Computing weak packets for 21 dual orbits of compact connected real group with Lie algebra 'e6' Initializing CharacterTable for Lie type 'E6' Step 1/6 Step 2/6 Step 3/6 Step 4/6 Step 5/6 Step 6/6 Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 16, 22, 30, 42, 30, 16 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 0, 0, 0, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 12, 16, 22, 30, 22, 12 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 1, 0, 0, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 10, 14, 18, 26, 18, 10 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 1, 0, 0, 0, 0, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 8, 10, 14, 20, 14, 8 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 0, 0, 1, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 8, 10, 14, 20, 14, 8 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 2, 0, 0, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 7, 10, 13, 18, 13, 7 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 1, 1, 0, 0, 0, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 6, 10, 12, 18, 12, 6 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 2, 0, 0, 0, 0, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 6, 8, 11, 15, 11, 6 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 0, 1, 0, 1, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 6, 8, 10, 14, 10, 6 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 1, 2, 0, 0, 0, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 4, 6, 8, 12, 8, 4 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 1, 0, 0, 1, 0, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 4, 6, 8, 12, 8, 4 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 1, 1, 0, 1, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 4, 6, 8, 12, 8, 4 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 0, 0, 2, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 4, 6, 7, 10, 7, 4 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 2, 2, 0, 0, 0, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 4, 4, 6, 8, 6, 4 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 0, 2, 0, 2, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 4, 5, 7, 10, 7, 4 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 1, 1, 1, 0, 1, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 3, 4, 5, 7, 5, 3 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 1, 2, 1, 0, 1, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 2, 4, 4, 6, 4, 2 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 2, 1, 1, 0, 1, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 2, 4, 4, 6, 4, 2 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 2, 0, 0, 2, 0, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 2, 2, 3, 4, 3, 2 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 2, 2, 0, 2, 0, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 1, 2, 2, 3, 2, 1 ]) Skipping dual orbit (adjoint root datum of Lie type 'E6',(),[ 2, 2, 2, 0, 2, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected root datum of Lie type 'E6',(),[ 0, 0, 0, 0, 0, 0 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'E6' [ 2, 2, 2, 2, 2, 2 ] dim=72 Computing weak packets for compact connected real group with Lie algebra 'e6' gamma:[ 3, 3, 3, 3, 3, 3 ]/1 gamma_final:[ 1, 1, 1, 1, 1, 1 ]/1 Initializing CharacterTable for Lie type 'E6' Step 1/6 Step 2/6 Step 3/6 Step 4/6 Step 5/6 Step 6/6 integral data: st_int rd_int:simply connected root datum of Lie type 'E6' st_int.rd: simply connected root datum of Lie type 'E6' O_check_int:(adjoint root datum of Lie type 'E6',(),[ 2, 2, 2, 2, 2, 2 ]) computing packet for:(adjoint root datum of Lie type 'E6',(),[ 2, 2, 2, 2, 2, 2 ]) computing springer map of[0,0,0,0,0,0] O: (adjoint root datum of Lie type 'E6',(),[ 0, 0, 0, 0, 0, 0 ]) survive:final parameter(x=0,lambda=[3,3,3,3,3,3]/1,nu=[0,0,0,0,0,0]/1) [ 1, 1, 1, 1, 1, 1 ]/1 cell character: 1 springer_O:1 =============================================================================== Orbits for the dual group: connected split real group with Lie algebra 'e6(R)' 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 i diagram dim BC Levi Cent_0 Z C_2 A(O) #RF #AP 0 [0,0,0,0,0,0] 0 6T1 E6 1 3 [1] [1,1] 2 1 [0,1,0,0,0,0] 22 A1+5T1 A5 2 4 [1] [] 2 2 [1,0,0,0,0,1] 32 2A1+4T1 B3+T1 2 5 [1] [] 2 3 [0,0,0,1,0,0] 40 3A1+3T1 A1+A2 2 4 [1] [] 2 4 [0,2,0,0,0,0] 42 A2+4T1 2A2 3 4 [1,2] [0,2] 2 5 [1,1,0,0,0,1] 46 A1+A2+3T1 A2+T1 3 4 [1] [] 0 6 [2,0,0,0,0,2] 48 2A2+2T1 G2 1 2 [1] [1,1] 2 7 [0,0,1,0,1,0] 50 2A1+A2+2T1 A1+T1 2 3 [1] [] 1 8 [1,2,0,0,0,1] 52 A3+3T1 B2+T1 2 4 [1] [] 2 9 [1,0,0,1,0,1] 54 A1+2A2+T1 A1 2 2 [1] [] 2 10 [0,1,1,0,1,0] 56 A1+A3+2T1 A1+T1 2 4 [1] [] 2 11 [0,0,0,2,0,0] 58 D4+2T1 2T1 1 4 [1,2,3] [0,2] 2 12 [2,2,0,0,0,2] 60 A4+2T1 A1+T1 2 3 [1] [0,1] 1 13 [0,2,0,2,0,0] 60 D4+2T1 A2 1 2 [1] [0,1] 1 14 [1,1,1,0,1,1] 62 A1+A4+T1 T1 1 2 [1] [] 0 15 [1,2,1,0,1,1] 64 D5+T1 T1 1 2 [1] [] 0 16 [2,1,1,0,1,2] 64 A5+T1 A1 2 2 [1] [] 2 17 [2,0,0,2,0,2] 66 E6 e 1 1 [1,2] [0,2] 2 18 [2,2,0,2,0,2] 68 D5+T1 T1 1 2 [1] [0,1] 1 19 [2,2,2,0,2,2] 70 E6 e 1 1 [1] [0,1] 1 20 [2,2,2,2,2,2] 72 E6 e 1 1 [1] [0,1] 1 Information about orbit centralizers: orbit#: 0 diagram: [0,0,0,0,0,0] isogeny information: Centralizer: E6 Center is trivial adjoint root datum of Lie type 'E6' ------------- orbit#: 1 diagram: [0,1,0,0,0,0] isogeny information: Centralizer: A5 Group is semisimple center=Z/2Z root datum of Lie type 'A5' ------------- orbit#: 2 diagram: [1,0,0,0,0,1] isogeny information: Centralizer: B3+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'B3' ------------- orbit#: 3 diagram: [0,0,0,1,0,0] isogeny information: Centralizer: A1+A2 Group is semisimple center=Z/2Z adjoint root datum of Lie type 'A2' simply connected root datum of Lie type 'A1' ------------- orbit#: 4 diagram: [0,2,0,0,0,0] isogeny information: Centralizer: 2A2 Group is semisimple center=Z/3Z simply connected root datum of Lie type 'A2' simply connected root datum of Lie type 'A2' ------------- orbit#: 5 diagram: [1,1,0,0,0,1] isogeny information: Centralizer: A2+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'A2' ------------- orbit#: 6 diagram: [2,0,0,0,0,2] isogeny information: Centralizer: G2 Center is trivial simply connected adjoint root datum of Lie type 'G2' ------------- orbit#: 7 diagram: [0,0,1,0,1,0] isogeny information: Centralizer: A1+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'A1' ------------- orbit#: 8 diagram: [1,2,0,0,0,1] isogeny information: Centralizer: B2+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'B2' ------------- orbit#: 9 diagram: [1,0,0,1,0,1] isogeny information: Centralizer: A1 Group is semisimple center=Z/2Z simply connected root datum of Lie type 'A1' ------------- orbit#: 10 diagram: [0,1,1,0,1,0] isogeny information: Centralizer: A1+T1 Split exact sequence: 1->S->Z->Z/S->1 S=complex torus of rank 1 Z/S=Center(G/S)=Z/2Z simply connected root datum of Lie type 'A1' ------------- orbit#: 11 diagram: [0,0,0,2,0,0] isogeny information: Centralizer: 2T1 Center is a connected complex torus of rank 2 ------------- orbit#: 12 diagram: [2,2,0,0,0,2] isogeny information: Centralizer: A1+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'A1' ------------- orbit#: 13 diagram: [0,2,0,2,0,0] isogeny information: Centralizer: A2 Center is trivial adjoint root datum of Lie type 'A2' ------------- orbit#: 14 diagram: [1,1,1,0,1,1] isogeny information: Centralizer: T1 Center is a connected complex torus of rank 1 ------------- orbit#: 15 diagram: [1,2,1,0,1,1] isogeny information: Centralizer: T1 Center is a connected complex torus of rank 1 ------------- orbit#: 16 diagram: [2,1,1,0,1,2] isogeny information: Centralizer: A1 Group is semisimple center=Z/2Z simply connected root datum of Lie type 'A1' ------------- orbit#: 17 diagram: [2,0,0,2,0,2] isogeny information: Centralizer: e Center is trivial ------------- orbit#: 18 diagram: [2,2,0,2,0,2] isogeny information: Centralizer: T1 Center is a connected complex torus of rank 1 ------------- orbit#: 19 diagram: [2,2,2,0,2,2] isogeny information: Centralizer: e Center is trivial ------------- orbit#: 20 diagram: [2,2,2,2,2,2] isogeny information: Centralizer: e Center is trivial ------------- Arthur parameters listed by orbit: #parameters by orbit: [2,2,2,2,2,0,2,1,2,2,2,2,1,1,0,0,2,2,1,1,1] Total: 30 orbit #0 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 0, 0 ] [0,0,0,0] 0 1 adjoint root datum of Lie type 'C4' [ 0, 0, 0, 0 ] [0,0,0,0] 0 1 orbit #1 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 1, 0, 0 ] [1,0,0,0] 16 1 adjoint root datum of Lie type 'C4' [ 0, -1, 0, 1 ] [0,0,0,1] 8 1 orbit #2 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 1, 0, 0, 0 ] [0,0,1,0] 22 1 adjoint root datum of Lie type 'C4' [ 1, 0, 0, 0 ] [0,1,0,0] 14 1 orbit #3 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 0, 1 ] [0,1,0,0] 28 1 adjoint root datum of Lie type 'C4' [ 0, 1, 1, -1 ] [0,0,1,0] 18 1 orbit #4 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 2, 0, 0 ] [2,0,0,0] 30 1 adjoint root datum of Lie type 'C4' [ 0, 2, 0, 0 ] [2,0,0,0] 20 1 orbit #5 for G #orbits for (disconnected) Cent(O): 0 K_0 H diagram dim mult orbit #6 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 2, 0, 0, 0 ] [0,0,2,0] 30 1 adjoint root datum of Lie type 'C4' [ 2, 0, 0, 0 ] [0,2,0,0] 22 1 orbit #7 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 1, 0 ] [0,0,0,1] 34 1 orbit #8 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 1, 2, 0, 0 ] [2,0,1,0] 36 1 adjoint root datum of Lie type 'C4' [ 1, -2, 0, 2 ] [0,1,0,2] 20 1 orbit #9 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 1, 0, 0, 1 ] [0,1,1,0] 36 1 adjoint root datum of Lie type 'C4' [ 1, 1, 1, -1 ] [0,1,1,0] 24 1 orbit #10 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 1, 1, 0 ] [1,0,0,1] 38 1 adjoint root datum of Lie type 'C4' [ 0, -1, 1, 1 ] [0,0,1,2] 24 1 orbit #11 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 0, 2 ] [0,2,0,0] 40 1 adjoint root datum of Lie type 'C4' [ 0, 0, 0, 2 ] [2,0,0,2] 26 1 orbit #12 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult adjoint root datum of Lie type 'C4' [ 2, 2, 0, 0 ] [2,2,0,0] 28 1 orbit #13 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 0, 2, 0, 2 ] [2,2,0,0] 42 1 orbit #14 for G #orbits for (disconnected) Cent(O): 0 K_0 H diagram dim mult orbit #15 for G #orbits for (disconnected) Cent(O): 0 K_0 H diagram dim mult orbit #16 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 2, 1, 1, 0 ] [1,0,2,1] 42 1 adjoint root datum of Lie type 'C4' [ 2, -1, 1, 1 ] [0,2,1,2] 28 1 orbit #17 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 2, 0, 0, 2 ] [0,2,2,0] 44 1 adjoint root datum of Lie type 'C4' [ 2, 0, 0, 2 ] [2,2,0,2] 30 1 orbit #18 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 2, 2, 0, 2 ] [2,2,2,0] 46 1 orbit #19 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult adjoint root datum of Lie type 'C4' [ 2, 2, 2, 0 ] [2,2,2,2] 32 1 orbit #20 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'F4' [ 2, 2, 2, 2 ] [2,2,2,2] 48 1 orbit |packet| 20 1 Total 1 *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 20 0 0 1 Total 1 orbit# block# cell# parameters inf. char. 20 0 0 final parameter(x=0,lambda=[1,1,1,1,1,1]/1,nu=[0,0,0,0,0,0]/1) [ 1, 1, 1, 1, 1, 1 ]/1 Total 1 Induced 0 set parameters=[ parameter(G,0,[ 1, 1, 1, 1, 1, 1 ]/1,[ 0, 0, 0, 0, 0, 0 ]/1) ]