atlas> G:=SL(2,R) Value: connected split real group with Lie algebra 'sl(2,R)' atlas> atlas> atlas> print_KGB (G) kgbsize: 3 Base grading: [1]. 0: 0 [n] 1 2 (0)#0 e 1: 0 [n] 0 2 (1)#0 e 2: 1 [r] 2 * (0)#1 1^e atlas> atlas> atlas> atlas> atlas> set x=KGB(G,0) Identifier x: KGBElt (hiding previous one of type int) atlas> set B=block_of (trivial(G)) Identifier B: [Param] (hiding previous one of type (mat,string,int)) atlas> for p in B do prints(p) od final parameter (x=0,lambda=[1]/1,nu=[0]/1) final parameter (x=1,lambda=[1]/1,nu=[0]/1) final parameter (x=2,lambda=[1]/1,nu=[1]/1) Value: [(),(),()] atlas> atlas> atlas> atlas> set show([Param] params)=void:for p in params do prints(p) od Added definition [6] of show: ([Param]->) atlas> show(B) final parameter (x=0,lambda=[1]/1,nu=[0]/1) final parameter (x=1,lambda=[1]/1,nu=[0]/1) final parameter (x=2,lambda=[1]/1,nu=[1]/1) atlas> atlas> atlas> atlas> B[0] Value: final parameter (x=0,lambda=[1]/1,nu=[0]/1) atlas> atlas> atlas> atlas> B[1] Value: final parameter (x=1,lambda=[1]/1,nu=[0]/1) atlas> atlas> x Value: KGB element #0 atlas> set x_b=KGB(G,0) Identifier x_b: KGBElt atlas> atlas> atlas> atlas> x_b Value: KGB element #0 atlas> atlas> atlas> B[0] Value: final parameter (x=0,lambda=[1]/1,nu=[0]/1) atlas> hc_parameter(B[0],x_b) Value: [ 1 ]/1 atlas> B[1] Value: final parameter (x=1,lambda=[1]/1,nu=[0]/1) atlas> hc_parameter(B[1],x_b) Value: [ -1 ]/1 atlas> atlas> G:=Sp(4,R) Value: connected split real group with Lie algebra 'sp(4,R)' atlas> whattype discrete_series Display all 1168 possibilities? (y or n) atlas> whattype discrete_series ? Overloaded instances of 'discrete_series' (KGBElt,ratvec)->Param (RealForm,ratvec)->Param atlas> all_d all_discrete_series_gamma all_dominant_norm_upto atlas> all_d all_discrete_series_gamma all_dominant_norm_upto atlas> whattype all_discrete_series_gamma ? Overloaded instances of 'all_discrete_series_gamma' (RealForm,ratvec)->[Param] atlas> set ds=all_discrete_series_gamma (G,rho(G)) Identifier ds: [Param] atlas> show(ds) final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) atlas> {choose x_b} atlas> x_b=KGB(G,0) Value: false atlas> x_b:=KGB(G,0) Value: KGB element #0 atlas> K_0(x_b) Value: compact connected real group with Lie algebra 'su(2).u(1)' atlas> set K=K_0(x_b) Identifier K: RealForm atlas> simple_roots (K) Value: | 1 | | 1 | atlas> x_b:=KGB(G,1) Value: KGB element #1 atlas> simple_roots (K_0(x_b)) Value: | 1 | | 1 | atlas> x_b:=KGB(G,2) Value: KGB element #2 atlas> simple_roots (K_0(x_b)) Value: | 1 | | -1 | atlas> x_b Value: KGB element #2 atlas> for p in ds do prints(p, " ", hc_parameter (p,x_b)) od final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) [ 2, -1 ]/1 final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) [ 1, -2 ]/1 final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) [ 2, 1 ]/1 final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) [ -1, -2 ]/1 Value: [(),(),(),()] atlas> for p in ds do prints(p, " ", hc_parameter (p,x_b), " ", status_texts(x(p))) od final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) [ 2, -1 ]/1 ["nc","nc"] final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) [ 1, -2 ]/1 ["nc","nc"] final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) [ 2, 1 ]/1 ["ic","nc"] final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) [ -1, -2 ]/1 ["ic","nc"] Value: [(),(),(),()] atlas> atlas> atlas> atlas> whattype discrete_series ? Overloaded instances of 'discrete_series' (KGBElt,ratvec)->Param (RealForm,ratvec)->Param atlas> p:=discrete_series (x_b,[2,1]) Value: final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) atlas> p:=discrete_series (x_b,[2,-1]) Value: final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) atlas> atlas> atlas> atlas> atlas> set W=generate_W (G) Identifier W: [(RootDatum,[int])] atlas> #W Value: 8 atlas> for w in W do prints(w) od simply connected root datum of Lie type 'C2'[] simply connected root datum of Lie type 'C2'[0] simply connected root datum of Lie type 'C2'[1] simply connected root datum of Lie type 'C2'[1,0] simply connected root datum of Lie type 'C2'[0,1] simply connected root datum of Lie type 'C2'[0,1,0] simply connected root datum of Lie type 'C2'[1,0,1] simply connected root datum of Lie type 'C2'[1,0,1,0] Value: [(),(),(),(),(),(),(),()] atlas> for w in W do prints(cross(w,x_b)) od KGB element #2 KGB element #2 KGB element #0 KGB element #0 KGB element #1 KGB element #1 KGB element #3 KGB element #3 Value: [(),(),(),(),(),(),(),()] atlas> for w in W do prints(cross(w,KGB(G,10))) od KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 Value: [(),(),(),(),(),(),(),()] atlas> for p in ds do prints(p, " ", hc_parameter (p,x_b)) od final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) [ 2, -1 ]/1 final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) [ 1, -2 ]/1 final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) [ 2, 1 ]/1 final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) [ -1, -2 ]/1 Value: [(),(),(),()] atlas> for p in ds do prints(p, " ", hc_parameter (p)) od final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) [ 2, 1 ]/1 final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) [ 1, 2 ]/1 final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) [ 2, -1 ]/1 final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) [ -1, 2 ]/1 Value: [(),(),(),()] atlas> for p in ds do prints(p, " ", hc_parameter (p,KGB(G,0))) od final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1) [ 2, 1 ]/1 final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1) [ 1, 2 ]/1 final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1) [ 2, -1 ]/1 final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1) [ -1, 2 ]/1 Value: [(),(),(),()] atlas> atlas> atlas> G:=Sp(6,R) Value: connected split real group with Lie algebra 'sp(6,R)' atlas> set F=distinguished_fiber (G) Identifier F: [int] atlas> F Value: [0,1,2,3,4,5,6,7] atlas> ds:=all_discrete_series_gamma (G,rho(G)) Value: [final parameter (x=0,lambda=[3,2,1]/1,nu=[0,0,0]/1),final parameter (x=1,lambda=[3,2,1]/1,nu=[0,0,0]/1),final parameter (x=2,lambda=[3,2,1]/1,nu=[0,0,0]/1),final parameter (x=3,lambda=[3,2,1]/1,nu=[0,0,0]/1),final parameter (x=4,lambda=[3,2,1]/1,nu=[0,0,0]/1),final parameter (x=5,lambda=[3,2,1]/1,nu=[0,0,0]/1),final parameter (x=6,lambda=[3,2,1]/1,nu=[0,0,0]/1),final parameter (x=7,lambda=[3,2,1]/1,nu=[0,0,0]/1)] atlas> #s Error in expression #(s) at :239:0-2 Failed to match '#' with argument type Split Type check failed atlas> #ds Value: 8 atlas> show(ds) final parameter (x=0,lambda=[3,2,1]/1,nu=[0,0,0]/1) final parameter (x=1,lambda=[3,2,1]/1,nu=[0,0,0]/1) final parameter (x=2,lambda=[3,2,1]/1,nu=[0,0,0]/1) final parameter (x=3,lambda=[3,2,1]/1,nu=[0,0,0]/1) final parameter (x=4,lambda=[3,2,1]/1,nu=[0,0,0]/1) final parameter (x=5,lambda=[3,2,1]/1,nu=[0,0,0]/1) final parameter (x=6,lambda=[3,2,1]/1,nu=[0,0,0]/1) final parameter (x=7,lambda=[3,2,1]/1,nu=[0,0,0]/1) atlas> for i in F do prints(i, " ", rho_K(KGB(G,i))) od 0 [ 1, 1, 0 ]/1 1 [ 1, 1, 0 ]/1 2 [ 1, 0, 1 ]/1 3 [ 1, 1, 0 ]/1 4 [ 1, 1, 0 ]/1 5 [ 1, 0, -1 ]/1 6 [ 1, 0, 1 ]/1 7 [ 1, 0, -1 ]/1 Value: [(),(),(),(),(),(),(),()] atlas> x_b:=KGB(G,5) Value: KGB element #5 atlas> simple_roots (K_0(x_b)) Value: | 1, 0 | | -1, 1 | | 0, -1 | atlas> for p in ds prints(x(p), " ", hc_parameter (p,x_b)) od ^^^^^^ syntax error, unexpected IDENT, expecting DO atlas> for p in ds do prints(x(p), " ", hc_parameter (p,x_b)) od KGB element #0 [ 3, 1, -2 ]/1 KGB element #1 [ 2, 1, -3 ]/1 KGB element #2 [ 3, 2, -1 ]/1 KGB element #3 [ 3, -1, -2 ]/1 KGB element #4 [ 2, -1, -3 ]/1 KGB element #5 [ 3, 2, 1 ]/1 KGB element #6 [ 1, -2, -3 ]/1 KGB element #7 [ -1, -2, -3 ]/1 Value: [(),(),(),(),(),(),(),()] atlas> atlas> atlas> p:=discrete_series (x_b,[-1,-2,-3]) Value: final parameter (x=7,lambda=[3,2,1]/1,nu=[0,0,0]/1) atlas> atlas> atlas> print_KGB (G) kgbsize: 45 Base grading: [111]. 0: 0 [n,n,n] 1 2 3 8 10 12 (0,0,0)#0 e 1: 0 [n,c,n] 0 1 4 8 * 13 (1,1,0)#0 e 2: 0 [c,n,n] 2 0 5 * 10 14 (0,1,1)#0 e 3: 0 [n,c,n] 4 3 0 9 * 12 (0,0,1)#0 e 4: 0 [n,n,n] 3 6 1 9 11 13 (1,1,1)#0 e 5: 0 [c,c,n] 5 5 2 * * 14 (0,1,0)#0 e 6: 0 [c,n,n] 6 4 7 * 11 15 (1,0,0)#0 e 7: 0 [c,c,n] 7 7 6 * * 15 (1,0,1)#0 e 8: 1 [r,C,n] 8 17 9 * * 16 (0,0,0) 1 1^e 9: 1 [r,C,n] 9 18 8 * * 16 (0,0,1) 1 1^e 10: 1 [C,r,C] 17 10 23 * * * (0,0,0) 1 2^e 11: 1 [C,r,C] 18 11 24 * * * (1,0,0) 1 2^e 12: 1 [n,C,r] 13 19 12 16 * * (0,0,0) 2 3^e 13: 1 [n,C,r] 12 20 13 16 * * (1,1,0) 2 3^e 14: 1 [c,C,r] 14 21 14 * * * (0,1,0) 2 3^e 15: 1 [c,C,r] 15 22 15 * * * (1,0,0) 2 3^e 16: 2 [r,C,r] 16 25 16 * * * (0,0,0) 4 1^3^e 17: 2 [C,C,C] 10 8 32 * * * (0,0,0) 1 1x2^e 18: 2 [C,C,C] 11 9 33 * * * (0,1,0) 1 1x2^e 19: 2 [C,C,n] 28 12 21 * * 26 (0,0,0) 2 2x3^e 20: 2 [C,C,n] 29 13 22 * * 27 (1,0,1) 2 2x3^e 21: 2 [C,C,n] 30 14 19 * * 26 (0,0,1) 2 2x3^e 22: 2 [C,C,n] 31 15 20 * * 27 (1,0,0) 2 2x3^e 23: 2 [C,n,C] 32 23 10 * 26 * (0,0,0) 1 3x2^e 24: 2 [C,n,C] 33 24 11 * 27 * (1,0,0) 1 3x2^e 25: 3 [C,C,C] 34 16 37 * * * (0,0,0) 4 2x1^3^e 26: 3 [C,r,r] 35 26 26 * * * (0,0,0) 3 2^3x2^e 27: 3 [C,r,r] 36 27 27 * * * (1,0,0) 3 2^3x2^e 28: 3 [C,n,n] 19 29 30 * 34 35 (0,0,0)#2 1x2x3^e 29: 3 [C,n,n] 20 28 31 * 34 36 (0,1,1)#2 1x2x3^e 30: 3 [C,c,n] 21 30 28 * * 35 (0,0,1)#2 1x2x3^e 31: 3 [C,c,n] 22 31 29 * * 36 (0,1,0)#2 1x2x3^e 32: 3 [C,C,C] 23 38 17 * * * (0,0,0) 1 1x3x2^e 33: 3 [C,C,C] 24 39 18 * * * (0,1,0) 1 1x3x2^e 34: 4 [C,r,C] 25 34 42 * * * (0,0,0) 4 1x2x1^3^e 35: 4 [C,C,r] 26 40 35 * * * (0,0,0) 3 1x2^3x2^e 36: 4 [C,C,r] 27 41 36 * * * (0,1,0) 3 1x2^3x2^e 37: 4 [C,C,C] 42 43 25 * * * (0,0,0) 4 3x2x1^3^e 38: 4 [n,C,n] 38 32 39 40 * 43 (0,0,0)#1 2x1x3x2^e 39: 4 [n,C,n] 39 33 38 41 * 43 (0,0,1)#1 2x1x3x2^e 40: 5 [r,C,n] 40 35 41 * * 44 (0,0,0)#3 1^2x1x3x2^e 41: 5 [r,C,n] 41 36 40 * * 44 (0,0,1)#3 1^2x1x3x2^e 42: 5 [C,n,C] 37 42 34 * 44 * (0,0,0)#4 1x3x2x1^3^e 43: 5 [n,C,r] 43 37 43 44 * * (0,0,0) 4 2x3x2x1^3^e 44: 6 [r,r,r] 44 44 44 * * * (0,0,0)#5 1^2x3x2x1^3^e atlas> G:=Sp(4,R) Value: connected split real group with Lie algebra 'sp(4,R)' atlas> print_KGB (G) kgbsize: 11 Base grading: [11]. 0: 0 [n,n] 1 2 4 5 (0,0)#0 e 1: 0 [n,n] 0 3 4 6 (1,1)#0 e 2: 0 [c,n] 2 0 * 5 (0,1)#0 e 3: 0 [c,n] 3 1 * 6 (1,0)#0 e 4: 1 [r,C] 4 9 * * (0,0) 1 1^e 5: 1 [C,r] 7 5 * * (0,0) 2 2^e 6: 1 [C,r] 8 6 * * (1,0) 2 2^e 7: 2 [C,n] 5 8 * 10 (0,0)#2 1x2^e 8: 2 [C,n] 6 7 * 10 (0,1)#2 1x2^e 9: 2 [n,C] 9 4 10 * (0,0)#1 2x1^e 10: 3 [r,r] 10 10 * * (0,0)#3 1^2x1^e atlas> atlas> atlas> set H=Cartan_class(G,0) Identifier H: CartanClass (hiding previous one of type CartanClass) atlas> print_Cartan_info (H) compact: 2, complex: 0, split: 0 canonical twisted involution: e twisted involution orbit size: 1; fiber size: 4; strong inv: 4 imaginary root system: C2 real root system: empty complex factor: empty atlas> atlas> atlas> atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is a Weyl group of type A1 W^R is trivial generators for W_ic: 2,1,2 atlas> atlas> atlas> atlas> atlas> atlas> atlas> {number of KGB orbits for compact Cartan: 8/2=4} atlas> set H=Cartan_class(G,1) Identifier H: CartanClass (hiding previous one of type CartanClass) atlas> print_Cartan_info (H) compact: 0, complex: 1, split: 0 canonical twisted involution: 2,1,2 twisted involution orbit size: 2; fiber size: 1; strong inv: 2 imaginary root system: A1 real root system: A1 complex factor: empty atlas> atlas> atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is an elementary abelian 2-group of rank 1 W_ic is trivial W^R is a Weyl group of type A1 generators for A 1 generators for W^R: 2,1,2 atlas> {number of KGB orbits for C^* Cartan: 8/4=2} atlas> print_KGB (G) kgbsize: 11 Base grading: [11]. 0: 0 [n,n] 1 2 4 5 (0,0)#0 e 1: 0 [n,n] 0 3 4 6 (1,1)#0 e 2: 0 [c,n] 2 0 * 5 (0,1)#0 e 3: 0 [c,n] 3 1 * 6 (1,0)#0 e 4: 1 [r,C] 4 9 * * (0,0) 1 1^e 5: 1 [C,r] 7 5 * * (0,0) 2 2^e 6: 1 [C,r] 8 6 * * (1,0) 2 2^e 7: 2 [C,n] 5 8 * 10 (0,0)#2 1x2^e 8: 2 [C,n] 6 7 * 10 (0,1)#2 1x2^e 9: 2 [n,C] 9 4 10 * (0,0)#1 2x1^e 10: 3 [r,r] 10 10 * * (0,0)#3 1^2x1^e atlas> set x=KGB(G,4) Identifier x: KGBElt (hiding previous one of type KGBElt) atlas> for w in W do prints(cross(w,x)) od KGB element #4 KGB element #4 KGB element #9 KGB element #9 KGB element #9 KGB element #9 KGB element #4 KGB element #4 Value: [(),(),(),(),(),(),(),()] atlas> for (,w) in W do prints(w) od [] [0] [1] [1,0] [0,1] [0,1,0] [1,0,1] [1,0,1,0] Value: [(),(),(),(),(),(),(),()] atlas> set H=Cartan_class(G,2) Identifier H: CartanClass (hiding previous one of type CartanClass) atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is trivial W^R is a Weyl group of type A1 generators for W^R: 1,2,1 atlas> {number of KGB orbits for C^* Cartan: 8/2=4} atlas> print_KGB (G) kgbsize: 11 Base grading: [11]. 0: 0 [n,n] 1 2 4 5 (0,0)#0 e 1: 0 [n,n] 0 3 4 6 (1,1)#0 e 2: 0 [c,n] 2 0 * 5 (0,1)#0 e 3: 0 [c,n] 3 1 * 6 (1,0)#0 e 4: 1 [r,C] 4 9 * * (0,0) 1 1^e 5: 1 [C,r] 7 5 * * (0,0) 2 2^e 6: 1 [C,r] 8 6 * * (1,0) 2 2^e 7: 2 [C,n] 5 8 * 10 (0,0)#2 1x2^e 8: 2 [C,n] 6 7 * 10 (0,1)#2 1x2^e 9: 2 [n,C] 9 4 10 * (0,0)#1 2x1^e 10: 3 [r,r] 10 10 * * (0,0)#3 1^2x1^e atlas> set x=KGB(G,5) Identifier x: KGBElt (hiding previous one of type KGBElt) atlas> for w in W do prints(cross(w,x)) od KGB element #5 KGB element #7 KGB element #5 KGB element #8 KGB element #7 KGB element #6 KGB element #8 KGB element #6 Value: [(),(),(),(),(),(),(),()] atlas> set H=Cartan_class(G,3) Identifier H: CartanClass (hiding previous one of type CartanClass) atlas> print_real_Weyl (G,H) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is trivial W^R is a Weyl group of type B2 generators for W^R: 1 2 atlas> {number of KGB orbits for split Cartan: 8/8=1} atlas> set x=KGB(G,10) Identifier x: KGBElt (hiding previous one of type KGBElt) atlas> for w in W do prints(cross(w,x)) od KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 KGB element #10 atlas>