atlas> set G=SL(2,R)
Identifier G: RealForm
atlas> set p=trivial(G)
Identifier p: Param
atlas> p
Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas> set x=x(p)
Identifier x: KGBElt
atlas> 
atlas> involution (x)
Value: 
| -1 |

atlas> infinitesimal_character (p)
Value: [ 1 ]/1
atlas> rho(G)
Value: [ 1 ]/1
atlas> whattype discrete_series  ?
Overloaded instances of 'discrete_series'
  (KGBElt,ratvec)->Param
  (RealForm,ratvec)->Param
atlas> set q=discrete_series (KGB(G,0),rho(G))
Identifier q: Param (hiding previous one of type vec (constant))
atlas> q
Value: final parameter (x=0,lambda=[1]/1,nu=[0]/1)
atlas> print_block(p)
Parameter defines element 2 of the following block:
0:  0  [i1]  1   (2,*)  *(x=0,lam=rho+  [0], nu=  [0]/1)  e
1:  0  [i1]  0   (2,*)  *(x=1,lam=rho+  [0], nu=  [0]/1)  e
2:  1  [r1]  2   (0,1)  *(x=2,lam=rho+  [0], nu=  [1]/1)  1^e
atlas> set r=discrete_series (KGB(G,1),rho(G))
Identifier r: Param
atlas> set x_b=KGB(G,0)
Identifier x_b: KGBElt
atlas> hc_parameter (q,x)
Runtime error:
  index -1 out of range (0<= . <1) in subscription alpha[s]
(in call of act@((RootDatum,[int]),vec), defined at /home/jda/atlasSoftware/atlasofliegroups/atlas-scripts/Weylgroup.at:14:4--16:51)
(in call of act@((RootDatum,[int]),ratvec), defined at /home/jda/atlasSoftware/atlasofliegroups/atlas-scripts/Weylgroup.at:18:4-74)
(in call of hc_parameter@(Param,KGBElt), defined at /home/jda/atlasSoftware/atlasofliegroups/atlas-scripts/representations.at:44:4--47:49)
Evaluation aborted.
atlas> hc_parameter (q,x_b)
Value: [ 1 ]/1
atlas> hc_parameter (r,x_b)
Value: [ -1 ]/1
atlas> 
atlas> 
atlas> G:=PGL(2,R)
Value: disconnected split real group with Lie algebra 'sl(2,R)'
atlas> p:=trivial(G)
Value: final parameter (x=1,lambda=[1]/2,nu=[1]/2)
atlas> print_block(p)
Parameter defines element 1 of the following block:
0:  0  [i2]  0   (1,2)  *(x=0,lam=rho+  [0], nu=  [0]/1)  e
1:  1  [r2]  2   (0,*)  *(x=1,lam=rho+  [0], nu=  [1]/2)  1^e
2:  1  [r2]  1   (0,*)  *(x=1,lam=rho+ [-1], nu=  [1]/2)  1^e
atlas> q:=discrete_series (KGB(G,0),rho(G))
Value: final parameter (x=0,lambda=[1]/2,nu=[0]/1)
atlas> q
Value: final parameter (x=0,lambda=[1]/2,nu=[0]/1)
atlas> rho(G)
Value: [ 1 ]/2
atlas> hc_parameter (q)
Value: [ 1 ]/2
atlas> print_KGB (G)
kgbsize: 2
Base grading: [1].
0:  0  [n]   0    1  (0)#0 e
1:  1  [r]   1    *  (0)#1 1^e
atlas> print_block(trivial(G))
Parameter defines element 1 of the following block:
0:  0  [i2]  0   (1,2)  *(x=0,lam=rho+  [0], nu=  [0]/1)  e
1:  1  [r2]  2   (0,*)  *(x=1,lam=rho+  [0], nu=  [1]/2)  1^e
2:  1  [r2]  1   (0,*)  *(x=1,lam=rho+ [-1], nu=  [1]/2)  1^e
atlas> 
atlas> 
atlas> 
atlas> p
Value: final parameter (x=1,lambda=[1]/2,nu=[1]/2)
atlas> all_p
all_parameters               all_parameters_KGB_gamma     all_principal_series
all_parameters_Cartan_gamma  all_parameters_gamma         
atlas> print_block(trivial(G))
Parameter defines element 1 of the following block:
0:  0  [i2]  0   (1,2)  *(x=0,lam=rho+  [0], nu=  [0]/1)  e
1:  1  [r2]  2   (0,*)  *(x=1,lam=rho+  [0], nu=  [1]/2)  1^e
2:  1  [r2]  1   (0,*)  *(x=1,lam=rho+ [-1], nu=  [1]/2)  1^e
atlas> block_sizes (G)
Value: 
| 0, 1 |
| 1, 3 |

atlas> 
atlas> 
atlas> 
atlas> 
atlas> G:=Sp(4,R)
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> set B=block_of (trivial(G))
Identifier B: [Param]
atlas> print_block(trivial(G))
Parameter defines element 10 of the following block:
 0:  0  [i1,i1]   1   2   ( 6, *)  ( 4, *)  *(x= 0,lam=rho+  [0,0], nu=  [0,0]/1)  e
 1:  0  [i1,i1]   0   3   ( 6, *)  ( 5, *)  *(x= 1,lam=rho+  [0,0], nu=  [0,0]/1)  e
 2:  0  [ic,i1]   2   0   ( *, *)  ( 4, *)  *(x= 2,lam=rho+  [0,0], nu=  [0,0]/1)  e
 3:  0  [ic,i1]   3   1   ( *, *)  ( 5, *)  *(x= 3,lam=rho+  [0,0], nu=  [0,0]/1)  e
 4:  1  [C+,r1]   7   4   ( *, *)  ( 0, 2)  *(x= 5,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 5:  1  [C+,r1]   8   5   ( *, *)  ( 1, 3)  *(x= 6,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 6:  1  [r1,C+]   6   9   ( 0, 1)  ( *, *)  *(x= 4,lam=rho+  [0,0], nu= [1,-1]/2)  1^e
 7:  2  [C-,i1]   4   8   ( *, *)  (10, *)  *(x= 7,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 8:  2  [C-,i1]   5   7   ( *, *)  (10, *)  *(x= 8,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 9:  2  [i2,C-]   9   6   (10,11)  ( *, *)  *(x= 9,lam=rho+  [0,0], nu=  [3,3]/2)  2x1^e
10:  3  [r2,r1]  11  10   ( 9, *)  ( 7, 8)  *(x=10,lam=rho+  [0,0], nu=  [2,1]/1)  1^2x1^e
11:  3  [r2,rn]  10  11   ( 9, *)  ( *, *)  *(x=10,lam=rho+[-1,-1], nu=  [2,1]/1)  1^2x1^e
atlas> 
atlas> 
atlas> p:=B[4]
Value: final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas> infinitesimal_character (p)
Value: [ 2, 1 ]/1
atlas> x:=x(p)
Value: KGB element #5
atlas> involution (x)
Value: 
| 1,  0 |
| 0, -1 |

atlas> set theta=involution (x)
Identifier theta: mat
atlas> theta
Value: 
| 1,  0 |
| 0, -1 |

atlas> nu(p)
Value: [ 0, 1 ]/1
atlas> set (P,sigma)=cuspidal_data (p)
Identifiers P: ([int],KGBElt), sigma: Param
atlas> 
atlas> 
atlas> 
atlas> {P=real parabolic, sigma=discrete series of M=Levi(P)}
atlas> {I(p)= induced from sigma P->G}
atlas> 
atlas> P
Value: ([1],KGB element #7)
atlas> set M=Levi(P)
Identifier M: RealForm
atlas> M
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas> simple_roots (M)
Value: 
| 0 |
| 2 |

atlas> sigma
Value: final parameter (x=0,lambda=[1,2]/1,nu=[1,0]/1)
atlas> real_form(sigma)
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas> infinitesimal_character (sigma)
Value: [ 1, 2 ]/1
atlas> rho(M)
Value: [ 0, 1 ]/1
atlas> hc_parameter (sigma)
Value: [ 1, 2 ]/1
atlas> {discrete series: relative discrete series (discrete series modulo center)}
atlas> induce_standard (sigma,P,G)
Value: 
1*final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas> p
Value: final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas> 
atlas> 
atlas> induce_irreducible (sigma,P,G)
Value: 
1*final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1)
1*final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1)
1*final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas> 
atlas> 
atlas> 
atlas> induce_standard (sigma,P,G)
Value: 
1*final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas> 
atlas> {Ind_P^G(discrete series) = single standard module (x=5,[2,1],[0,1])}
atlas> 
atlas> induce_irreducible (sigma,P,G)
Value: 
1*final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1)
1*final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1)
1*final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas> {Ind_P^G(discrete series) = as a sum of irreducible modules for G}
atlas> 
atlas> p
Value: final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas> 
atlas> 
atlas> 
atlas> { p -> I(p)  standard module   and J(p) irreducible }
atlas> set irr=J(p)
Identifier irr: (Param,string) (hiding previous one of type string (constant))
atlas> show(irr)
J(x=5,lambda=[2/1,1/1],nu=[0/1,1/1])
atlas> 
atlas> 
atlas> set ind=I(p)
Identifier ind: (Param,string)
atlas> 
atlas> show(ind)
I(x=5,lambda=[2/1,1/1],nu=[0/1,1/1])
atlas> 
atlas> show(composition_series (ind))
1*J(x=0,lambda=[2/1,1/1],nu=[0/1,0/1])
1*J(x=2,lambda=[2/1,1/1],nu=[0/1,0/1])
1*J(x=5,lambda=[2/1,1/1],nu=[0/1,1/1])
atlas>  
atlas> 
atlas> show(character_formula (irr))
-1*I(x=0,lambda=[2/1,1/1],nu=[0/1,0/1])
-1*I(x=2,lambda=[2/1,1/1],nu=[0/1,0/1])
1*I(x=5,lambda=[2/1,1/1],nu=[0/1,1/1])
atlas> 
atlas> 
atlas> character_formula (std)
Error in expression character_formula(std) at <standard input>:104:0-23
  Failed to match 'character_formula' with argument type string
Type check failed
atlas> composition_series (irr)
Runtime error:
  Composition series not defined for irreducible modules
(in call of error, built-in)
(in call of assert@(bool,string), defined at /home/jda/atlasSoftware/atlasofliegroups/atlas-scripts/basic.at:7:4-74)
(in call of composition_series@(Param,string), defined at /home/jda/atlasSoftware/atlasofliegroups/atlas-scripts/modules.at:50:4--52:29)
Evaluation aborted.
atlas> print_block(trivial(G))
Parameter defines element 10 of the following block:
 0:  0  [i1,i1]   1   2   ( 6, *)  ( 4, *)  *(x= 0,lam=rho+  [0,0], nu=  [0,0]/1)  e
 1:  0  [i1,i1]   0   3   ( 6, *)  ( 5, *)  *(x= 1,lam=rho+  [0,0], nu=  [0,0]/1)  e
 2:  0  [ic,i1]   2   0   ( *, *)  ( 4, *)  *(x= 2,lam=rho+  [0,0], nu=  [0,0]/1)  e
 3:  0  [ic,i1]   3   1   ( *, *)  ( 5, *)  *(x= 3,lam=rho+  [0,0], nu=  [0,0]/1)  e
 4:  1  [C+,r1]   7   4   ( *, *)  ( 0, 2)  *(x= 5,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 5:  1  [C+,r1]   8   5   ( *, *)  ( 1, 3)  *(x= 6,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 6:  1  [r1,C+]   6   9   ( 0, 1)  ( *, *)  *(x= 4,lam=rho+  [0,0], nu= [1,-1]/2)  1^e
 7:  2  [C-,i1]   4   8   ( *, *)  (10, *)  *(x= 7,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 8:  2  [C-,i1]   5   7   ( *, *)  (10, *)  *(x= 8,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 9:  2  [i2,C-]   9   6   (10,11)  ( *, *)  *(x= 9,lam=rho+  [0,0], nu=  [3,3]/2)  2x1^e
10:  3  [r2,r1]  11  10   ( 9, *)  ( 7, 8)  *(x=10,lam=rho+  [0,0], nu=  [2,1]/1)  1^2x1^e
11:  3  [r2,rn]  10  11   ( 9, *)  ( *, *)  *(x=10,lam=rho+[-1,-1], nu=  [2,1]/1)  1^2x1^e
atlas> {representations 4,5,7,8 all from same Cartan/Levi=SL(2,R)xGL(1,R)}
atlas> 
atlas> set p4=B[4]
Identifier p4: Param
atlas> set p5=B[5]
Identifier p5: Param
atlas> set p7=B[7]
Identifier p7: Param
atlas> set p8=B[8]
Identifier p8: Param
atlas> set (P4,sigma4)=cuspidal_data (p4)
Identifiers P4: ([int],KGBElt), sigma4: Param
atlas> Levi(P4)
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas> sigma4
Value: final parameter (x=0,lambda=[1,2]/1,nu=[1,0]/1)
atlas> set (P5,sigma5)=cuspidal_data (p5)
Identifiers P5: ([int],KGBElt), sigma5: Param
atlas> set (P7,sigma7)=cuspidal_data (p7)
Identifiers P7: ([int],KGBElt), sigma7: Param
atlas> set (P8,sigma8)=cuspidal_data (p8)
Identifiers P8: ([int],KGBElt), sigma8: Param
atlas> sigma4
Value: final parameter (x=0,lambda=[1,2]/1,nu=[1,0]/1)
atlas> sigma5
Value: final parameter (x=1,lambda=[1,2]/1,nu=[1,0]/1)
atlas> sigma7
Value: final parameter (x=0,lambda=[0,1]/1,nu=[2,0]/1)
atlas> sigma8
Value: final parameter (x=1,lambda=[0,1]/1,nu=[2,0]/1)
atlas> hc_parameter (sigma4, x(sigma4))
Value: [ 1, 2 ]/1
atlas> 
atlas> {sigma4:  1 on GL(1) x [2] on SL(2,R)}
atlas> {sigma5:  1 on GL(1) x [-2] on SL(2,R)}
atlas> sigma7
Value: final parameter (x=0,lambda=[0,1]/1,nu=[2,0]/1)
atlas> sigma8
Value: final parameter (x=1,lambda=[0,1]/1,nu=[2,0]/1)
atlas> {sigma7:  2 on GL(1) x [1] on SL(2,R)}
atlas> {sigma8:  2 on GL(1) x [-1] on SL(2,R)}
atlas> {note: on GL(1)   |x|^1 or 2, sgn on \pm 1 =?}
atlas> 
atlas> 
atlas> print_block(trivial(G))
Parameter defines element 10 of the following block:
 0:  0  [i1,i1]   1   2   ( 6, *)  ( 4, *)  *(x= 0,lam=rho+  [0,0], nu=  [0,0]/1)  e
 1:  0  [i1,i1]   0   3   ( 6, *)  ( 5, *)  *(x= 1,lam=rho+  [0,0], nu=  [0,0]/1)  e
 2:  0  [ic,i1]   2   0   ( *, *)  ( 4, *)  *(x= 2,lam=rho+  [0,0], nu=  [0,0]/1)  e
 3:  0  [ic,i1]   3   1   ( *, *)  ( 5, *)  *(x= 3,lam=rho+  [0,0], nu=  [0,0]/1)  e
 4:  1  [C+,r1]   7   4   ( *, *)  ( 0, 2)  *(x= 5,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 5:  1  [C+,r1]   8   5   ( *, *)  ( 1, 3)  *(x= 6,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 6:  1  [r1,C+]   6   9   ( 0, 1)  ( *, *)  *(x= 4,lam=rho+  [0,0], nu= [1,-1]/2)  1^e
 7:  2  [C-,i1]   4   8   ( *, *)  (10, *)  *(x= 7,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 8:  2  [C-,i1]   5   7   ( *, *)  (10, *)  *(x= 8,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 9:  2  [i2,C-]   9   6   (10,11)  ( *, *)  *(x= 9,lam=rho+  [0,0], nu=  [3,3]/2)  2x1^e
10:  3  [r2,r1]  11  10   ( 9, *)  ( 7, 8)  *(x=10,lam=rho+  [0,0], nu=  [2,1]/1)  1^2x1^e
11:  3  [r2,rn]  10  11   ( 9, *)  ( *, *)  *(x=10,lam=rho+[-1,-1], nu=  [2,1]/1)  1^2x1^e
atlas> set p6=B[6]
Identifier p6: Param
atlas> set (P6,sigma6)=cuspidal_data (p6)
Identifiers P6: ([int],KGBElt), sigma6: Param
atlas> Levi(P6)
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas> set L6=Levi(P6)
Identifier L6: RealForm
atlas> simple_roots (L6)
Value: 
|  1 |
| -1 |

atlas> {M=GL(2,R)}
atlas> 
atlas> print_KGB (L6)
kgbsize: 2
Base grading: [1].
0:  0  [n]   0    1  (0,0)#0 e
1:  1  [r]   1    *  (0,0)#1 1^e
atlas> sigma6
Value: final parameter (x=0,lambda=[3,-3]/2,nu=[1,1]/2)
atlas> set p9=B[9]
Identifier p9: Param
atlas> set (P9,sigma9)=cuspidal_data (p9)
Identifiers P9: ([int],KGBElt), sigma9: Param
atlas> sigma6
Value: final parameter (x=0,lambda=[3,-3]/2,nu=[1,1]/2)
atlas> sigma9
Value: final parameter (x=0,lambda=[1,-1]/2,nu=[3,3]/2)
atlas> rho(L6)
Value: [  1, -1 ]/2
atlas> {block: 4 DS, 4 reps from SL(2,R)xGL(1,R),  2 from GL(2,R), 2 from GL(1,R)^2}
atlas> print_block(trivial(G))
Parameter defines element 10 of the following block:
 0:  0  [i1,i1]   1   2   ( 6, *)  ( 4, *)  *(x= 0,lam=rho+  [0,0], nu=  [0,0]/1)  e
 1:  0  [i1,i1]   0   3   ( 6, *)  ( 5, *)  *(x= 1,lam=rho+  [0,0], nu=  [0,0]/1)  e
 2:  0  [ic,i1]   2   0   ( *, *)  ( 4, *)  *(x= 2,lam=rho+  [0,0], nu=  [0,0]/1)  e
 3:  0  [ic,i1]   3   1   ( *, *)  ( 5, *)  *(x= 3,lam=rho+  [0,0], nu=  [0,0]/1)  e
 4:  1  [C+,r1]   7   4   ( *, *)  ( 0, 2)  *(x= 5,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 5:  1  [C+,r1]   8   5   ( *, *)  ( 1, 3)  *(x= 6,lam=rho+  [0,0], nu=  [0,1]/1)  2^e
 6:  1  [r1,C+]   6   9   ( 0, 1)  ( *, *)  *(x= 4,lam=rho+  [0,0], nu= [1,-1]/2)  1^e
 7:  2  [C-,i1]   4   8   ( *, *)  (10, *)  *(x= 7,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 8:  2  [C-,i1]   5   7   ( *, *)  (10, *)  *(x= 8,lam=rho+  [0,0], nu=  [2,0]/1)  1x2^e
 9:  2  [i2,C-]   9   6   (10,11)  ( *, *)  *(x= 9,lam=rho+  [0,0], nu=  [3,3]/2)  2x1^e
10:  3  [r2,r1]  11  10   ( 9, *)  ( 7, 8)  *(x=10,lam=rho+  [0,0], nu=  [2,1]/1)  1^2x1^e
11:  3  [r2,rn]  10  11   ( 9, *)  ( *, *)  *(x=10,lam=rho+[-1,-1], nu=  [2,1]/1)  1^2x1^e
atlas> 
atlas> 
atlas> 
atlas> {reference: algorithms paper}
atlas> 
atlas> {number: length/ types of simple roots/ cross-action/ Cayley-transforms/parameter/ Weyl elt}
atlas> 
atlas> 
atlas> {root: imaginary/real/complex  i/r/C with respect to theta_x}
atlas> {imaginary: theta_x(alpha)=alpha}
atlas> {real: theta_x(alpha)=-alpha}
atlas> {complex: neither}
atlas> 
atlas> {imaginary root: i -> i1 or i2: noncompact or ic compact}
atlas> 
atlas> 
atlas> 
atlas> {  2^w= s_2*w}
atlas> {  2xw = s_2*w*s_2}
atlas> 
atlas> 
atlas> {1^2x1^e =  s_1*s_2* s_1*s^2 =long elt of W = -I}
atlas> 
atlas> 
atlas> {mathematical background: Vogan's big green book}
atlas> 
atlas> 
atlas> 
atlas> {5 minute break - 3:10}
atlas> 
atlas> 
atlas> {any questions? post them to the questions column...}
atlas>