Arthur parameters for simply connected, equal rank inner class of E6 G=E6_h, E6_q or E6_c (these are simply connected) dual group is adjoint (split inner class on the dual side) complex nilpotent orbits for inner class Complex reductive group of type E6, with involution defining inner class of type 's', with 2 real forms and 3 dual real forms root datum of inner class: adjoint 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 Note: Since the innern class on the G side is not split, the inner class on the dual side is not equal rank. Therefore we only lcompute the #RF column entry (a list of orbits for the integrality datum) if the orbit is even. i diagram dim BC Levi Cent_0 Z C_2 A(O) #RF #AP Cent #reps(E6_q) E6_h E6_c 0 [0,0,0,0,0,0] 0 6T1 E6 1 3 [1] [1,1] 2 E6_ad 2*=1+1 0 0 1 [0,1,0,0,0,0] 22 A1+5T1 A5 2 4 [1] [] 2 SL(6)/Z3 2*=1+1 0 0 2 [1,0,0,0,0,1] 32 2A1+4T1 B3+T1 2 5 [1] [] 2 GSpin(7) [a] 3 0 0 3 [0,0,0,1,0,0] 40 3A1+3T1 A1+A2 2 4 [1] [] 2 SL(2)xPSL(3) [b] 2*=1+1 0 0 4 [0,2,0,0,0,0] 42 A2+4T1 2A2 3 4 [1,2] [0,2] 2 [SL(3)xSL(3)]/diag]|2 [c] 4*=2+2 0 0 5 [1,1,0,0,0,1] 46 A1+A2+3T1 A2+T1 3 4 [1] [] 0 GL(3) [d] 0 0 0 6 [2,0,0,0,0,2] 48 2A2+2T1 G2 1 2 [1] [1,1] 2 G2 3 0 0 7 [0,0,1,0,1,0] 50 2A1+A2+2T1 A1+T1 2 3 [1] [] 1 GL(2) [e] 1=1 0 0 8 [1,2,0,0,0,1] 52 A3+3T1 B2+T1 2 4 [1] [] 2 GSpin(5) [f] 3 1 0 9 [1,0,0,1,0,1] 54 A1+2A2+T1 A1 2 2 [1] [] 2 SL(2) 2=1+1 0 0 10 [0,1,1,0,1,0] 56 A1+A3+2T1 A1+T1 2 4 [1] [] 2 SL(2)xGL(1) 3 0 0 11 [0,0,0,2,0,0] 58 D4+2T1 2T1 1 4 [1,2,3] [0,2] 2 [GL(1)^3_0]|S3 [g] 7 0 0 12 [2,2,0,0,0,2] 60 A4+2T1 A1+T1 2 3 [1] [0,1] 1 GL(2) [h] 2*=2 2 0 13 [0,2,0,2,0,0] 60 D4+2T1 A2 1 2 [1] [0,1] 1 PSL(3) 1*=1 1 0 14 [1,1,1,0,1,1] 62 A1+A4+T1 T1 1 2 [1] [] 0 GL(1) 0 0 0 15 [1,2,1,0,1,1] 64 D5+T1 T1 1 2 [1] [] 0 GL(1) 0 0 0 16 [2,1,1,0,1,2] 64 A5+T1 A1 2 2 [1] [] 2 SL(2) 2=1+1 1 0 17 [2,0,0,2,0,2] 66 E6 e 1 1 [1,2] [0,2] 2 Z2 6 1 0 18 [2,2,0,2,0,2] 68 D5+T1 T1 1 2 [1] [0,1] 1 GL(1) 2=2 3 0 19 [2,2,2,0,2,2] 70 E6 e 1 1 [1] [0,1] 1 e 1=1 2 0 20 [2,2,2,2,2,2] 72 E6 e 1 1 [1] [0,1] 1 e 1=1 1 1 [a] Center is GL(1) [b] E6_ad_centralizer_isogenies.txt [c] extension splits (E6_ad_components.txt), C_2=#AP=> switches factors [SL(3)xSL(3)]\rtimes Z2, action by switching factors [d,e,f] Center is GL(1) [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.