Arthur parameters for adjoint, split inner class of E6 G=adjoint(E6_s) or adjoint(E6_F4) dual group is simply connected complex nilpotent orbits for inner class (E6_c simply connected) Complex reductive group of type E6, with involution defining inner class of type 'c', with 3 real forms and 2 dual real forms root datum of inner class: simply connected root datum of Lie type 'E6' 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(E6_s) E6_F4 0 [0,0,0,0,0,0] 0 6T1 E6 3 3 [1] [1,1,1] 3 E6_sc 3*=1+1+1 0 1 [0,1,0,0,0,0] 22 A1+5T1 A5 6 4 [1] [0,0,2,0,2] 4 SL(6) 4*=1+1+1+1 0 2 [1,0,0,0,0,1] 32 2A1+4T1 B3+T1 2 5 [1] [0,0,0,0,3,2] 5 GSpin(7) [a] 5*=1+1+1+1+1 0 3 [0,0,0,1,0,0] 40 3A1+3T1 A1+A2 6 4 [1] [0,0,0,0,4] 4 SL(2)xSL(3) 4=1+1+1+1 0 4 [0,2,0,0,0,0] 42 A2+4T1 2A2 9 4 [1,2] [0,1,3] 4 [SL(3)xSL(3)]|2 [b] 7*=3+2+2 0 5 [1,1,0,0,0,1] 46 A1+A2+3T1 A2+T1 3 4 [1] [0,0,0,0,4] 4 SL(3)xGL(1) [c] 4*=1+1+1+1 0 6 [2,0,0,0,0,2] 48 2A2+2T1 G2 1 2 [1,3,3] [0,1,1] 2 G2 2*=1+1 0 7 [0,0,1,0,1,0] 50 2A1+A2+2T1 A1+T1 2 3 [1] [0,0,0,0,0,3] 3 GL(2) [d] 7=1+1+1 0 8 [1,2,0,0,0,1] 52 A3+3T1 B2+T1 2 4 [1] [0,0,0,0,2,2] 4 GSpin(5) [e] 4=1+1+1+1 0 9 [1,0,0,1,0,1] 54 A1+2A2+T1 A1 2 2 [1,3,3] [0,0,0,0,2] 2 SL(2)xZ3 2=1+1 0 10 [0,1,1,0,1,0] 56 A1+A3+2T1 A1+T1 2 4 [1] [0,0,0,0,4] 4 SL(2)xGL(1) [f] 4=1+1+1+1 0 11 [0,0,0,2,0,0] 58 D4+2T1 2T1 1 4 [1,2,3] [0,0,3] 3 [GL(1)^3_0]|S3 [g] 7=3+2+2 0 12 [2,2,0,0,0,2] 60 A4+2T1 A1+T1 2 3 [1] [0,1,2] 3 GL(2) 3*=1+1+1 0 13 [0,2,0,2,0,0] 60 D4+2T1 A2 3 2 [1] [0,0,2] 2 SL(3) 2*=1+1 1 14 [1,1,1,0,1,1] 62 A1+A4+T1 T1 1 2 [1] [0,0,0,0,2] 2 GL(1) 2=1+1 0 15 [1,2,1,0,1,1] 64 D5+T1 T1 1 2 [1] [0,0,0,0,0,2] 2 GL(1) 2=1+1 0 16 [2,1,1,0,1,2] 64 A5+T1 A1 2 2 [1,3,3] [0,0,0,0,2] 2 SL(2)xZ3 2=1+1 0 17 [2,0,0,2,0,2] 66 E6 e 1 1 [1,2,3,3,6,6] [0,0,2] 2 Z2xZ3 4+2+2 0 18 [2,2,0,2,0,2] 68 D5+T1 T1 1 2 [1] [0,0,2] 2 GL(1) 2=1+1 1 19 [2,2,2,0,2,2] 70 E6 e 1 1 [1,3,3] [0,0,1] 1 Z3 1=1 0 20 [2,2,2,2,2,2] 72 E6 e 1 1 [1,3,3] [0,0,1] 1 Z3 1=1 1 [a] Center is a connected torus [b] extension splits (E6_sc_components.txt), C_2=#AP=> switches factors [SL(3)xSL(3)]\rtimes Z2, action by switching factors [c] Center is GL(1)xZ3 (note: G is not GL(3)) [d,e] Center is GL(1) (E6_sc_centralizer_isogenies.txt) [f] Center is GL(1)xZ2 (E6_sc_centralizer_isogenies.txt) [g] 1 -> GL(1)^3_0 -> Cent -> S3 -> 1 GL(1)^2=GL(3)^3_0={(x,y,z)| xyz=1}\simeq GL(1)^2 This is an interesting case. Claim: S_3 is acting by permuting the factors (assume this) According to E6_sc_components.txt, the 3 chosen generators of S_3 have orders 1,4 and 3: Component info for orbit: H=[ 4, 6, 8, 12, 8, 4 ] diagram:[0,0,0,2,0,0] dim:58 orders:[1,2,3] pseudo_Levi Generators 2A1+A3 [[ -3, -4, -6, -8, -6, -3 ]/4] 3A2 [[ -2, -3, -4, -6, -3, 0 ]/3] D4 [[ 0, 0, 0, 0, 0, 0 ]/1] However the exact sequence necessarily splits. Assume not: Cent = where a,b have order 2,3 in the component group. The only thing to worry about is say a=(1,2)\in S_3, a^2\ne 1. Then a^2 has to be fixed by the action of a, so a^2=(x,x,z) Then [(r,s,t)a]^2=(rs,rs,t^2)a^2=(xrs,xrs,zt^2). You can choose rs=1/x, t^2=z to make this trivial.