Arthur packets for the unique inner class of real forms of F4 complex nilpotent orbits for inner class Complex reductive group of type F4, with involution defining inner class of type 'c', with 3 real forms and 3 dual real forms root datum of inner class: simply connected adjoint root datum of Lie type 'F4' 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(O) #reps(F4_s) F4_B4 F4_c 0 [0,0,0,0] 0 4T1 F4 1 3 [1] [1,1,1] 3 F4 3*=1+1+1 0 0 1 [0,0,0,1] 16 A1+3T1 C3 2 4 [1] [2,2,0] 4 Sp(6) 4*=1+1+1+1 0 0 2 [1,0,0,0] 22 A1+3T1 A3 4 3 [1,2] [3,2] 5 SL(4).2 [a] 10*=2+2+2+2+2 0 0 3 [0,0,1,0] 28 2A1+2T1 2A1 2 4 [1] [0,0,4] 4 SL(2)xPSL(2) [b] 5 0 0 4 [2,0,0,0] 30 A2+2T1 G2 1 2 [1] [0,1,1] 2 SL(3)|2 [c] 2*=1+1 0 0 5 [0,0,0,2] 30 A2+2T1 A2 3 2 [1,2] [0,0,3] 3 G2 4 0 0 6 [0,1,0,0] 34 A1+A2+T1 A1 2 2 [1] [0,2] 2 SL(2) 2*=1+1 0 0 7 [1,0,0,2] 36 C2+2T1 2A1 4 4 [1,2] [2,2] 4 [d] 7 0 0 8 [1,0,1,0] 36 A1+A2+T1 A1 2 2 [1] [0,2,0] 2 SL(2) 2 0 0 9 [0,1,0,1] 38 C3+T1 A1 2 2 [1,2] [0,0,4] 4 SL(2)xZ2 [e] 8 0 0 10 [0,0,2,0] 40 F4 e 1 1 [1,2,2,3,4] [0,0,3] 3 S4 14*=4+5+5 0 0 11 [2,1,0,1] 42 C3+T1 A1 2 2 [1] [0,0,2] 2 PSL(2) 2=1+1 1 0 12 [0,0,2,2] 42 B3+T1 A1 1 2 [1] [0,0,2] 2 SL(2) 4 0 0 13 [2,0,2,0] 44 F4 e 1 1 [1,2] [0,0,2] 2 Z2 3 0 0 14 [2,0,2,2] 46 F4 e 1 1 [1,2] [0,0,2] 2 Z2 4=2+2 1 0 15 [2,2,2,2] 48 F4 e 1 1 [1] [0,0,1] 1 1 1 1 1 [a] 1->SL(4)->Cent->Z2->1 sequence splits (set F4_centralizer_components.txt) action is non-trivial (#AP=5 not 6) [b]: from F4_centralizer_isogenies: orbit: H:[ 3, 6, 4, 2 ] diagram:[0,1,0,0] dim:28 Centralizer: 2A1 simply connected root datum of Lie type 'A1' adjoint root datum of Lie type 'A1' [c] SL(2)|2: semidirect product sequence obviously splits action must be outer (#AP=2 not 4) [d] if action is trivial => >4 elements of order 2 in Cent## => action is non-trivial sequence does NOT split F4_centralizer_components.txt: Component info for orbit: H=[ 6, 10, 7, 4 ] diagram:[2,0,0,1] dim:36 orders:[1,2] pseudo_Levi Generators B2 [[ 0, 0, 0, 0 ]/1] A3 [[ -4, -8, -6, -3 ]/4] Cent= x^2=(-I,-I) This group has 4 conjugacy classes of order 1 or 2: (I,I), (I,-I), (-I, I) (conjugate to (I,-I)) and (I,-I)x [all (a,-1/a)x are conjguate] [e] center of derived group is Z2 sequence splits (F4_centralizer_components.at or #AP=4)