G = E7, adjoint, quaternionic G^v = E7, simply connected, split Note: G is connected type: E7 ad s real form: 2 dual real form: 3 Dynkin Diagram: 2 | 1-3-4-5-6-7 Infinitesimal character of block: rho Special Special Orbit Cells Diagram #R A DualOrbit Cells Diagram #R A 1 31 0000000 1 1 E7 0 2222222 2 1 A1 30 1000000 1 1 E7(a1) 1 2220222 2 2/1 2A1 28,29 0000010 2 1 E7(a2) 2,3 2220202 2 1 A2 18,26,27 2000000 3 2/2 E7(a3) 4,5,13 2002022 4 2/2 A2+2A1 23,24 0001000 2 1 E7(a4) 7,9 2002002 4 2/1 2A2 16,22 0000020 2 1 D5+A1 8,15 2110110 2 1 A3 17,21 2000010 2 1 D6(a1) 10,14 2110102 2 1 D4(a1) 9,19,20 0020000 3 S3 E7(a5) 11,12,21 0002002 4 S3 A3+A2 15 0001010 1 2/1 D5(a1)+A1 16 2000200 4 1 A4 7,14 2000020 2 2/2 D5(a1) 17,23 2001010 3 2/2 A4+A1 13 1001010 1 2/2 A4+A1 18 1001010 3 2/2 D4 3,8 2020000 2 S3/1 A5'' 22,29 2000022 2 1 A4+A2 12 0002000 1 1 A3+A2+A1 19 0000200 4 1 E6(a3) 5,11 0020020 2 2/2 D4(a1)+A1 20,25 0110001 4 2/2 D5 2,4 2020020 2 1 (A3+A1)'' 26,30 2000002 2 1 A6 10 0002020 1 1 A2+3A1 27 0200000 4 1 E6(a1) 1 2002020 1 2/2 A2+A1 28 1000010 3 2/2 E6 0 2022020 1 1 (3A1)'' 31 0000002 2 1 Fundamental weights lambda_2,lambda_5,lambda_7 are not in root lattice Infinitesimal character of block = rho => allowed infinitesimal characters: #0s on 2,5,7 is even not allowed: A2+A1 25 1000010 1 2/2 E6(a1) 6 2002020 2 2/2 D5(a1) 6 2001010 1 2/2 A4 24 2000020 2 2/2