Computing weak packets for 5 dual orbits of connected split real group with Lie algebra 'g2(R)' Initializing CharacterTable for Lie type 'G2' Step 1/6 Step 2/6 Step 3/6 Step 4/6 Step 5/6 Step 6/6 Orbit by diagram: (simply connected adjoint root datum of Lie type 'G2',(),[ 6, 10 ]) Computing weak packet for orbit: simply connected adjoint root datum of Lie type 'G2' [ 0, 0 ] dim=0 Computing weak packets for connected split real group with Lie algebra 'g2(R)' gamma:[ 2, 2 ]/1 gamma_final:[ 0, 0 ]/1 Initializing CharacterTable for Lie type 'G2' 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 adjoint root datum of Lie type 'G2' st_int.rd: simply connected adjoint root datum of Lie type 'G2' O_check_int:(simply connected adjoint root datum of Lie type 'G2',(),[ 0, 0 ]) computing packet for:(simply connected adjoint root datum of Lie type 'G2',(),[ 0, 0 ]) computing springer map of[2,2] O: (simply connected adjoint root datum of Lie type 'G2',(),[ 2, 2 ]) survive:final parameter(x=9,lambda=[1,1]/1,nu=[2,2]/1) [ 0, 0 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=0,lambda=[2,2]/1,nu=[0,0]/1) [ 0, 0 ]/1 cell character: 0 springer_O:0 Orbit by diagram: (simply connected adjoint root datum of Lie type 'G2',(),[ 2, 4 ]) Computing weak packet for orbit: simply connected adjoint root datum of Lie type 'G2' [ 1, 0 ] dim=6 Computing weak packets for connected split real group with Lie algebra 'g2(R)' gamma:[ 5, 4 ]/2 gamma_final:[ 1, 0 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.A1' st_int.rd: simply connected root datum of Lie type 'A1.A1' O_check_int:(adjoint root datum of Lie type 'A1.A1',(),[ 0, 2 ]) computing packet for:(adjoint root datum of Lie type 'A1.A1',(),[ 0, 2 ]) computing springer map of[2,0] O: (simply connected root datum of Lie type 'A1.A1',(),[ 1, 0 ]) survive:final parameter(x=3,lambda=[4,1]/1,nu=[-3,2]/1) [ 1, 0 ]/2 survive:final parameter(x=9,lambda=[1,1]/1,nu=[5,4]/2) [ 1, 0 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[2,1]/1,nu=[5,4]/2) [ 1, 0 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=8,lambda=[0,2]/1,nu=[11,0]/2) [ 1, 0 ]/2 Orbit by diagram: (simply connected adjoint root datum of Lie type 'G2',(),[ 2, 4 ]) Computing weak packet for orbit: simply connected adjoint root datum of Lie type 'G2' [ 0, 1 ] dim=8 Computing weak packets for connected split real group with Lie algebra 'g2(R)' gamma:[ 4, 5 ]/2 gamma_final:[ 0, 1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.A1' st_int.rd: simply connected root datum of Lie type 'A1.A1' O_check_int:(adjoint root datum of Lie type 'A1.A1',(),[ 0, 2 ]) computing packet for:(adjoint root datum of Lie type 'A1.A1',(),[ 0, 2 ]) computing springer map of[2,0] O: (simply connected root datum of Lie type 'A1.A1',(),[ 1, 0 ]) survive:final parameter(x=4,lambda=[1,3]/1,nu=[2,-1]/1) [ 0, 1 ]/2 survive:final parameter(x=9,lambda=[1,2]/1,nu=[4,5]/2) [ 0, 1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[1,1]/1,nu=[4,5]/2) [ 0, 1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=7,lambda=[2,1]/1,nu=[0,7]/2) [ 0, 1 ]/2 Orbit by diagram: (simply connected adjoint root datum of Lie type 'G2',(),[ 2, 4 ]) Computing weak packet for orbit: simply connected adjoint root datum of Lie type 'G2' [ 2, 0 ] dim=10 Computing weak packets for connected split real group with Lie algebra 'g2(R)' gamma:[ 3, 2 ]/1 gamma_final:[ 1, 0 ]/1 Initializing CharacterTable for Lie type 'G2' 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 adjoint root datum of Lie type 'G2' st_int.rd: simply connected adjoint root datum of Lie type 'G2' O_check_int:(simply connected adjoint root datum of Lie type 'G2',(),[ 2, 0 ]) computing packet for:(simply connected adjoint root datum of Lie type 'G2',(),[ 2, 0 ]) computing springer map of[2,0] O: (simply connected adjoint root datum of Lie type 'G2',(),[ 2, 0 ]) dim: 1 2 survive:final parameter(x=4,lambda=[1,3]/1,nu=[6,-3]/2) [ 1, 0 ]/1 survive:final parameter(x=8,lambda=[0,2]/1,nu=[6,0]/1) [ 1, 0 ]/1 cell character: 5 springer_O:5 survive:final parameter(x=2,lambda=[3,2]/1,nu=[0,0]/1) [ 1, 0 ]/1 survive:final parameter(x=6,lambda=[0,3]/1,nu=[15,-5]/2) [ 1, 0 ]/1 survive:final parameter(x=9,lambda=[1,1]/1,nu=[3,2]/1) [ 1, 0 ]/1 cell character: 5 springer_O:5 dim: 1 2 dim: 1 2 Orbit by diagram: (simply connected adjoint root datum of Lie type 'G2',(),[ 0, 0 ]) Computing weak packet for orbit: simply connected adjoint root datum of Lie type 'G2' [ 2, 2 ] dim=12 Computing weak packets for connected split real group with Lie algebra 'g2(R)' gamma:[ 3, 3 ]/1 gamma_final:[ 1, 1 ]/1 Initializing CharacterTable for Lie type 'G2' 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 adjoint root datum of Lie type 'G2' st_int.rd: simply connected adjoint root datum of Lie type 'G2' O_check_int:(simply connected adjoint root datum of Lie type 'G2',(),[ 2, 2 ]) computing packet for:(simply connected adjoint root datum of Lie type 'G2',(),[ 2, 2 ]) computing springer map of[0,0] O: (simply connected adjoint root datum of Lie type 'G2',(),[ 0, 0 ]) survive:final parameter(x=0,lambda=[3,3]/1,nu=[0,0]/1) [ 1, 1 ]/1 survive:final parameter(x=1,lambda=[3,3]/1,nu=[0,0]/1) [ 1, 1 ]/1 survive:final parameter(x=4,lambda=[1,4]/1,nu=[6,-3]/2) [ 1, 1 ]/1 survive:final parameter(x=5,lambda=[7,-1]/1,nu=[-6,6]/1) [ 1, 1 ]/1 survive:final parameter(x=8,lambda=[-2,3]/1,nu=[15,0]/2) [ 1, 1 ]/1 survive:final parameter(x=9,lambda=[2,1]/1,nu=[3,3]/1) [ 1, 1 ]/1 survive:final parameter(x=2,lambda=[3,3]/1,nu=[0,0]/1) [ 1, 1 ]/1 survive:final parameter(x=3,lambda=[6,1]/1,nu=[-9,6]/2) [ 1, 1 ]/1 survive:final parameter(x=6,lambda=[-3,5]/1,nu=[9,-3]/1) [ 1, 1 ]/1 survive:final parameter(x=7,lambda=[3,0]/1,nu=[0,9]/2) [ 1, 1 ]/1 survive:final parameter(x=9,lambda=[1,2]/1,nu=[3,3]/1) [ 1, 1 ]/1 survive:final parameter(x=9,lambda=[1,1]/1,nu=[3,3]/1) [ 1, 1 ]/1 cell character: 3 springer_O:3 survive:final parameter(x=9,lambda=[2,2]/1,nu=[3,3]/1) [ 1, 1 ]/1 Computing weak packets for 2 dual orbits of disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A1.T1',(),[ 0, 1 ]) Computing weak packet for orbit: root datum of Lie type 'A1.T1' [ 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' gamma:[ -3, 2 ]/1 gamma_final:[ 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 0 ]) computing packet for:(adjoint root datum of Lie type 'A1',(),[ 0 ]) computing springer map of[2] O: (simply connected root datum of Lie type 'A1',(),[ 1 ]) survive:final parameter(x=1,lambda=[-3,2]/2,nu=[-3,2]/1) [ 0, 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=1,lambda=[-1,2]/2,nu=[-3,2]/1) [ 0, 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=0,lambda=[-5,4]/2,nu=[0,0]/1) [ 0, 0 ]/1 cell character: 1 springer_O:1 Orbit by diagram: (root datum of Lie type 'A1.T1',(),[ 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A1.T1' [ -3, 2 ] dim=2 Computing weak packets for disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' gamma:[ -9, 6 ]/2 gamma_final:[ -3, 2 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) computing packet for:(adjoint root datum of Lie type 'A1',(),[ 2 ]) computing springer map of[0] O: (simply connected root datum of Lie type 'A1',(),[ 0 ]) survive:final parameter(x=0,lambda=[-9,6]/2,nu=[0,0]/1) [ -3, 2 ]/2 survive:final parameter(x=1,lambda=[-1,2]/2,nu=[-9,6]/2) [ -3, 2 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=1,lambda=[-3,2]/2,nu=[-9,6]/2) [ -3, 2 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=1,lambda=[-3,4]/2,nu=[-9,6]/2) [ -3, 2 ]/2 survive:final parameter(x=1,lambda=[-1,4]/2,nu=[-9,6]/2) [ -3, 2 ]/2 Computing weak packets for 2 dual orbits of disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A1.T1',(),[ 1, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A1.T1' [ 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' gamma:[ 2, -1 ]/1 gamma_final:[ 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 0 ]) computing packet for:(adjoint root datum of Lie type 'A1',(),[ 0 ]) computing springer map of[2] O: (simply connected root datum of Lie type 'A1',(),[ 1 ]) survive:final parameter(x=1,lambda=[2,-1]/2,nu=[2,-1]/1) [ 0, 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=0,lambda=[4,-1]/2,nu=[0,0]/1) [ 0, 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=1,lambda=[2,1]/2,nu=[2,-1]/1) [ 0, 0 ]/1 cell character: 1 springer_O:1 Orbit by diagram: (root datum of Lie type 'A1.T1',(),[ 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A1.T1' [ 2, -1 ] dim=2 Computing weak packets for disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' gamma:[ 6, -3 ]/2 gamma_final:[ 2, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) computing packet for:(adjoint root datum of Lie type 'A1',(),[ 2 ]) computing springer map of[0] O: (simply connected root datum of Lie type 'A1',(),[ 0 ]) survive:final parameter(x=0,lambda=[6,-3]/2,nu=[0,0]/1) [ 2, -1 ]/2 survive:final parameter(x=1,lambda=[2,1]/2,nu=[6,-3]/2) [ 2, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=1,lambda=[2,-1]/2,nu=[6,-3]/2) [ 2, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=1,lambda=[4,-1]/2,nu=[6,-3]/2) [ 2, -1 ]/2 survive:final parameter(x=1,lambda=[4,1]/2,nu=[6,-3]/2) [ 2, -1 ]/2 =============================================================================== Orbits for the dual group: connected split real group with Lie algebra 'g2(R)' complex nilpotent orbits for inner class Complex reductive group of type G2, with involution defining inner class of type 'c', with 2 real forms and 2 dual real forms root datum of inner class: simply connected adjoint root datum of Lie type 'G2' i: orbit number H: semisimple element BC Levi: Bala-Carter Levi Cent: 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 Z C_2 A(O) #RF #AP 0 [0,0] 0 2T1 G2 1 2 [1] [1,1] 2 1 [1,0] 6 A1+T1 A1 2 2 [1] [0,2] 2 2 [0,1] 8 A1+T1 A1 2 2 [1] [0,2] 2 3 [2,0] 10 G2 e 1 1 [1,2,3] [0,2] 2 4 [2,2] 12 G2 e 1 1 [1] [0,1] 1 Information about orbit centralizers: orbit#: 0 diagram: [0,0] isogeny information: Centralizer: G2 Center is trivial simply connected adjoint root datum of Lie type 'G2' ------------- orbit#: 1 diagram: [1,0] isogeny information: Centralizer: A1 Group is semisimple center=Z/2Z simply connected root datum of Lie type 'A1' ------------- orbit#: 2 diagram: [0,1] isogeny information: Centralizer: A1 Group is semisimple center=Z/2Z simply connected root datum of Lie type 'A1' ------------- orbit#: 3 diagram: [2,0] isogeny information: Centralizer: e Center is trivial ------------- orbit#: 4 diagram: [2,2] isogeny information: Centralizer: e Center is trivial ------------- Arthur parameters listed by orbit: #parameters by orbit: [2,2,2,2,1] Total: 9 orbit #0 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'G2' [ 0, 0 ] [0,0] 0 1 root datum of Lie type 'A1.A1' [ 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 'G2' [ 1, 0 ] [1,0] 6 1 root datum of Lie type 'A1.A1' [ -1, 1 ] [0,2] 2 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 'G2' [ 0, 1 ] [0,1] 8 1 root datum of Lie type 'A1.A1' [ 3, -1 ] [2,0] 2 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 'G2' [ 2, 0 ] [2,0] 10 1 root datum of Lie type 'A1.A1' [ 2, 0 ] [2,2] 4 1 orbit #4 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult simply connected adjoint root datum of Lie type 'G2' [ 2, 2 ] [2,2] 12 1 orbit |packet| 0 2 1 2 2 2 3 5 4 1 Total 12 *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 0 0 0* 1 0 1 0* 1 1 0 1 1 1 0 2 1 2 0 1 1 2 0 2 1 3 0 1* 2 3 0 2* 3 4 0 3* 1 Total 12 orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=9,lambda=[1,1]/1,nu=[0,0]/1)*(I) [ 0, 0 ]/1 0 1 0* final parameter(x=0,lambda=[0,0]/1,nu=[0,0]/1)*(I) [ 0, 0 ]/1 1 0 1 final parameter(x=9,lambda=[1,1]/1,nu=[1,0]/2)(I) [ 1, 0 ]/2 1 0 2 final parameter(x=9,lambda=[2,1]/1,nu=[1,0]/2)(I) [ 1, 0 ]/2 2 0 1 final parameter(x=9,lambda=[1,2]/1,nu=[0,1]/2)(I) [ 0, 1 ]/2 2 0 2 final parameter(x=9,lambda=[1,1]/1,nu=[0,1]/2)(I) [ 0, 1 ]/2 3 0 1* final parameter(x=4,lambda=[1,0]/1,nu=[2,-1]/2) [ 1, 0 ]/1 3 0 1* final parameter(x=8,lambda=[3,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 3 0 2* final parameter(x=2,lambda=[1,0]/1,nu=[0,0]/1) [ 1, 0 ]/1 3 0 2* final parameter(x=6,lambda=[4,-1]/1,nu=[3,-1]/2) [ 1, 0 ]/1 3 0 2* final parameter(x=9,lambda=[1,1]/1,nu=[1,0]/1)* [ 1, 0 ]/1 4 0 3* final parameter(x=9,lambda=[1,1]/1,nu=[1,1]/1)* [ 1, 1 ]/1 Total 12 Induced 6 Testing conjecture about size of weak Arthur packets for connected split real group with Lie algebra 'g2(R)' i: number of orbit (with A(O)=1) data: combinatorial data derived from the orbit guess: conjectural size of weak Arthur packet actual: size of weak Arthur packet A: A(O), if it isn't 1 the conjecture doesn't apply disjoint: Arthur packets are disjoint, if false the conjecture doesn't apply conjecture: validity for given orbit Orbits for G with A(O)=1: i H diagram dim BC Levi Cent Z C_2 A(O) 0 [0,0] [0,0] 0 2T1 G2 1 2 [1] 1 [1,0] [1,0] 6 A1+T1 A1 2 2 [1] 2 [0,1] [0,1] 8 A1+T1 A1 2 2 [1] 3 [2,2] [2,2] 12 G2 e 1 1 [1] i data guess actual A disjoint conjecture 0 [1,1] 2 2 1 true true 1 [1,1] 2 2 1 false N/A 2 [1,1] 2 2 1 false N/A 3 [1] 1 1 1 true true ------------------------------------------------------------- set parameters=[ parameter(G,9,[ 1, 1 ]/1,[ 0, 0 ]/1), parameter(G,0,[ 0, 0 ]/1,[ 0, 0 ]/1), parameter(G,9,[ 1, 1 ]/1,[ 1, 0 ]/2), parameter(G,9,[ 2, 1 ]/1,[ 1, 0 ]/2), parameter(G,9,[ 1, 2 ]/1,[ 0, 1 ]/2), parameter(G,9,[ 1, 1 ]/1,[ 0, 1 ]/2), parameter(G,4,[ 1, 0 ]/1,[ 2, -1 ]/2), parameter(G,8,[ 3, 0 ]/1,[ 1, 0 ]/1), parameter(G,2,[ 1, 0 ]/1,[ 0, 0 ]/1), parameter(G,6,[ 4, -1 ]/1,[ 3, -1 ]/2), parameter(G,9,[ 1, 1 ]/1,[ 1, 0 ]/1), parameter(G,9,[ 1, 1 ]/1,[ 1, 1 ]/1) ]