Computing weak packets for 16 dual orbits of compact connected real group with Lie algebra 'f4' Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 22, 42, 30, 16 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 14, 26, 18, 10 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 0, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 14, 26, 18, 10 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 1, 0, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 10, 20, 14, 8 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 1, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 10, 19, 14, 8 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 2, 0, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 10, 18, 12, 6 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 0, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 6, 12, 8, 4 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 1, 0, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 6, 12, 8, 4 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 1, 0, 0, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 6, 12, 8, 4 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 1, 0, 1, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 6, 12, 8, 4 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 1, 0, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 6, 12, 8, 4 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 2, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 4, 8, 6, 4 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 2, 1, 0, 1 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 4, 6, 4, 2 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 2, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 3, 6, 4, 2 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 2, 0, 2, 0 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 2, 4, 3, 2 ]) Skipping dual orbit (simply connected adjoint root datum of Lie type 'F4',(),[ 2, 0, 2, 2 ]) (d(O_check) has no real forms for G) Orbit by diagram: (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 0, 0 ]) Computing weak packet for orbit: simply connected adjoint root datum of Lie type 'F4' [ 2, 2, 2, 2 ] dim=48 Computing weak packets for compact connected real group with Lie algebra 'f4' gamma:[ 3, 3, 3, 3 ]/1 gamma_final:[ 1, 1, 1, 1 ]/1 integral data: st_int rd_int:simply connected adjoint root datum of Lie type 'F4' st_int.rd: simply connected adjoint root datum of Lie type 'F4' O_check_int:(simply connected adjoint root datum of Lie type 'F4',(),[ 2, 2, 2, 2 ]) permutation: | 1, 0, 0, 0 | | 0, 1, 0, 0 | | 0, 0, 1, 0 | | 0, 0, 0, 1 | computing packet for:(simply connected adjoint root datum of Lie type 'F4',(),[ 2, 2, 2, 2 ]) computing springer map of[0,0,0,0] O: (simply connected adjoint root datum of Lie type 'F4',(),[ 0, 0, 0, 0 ]) survive:final parameter(x=0,lambda=[3,3,3,3]/1,nu=[0,0,0,0]/1) [ 1, 1, 1, 1 ]/1 cell character: 3 springer_O:3 =============================================================================== complex nilpotent orbits for compact connected real group with Lie algebra '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 A(O): orders of conj. classes in component group of centralizer #RF(O): number of real forms of O C_2: conjugacy classes in Cent(SL(2))_0 with square 1 i H diagram dim BC Levi Cent_0 Z(Cent^0) A(O) #RF(O) C_2 0 [0,0,0,0] [0,0,0,0] 0 4T1 F4 1 [1] 1 3 1 [2,3,2,1] [1,0,0,0] 16 A1+3T1 C3 2 [1] 0 4 2 [2,4,3,2] [0,0,0,1] 22 A1+3T1 A3 4 [1,2] 0 3 3 [3,6,4,2] [0,1,0,0] 28 2A1+2T1 2A1 2 [1] 0 4 4 [4,6,4,2] [2,0,0,0] 30 A2+2T1 A2 3 [1,2] 0 2 5 [4,8,6,4] [0,0,0,2] 30 A2+2T1 G2 1 [1] 0 2 6 [4,8,6,3] [0,0,1,0] 34 A1+A2+T1 A1 2 [1] 0 2 7 [6,10,7,4] [2,0,0,1] 36 B2+2T1 2A1 4 [1,2] 0 4 8 [5,10,7,4] [0,1,0,1] 36 A1+A2+T1 A1 2 [1] 0 2 9 [6,11,8,4] [1,0,1,0] 38 C3+T1 A1 2 [1,2] 0 2 10 [6,12,8,4] [0,2,0,0] 40 F4 e 1 [1,2,2,3,4] 0 1 11 [10,18,12,6] [2,2,0,0] 42 B3+T1 A1 1 [1] 0 2 12 [10,19,14,8] [1,0,1,2] 42 C3+T1 A1 2 [1] 0 2 13 [10,20,14,8] [0,2,0,2] 44 F4 e 1 [1,2] 0 1 14 [14,26,18,10] [2,2,0,2] 46 F4 e 1 [1,2] 0 1 15 [22,42,30,16] [2,2,2,2] 48 F4 e 1 [1] 0 1 Arthur parameters listed by orbit: #parameters by orbit: [3,4,5,4,2,3,2,4,2,4,3,2,2,2,2,1] Total: 45 orbit #0 for G #orbits for (disconnected) Cent(O): 3 K_0 H mult root datum of Lie type 'C3.A1' [ 0, 0, 0, 0 ] 1 simply connected root datum of Lie type 'B4' [ 0, 0, 0, 0 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 0, 0 ] 1 orbit #1 for G #orbits for (disconnected) Cent(O): 4 K_0 H mult root datum of Lie type 'C3.A1' [ -1, 1, -1, 1 ] 1 root datum of Lie type 'C3.A1' [ 0, 0, 1, -1 ] 1 simply connected root datum of Lie type 'B4' [ 0, 0, 0, 1 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 0, 1 ] 1 orbit #2 for G #orbits for (disconnected) Cent(O): 5 K_0 H mult root datum of Lie type 'C3.A1' [ -1, 1, 0, 0 ] 1 root datum of Lie type 'C3.A1' [ 1, 0, 0, 0 ] 1 simply connected root datum of Lie type 'B4' [ -1, 1, 0, 0 ] 1 simply connected root datum of Lie type 'B4' [ 1, 0, 0, 0 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 1, 0, 0, 0 ] 1 orbit #3 for G #orbits for (disconnected) Cent(O): 4 K_0 H mult root datum of Lie type 'C3.A1' [ 1, -1, 1, 1 ] 1 root datum of Lie type 'C3.A1' [ 0, 1, -1, 1 ] 1 simply connected root datum of Lie type 'B4' [ 0, 0, 1, 0 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 1, 0 ] 1 orbit #4 for G #orbits for (disconnected) Cent(O): 2 K_0 H mult root datum of Lie type 'C3.A1' [ 2, 0, 0, 0 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 2, 0, 0, 0 ] 1 orbit #5 for G #orbits for (disconnected) Cent(O): 3 K_0 H mult root datum of Lie type 'C3.A1' [ 0, 0, 0, 2 ] 1 simply connected root datum of Lie type 'B4' [ 0, 0, 0, 2 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 0, 2 ] 1 orbit #6 for G #orbits for (disconnected) Cent(O): 2 K_0 H mult simply connected root datum of Lie type 'B4' [ 1, -1, 2, 0 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 1, 0, 0 ] 1 orbit #7 for G #orbits for (disconnected) Cent(O): 4 K_0 H mult root datum of Lie type 'C3.A1' [ 1, 0, 2, -2 ] 1 simply connected root datum of Lie type 'B4' [ -1, 1, 0, 2 ] 1 simply connected root datum of Lie type 'B4' [ 1, 0, 0, 2 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 1, 0, 0, 2 ] 1 orbit #8 for G #orbits for (disconnected) Cent(O): 2 K_0 H mult root datum of Lie type 'C3.A1' [ 1, 1, -1, 1 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 1, 0, 1, 0 ] 1 orbit #9 for G #orbits for (disconnected) Cent(O): 4 K_0 H mult root datum of Lie type 'C3.A1' [ 1, -1, 3, -1 ] 1 root datum of Lie type 'C3.A1' [ 0, 1, 1, -1 ] 1 simply connected root datum of Lie type 'B4' [ 0, 1, 0, 1 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 1, 0, 1 ] 1 orbit #10 for G #orbits for (disconnected) Cent(O): 3 K_0 H mult root datum of Lie type 'C3.A1' [ 0, 0, 2, 0 ] 1 simply connected root datum of Lie type 'B4' [ 0, 0, 2, 0 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 2, 0 ] 1 orbit #11 for G #orbits for (disconnected) Cent(O): 2 K_0 H mult root datum of Lie type 'C3.A1' [ 3, -1, 3, -1 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 2, 1, 0, 1 ] 1 orbit #12 for G #orbits for (disconnected) Cent(O): 2 K_0 H mult simply connected root datum of Lie type 'B4' [ 0, 0, 2, 2 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 0, 0, 2, 2 ] 1 orbit #13 for G #orbits for (disconnected) Cent(O): 2 K_0 H mult root datum of Lie type 'C3.A1' [ 2, 0, 2, 0 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 2, 0, 2, 0 ] 1 orbit #14 for G #orbits for (disconnected) Cent(O): 2 K_0 H mult simply connected root datum of Lie type 'B4' [ 2, 0, 2, 2 ] 1 simply connected adjoint root datum of Lie type 'F4' [ 2, 0, 2, 2 ] 1 orbit #15 for G #orbits for (disconnected) Cent(O): 1 K_0 H mult simply connected adjoint root datum of Lie type 'F4' [ 2, 2, 2, 2 ] 1 orbit |packet| 15 1 Total 1 *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 15 0 0 1 Total 1 *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 15 0 0 final parameter(x=0,lambda=[1,1,1,1]/1,nu=[0,0,0,0]/1) [ 1, 1, 1, 1 ]/1 Total 1 set parameters=[ parameter(G,0,[ 1, 1, 1, 1 ]/1,[ 0, 0, 0, 0 ]/1) ] Value: [([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([],[]),([([final parameter(x=0,lambda=[3,3,3,3]/1,nu=[0,0,0,0]/1)],[([0],[([0,1,2,3],[])])])],[(0,0,final parameter(x=0,lambda=[1,1,1,1]/1,nu=[0,0,0,0]/1))])]