atlas> atlas> set G=SL(2,R) Variable G: RealForm atlas> set wp=weak_packets(G) Computing weak packets for 2 dual orbits of connected split real group with Lie algebra 'sl(2,R)' Orbit by diagram: (simply connected root datum of Lie type 'A1',(),[ 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'A1' [ 0 ] dim=0 Computing weak packets for connected split real group with Lie algebra 'sl(2,R)' gamma:[ 2 ]/1 gamma_final:[ 0 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'A1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 0 ]) permutation: | 1 | 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=2,lambda=[1]/1,nu=[2]/1) [ 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=0,lambda=[2]/1,nu=[0]/1) [ 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=1,lambda=[2]/1,nu=[0]/1) [ 0 ]/1 cell character: 1 springer_O:1 Orbit by diagram: (simply connected root datum of Lie type 'A1',(),[ 0 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'A1' [ 2 ] dim=2 Computing weak packets for connected split real group with Lie algebra 'sl(2,R)' gamma:[ 3 ]/1 gamma_final:[ 1 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'A1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) permutation: | 1 | 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=[3]/1,nu=[0]/1) [ 1 ]/1 survive:final parameter(x=1,lambda=[3]/1,nu=[0]/1) [ 1 ]/1 survive:final parameter(x=2,lambda=[1]/1,nu=[3]/1) [ 1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=2,lambda=[2]/1,nu=[3]/1) [ 1 ]/1 Variable wp: [([([Param],[WCell])],[(int,int,Param)])] atlas> atlas> atlas> atlas> atlas> atlas> atlas> wp Value: [([([final parameter(x=2,lambda=[1]/1,nu=[2]/1)],[([0],[([],[])])]),([final parameter(x=0,lambda=[2]/1,nu=[0]/1),final parameter(x=1,lambda=[2]/1,nu=[0]/1),final parameter(x=2,lambda=[2]/1,nu=[2]/1)],[([0],[([],[])]),([1],[([],[])]),([2],[([0],[])])])],[(0,0,final parameter(x=2,lambda=[1]/1,nu=[0]/1)),(1,0,final parameter(x=0,lambda=[0]/1,nu=[0]/1)),(1,1,final parameter(x=1,lambda=[0]/1,nu=[0]/1))]),([([final parameter(x=0,lambda=[3]/1,nu=[0]/1),final parameter(x=1,lambda=[3]/1,nu=[0]/1),final parameter(x=2,lambda=[1]/1,nu=[3]/1)],[([0],[([],[])]),([1],[([],[])]),([2],[([0],[])])]),([final parameter(x=2,lambda=[2]/1,nu=[3]/1)],[([0],[([],[])])])],[(0,2,final parameter(x=2,lambda=[1]/1,nu=[1]/1))])] atlas> atlas> atlas> atlas> atlas> show(wp) *: dual(cell) contains an Aq(lambda) orbit#  block#  cell# parameters 0 0 0* 1 0 1 0* 1 0 1 1* 1 1 0 2* 1 Total 4 atlas> show_short(wp) orbit |packet| 0 3 1 1 Total 4 atlas> show(wp) *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 0 0 0* 1 0 1 0* 1 0 1 1* 1 1 0 2* 1 Total 4 atlas> show_nilpotent_orbits(G.dual) i H diagram dim BC Levi Cent A(O) 0 [0] [0] 0 T1 A1 [1] 1 [2] [2] 2 A1 e [1] atlas> show_long (wp) *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=2,lambda=[1]/1,nu=[0]/1)* [ 0 ]/1 0 1 0* final parameter(x=0,lambda=[0]/1,nu=[0]/1)* [ 0 ]/1 0 1 1* final parameter(x=1,lambda=[0]/1,nu=[0]/1)* [ 0 ]/1 1 0 2* final parameter(x=2,lambda=[1]/1,nu=[1]/1)* [ 1 ]/1 Total 4 atlas> show_nilpotent_orbits_ show_nilpotent_orbits_long show_nilpotent_orbits_long_plus show_nilpotent_orbits_short show_nilpotent_orbits_very_long atlas> atlas> atlas> atlas> atlas> atlas> atlas> whattype weak_packets ? Overloaded instances of 'weak_packets' ([ComplexNilpotent],RealForm)->[([([Param],[WCell])],[(int,int,Param)])] RealForm->[([([Param],[WCell])],[(int,int,Param)])] [RealForm]->[([([Param],[WCell])],[(int,int,Param)])] ([ComplexNilpotent],RealForm,ratvec)->[([([Param],[WCell])],[(int,int,Param)])] (SpringerTable,[ComplexNilpotent],[(([Param],[WCell]),[([int],[int])])])->[([([Param],[WCell])],[(int,int,Param)])] (SpringerTable,[ComplexNilpotent],(([Param],[WCell]),[([int],[int])]))->[([([Param],[WCell])],[(int,int,Param)])] (SpringerTable,[(([Param],[WCell]),[([int],[int])])])->[([([Param],[WCell])],[(int,int,Param)])] (RealForm,[ComplexNilpotent],[(([Param],[WCell]),[([int],[int])])])->[([([Param],[WCell])],[(int,int,Param)])] (RealForm,[ComplexNilpotent],(([Param],[WCell]),[([int],[int])]))->[([([Param],[WCell])],[(int,int,Param)])] atlas> whattype weak_packet ? Overloaded instances of 'weak_packet' (ComplexNilpotent,RealForm)->([([Param],[WCell])],[(int,int,Param)]) (ComplexNilpotent,RealForm,ratvec)->([([Param],[WCell])],[(int,int,Param)]) (SpringerTable,ComplexNilpotent,[(([Param],[WCell]),[([int],[int])])])->([([Param],[WCell])],[(int,int,Param)]) atlas> void:weak_packet_report(SL(2,R)) Computing weak packets for 2 dual orbits of connected split real group with Lie algebra 'sl(2,R)' Orbit by diagram: (simply connected root datum of Lie type 'A1',(),[ 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'A1' [ 0 ] dim=0 Computing weak packets for connected split real group with Lie algebra 'sl(2,R)' gamma:[ 2 ]/1 gamma_final:[ 0 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'A1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 0 ]) permutation: | 1 | 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=2,lambda=[1]/1,nu=[2]/1) [ 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=0,lambda=[2]/1,nu=[0]/1) [ 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=1,lambda=[2]/1,nu=[0]/1) [ 0 ]/1 cell character: 1 springer_O:1 Orbit by diagram: (simply connected root datum of Lie type 'A1',(),[ 0 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'A1' [ 2 ] dim=2 Computing weak packets for connected split real group with Lie algebra 'sl(2,R)' gamma:[ 3 ]/1 gamma_final:[ 1 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'A1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) permutation: | 1 | 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=[3]/1,nu=[0]/1) [ 1 ]/1 survive:final parameter(x=1,lambda=[3]/1,nu=[0]/1) [ 1 ]/1 survive:final parameter(x=2,lambda=[1]/1,nu=[3]/1) [ 1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=2,lambda=[2]/1,nu=[3]/1) [ 1 ]/1 =============================================================================== Orbits for the dual group: disconnected split real group with Lie algebra 'sl(2,R)' complex nilpotent orbits for inner class Complex reductive group of type A1, with involution defining inner class of type 'c', with 2 real forms and 2 dual real forms root datum of inner class: adjoint root datum of Lie type 'A1' 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(O): number of real forms of O for all real forms in inner class #AP(O): number of Arthur parameters for O i diagram dim BC Levi Cent Z C_2 A(O) #RF(O) #ap 0 [0] 0 T1 A1 1 2 [1] [1,1] 2 1 [2] 2 A1 e 1 1 [1] [0,1] 1 Information about orbit centralizers: orbit#: 0 diagram: [0] isogeny information: Centralizer: A1 Center is trivial adjoint root datum of Lie type 'A1' ------------- orbit#: 1 diagram: [2] isogeny information: Centralizer: e Center is trivial ------------- Arthur parameters listed by orbit: #parameters by orbit: [2,1] Total: 3 orbit #0 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult root datum of Lie type 'T1' [ 0 ] [] 0 1 adjoint root datum of Lie type 'A1' [ 0 ] [0] 0 1 orbit #1 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult adjoint root datum of Lie type 'A1' [ 2 ] [2] 2 1 orbit |packet| 0 3 1 1 Total 4 *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 0 0 0* 1 0 1 0* 1 0 1 1* 1 1 0 2* 1 Total 4 *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=2,lambda=[1]/1,nu=[0]/1)* [ 0 ]/1 0 1 0* final parameter(x=0,lambda=[0]/1,nu=[0]/1)* [ 0 ]/1 0 1 1* final parameter(x=1,lambda=[0]/1,nu=[0]/1)* [ 0 ]/1 1 0 2* final parameter(x=2,lambda=[1]/1,nu=[1]/1)* [ 1 ]/1 Total 4 Testing conjecture about size of weak Arthur packets for connected split real group with Lie algebra 'sl(2,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 T1 A1 1 2 [1] 1 [2] [2] 2 A1 e 1 1 [1] i data guess actual A disjoint conjecture 0 [2,1] 3 3 1 true true 1 [1] 1 1 1 true true ------------------------------------------------------------- set parameters=[ parameter(G,2,[ 1 ]/1,[ 0 ]/1), parameter(G,0,[ 0 ]/1,[ 0 ]/1), parameter(G,1,[ 0 ]/1,[ 0 ]/1), parameter(G,2,[ 1 ]/1,[ 1 ]/1) ] atlas> set parameters=[ [ > parameter(G,2,[ 1 ]/1,[ 0 ]/1), [, > parameter(G,0,[ 0 ]/1,[ 0 ]/1), [, > parameter(G,1,[ 0 ]/1,[ 0 ]/1), [, > parameter(G,2,[ 1 ]/1,[ 1 ]/1) [ > ] Variable parameters: [Param] atlas> for p in parameters do prints(p) od final parameter(x=2,lambda=[1]/1,nu=[0]/1) final parameter(x=0,lambda=[0]/1,nu=[0]/1) final parameter(x=1,lambda=[0]/1,nu=[0]/1) final parameter(x=2,lambda=[1]/1,nu=[1]/1) Value: [(),(),(),()] atlas> atlas> atlas> atlas> atlas> set wp=weak_packet(PSL(2,R)) Error in expression weak_packet(PSL(2,R)) at :47:7-28 Failed to match 'weak_packet' with argument type RealForm Error in 'set' command at :47:0-29: Expression analysis failed Command 'set wp' not executed, nothing defined. atlas> set wp=weak_packets(PSL(2,R)) Computing weak packets for 2 dual orbits of disconnected split real group with Lie algebra 'sl(2,R)' Orbit by diagram: (adjoint root datum of Lie type 'A1',(),[ 2 ]) Computing weak packet for orbit: simply connected root datum of Lie type 'A1' [ 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(2,R)' gamma:[ 1 ]/1 gamma_final:[ 0 ]/1 integral data: st_int rd_int:adjoint root datum of Lie type 'A1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 0 ]) permutation: | 1 | 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=[1]/2,nu=[1]/1) [ 0 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=1,lambda=[3]/2,nu=[1]/1) [ 0 ]/1 cell character: 1 springer_O:1 Orbit by diagram: (adjoint root datum of Lie type 'A1',(),[ 0 ]) Computing weak packet for orbit: simply connected root datum of Lie type 'A1' [ 1 ] dim=2 Computing weak packets for disconnected split real group with Lie algebra 'sl(2,R)' gamma:[ 3 ]/2 gamma_final:[ 1 ]/2 integral data: st_int rd_int:adjoint root datum of Lie type 'A1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) permutation: | 1 | 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=[3]/2,nu=[0]/1) [ 1 ]/2 survive:final parameter(x=1,lambda=[3]/2,nu=[3]/2) [ 1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=1,lambda=[1]/2,nu=[3]/2) [ 1 ]/2 cell character: 0 springer_O:0 Variable wp: [([([Param],[WCell])],[(int,int,Param)])] (overriding previous instance, which had type [([([Param],[WCell])],[(int,int,Param)])]) atlas> show_long(wp) *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=1,lambda=[3]/2,nu=[0]/1)* [ 0 ]/1 0 1 0* final parameter(x=1,lambda=[1]/2,nu=[0]/1)* [ 0 ]/1 1 0 1 final parameter(x=1,lambda=[1]/2,nu=[1]/2) [ 1 ]/2 1 0 2 final parameter(x=1,lambda=[3]/2,nu=[1]/2) [ 1 ]/2 Total 4 atlas> atlas> atlas> show_nilpotent_orbits(dual(PSL(2,R))) i H diagram dim BC Levi Cent A(O) 0 [0] [0] 0 T1 A1 [1] 1 [1] [2] 2 A1 e [1,2] atlas> show_short(wp) orbit |packet| 0 2 1 2 Total 4 atlas> set G=Sp(4,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set wp=weak_packets(G) Computing weak packets for 4 dual orbits of connected split real group with Lie algebra 'sp(4,R)' Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 3, 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 0, 0 ] dim=0 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 4, 2 ]/1 gamma_final:[ 0, 0 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'C2' st_int.rd: simply connected root datum of Lie type 'C2' O_check_int:(adjoint root datum of Lie type 'B2',(),[ 0, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'B2',(),[ 0, 0 ]) computing springer map of[2,2] O: (simply connected root datum of Lie type 'C2',(),[ 3, 4 ]) survive:final parameter(x=10,lambda=[2,1]/1,nu=[4,2]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=0,lambda=[4,2]/1,nu=[0,0]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=1,lambda=[4,2]/1,nu=[0,0]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=5,lambda=[4,1]/1,nu=[0,2]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=6,lambda=[4,1]/1,nu=[0,2]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 1, 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 1, 1 ] dim=4 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 9, 5 ]/2 gamma_final:[ 1, 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 ]) permutation: | 1, 0 | | 0, 1 | 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=[4,3]/1,nu=[1,-1]/1) [ 1, 1 ]/2 survive:final parameter(x=10,lambda=[2,1]/1,nu=[9,5]/2) [ 1, 1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=10,lambda=[3,2]/1,nu=[9,5]/2) [ 1, 1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[3,1]/1,nu=[7,7]/2) [ 1, 1 ]/2 Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 1, 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 2, 0 ] dim=6 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 5, 2 ]/1 gamma_final:[ 1, 0 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'C2' st_int.rd: simply connected root datum of Lie type 'C2' O_check_int:(adjoint root datum of Lie type 'B2',(),[ 2, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'B2',(),[ 2, 0 ]) computing springer map of[0,2] O: (simply connected root datum of Lie type 'C2',(),[ 1, 2 ]) dim: 1 2 dim: 1 2 survive:final parameter(x=2,lambda=[5,2]/1,nu=[0,0]/1) [ 1, 0 ]/1 survive:final parameter(x=7,lambda=[3,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=3,lambda=[5,2]/1,nu=[0,0]/1) [ 1, 0 ]/1 survive:final parameter(x=8,lambda=[3,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=4,lambda=[4,3]/1,nu=[3,-3]/2) [ 1, 0 ]/1 survive:final parameter(x=10,lambda=[2,1]/1,nu=[5,2]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 dim: 1 2 dim: 1 2 dim: 1 2 dim: 1 2 survive:final parameter(x=7,lambda=[2,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 survive:final parameter(x=8,lambda=[2,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 0, 0 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 4, 2 ] dim=8 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 6, 3 ]/1 gamma_final:[ 2, 1 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'C2' st_int.rd: simply connected root datum of Lie type 'C2' O_check_int:(adjoint root datum of Lie type 'B2',(),[ 2, 2 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'B2',(),[ 2, 2 ]) computing springer map of[0,0] O: (simply connected root datum of Lie type 'C2',(),[ 0, 0 ]) survive:final parameter(x=0,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=1,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=2,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=5,lambda=[6,1]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=7,lambda=[2,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=3,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=6,lambda=[6,1]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=8,lambda=[2,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=4,lambda=[5,4]/1,nu=[3,-3]/2) [ 2, 1 ]/1 survive:final parameter(x=9,lambda=[3,0]/1,nu=[9,9]/2) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[3,2]/1,nu=[6,3]/1) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[2,1]/1,nu=[6,3]/1) [ 2, 1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=5,lambda=[6,2]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=6,lambda=[6,2]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=7,lambda=[3,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=8,lambda=[3,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[3,1]/1,nu=[6,3]/1) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[2,2]/1,nu=[6,3]/1) [ 2, 1 ]/1 Variable wp: [([([Param],[WCell])],[(int,int,Param)])] (overriding previous instance, which had type [([([Param],[WCell])],[(int,int,Param)])]) atlas> show_nilpotent_orbits(G.dual) i H diagram dim BC Levi Cent A(O) 0 [0,0] [0,0] 0 2T1 C2 [1] 1 [1,1] [0,1] 4 A1+T1 A1 [1] 2 [2,0] [2,0] 6 A1+T1 T1 [1,2] 3 [4,2] [2,2] 8 B2 e [1] atlas> G.dual Value: Complex reductive group of type B2, with involution defining inner class of type 'c', with 3 real forms and 3 dual real forms atlas> {SO(5,C)} atlas> show_short(wp) orbit |packet| 0 5 1 2 2 8 3 1 Total 16 atlas> show_long (wp) *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=10,lambda=[2,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 1 0* final parameter(x=0,lambda=[0,0]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 1 1* final parameter(x=1,lambda=[0,0]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 2 0* final parameter(x=5,lambda=[0,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 2 1* final parameter(x=6,lambda=[0,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 1 0 1* final parameter(x=10,lambda=[2,1]/1,nu=[1,1]/2) [ 1, 1 ]/2 1 0 2* final parameter(x=10,lambda=[3,2]/1,nu=[1,1]/2) [ 1, 1 ]/2 2 0 2* final parameter(x=2,lambda=[1,0]/1,nu=[0,0]/1) [ 1, 0 ]/1 2 0 2* final parameter(x=7,lambda=[3,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 0 3* final parameter(x=3,lambda=[1,0]/1,nu=[0,0]/1) [ 1, 0 ]/1 2 0 3* final parameter(x=8,lambda=[3,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 0 4* final parameter(x=4,lambda=[1,0]/1,nu=[1,-1]/2) [ 1, 0 ]/1 2 0 4* final parameter(x=10,lambda=[2,1]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 2 2* final parameter(x=7,lambda=[2,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 2 2* final parameter(x=8,lambda=[2,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 3 0 5* final parameter(x=10,lambda=[2,1]/1,nu=[2,1]/1)* [ 2, 1 ]/1 Total 16 atlas> void:weak_packet_report(Sp(4,R)) Computing weak packets for 4 dual orbits of connected split real group with Lie algebra 'sp(4,R)' Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 3, 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 0, 0 ] dim=0 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 4, 2 ]/1 gamma_final:[ 0, 0 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'C2' st_int.rd: simply connected root datum of Lie type 'C2' O_check_int:(adjoint root datum of Lie type 'B2',(),[ 0, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'B2',(),[ 0, 0 ]) computing springer map of[2,2] O: (simply connected root datum of Lie type 'C2',(),[ 3, 4 ]) survive:final parameter(x=10,lambda=[2,1]/1,nu=[4,2]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=0,lambda=[4,2]/1,nu=[0,0]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=1,lambda=[4,2]/1,nu=[0,0]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=5,lambda=[4,1]/1,nu=[0,2]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=6,lambda=[4,1]/1,nu=[0,2]/1) [ 0, 0 ]/1 cell character: 4 springer_O:4 Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 1, 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 1, 1 ] dim=4 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 9, 5 ]/2 gamma_final:[ 1, 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 ]) permutation: | 1, 0 | | 0, 1 | 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=[4,3]/1,nu=[1,-1]/1) [ 1, 1 ]/2 survive:final parameter(x=10,lambda=[2,1]/1,nu=[9,5]/2) [ 1, 1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=10,lambda=[3,2]/1,nu=[9,5]/2) [ 1, 1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[3,1]/1,nu=[7,7]/2) [ 1, 1 ]/2 Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 1, 1 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 2, 0 ] dim=6 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 5, 2 ]/1 gamma_final:[ 1, 0 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'C2' st_int.rd: simply connected root datum of Lie type 'C2' O_check_int:(adjoint root datum of Lie type 'B2',(),[ 2, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'B2',(),[ 2, 0 ]) computing springer map of[0,2] O: (simply connected root datum of Lie type 'C2',(),[ 1, 2 ]) dim: 1 2 dim: 1 2 survive:final parameter(x=2,lambda=[5,2]/1,nu=[0,0]/1) [ 1, 0 ]/1 survive:final parameter(x=7,lambda=[3,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=3,lambda=[5,2]/1,nu=[0,0]/1) [ 1, 0 ]/1 survive:final parameter(x=8,lambda=[3,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=4,lambda=[4,3]/1,nu=[3,-3]/2) [ 1, 0 ]/1 survive:final parameter(x=10,lambda=[2,1]/1,nu=[5,2]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 dim: 1 2 dim: 1 2 dim: 1 2 dim: 1 2 survive:final parameter(x=7,lambda=[2,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 survive:final parameter(x=8,lambda=[2,2]/1,nu=[5,0]/1) [ 1, 0 ]/1 cell character: 2 springer_O:2 Orbit by diagram: (simply connected root datum of Lie type 'C2',(),[ 0, 0 ]) Computing weak packet for orbit: adjoint root datum of Lie type 'B2' [ 4, 2 ] dim=8 Computing weak packets for connected split real group with Lie algebra 'sp(4,R)' gamma:[ 6, 3 ]/1 gamma_final:[ 2, 1 ]/1 integral data: st_int rd_int:simply connected root datum of Lie type 'C2' st_int.rd: simply connected root datum of Lie type 'C2' O_check_int:(adjoint root datum of Lie type 'B2',(),[ 2, 2 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'B2',(),[ 2, 2 ]) computing springer map of[0,0] O: (simply connected root datum of Lie type 'C2',(),[ 0, 0 ]) survive:final parameter(x=0,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=1,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=2,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=5,lambda=[6,1]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=7,lambda=[2,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=3,lambda=[6,3]/1,nu=[0,0]/1) [ 2, 1 ]/1 survive:final parameter(x=6,lambda=[6,1]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=8,lambda=[2,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=4,lambda=[5,4]/1,nu=[3,-3]/2) [ 2, 1 ]/1 survive:final parameter(x=9,lambda=[3,0]/1,nu=[9,9]/2) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[3,2]/1,nu=[6,3]/1) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[2,1]/1,nu=[6,3]/1) [ 2, 1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=5,lambda=[6,2]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=6,lambda=[6,2]/1,nu=[0,3]/1) [ 2, 1 ]/1 survive:final parameter(x=7,lambda=[3,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=8,lambda=[3,3]/1,nu=[6,0]/1) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[3,1]/1,nu=[6,3]/1) [ 2, 1 ]/1 survive:final parameter(x=10,lambda=[2,2]/1,nu=[6,3]/1) [ 2, 1 ]/1 =============================================================================== Orbits for the dual group: disconnected split real group with Lie algebra 'so(3,2)' complex nilpotent orbits for inner class Complex reductive group of type B2, with involution defining inner class of type 'c', with 3 real forms and 3 dual real forms root datum of inner class: adjoint root datum of Lie type 'B2' 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(O): number of real forms of O for all real forms in inner class #AP(O): number of Arthur parameters for O i diagram dim BC Levi Cent Z C_2 A(O) #RF(O) #ap 0 [0,0] 0 2T1 C2 1 3 [1] [1,1,1] 3 1 [0,1] 4 A1+T1 A1 2 2 [1] [0,0,1] 2 2 [2,0] 6 A1+T1 T1 1 2 [1,2] [0,1,2] 3 3 [2,2] 8 B2 e 1 1 [1] [0,0,1] 1 Information about orbit centralizers: orbit#: 0 diagram: [0,0] isogeny information: Centralizer: C2 Center is trivial adjoint root datum of Lie type 'C2' ------------- orbit#: 1 diagram: [0,1] isogeny information: Centralizer: A1 Group is semisimple center=Z/2Z simply connected root datum of Lie type 'A1' ------------- orbit#: 2 diagram: [2,0] isogeny information: Centralizer: T1 Center is a connected complex torus of rank 1 ------------- orbit#: 3 diagram: [2,2] isogeny information: Centralizer: e Center is trivial ------------- Arthur parameters listed by orbit: #parameters by orbit: [3,2,3,1] Total: 9 orbit #0 for G #orbits for (disconnected) Cent(O): 3 K_0 H diagram dim mult root datum of Lie type 'A1.T1' [ 0, 0 ] [0] 0 1 root datum of Lie type 'A1.A1' [ 0, 0 ] [0,0] 0 1 adjoint root datum of Lie type 'B2' [ 0, 0 ] [0,0] 0 1 orbit #1 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult root datum of Lie type 'A1.A1' [ 1, -1 ] [2,0] 2 2 adjoint root datum of Lie type 'B2' [ 1, 1 ] [0,1] 4 1 orbit #2 for G #orbits for (disconnected) Cent(O): 3 K_0 H diagram dim mult root datum of Lie type 'A1.T1' [ 2, 0 ] [2] 2 1 root datum of Lie type 'A1.A1' [ 2, 0 ] [2,2] 4 1 adjoint root datum of Lie type 'B2' [ 2, 0 ] [2,0] 6 1 orbit #3 for G #orbits for (disconnected) Cent(O): 1 K_0 H diagram dim mult adjoint root datum of Lie type 'B2' [ 4, 2 ] [2,2] 8 1 orbit |packet| 0 5 1 2 2 8 3 1 Total 16 *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 0 0 0* 1 0 1 0* 1 0 1 1* 1 0 2 0* 1 0 2 1* 1 1 0 1* 1 1 0 2* 1 2 0 2* 2 2 0 3* 2 2 0 4* 2 2 2 2* 2 3 0 5* 1 Total 16 *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=10,lambda=[2,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 1 0* final parameter(x=0,lambda=[0,0]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 1 1* final parameter(x=1,lambda=[0,0]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 2 0* final parameter(x=5,lambda=[0,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 2 1* final parameter(x=6,lambda=[0,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 1 0 1* final parameter(x=10,lambda=[2,1]/1,nu=[1,1]/2) [ 1, 1 ]/2 1 0 2* final parameter(x=10,lambda=[3,2]/1,nu=[1,1]/2) [ 1, 1 ]/2 2 0 2* final parameter(x=2,lambda=[1,0]/1,nu=[0,0]/1) [ 1, 0 ]/1 2 0 2* final parameter(x=7,lambda=[3,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 0 3* final parameter(x=3,lambda=[1,0]/1,nu=[0,0]/1) [ 1, 0 ]/1 2 0 3* final parameter(x=8,lambda=[3,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 0 4* final parameter(x=4,lambda=[1,0]/1,nu=[1,-1]/2) [ 1, 0 ]/1 2 0 4* final parameter(x=10,lambda=[2,1]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 2 2* final parameter(x=7,lambda=[2,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 2 2* final parameter(x=8,lambda=[2,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 3 0 5* final parameter(x=10,lambda=[2,1]/1,nu=[2,1]/1)* [ 2, 1 ]/1 Total 16 Testing conjecture about size of weak Arthur packets for connected split real group with Lie algebra 'sp(4,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 C2 1 3 [1] 1 [1,1] [0,1] 4 A1+T1 A1 2 2 [1] 2 [4,2] [2,2] 8 B2 e 1 1 [1] i data guess actual A disjoint conjecture 0 [2,2,1] 5 5 1 true true 1 [1,1] 2 2 1 true true 2 [1] 1 1 1 true true ------------------------------------------------------------- set parameters=[ parameter(G,10,[ 2, 1 ]/1,[ 0, 0 ]/1), parameter(G,0,[ 0, 0 ]/1,[ 0, 0 ]/1), parameter(G,1,[ 0, 0 ]/1,[ 0, 0 ]/1), parameter(G,5,[ 0, 1 ]/1,[ 0, 0 ]/1), parameter(G,6,[ 0, 1 ]/1,[ 0, 0 ]/1), parameter(G,10,[ 2, 1 ]/1,[ 1, 1 ]/2), parameter(G,10,[ 3, 2 ]/1,[ 1, 1 ]/2), parameter(G,2,[ 1, 0 ]/1,[ 0, 0 ]/1), parameter(G,7,[ 3, 0 ]/1,[ 1, 0 ]/1), parameter(G,3,[ 1, 0 ]/1,[ 0, 0 ]/1), parameter(G,8,[ 3, 0 ]/1,[ 1, 0 ]/1), parameter(G,4,[ 1, 0 ]/1,[ 1, -1 ]/2), parameter(G,10,[ 2, 1 ]/1,[ 1, 0 ]/1), parameter(G,7,[ 2, 0 ]/1,[ 1, 0 ]/1), parameter(G,8,[ 2, 0 ]/1,[ 1, 0 ]/1), parameter(G,10,[ 2, 1 ]/1,[ 2, 1 ]/1) ] atlas> show_long(wp) *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=10,lambda=[2,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 1 0* final parameter(x=0,lambda=[0,0]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 1 1* final parameter(x=1,lambda=[0,0]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 2 0* final parameter(x=5,lambda=[0,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 0 2 1* final parameter(x=6,lambda=[0,1]/1,nu=[0,0]/1)* [ 0, 0 ]/1 1 0 1* final parameter(x=10,lambda=[2,1]/1,nu=[1,1]/2) [ 1, 1 ]/2 1 0 2* final parameter(x=10,lambda=[3,2]/1,nu=[1,1]/2) [ 1, 1 ]/2 2 0 2* final parameter(x=2,lambda=[1,0]/1,nu=[0,0]/1) [ 1, 0 ]/1 2 0 2* final parameter(x=7,lambda=[3,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 0 3* final parameter(x=3,lambda=[1,0]/1,nu=[0,0]/1) [ 1, 0 ]/1 2 0 3* final parameter(x=8,lambda=[3,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 0 4* final parameter(x=4,lambda=[1,0]/1,nu=[1,-1]/2) [ 1, 0 ]/1 2 0 4* final parameter(x=10,lambda=[2,1]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 2 2* final parameter(x=7,lambda=[2,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 2 2 2* final parameter(x=8,lambda=[2,0]/1,nu=[1,0]/1)* [ 1, 0 ]/1 3 0 5* final parameter(x=10,lambda=[2,1]/1,nu=[2,1]/1)* [ 2, 1 ]/1 Total 16 atlas> weak_packets(GL(3,R)) Computing weak packets for 3 dual orbits of disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 2, 0, -2 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 0, 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 2, 0, -2 ]/1 gamma_final:[ 0, 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) computing springer map of[2,2] O: (simply connected root datum of Lie type 'A2',(),[ 2, 0 ]) survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,1,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,0,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 1, 0, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 1, 0, -1 ] dim=4 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 5, 0, -5 ]/2 gamma_final:[ 1, 0, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.T1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) permutation: | 1 | 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=[3,1,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=0,lambda=[3,0,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 0, 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 2, 0, -2 ] dim=6 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 3, 0, -3 ]/1 gamma_final:[ 1, 0, -1 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) computing springer map of[0,0] O: (simply connected root datum of Lie type 'A2',(),[ 0, 0 ]) survive:final parameter(x=0,lambda=[3,0,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,-1]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[1,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=0,lambda=[3,1,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,0]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[2,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 Value: [([([final parameter(x=3,lambda=[1,0,-1]/1,nu=[2,0,-2]/1)],[([0],[([],[])])]),([final parameter(x=0,lambda=[2,1,-2]/1,nu=[0,0,0]/1),final parameter(x=1,lambda=[2,0,-1]/1,nu=[1,1,-2]/1),final parameter(x=2,lambda=[1,1,-1]/1,nu=[2,-1,-1]/1),final parameter(x=3,lambda=[1,0,0]/1,nu=[2,0,-2]/1),final parameter(x=3,lambda=[1,1,-1]/1,nu=[2,0,-2]/1),final parameter(x=3,lambda=[2,0,-1]/1,nu=[2,0,-2]/1)],[([0],[([],[])]),([1,5],[([1],[(1,1)]),([0],[(0,1)])]),([2,3],[([0],[(1,1)]),([1],[(0,1)])]),([4],[([0,1],[])])]),([final parameter(x=0,lambda=[2,0,-2]/1,nu=[0,0,0]/1),final parameter(x=1,lambda=[2,0,0]/1,nu=[1,1,-2]/1),final parameter(x=2,lambda=[2,1,-1]/1,nu=[2,-1,-1]/1),final parameter(x=3,lambda=[2,0,0]/1,nu=[2,0,-2]/1),final parameter(x=3,lambda=[1,1,0]/1,nu=[2,0,-2]/1),final parameter(x=3,lambda=[2,1,-1]/1,nu=[2,0,-2]/1)],[([0],[([],[])]),([1,4],[([1],[(1,1)]),([0],[(0,1)])]),([2,5],[([0],[(1,1)]),([1],[(0,1)])]),([3],[([0,1],[])])]),([final parameter(x=3,lambda=[2,1,0]/1,nu=[2,0,-2]/1)],[([0],[([],[])])])],[(0,0,final parameter(x=3,lambda=[1,0,-1]/1,nu=[0,0,0]/1)),(1,0,final parameter(x=0,lambda=[0,1,0]/1,nu=[0,0,0]/1)),(2,0,final parameter(x=0,lambda=[0,0,0]/1,nu=[0,0,0]/1)),(3,0,final parameter(x=3,lambda=[2,1,0]/1,nu=[0,0,0]/1))]),([([final parameter(x=0,lambda=[3,1,-2]/1,nu=[0,0,0]/1),final parameter(x=3,lambda=[2,0,0]/1,nu=[5,0,-5]/2),final parameter(x=3,lambda=[1,0,-1]/1,nu=[5,0,-5]/2)],[([0],[([],[])]),([1],[([0],[])]),([2],[([0],[])])]),([final parameter(x=3,lambda=[2,0,-1]/1,nu=[5,0,-5]/2)],[([0],[([],[])])]),([final parameter(x=0,lambda=[3,0,-2]/1,nu=[0,0,0]/1),final parameter(x=3,lambda=[2,1,0]/1,nu=[5,0,-5]/2),final parameter(x=3,lambda=[1,1,-1]/1,nu=[5,0,-5]/2)],[([0],[([],[])]),([1],[([0],[])]),([2],[([0],[])])]),([final parameter(x=3,lambda=[2,1,-1]/1,nu=[5,0,-5]/2)],[([0],[([],[])])]),([final parameter(x=3,lambda=[1,0,0]/1,nu=[5,0,-5]/2)],[([0],[([],[])])]),([final parameter(x=3,lambda=[1,1,0]/1,nu=[5,0,-5]/2)],[([0],[([],[])])])],[(0,1,final parameter(x=3,lambda=[2,0,0]/1,nu=[1,0,-1]/2)),(0,2,final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/2)),(2,1,final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/2)),(2,2,final parameter(x=3,lambda=[1,1,-1]/1,nu=[1,0,-1]/2))]),([([final parameter(x=0,lambda=[3,0,-3]/1,nu=[0,0,0]/1),final parameter(x=1,lambda=[2,-1,-1]/1,nu=[3,3,-6]/2),final parameter(x=2,lambda=[1,1,-2]/1,nu=[6,-3,-3]/2),final parameter(x=3,lambda=[1,0,-1]/1,nu=[3,0,-3]/1),final parameter(x=3,lambda=[2,1,-1]/1,nu=[3,0,-3]/1),final parameter(x=3,lambda=[1,1,0]/1,nu=[3,0,-3]/1)],[([0],[([],[])]),([1,4],[([1],[(1,1)]),([0],[(0,1)])]),([2,5],[([0],[(1,1)]),([1],[(0,1)])]),([3],[([0,1],[])])]),([final parameter(x=0,lambda=[3,1,-3]/1,nu=[0,0,0]/1),final parameter(x=1,lambda=[2,-1,0]/1,nu=[3,3,-6]/2),final parameter(x=2,lambda=[2,1,-2]/1,nu=[6,-3,-3]/2),final parameter(x=3,lambda=[2,0,-1]/1,nu=[3,0,-3]/1),final parameter(x=3,lambda=[2,1,0]/1,nu=[3,0,-3]/1),final parameter(x=3,lambda=[1,0,0]/1,nu=[3,0,-3]/1)],[([0],[([],[])]),([1,5],[([1],[(1,1)]),([0],[(0,1)])]),([2,3],[([0],[(1,1)]),([1],[(0,1)])]),([4],[([0,1],[])])]),([final parameter(x=3,lambda=[1,1,-1]/1,nu=[3,0,-3]/1)],[([0],[([],[])])]),([final parameter(x=3,lambda=[2,0,0]/1,nu=[3,0,-3]/1)],[([0],[([],[])])])],[(0,3,final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/1)),(1,3,final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/1))])] atlas> set sp=weak_packets(GL(3,R)) Computing weak packets for 3 dual orbits of disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 2, 0, -2 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 0, 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 2, 0, -2 ]/1 gamma_final:[ 0, 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) computing springer map of[2,2] O: (simply connected root datum of Lie type 'A2',(),[ 2, 0 ]) survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,1,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,0,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 1, 0, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 1, 0, -1 ] dim=4 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 5, 0, -5 ]/2 gamma_final:[ 1, 0, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.T1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) permutation: | 1 | 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=[3,1,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=0,lambda=[3,0,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 0, 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 2, 0, -2 ] dim=6 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 3, 0, -3 ]/1 gamma_final:[ 1, 0, -1 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) computing springer map of[0,0] O: (simply connected root datum of Lie type 'A2',(),[ 0, 0 ]) survive:final parameter(x=0,lambda=[3,0,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,-1]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[1,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=0,lambda=[3,1,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,0]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[2,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 Variable sp: [([([Param],[WCell])],[(int,int,Param)])] (overriding previous instance, which had type string) atlas> set wp=weak_packets(GL(3,R)) Computing weak packets for 3 dual orbits of disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 2, 0, -2 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 0, 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 2, 0, -2 ]/1 gamma_final:[ 0, 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) computing springer map of[2,2] O: (simply connected root datum of Lie type 'A2',(),[ 2, 0 ]) survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,1,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,0,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 1, 0, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 1, 0, -1 ] dim=4 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 5, 0, -5 ]/2 gamma_final:[ 1, 0, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.T1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) permutation: | 1 | 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=[3,1,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=0,lambda=[3,0,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 0, 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 2, 0, -2 ] dim=6 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 3, 0, -3 ]/1 gamma_final:[ 1, 0, -1 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) computing springer map of[0,0] O: (simply connected root datum of Lie type 'A2',(),[ 0, 0 ]) survive:final parameter(x=0,lambda=[3,0,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,-1]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[1,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=0,lambda=[3,1,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,0]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[2,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 Variable wp: [([([Param],[WCell])],[(int,int,Param)])] (overriding previous instance, which had type [([([Param],[WCell])],[(int,int,Param)])]) atlas> show_short(wp) orbit |packet| 0 4 1 4 2 2 Total 10 atlas> show_nilpotent_orbits(GL(3).dual) i H diagram dim BC Levi Cent A(O) 0 [0,0,0] [0,0] 0 3T1 A2+T1 [1] 1 [1,0,-1] [1,1] 4 A1+2T1 2T1 [1] 2 [2,0,-2] [2,2] 6 A2+T1 T1 [1] atlas> show_long (wp) *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=3,lambda=[1,0,-1]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 1 0* final parameter(x=0,lambda=[0,1,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 2 0* final parameter(x=0,lambda=[0,0,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 3 0* final parameter(x=3,lambda=[2,1,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 1 0 1 final parameter(x=3,lambda=[2,0,0]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 0 2 final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 2 1 final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 2 2 final parameter(x=3,lambda=[1,1,-1]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 2 0 3* final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/1)* [ 1, 0, -1 ]/1 2 1 3* final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/1)* [ 1, 0, -1 ]/1 Total 10 atlas> void:weak_packet_report(GL(3,R)) Computing weak packets for 3 dual orbits of disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 2, 0, -2 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 0, 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 2, 0, -2 ]/1 gamma_final:[ 0, 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 0, 0 ]) computing springer map of[2,2] O: (simply connected root datum of Lie type 'A2',(),[ 2, 0 ]) survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,1,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=0,lambda=[2,0,-2]/1,nu=[0,0,0]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[2,0,-2]/1) [ 0, 0, 0 ]/1 cell character: 2 springer_O:2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 1, 0, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 1, 0, -1 ] dim=4 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 5, 0, -5 ]/2 gamma_final:[ 1, 0, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.T1.T1' st_int.rd: simply connected root datum of Lie type 'A1' O_check_int:(adjoint root datum of Lie type 'A1',(),[ 2 ]) permutation: | 1 | 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=[3,1,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=0,lambda=[3,0,-2]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[5,0,-5]/2) [ 1, 0, -1 ]/2 Orbit by diagram: (root datum of Lie type 'A2.T1',(),[ 0, 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A2.T1' [ 2, 0, -2 ] dim=6 Computing weak packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,R)' gamma:[ 3, 0, -3 ]/1 gamma_final:[ 1, 0, -1 ]/1 integral data: st_int rd_int:root datum of Lie type 'A2.T1' st_int.rd: simply connected root datum of Lie type 'A2' O_check_int:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) permutation: | 1, 0 | | 0, 1 | computing packet for:(adjoint root datum of Lie type 'A2',(),[ 2, 2 ]) computing springer map of[0,0] O: (simply connected root datum of Lie type 'A2',(),[ 0, 0 ]) survive:final parameter(x=0,lambda=[3,0,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,-1]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[1,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=0,lambda=[3,1,-3]/1,nu=[0,0,0]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=1,lambda=[2,-1,0]/1,nu=[3,3,-6]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[1,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=2,lambda=[2,1,-2]/1,nu=[6,-3,-3]/2) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,1,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 cell character: 0 springer_O:0 survive:final parameter(x=3,lambda=[1,1,-1]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 survive:final parameter(x=3,lambda=[2,0,0]/1,nu=[3,0,-3]/1) [ 1, 0, -1 ]/1 =============================================================================== Orbits for the dual group: connected quasisplit real group with Lie algebra 'su(2,1).u(1)' complex nilpotent orbits for inner class Complex reductive group of type A2.T1, with involution defining inner class of type 'cc', with 2 real forms and 1 dual real form root datum of inner class: root datum of Lie type 'A2.T1' 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(O): number of real forms of O for all real forms in inner class #AP(O): number of Arthur parameters for O i diagram dim BC Levi Cent Z C_2 A(O) #RF(O) #ap 0 [0,0] 0 3T1 A2+T1 3 4 [1] [1,1] 4 1 [1,1] 4 A1+2T1 2T1 1 4 [1] [0,2] 4 2 [2,2] 6 A2+T1 T1 1 2 [1] [0,1] 2 Information about orbit centralizers: orbit#: 0 diagram: [0,0] isogeny information: Centralizer: A2+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'A2' ------------- orbit#: 1 diagram: [1,1] isogeny information: Centralizer: 2T1 Center is a connected complex torus of rank 2 ------------- orbit#: 2 diagram: [2,2] isogeny information: Centralizer: T1 Center is a connected complex torus of rank 1 ------------- Arthur parameters listed by orbit: #parameters by orbit: [4,4,2] Total: 10 orbit #0 for G #orbits for (disconnected) Cent(O): 4 K_0 H diagram dim mult root datum of Lie type 'A1.T1.T1' [ 0, 0, 0 ] [0] 0 1 root datum of Lie type 'A1.T1.T1' [ 0, 0, 0 ] [0] 0 1 root datum of Lie type 'A2.T1' [ 0, 0, 0 ] [0,0] 0 1 root datum of Lie type 'A2.T1' [ 0, 0, 0 ] [0,0] 0 1 orbit #1 for G #orbits for (disconnected) Cent(O): 4 K_0 H diagram dim mult root datum of Lie type 'A1.T1.T1' [ 1, 0, -1 ] [2] 2 1 root datum of Lie type 'A1.T1.T1' [ 1, 0, -1 ] [2] 2 1 root datum of Lie type 'A2.T1' [ 1, 0, -1 ] [1,1] 4 1 root datum of Lie type 'A2.T1' [ 1, 0, -1 ] [1,1] 4 1 orbit #2 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult root datum of Lie type 'A2.T1' [ 2, 0, -2 ] [2,2] 6 1 root datum of Lie type 'A2.T1' [ 2, 0, -2 ] [2,2] 6 1 orbit |packet| 0 4 1 4 2 2 Total 10 *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 0 0 0* 1 0 1 0* 1 0 2 0* 1 0 3 0* 1 1 0 1 1 1 0 2 1 1 2 1 1 1 2 2 1 2 0 3* 1 2 1 3* 1 Total 10 *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=3,lambda=[1,0,-1]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 1 0* final parameter(x=0,lambda=[0,1,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 2 0* final parameter(x=0,lambda=[0,0,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 3 0* final parameter(x=3,lambda=[2,1,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 1 0 1 final parameter(x=3,lambda=[2,0,0]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 0 2 final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 2 1 final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 2 2 final parameter(x=3,lambda=[1,1,-1]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 2 0 3* final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/1)* [ 1, 0, -1 ]/1 2 1 3* final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/1)* [ 1, 0, -1 ]/1 Total 10 Testing conjecture about size of weak Arthur packets for disconnected split real group with Lie algebra 'sl(3,R).gl(1,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] 0 3T1 A2+T1 3 4 [1] 1 [1,0,-1] [1,1] 4 A1+2T1 2T1 1 4 [1] 2 [2,0,-2] [2,2] 6 A2+T1 T1 1 2 [1] i data guess actual A disjoint conjecture 0 [1,1,1,1] 4 4 1 true true 1 [1,1,1,1] 4 4 1 false N/A 2 [1,1] 2 2 1 true true ------------------------------------------------------------- set parameters=[ parameter(G,3,[ 1, 0, -1 ]/1,[ 0, 0, 0 ]/1), parameter(G,0,[ 0, 1, 0 ]/1,[ 0, 0, 0 ]/1), parameter(G,0,[ 0, 0, 0 ]/1,[ 0, 0, 0 ]/1), parameter(G,3,[ 2, 1, 0 ]/1,[ 0, 0, 0 ]/1), parameter(G,3,[ 2, 0, 0 ]/1,[ 1, 0, -1 ]/2), parameter(G,3,[ 1, 0, -1 ]/1,[ 1, 0, -1 ]/2), parameter(G,3,[ 2, 1, 0 ]/1,[ 1, 0, -1 ]/2), parameter(G,3,[ 1, 1, -1 ]/1,[ 1, 0, -1 ]/2), parameter(G,3,[ 1, 0, -1 ]/1,[ 1, 0, -1 ]/1), parameter(G,3,[ 2, 1, 0 ]/1,[ 1, 0, -1 ]/1) ] atlas> show_long (wp) *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=3,lambda=[1,0,-1]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 1 0* final parameter(x=0,lambda=[0,1,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 2 0* final parameter(x=0,lambda=[0,0,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 0 3 0* final parameter(x=3,lambda=[2,1,0]/1,nu=[0,0,0]/1)* [ 0, 0, 0 ]/1 1 0 1 final parameter(x=3,lambda=[2,0,0]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 0 2 final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 2 1 final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 1 2 2 final parameter(x=3,lambda=[1,1,-1]/1,nu=[1,0,-1]/2) [ 1, 0, -1 ]/2 2 0 3* final parameter(x=3,lambda=[1,0,-1]/1,nu=[1,0,-1]/1)* [ 1, 0, -1 ]/1 2 1 3* final parameter(x=3,lambda=[2,1,0]/1,nu=[1,0,-1]/1)* [ 1, 0, -1 ]/1 Total 10 atlas> set wp=weak_packets(GL(4,R)) Computing weak packets for 5 dual orbits of disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 3, 1, -1, -3 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 0, 0, 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 3, 1, -1, -3 ]/1 gamma_final:[ 0, 0, 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 0, 0, 0 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 0, 0, 0 ]) computing springer map of[2,2,2] O: (simply connected root datum of Lie type 'A3',(),[ 3, 1, -1 ]) survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=1,lambda=[7,3,1,-5]/2,nu=[0,1,-1,0]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=0,lambda=[7,3,-1,-5]/2,nu=[0,0,0,0]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=1,lambda=[7,1,-1,-5]/2,nu=[0,1,-1,0]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[3,1,-1,-3]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 2, 0, 0, -2 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 1, 0, 0, -1 ] dim=6 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 7, 2, -2, -7 ]/2 gamma_final:[ 1, 0, 0, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.A1.T1.T1' 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 ]) permutation: | 1, 0 | | 0, 1 | 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=1,lambda=[7,3,1,-7]/2,nu=[0,1,-1,0]/1) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,1,-1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,1,-1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=0,lambda=[7,3,-1,-7]/2,nu=[0,0,0,0]/1) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=7,lambda=[3,3,-1,-3]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=7,lambda=[5,3,-1,-1]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=7,lambda=[5,3,-1,-3]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=1,lambda=[7,1,-1,-7]/2,nu=[0,1,-1,0]/1) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,3,1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,1,-1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=7,lambda=[3,3,-1,-1]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 1, 1, -1, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 1, 1, -1, -1 ] dim=8 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 7, 3, -3, -7 ]/2 gamma_final:[ 1, 1, -1, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 0, 2, 0 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 0, 2, 0 ]) computing springer map of[0,2,0] O: (simply connected root datum of Lie type 'A3',(),[ 1, 1, -1 ]) dim: 1 2 survive:final parameter(x=1,lambda=[7,3,1,-7]/2,nu=[0,3,-3,0]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=1,lambda=[7,1,-1,-7]/2,nu=[0,3,-3,0]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=5,lambda=[5,1,1,-3]/2,nu=[5,5,-5,-5]/2) [ 1, 1, -1, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[7,3,-3,-7]/2) [ 1, 1, -1, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[7,3,-3,-7]/2) [ 1, 1, -1, -1 ]/2 cell character: 2 springer_O:2 dim: 1 2 dim: 1 2 dim: 1 2 survive:final parameter(x=6,lambda=[5,1,-1,-3]/2,nu=[5,5,-3,-7]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=8,lambda=[5,3,1,-3]/2,nu=[7,3,-5,-5]/2) [ 1, 1, -1, -1 ]/2 dim: 1 2 dim: 1 2 survive:final parameter(x=6,lambda=[5,1,1,-1]/2,nu=[5,5,-3,-7]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=8,lambda=[3,1,1,-3]/2,nu=[7,3,-5,-5]/2) [ 1, 1, -1, -1 ]/2 dim: 1 2 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 1, 0, 0, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 2, 0, 0, -2 ] dim=10 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 4, 1, -1, -4 ]/1 gamma_final:[ 1, 0, 0, -1 ]/1 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 2, 0, 2 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 2, 0, 2 ]) computing springer map of[1,0,1] O: (simply connected root datum of Lie type 'A3',(),[ 1, 0, 0 ]) dim: 1 3 survive:final parameter(x=3,lambda=[7,3,-3,-3]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[5,5,-1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=3,lambda=[7,1,-3,-1]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[3,5,1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 dim: 2 3 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 dim: 2 3 dim: 2 3 dim: 1 3 dim: 1 3 dim: 1 3 survive:final parameter(x=3,lambda=[7,3,-3,-1]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[3,1,-1,-1]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[5,5,1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[5,1,-1,-3]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=7,lambda=[5,3,-1,-1]/2,nu=[4,0,0,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 dim: 1 3 survive:final parameter(x=3,lambda=[7,1,-3,-3]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[5,3,1,-3]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[3,5,-1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[3,3,1,-1]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=7,lambda=[3,3,-1,-3]/2,nu=[4,0,0,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 dim: 1 3 dim: 1 3 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 0, 0, 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 3, 1, -1, -3 ] dim=12 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 9, 3, -3, -9 ]/2 gamma_final:[ 3, 1, -1, -3 ]/2 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 2, 2, 2 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 2, 2, 2 ]) computing springer map of[0,0,0] O: (simply connected root datum of Lie type 'A3',(),[ 0, 0, 0 ]) survive:final parameter(x=0,lambda=[9,3,-3,-9]/2,nu=[0,0,0,0]/1) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=1,lambda=[9,3,1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,3,-5,-1]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[5,5,1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=1,lambda=[9,1,-1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,1,-5,-3]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[3,5,-1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=2,lambda=[7,5,-5,-7]/2,nu=[3,-3,3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=5,lambda=[5,-1,1,-5]/2,nu=[3,3,-3,-3]/1) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,1,-1]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[3,1,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,1,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,-1,-3]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[5,3,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[5,3,-3,-1]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[3,3,-3,-3]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=1,lambda=[9,3,-1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,3,-5,-3]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,1,-3]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,1,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[5,5,-1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[5,1,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[5,3,-3,-3]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=1,lambda=[9,1,1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,1,-5,-1]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,-1,-1]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[3,5,1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[3,3,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[3,3,-3,-1]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,1,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 Variable wp: [([([Param],[WCell])],[(int,int,Param)])] (overriding previous instance, which had type [([([Param],[WCell])],[(int,int,Param)])]) atlas> weak_packet_report(GL(4,R)) Computing weak packets for 5 dual orbits of disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 3, 1, -1, -3 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 0, 0, 0, 0 ] dim=0 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 3, 1, -1, -3 ]/1 gamma_final:[ 0, 0, 0, 0 ]/1 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 0, 0, 0 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 0, 0, 0 ]) computing springer map of[2,2,2] O: (simply connected root datum of Lie type 'A3',(),[ 3, 1, -1 ]) survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=1,lambda=[7,3,1,-5]/2,nu=[0,1,-1,0]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=0,lambda=[7,3,-1,-5]/2,nu=[0,0,0,0]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=1,lambda=[7,1,-1,-5]/2,nu=[0,1,-1,0]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[3,1,-1,-3]/1) [ 0, 0, 0, 0 ]/1 cell character: 4 springer_O:4 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 2, 0, 0, -2 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 1, 0, 0, -1 ] dim=6 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 7, 2, -2, -7 ]/2 gamma_final:[ 1, 0, 0, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A1.A1.T1.T1' 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 ]) permutation: | 1, 0 | | 0, 1 | 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=1,lambda=[7,3,1,-7]/2,nu=[0,1,-1,0]/1) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,1,-1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,1,-1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=0,lambda=[7,3,-1,-7]/2,nu=[0,0,0,0]/1) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=7,lambda=[3,3,-1,-3]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=7,lambda=[5,3,-1,-1]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=7,lambda=[5,3,-1,-3]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=1,lambda=[7,1,-1,-7]/2,nu=[0,1,-1,0]/1) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,3,1,-3]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,1,-1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=7,lambda=[3,3,-1,-1]/2,nu=[7,0,0,-7]/2) [ 1, 0, 0, -1 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-1]/2,nu=[7,2,-2,-7]/2) [ 1, 0, 0, -1 ]/2 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 1, 1, -1, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 1, 1, -1, -1 ] dim=8 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 7, 3, -3, -7 ]/2 gamma_final:[ 1, 1, -1, -1 ]/2 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 0, 2, 0 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 0, 2, 0 ]) computing springer map of[0,2,0] O: (simply connected root datum of Lie type 'A3',(),[ 1, 1, -1 ]) dim: 1 2 survive:final parameter(x=1,lambda=[7,3,1,-7]/2,nu=[0,3,-3,0]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=1,lambda=[7,1,-1,-7]/2,nu=[0,3,-3,0]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=5,lambda=[5,1,1,-3]/2,nu=[5,5,-5,-5]/2) [ 1, 1, -1, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[7,3,-3,-7]/2) [ 1, 1, -1, -1 ]/2 cell character: 2 springer_O:2 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[7,3,-3,-7]/2) [ 1, 1, -1, -1 ]/2 cell character: 2 springer_O:2 dim: 1 2 dim: 1 2 dim: 1 2 survive:final parameter(x=6,lambda=[5,1,-1,-3]/2,nu=[5,5,-3,-7]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=8,lambda=[5,3,1,-3]/2,nu=[7,3,-5,-5]/2) [ 1, 1, -1, -1 ]/2 dim: 1 2 dim: 1 2 survive:final parameter(x=6,lambda=[5,1,1,-1]/2,nu=[5,5,-3,-7]/2) [ 1, 1, -1, -1 ]/2 survive:final parameter(x=8,lambda=[3,1,1,-3]/2,nu=[7,3,-5,-5]/2) [ 1, 1, -1, -1 ]/2 dim: 1 2 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 1, 0, 0, -1 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 2, 0, 0, -2 ] dim=10 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 4, 1, -1, -4 ]/1 gamma_final:[ 1, 0, 0, -1 ]/1 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 2, 0, 2 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 2, 0, 2 ]) computing springer map of[1,0,1] O: (simply connected root datum of Lie type 'A3',(),[ 1, 0, 0 ]) dim: 1 3 survive:final parameter(x=3,lambda=[7,3,-3,-3]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[5,5,-1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=3,lambda=[7,1,-3,-1]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[3,5,1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 dim: 2 3 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 dim: 2 3 dim: 2 3 dim: 1 3 dim: 1 3 dim: 1 3 survive:final parameter(x=3,lambda=[7,3,-3,-1]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[3,1,-1,-1]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[5,5,1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[5,1,-1,-3]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=7,lambda=[5,3,-1,-1]/2,nu=[4,0,0,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 dim: 1 3 survive:final parameter(x=3,lambda=[7,1,-3,-3]/2,nu=[3,2,3,-8]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[5,3,1,-3]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=4,lambda=[3,5,-1,-5]/2,nu=[8,-3,-2,-3]/2) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=9,lambda=[3,3,1,-1]/2,nu=[4,1,-1,-4]/1) [ 1, 0, 0, -1 ]/1 survive:final parameter(x=7,lambda=[3,3,-1,-3]/2,nu=[4,0,0,-4]/1) [ 1, 0, 0, -1 ]/1 cell character: 1 springer_O:1 dim: 1 3 dim: 1 3 Orbit by diagram: (root datum of Lie type 'A3.T1',(),[ 0, 0, 0, 0 ]) Computing weak packet for orbit: root datum of Lie type 'A3.T1' [ 3, 1, -1, -3 ] dim=12 Computing weak packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' gamma:[ 9, 3, -3, -9 ]/2 gamma_final:[ 3, 1, -1, -3 ]/2 integral data: st_int rd_int:root datum of Lie type 'A3.T1' st_int.rd: simply connected root datum of Lie type 'A3' O_check_int:(adjoint root datum of Lie type 'A3',(),[ 2, 2, 2 ]) permutation: | 1, 0, 0 | | 0, 1, 0 | | 0, 0, 1 | computing packet for:(adjoint root datum of Lie type 'A3',(),[ 2, 2, 2 ]) computing springer map of[0,0,0] O: (simply connected root datum of Lie type 'A3',(),[ 0, 0, 0 ]) survive:final parameter(x=0,lambda=[9,3,-3,-9]/2,nu=[0,0,0,0]/1) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=1,lambda=[9,3,1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,3,-5,-1]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[5,5,1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=1,lambda=[9,1,-1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,1,-5,-3]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[3,5,-1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=2,lambda=[7,5,-5,-7]/2,nu=[3,-3,3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=5,lambda=[5,-1,1,-5]/2,nu=[3,3,-3,-3]/1) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,1,-1]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[3,1,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,1,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,-1,-3]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[5,3,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[5,3,-3,-1]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[3,3,-3,-3]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 cell character: 0 springer_O:0 survive:final parameter(x=1,lambda=[9,3,-1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,3,-5,-3]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,1,-3]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,1,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[5,5,-1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[5,1,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[5,3,-3,-3]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=1,lambda=[9,1,1,-9]/2,nu=[0,3,-3,0]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=3,lambda=[7,1,-5,-1]/2,nu=[3,3,3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=6,lambda=[5,-1,-1,-1]/2,nu=[6,6,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,3,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=4,lambda=[3,5,1,-7]/2,nu=[9,-3,-3,-3]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=8,lambda=[3,3,1,-5]/2,nu=[9,3,-6,-6]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,-1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=7,lambda=[3,3,-3,-1]/2,nu=[9,0,0,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,1,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[5,1,1,-3]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 survive:final parameter(x=9,lambda=[3,3,-1,-1]/2,nu=[9,3,-3,-9]/2) [ 3, 1, -1, -3 ]/2 =============================================================================== Orbits for the dual group: connected quasisplit real group with Lie algebra 'su(2,2).u(1)' complex nilpotent orbits for inner class Complex reductive group of type A3.T1, with involution defining inner class of type 'cc', with 3 real forms and 2 dual real forms root datum of inner class: root datum of Lie type 'A3.T1' 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(O): number of real forms of O for all real forms in inner class #AP(O): number of Arthur parameters for O i diagram dim BC Levi Cent Z C_2 A(O) #RF(O) #ap 0 [0,0,0] 0 4T1 A3+T1 4 5 [1] [1,1,1] 5 1 [1,0,1] 6 A1+3T1 A1+2T1 2 6 [1] [0,2,2] 6 2 [0,2,0] 8 2A1+2T1 A1+T1 2 3 [1] [0,0,3] 3 3 [2,0,2] 10 A2+2T1 2T1 1 4 [1] [0,1,2] 4 4 [2,2,2] 12 A3+T1 T1 1 2 [1] [0,0,2] 2 Information about orbit centralizers: orbit#: 0 diagram: [0,0,0] isogeny information: Centralizer: A3+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'A3' ------------- orbit#: 1 diagram: [1,0,1] isogeny information: Centralizer: A1+2T1 Center is a connected complex torus of rank 2 simply connected root datum of Lie type 'A1' ------------- orbit#: 2 diagram: [0,2,0] isogeny information: Centralizer: A1+T1 Center is a connected complex torus of rank 1 simply connected root datum of Lie type 'A1' ------------- orbit#: 3 diagram: [2,0,2] isogeny information: Centralizer: 2T1 Center is a connected complex torus of rank 2 ------------- orbit#: 4 diagram: [2,2,2] isogeny information: Centralizer: T1 Center is a connected complex torus of rank 1 ------------- Arthur parameters listed by orbit: #parameters by orbit: [5,6,3,4,2] Total: 20 orbit #0 for G #orbits for (disconnected) Cent(O): 5 K_0 H diagram dim mult root datum of Lie type 'A1.A1.T1.T1' [ 0, 0, 0, 0 ] [0,0] 0 1 root datum of Lie type 'A2.T1.T1' [ 0, 0, 0, 0 ] [0,0] 0 1 root datum of Lie type 'A2.T1.T1' [ 0, 0, 0, 0 ] [0,0] 0 1 root datum of Lie type 'A3.T1' [ 0, 0, 0, 0 ] [0,0,0] 0 1 root datum of Lie type 'A3.T1' [ 0, 0, 0, 0 ] [0,0,0] 0 1 orbit #1 for G #orbits for (disconnected) Cent(O): 6 K_0 H diagram dim mult root datum of Lie type 'A1.A1.T1.T1' [ 1, 0, -1, 0 ] [2,0] 2 1 root datum of Lie type 'A1.A1.T1.T1' [ 0, 1, 0, -1 ] [0,2] 2 1 root datum of Lie type 'A2.T1.T1' [ 1, 0, 0, -1 ] [1,1] 4 1 root datum of Lie type 'A2.T1.T1' [ 1, 0, 0, -1 ] [1,1] 4 1 root datum of Lie type 'A3.T1' [ 1, 0, 0, -1 ] [1,0,1] 6 1 root datum of Lie type 'A3.T1' [ 1, 0, 0, -1 ] [1,0,1] 6 1 orbit #2 for G #orbits for (disconnected) Cent(O): 3 K_0 H diagram dim mult root datum of Lie type 'A1.A1.T1.T1' [ 1, 1, -1, -1 ] [2,2] 4 1 root datum of Lie type 'A3.T1' [ 1, 1, -1, -1 ] [0,2,0] 8 1 root datum of Lie type 'A3.T1' [ 1, 1, -1, -1 ] [0,2,0] 8 1 orbit #3 for G #orbits for (disconnected) Cent(O): 4 K_0 H diagram dim mult root datum of Lie type 'A2.T1.T1' [ 2, 0, 0, -2 ] [2,2] 6 1 root datum of Lie type 'A2.T1.T1' [ 2, 0, 0, -2 ] [2,2] 6 1 root datum of Lie type 'A3.T1' [ 2, 0, 0, -2 ] [2,0,2] 10 1 root datum of Lie type 'A3.T1' [ 2, 0, 0, -2 ] [2,0,2] 10 1 orbit #4 for G #orbits for (disconnected) Cent(O): 2 K_0 H diagram dim mult root datum of Lie type 'A3.T1' [ 3, 1, -1, -3 ] [2,2,2] 12 1 root datum of Lie type 'A3.T1' [ 3, 1, -1, -3 ] [2,2,2] 12 1 orbit |packet| 0 5 1 6 2 3 3 4 4 2 Total 20 *: dual(cell) contains an Aq(lambda) orbit# block# cell# parameters 0 0 0* 1 0 1 0* 1 0 2 0* 1 0 3 0* 1 0 4 0* 1 1 0 1 1 1 0 2 1 1 2 3 1 1 2 4 1 1 4 1 1 1 4 2 1 2 0 3* 1 2 0 6* 1 2 0 7* 1 3 0 4* 1 3 0 5* 1 3 1 3* 1 3 2 3* 1 4 0 8* 1 4 0 9* 1 Total 20 *: dual(cell) contains an Aq(lambda) *: dual(p) is an Aq(lambda) orbit# block# cell# parameters inf. char. 0 0 0* final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[0,0,0,0]/1)* [ 0, 0, 0, 0 ]/1 0 1 0* final parameter(x=1,lambda=[1,1,-1,1]/2,nu=[0,0,0,0]/1)* [ 0, 0, 0, 0 ]/1 0 2 0* final parameter(x=0,lambda=[1,1,1,1]/2,nu=[0,0,0,0]/1)* [ 0, 0, 0, 0 ]/1 0 3 0* final parameter(x=1,lambda=[1,3,1,1]/2,nu=[0,0,0,0]/1)* [ 0, 0, 0, 0 ]/1 0 4 0* final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1)* [ 0, 0, 0, 0 ]/1 1 0 1 final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[1,0,0,-1]/2) [ 1, 0, 0, -1 ]/2 1 0 2 final parameter(x=9,lambda=[3,3,1,-3]/2,nu=[1,0,0,-1]/2) [ 1, 0, 0, -1 ]/2 1 2 3 final parameter(x=7,lambda=[5,1,1,-1]/2,nu=[1,0,0,-1]/2) [ 1, 0, 0, -1 ]/2 1 2 4 final parameter(x=7,lambda=[3,1,1,-3]/2,nu=[1,0,0,-1]/2) [ 1, 0, 0, -1 ]/2 1 4 1 final parameter(x=9,lambda=[5,1,-1,-1]/2,nu=[1,0,0,-1]/2) [ 1, 0, 0, -1 ]/2 1 4 2 final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[1,0,0,-1]/2) [ 1, 0, 0, -1 ]/2 2 0 3* final parameter(x=5,lambda=[3,3,-1,-1]/2,nu=[1,1,-1,-1]/2)* [ 1, 1, -1, -1 ]/2 2 0 6* final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[1,1,-1,-1]/2)* [ 1, 1, -1, -1 ]/2 2 0 7* final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[1,1,-1,-1]/2)* [ 1, 1, -1, -1 ]/2 3 0 4* final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[1,0,0,-1]/1)* [ 1, 0, 0, -1 ]/1 3 0 5* final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[1,0,0,-1]/1)* [ 1, 0, 0, -1 ]/1 3 1 3* final parameter(x=7,lambda=[3,1,1,-3]/2,nu=[1,0,0,-1]/1)* [ 1, 0, 0, -1 ]/1 3 2 3* final parameter(x=7,lambda=[5,1,1,-1]/2,nu=[1,0,0,-1]/1)* [ 1, 0, 0, -1 ]/1 4 0 8* final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/2)* [ 3, 1, -1, -3 ]/2 4 0 9* final parameter(x=9,lambda=[5,3,1,-1]/2,nu=[3,1,-1,-3]/2)* [ 3, 1, -1, -3 ]/2 Total 20 Testing conjecture about size of weak Arthur packets for disconnected split real group with Lie algebra 'sl(4,R).gl(1,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] [0,0,0] 0 4T1 A3+T1 4 5 [1] 1 [1,0,0,-1] [1,0,1] 6 A1+3T1 A1+2T1 2 6 [1] 2 [1,1,-1,-1] [0,2,0] 8 2A1+2T1 A1+T1 2 3 [1] 3 [2,0,0,-2] [2,0,2] 10 A2+2T1 2T1 1 4 [1] 4 [3,1,-1,-3] [2,2,2] 12 A3+T1 T1 1 2 [1] i data guess actual A disjoint conjecture 0 [1,1,1,1,1] 5 5 1 true true 1 [1,1,1,1,1,1] 6 6 1 false N/A 2 [1,1,1] 3 3 1 true true 3 [1,1,1,1] 4 4 1 true true 4 [1,1] 2 2 1 true true ------------------------------------------------------------- set parameters=[ parameter(G,9,[ 5, 3, 1, -1 ]/2,[ 0, 0, 0, 0 ]/1), parameter(G,1,[ 1, 1, -1, 1 ]/2,[ 0, 0, 0, 0 ]/1), parameter(G,0,[ 1, 1, 1, 1 ]/2,[ 0, 0, 0, 0 ]/1), parameter(G,1,[ 1, 3, 1, 1 ]/2,[ 0, 0, 0, 0 ]/1), parameter(G,9,[ 3, 1, -1, -3 ]/2,[ 0, 0, 0, 0 ]/1), parameter(G,9,[ 5, 3, 1, -1 ]/2,[ 1, 0, 0, -1 ]/2), parameter(G,9,[ 3, 3, 1, -3 ]/2,[ 1, 0, 0, -1 ]/2), parameter(G,7,[ 5, 1, 1, -1 ]/2,[ 1, 0, 0, -1 ]/2), parameter(G,7,[ 3, 1, 1, -3 ]/2,[ 1, 0, 0, -1 ]/2), parameter(G,9,[ 5, 1, -1, -1 ]/2,[ 1, 0, 0, -1 ]/2), parameter(G,9,[ 3, 1, -1, -3 ]/2,[ 1, 0, 0, -1 ]/2), parameter(G,5,[ 3, 3, -1, -1 ]/2,[ 1, 1, -1, -1 ]/2), parameter(G,9,[ 5, 3, 1, -1 ]/2,[ 1, 1, -1, -1 ]/2), parameter(G,9,[ 3, 1, -1, -3 ]/2,[ 1, 1, -1, -1 ]/2), parameter(G,9,[ 5, 3, 1, -1 ]/2,[ 1, 0, 0, -1 ]/1), parameter(G,9,[ 3, 1, -1, -3 ]/2,[ 1, 0, 0, -1 ]/1), parameter(G,7,[ 3, 1, 1, -3 ]/2,[ 1, 0, 0, -1 ]/1), parameter(G,7,[ 5, 1, 1, -1 ]/2,[ 1, 0, 0, -1 ]/1), parameter(G,9,[ 3, 1, -1, -3 ]/2,[ 3, 1, -1, -3 ]/2), parameter(G,9,[ 5, 3, 1, -1 ]/2,[ 3, 1, -1, -3 ]/2) ]