | 2 | 8 | 8 |
| D₄(a₁) + A₁ | 36 × 36, 8 × 8 | 18 | 30 | 32 | 44 |
| A₃ + A₂ | 34 × 34, 5 × 5 | 20 | 29 | 30 | 39 |
| A₃ + A₂ + A₁ | 31 × 31, 8 × 8 | 20 | 34 | 25 | 39 |
| A₄ | 27 × 27 | 3 | 6 | 24 | 27 |
| A₅'' | 20 × 20, 3 × 3 | 2 | 4 | 20 | 23 |
| D₄ + A₁ | 24 × 24, 4 × 4 | 4 | 6 | 24 | 28 |
| A₄ + A₁ | 56 × 56, 10 × 10 | 6 | 14 | 50 | 66 |
| A₄ + A₂ | 28 × 28 | 4 | 6 | 24 | 28 |
| D₅(a₁) | 40 × 40, 5 × 5 | 6 | 10 | 36 | 45 |
| A₅' | 20 × 20 | 2 | 2 | 20 | 20 |
| D₅(a₁) + A₁ | 40 × 40, 10 × 10 | 8 | 14 | 35 | 50 |
| A₅ + A₁ | 40 × 40, 6 × 6 | 4 | 8 | 40 | 46 |
| D₆(a₂) | 55 × 55, 8 × 8 | 8 | 14 | 55 | 63 |
| E₆(a₃) | 39 × 39 | 6 | 8 | 35 | 39 |
| E₇(a₅) | 74 × 74, 15 × 15, 1 × 1 | 20 | 42 | 64 | 90 |
| D₅ | 26 × 26 | 2 | 2 | 26 | 26 |
| A₆ | 35 × 35 | 2 | 2 | 32 | 35 |
| D₅ + A₁ | 48 × 48, 8 × 8 | 4 | 6 | 48 | 56 |
| D₆(a₁) | 60 × 60, 8 × 8 | 4 | 6 | 60 | 68 |
| E₇(a₄) | 92 × 92, 20 × 20 | 12 | 24 | 84 | 112 |
| E₆(a₁) | 73 × 73 | 2 | 4 | 67 | 73 |
| D₆ | 64 × 64, 8 × 8 | 2 | 4 | 64 | 72 |
| E₆ | 40 × 40 | 2 | 2 | 40 | 40 |
| E₇(a₃) | 106 × 106, 20 × 20 | 4 | 12 | 98 | 126 |
| E₇(a₂) | 110 × 110, 16 × 16 | 4 | 6 | 110 | 126 |
| E₇(a₁) | 120 × 120, 16 × 16 | 2 | 4 | 120 | 136 |
| E₇ | 128 × 128, 16 × 16 | 2 | 4 | 128 | 144 |
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.