Arthur parameters for adjoint E7 one inner class G=adjoint(E7_s, E7_q, E7_h, E7_c) dual group is simply connected complex nilpotent orbits for inner class Complex reductive group of type E7, with involution defining inner class of type 'c', with 4 real forms and 4 dual real forms root datum of inner class: simply connected root datum of Lie type 'E7' i: orbit number H: semisimple element BC Levi: Bala-Carter Levi Cent_0: identity component of Cent(SL(2)) Z(Cent^0): order of center of derived group of id. comp. of Centralizer C_2: conjugacy classes in Cent(SL(2))_0 with square 1 A(O): orders of conj. classes in component group of centralizer of O #RF: number of real forms of O for all real forms (of integrality datum) in inner class #AP: number of Arthur parameters for O i diagram dim BC Levi Cent_0 Z C_2 A(O) #RF #AP Cent #reps q h c s 0 [0,0,0,0,0,0,0] 0 7T1 E7 2 4 [1] [1,1,1,1] 4 E7 0 0 0 4*=1+1+1+1 1 [1,0,0,0,0,0,0] 34 A1+6T1 D6 4 6 [1] [2,2,0,0,2] 6 Spin(12) 0 0 0 6*=1+1+1+1+1+1 2 [0,0,0,0,0,1,0] 52 2A1+5T1 A1+B4 4 8 [1] [3,3,0,2] 8 SL(2)xSpin(9) 0 0 0 8* 3 [0,0,0,0,0,0,2] 54 3A1+4T1 F4 1 3 [1,2] [4,2] 6 F4xZ2 2 0 0 6=2+2+2 4 [0,0,1,0,0,0,0] 64 3A1+4T1 A1+C3 4 8 [1] [0,0,4,4,0] 8 SL(2)xSp(6) 0 0 0 8* 5 [2,0,0,0,0,0,0] 66 A2+5T1 A5 6 4 [1,2] [0,0,3,3] 6 SL(6).2 [a] 0 0 0 12=2^6 6 [0,1,0,0,0,0,1] 70 4A1+3T1 C3 2 4 [1,2] [0,0,8,0] 8 Sp(6)xZ2 2 0 0 8* 7 [1,0,0,0,0,1,0] 76 A1+A2+4T1 A3+T1 4 6 [1,2] [0,0,0,0,8] 8 GL(4).2 [c] 2 0 0 16*=2^8 8 [0,0,0,1,0,0,0] 82 2A1+A2+3T1 3A1 4 6 [1] [0,0,0,6] 6 SL(2)^3/<-1,-1,-1> [d] 0 0 0 7* 9 [2,0,0,0,0,1,0] 84 A3+4T1 A1+B3 4 6 [1] [2,2,0,2] 6 SL(2)xSpin(7) 0 0 0 6=1^6 10 [0,0,0,0,0,2,0] 84 2A2+3T1 A1+G2 2 4 [1] [0,0,2,2] 4 SL(2)xG2 0 0 0 4*=1^4 11 [0,2,0,0,0,0,0] 84 3A1+A2+2T1 G2 1 2 [1,2] [0,4] 4 G2xZ2 1 0 0 4=1^4 12 [2,0,0,0,0,0,2] 86 A1+A3+3T1 B3 2 3 [1,2] [4,2] 6 Spin(7)xZ2 3 0 0 6=1^6 13 [0,0,1,0,0,1,0] 90 A1+2A2+2T1 2A1 4 4 [1] [0,0,0,0,4] 4 SL(2)xSL(2) 0 0 0 4=1^4 14 [1,0,0,1,0,0,0] 92 A1+A3+3T1 3A1 8 8 [1] [0,0,4,4,0] 8 SL(2)^3 0 0 0 8=1^8 15 [0,0,2,0,0,0,0] 94 D4+3T1 3A1 8 8 [1,2,3] [0,0,3,3] 6 SL(2)^3].S3 [e] 0 0 0 14*=2+2+3+2+3+2 16 [1,0,0,0,1,0,1] 94 2A1+A3+2T1 2A1 4 4 [1,2] [0,0,8,0] 8 SL(2)^2 2 0 0 8=1^8 17 [2,0,2,0,0,0,0] 96 D4+3T1 C3 2 4 [1] [0,0,2,2] 4 Sp(6) 0 2 0 4*=1^4 18 [0,1,1,0,0,0,1] 96 A1+D4+2T1 2A1 4 4 [1,2,2,2] [0,0,0,0,8] 8 [SL(2)^2].8 [f] 4 0 0 14 19 [0,0,0,1,0,1,0] 98 A2+A3+2T1 A1+T1 2 4 [1,2] [0,0,0,6] 6 [GL(2)].2 [g] 0 0 0 7 20 [2,0,0,0,0,2,0] 100 A4+3T1 A2+T1 3 4 [1,2] [0,0,2,2] 4 GL(2).2 [h] 1 0 0 8*=2^4 21 [0,0,0,0,2,0,0] 100 A1+A2+A3+T1 A1 1 2 [1,2] [0,4] 4 PSLS(2)xZ2 1 0 0 4=1^4 22 [2,1,1,0,0,0,1] 102 A1+D4+2T1 B2 2 3 [1,2] [0,0,0,0,6] 6 Spin(5)xZ2 3 2 0 6=1^6 23 [2,0,0,0,0,2,2] 102 A5+2T1 G2 1 2 [1,2] [2,2] 4 G2xZ2 3 0 0 4=1^4 24 [1,0,0,1,0,1,0] 104 A1+A4+2T1 2T1 1 4 [1,2] [0,0,0,0,4] 4 [GL(1)^2].x [i] 0 0 0 8=2^4 25 [2,0,0,1,0,1,0] 106 D5+2T1 A1+T1 2 4 [1,2] [0,0,0,6] 6 GL(2).2 [j] 2 2 0 12=2^6 26 [0,0,0,2,0,0,0] 106 A2+A4+T1 A1 2 2 [1] [0,0,1,1] 2 SL(2) 0 0 0 2=1+1 27 [1,0,0,1,0,2,0] 108 A5+2T1 2A1 4 4 [1] [0,0,2,2,0] 4 SL(2)^2 0 0 0 4 28 [2,0,0,0,2,0,0] 108 A1+D5+T1 A1 1 2 [1,2] [0,4] 4 PSL(2)xZ2 1 1 0 5 29 [1,0,0,1,0,1,2] 108 A1+A5+T1 A1 2 2 [1,2] [0,0,4,0] 4 SL(2)xZ2 2 0 0 4=1^4 30 [0,0,2,0,0,2,0] 110 E6+T1 A1 2 2 [1,2] [0,0,2,2] 4 SL(2)xZ2 0 0 0 8*=2^4 31 [0,1,1,0,1,0,2] 110 D6+T1 A1 2 2 [1,2] [0,0,0,4] 4 SL(2)xZ2 3 0 0 4=1^4 32 [2,0,2,0,0,2,0] 112 D5+2T1 2A1 4 4 [1] [0,0,2,2] 4 SL(2)^2 0 2 0 4*=1^4 33 [0,0,0,2,0,0,2] 112 E7 e 1 1 [1,2,2,2,3,6] [0,4] 4 S3xZ2 7 0 0 10=2+2+3+3 34 [2,1,1,0,1,1,0] 114 A1+D5+T1 A1 2 2 [1,2] [0,0,0,0,4] 4 SL(2)xZ2 1 2 0 4=1^4 35 [0,0,0,2,0,2,0] 114 A6+T1 A1 2 2 [1] [0,0,1,1] 2 SL(2 0 0 0 2*=1+1 36 [2,1,1,0,1,0,2] 114 D6+T1 A1 2 2 [1,2] [0,0,0,4] 4 SL(2)xZ2 1 1 0 4=1^4 37 [2,0,0,2,0,0,2] 116 E7 e 1 1 [1,2,2,2] [0,4] 4 Z2xZ2 2 0 0 5 38 [2,0,0,2,0,2,0] 118 E6+T1 T1 1 2 [1,2] [0,0,1,1] 2 GL(1).2 [k] 1 0 0 4*=2+2 39 [2,1,1,0,1,2,2] 118 D6+T1 A1 2 2 [1,2] [0,0,0,4] 4 SL(2) 2 2 0 4=1^4 40 [2,0,2,2,0,2,0] 120 E6+T1 A1 2 2 [1] [0,0,1,1] 2 SL(2) 0 2 0 2*=1+1 41 [2,0,0,2,0,2,2] 120 E7 e 1 1 [1,2,2,2] [0,4] 4 Z2xZ2 6 2 0 8=2^4 42 [2,2,2,0,2,0,2] 122 E7 e 1 1 [1,2] [0,2] 2 Z2 2 2 0 2=1+1 43 [2,2,2,0,2,2,2] 124 E7 e 1 1 [1,2] [0,2] 2 Z2 1 1 0 2=1+1 44 [2,2,2,2,2,2,2] 126 E7 e 1 1 [1,2] [0,2] 2 Z2 1 2 1 2=1+1