Arthur parameters for simply connected E7 one inner class G=E7_s, E7_q, E7_h, E7_c (all simply connected) dual group is adjoint complex nilpotent orbits for inner class (adjoint) 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: adjoint 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 1 4 [1] [1,1,1,1] 4 E7 0 0 0 6*=1+1+2+2 1 [1,0,0,0,0,0,0] 34 A1+6T1 D6 2 5 [1] [2,0,0,2,1,0] 5 Semispin(12) [a] 0 0 0 7*=1^7 2 [0,0,0,0,0,1,0] 52 2A1+5T1 A1+B4 2 6 [1] [0,3,0,0,2,1] 6 Spin(9)xSL(2)/<-1,-1> [b] 0 0 0 8* 3 [0,0,0,0,0,0,2] 54 3A1+4T1 F4 1 3 [1] [0,2,0,1] 3 F4 2 0 0 3*=1^3 4 [0,0,1,0,0,0,0] 64 3A1+4T1 A1+C3 2 6 [1] [0,0,4,0,0,2] 6 SL(2)xPSp(6) [c] 0 0 0 8* 5 [2,0,0,0,0,0,0] 66 A2+5T1 A5 3 4 [1,2] [0,1,3,2] 6 [SL(6)/Z2]|2 [d] 0 0 0 16=2^8 6 [0,1,0,0,0,0,1] 70 4A1+3T1 C3 2 4 [1] [0,0,0,4,0,0] 4 Sp(6) 2 0 0 4*=1^4 7 [1,0,0,0,0,1,0] 76 A1+A2+4T1 A3+T1 4 5 [1,2] [0,0,0,2,4,0] 6 GL(4)|2 [e] 2 0 0 12=2^6 8 [0,0,0,1,0,0,0] 82 2A1+A2+3T1 3A1 2 5 [1] [0,0,0,0,2,3] 5 SL(2)^3/<-1,-1,-1> [f] 0 0 0 9* 9 [2,0,0,0,0,1,0] 84 A3+4T1 A1+B3 2 5 [1] [0,2,0,0,2,1] 5 SL(2)xSpin(7)/<-1,-1> [g] 0 0 0 7 10 [0,0,0,0,0,2,0] 84 2A2+3T1 A1+G2 1 4 [1] [0,1,2,1] 4 PSL(2)xG2 [h] 0 0 0 6* 11 [0,2,0,0,0,0,0] 84 3A1+A2+2T1 G2 1 2 [1] [0,0,0,2] 2 G2 1 0 0 2*=1+1 12 [2,0,0,0,0,0,2] 86 A1+A3+3T1 B3 2 3 [1] [0,2,0,1] 3 Spin(7) 3 0 0 3=1+1+1 13 [0,0,1,0,0,1,0] 90 A1+2A2+2T1 2A1 2 4 [1] [0,0,0,2,2,0] 4 SL(2)xPSL(2) [i] 0 0 0 6* 14 [1,0,0,1,0,0,0] 92 A1+A3+3T1 3A1 4 6 [1] [0,0,4,0,0,2] 6 SL(2)^3/<-1,-1,-1> [j] 0 0 0 8 15 [0,0,2,0,0,0,0] 94 D4+3T1 3A1 4 5 [1,2,3] [0,0,3,2] 5 [SL(2)^3/<-1,-1,-1>].S3 [k] 0 0 0 17* 16 [1,0,0,0,1,0,1] 94 2A1+A3+2T1 2A1 4 4 [1] [0,0,0,4,0,0] 4 SL(2)^2/<-1,-1> [l] 2 0 0 4=1^4 17 [2,0,2,0,0,0,0] 96 D4+3T1 C3 1 3 [1] [0,0,2,1] 3 PSp(6) 0 3 0 4* 18 [0,1,1,0,0,0,1] 96 A1+D4+2T1 2A1 4 4 [1,2] [0,0,0,0,4,0] 4 [SL(2)xSL(2)]|2 [m] 4 0 0 7 19 [0,0,0,1,0,1,0] 98 A2+A3+2T1 A1+T1 2 3 [1,2] [0,0,0,0,1,3] 4 GL(2)|2 [n] 0 0 0 7 20 [2,0,0,0,0,2,0] 100 A4+3T1 A2+T1 3 4 [1,2] [0,1,2,2] 5 GL(3)|2 [o] 1 0 0 12*a 21 [0,0,0,0,2,0,0] 100 A1+A2+A3+T1 A1 1 2 [1] [0,0,0,2] 2 PSL(2) 1 0 0 2=1+1 22 [2,1,1,0,0,0,1] 102 A1+D4+2T1 C2 2 3 [1] [0,0,0,0,3,0] 3 Sp(4) 3 1 0 3=1+1+1 23 [2,0,0,0,0,2,2] 102 A5+2T1 G2 1 2 [1] [0,1,0,1] 2 G2 3 0 0 2=1+1 24 [1,0,0,1,0,1,0] 104 A1+A4+2T1 2T1 1 4 [1,2] [0,0,0,2,2,0] 4 [GL(1)xGL(1)].2 [p] 0 0 0 8 25 [2,0,0,1,0,1,0] 106 D5+2T1 A1+T1 2 3 [1,2] [0,0,0,0,1,3] 4 GL(2)|2 [q] 2 1 0 8=2^4 26 [0,0,0,2,0,0,0] 106 A2+A4+T1 A1 1 2 [1] [0,0,1,1] 2 PSL(2) 0 0 0 5 27 [1,0,0,1,0,2,0] 108 A5+2T1 2A1 2 4 [1] [0,0,2,0,0,2] 4 SL(2)xPSL(2) 0 0 0 6 28 [2,0,0,0,2,0,0] 108 A1+D5+T1 A1 1 2 [1] [0,0,0,2] 2 PSL(2) 1 0 0 4 29 [1,0,0,1,0,1,2] 108 A1+A5+T1 A1 2 2 [1] [0,0,0,2,0,0] 2 SL(2) 2 0 0 2=1+1 30 [0,0,2,0,0,2,0] 110 E6+T1 A1 1 2 [1,2] [0,0,2,2] 4 PSL(2)xZ2 0 0 0 12* 31 [0,1,1,0,1,0,2] 110 D6+T1 A1 2 2 [1] [0,0,0,0,0,2] 2 SL(2) 3 0 0 2=1+1 32 [2,0,2,0,0,2,0] 112 D5+2T1 2A1 2 3 [1] [0,0,2,1] 3 SO(4) [r] 0 5 0 4 33 [0,0,0,2,0,0,2] 112 E7 e 1 1 [1,2,3] [0,0,0,2] 2 S3 7 0 0 5 34 [2,1,1,0,1,1,0] 114 A1+D5+T1 A1 2 2 [1] [0,0,0,0,2,0] 2 SL(2) 1 1 0 2 35 [0,0,0,2,0,2,0] 114 A6+T1 A1 1 2 [1] [0,0,1,1] 2 PSL(2) 0 0 0 5*=2+3 36 [2,1,1,0,1,0,2] 114 D6+T1 A1 2 2 [1] [0,0,0,0,0,2] 2 SL(2) 1 2 0 2=1+1 37 [2,0,0,2,0,0,2] 116 E7 e 1 1 [1,2] [0,0,0,2] 2 Z2 2 0 0 4 38 [2,0,0,2,0,2,0] 118 E6+T1 T1 1 2 [1,2] [0,0,1,2] 3 GL(1)|2 [s] 1 2 0 8* 39 [2,1,1,0,1,2,2] 118 D6+T1 A1 2 2 [1] [0,0,0,0,0,2] 2 SL(2) 2 1 0 2=1+1 40 [2,0,2,2,0,2,0] 120 E6+T1 A1 1 2 [1] [0,0,1,1] 2 PSL(2) 0 5 0 3* 41 [2,0,0,2,0,2,2] 120 E7 e 1 1 [1,2] [0,0,0,2] 2 Z2 6 1 0 4*=2+2 42 [2,2,2,0,2,0,2] 122 E7 e 1 1 [1] [0,0,0,1] 1 e 2 3 0 1*=1 43 [2,2,2,0,2,2,2] 124 E7 e 1 1 [1] [0,0,0,1] 1 e 1 2 0 1*=1 44 [2,2,2,2,2,2,2] 126 E7 e 1 1 [1] [0,0,0,1] 1 e 1 1 1 1*=1 [a] center + number of involutions [b] center + centralizer isogeny [c] centralizer isogeny [d] centralizer isogeny => 1-> SL(6)/Z2 -> Cent -> Z2 -> 1 this splits (E7_ad_components.txt) #involutions => nontrivial action => [SL(6)/Z2]\rtimes Z2 [e] center + centralizer isogeny => Cent_0=SL(4)xGL(1)/* #involutions => Cent_0=GL(4) => 1 -> GL(4) -> Cent -> Z2 -> 1 this splits (E7_ad_components.txt) #involutions=5 => action not trivial on derived group SL(4) GL(1)\cap SL(4)=Z/4Z, action is by inverse on here => action is by inverse on GL(1) => GL(4)|2, semidirect product, action is transpose-inverse [f,g,h,i,j] centralizer isogeny [k] centralizer isogeny 1 -> SL(2)^3/<-1,-1,-1> -> Cent -> S3 -> 1 E7_ad_components.txt => extension splits S3 is acting by permutation of the factors [k,l] centralizer isogeny [m] center => Cent_0=SL(2)^2 E7_ad_components.txt => extension splits #AP=4 (not 4) => action is not trivial => action switches factors sequence splits (E7_ad_components.txt) [SL(2)xSL(2)]\rtimes Z2 [n,q] isogenies => Cent_0=GL(2) E7_ad_components.txt => extension splits #AP=4 (not 6) => non-trivial action => non-trivial on GL(1) GL(2)\rtimes Z2 (acting by transpose inverse) [o] isogenies => Cent_0=GL(3) E7_ad_components.txt => extension splits #AP=5 (not 8) => non-trivial action => GL(3)\rtimes Z2 (acting by transpose inverse) [p] 24 [1,0,0,1,0,1,0] 104 A1+A4+2T1 2T1 1 4 [1,2] [0,0,0,2,2,0] 4 [GL(1)xGL(1)].2 [p] 0 0 0 8 E7_ad_components.txt => extension does NOT split 1 -> GL(1)^2 -> Cent -> Z2 -> 1 #AP=4 => action is non-trivial, several cases action is NOT by switching factors (in which case the extension necessarily splits) two possibilities: acting by inverse on one or both factor Cent= = delta(z,w)=(z,1/w) and delta^2=(1,-1) or delta(z,w)=(1/z,1/w) and delta^2=(epsilon_1,epsilon_2)\ne (1,1) Claim: (T^delta)_0 is trivial: this orbit is quasi-distinguished, see Ciubotaru/He, Green Polynomials, page 14 Conclusion: Cent= \delta acting by inverse, delta^2 =(1,-1),(-1,1) or (-1,-1) [r] isogenies => SL(2)xSL(2)/<-I,-I>=SO(4) [s] E7_ad_components.txt => extension does splits, #AP=3 (not 4) => nontrivial action GL(1)\rtimes Z2 (acting by inverse)