{I tried to illustrate in atlas some of what Timothy talked about at
the end of his lecture concerning the Stein complementary series.}

atlas> set G=GL(4,C)
Variable G: RealForm

{Timothy wanted to use the (unique) real parabolic subgroup of GL(4,C)
having Levi subgroup GL(2,C) x GL(2,C). Rather than construct it
cleverly, I just asked atlas for a list of ALL real parabolic
subgroups}

atlas> set PP=all_real_parabolics (G)
Variable PP: [KGPElt]

{and then printed the Levi subgroups of each}

atlas> for P@i in PP do prints(i," ",P.Levi) od
0 connected quasisplit real group with Lie algebra 'gl(1,C).gl(1,C).gl(1,C).gl(1,C)'
1 connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C)'
2 connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C)'
3 connected quasisplit real group with Lie algebra 'sl(3,C).gl(1,C).gl(1,C)'
4 connected quasisplit real group with Lie algebra 'sl(2,C).gl(1,C).gl(1,C).gl(1,C)'
5 connected quasisplit real group with Lie algebra 'sl(2,C).sl(2,C).gl(1,C).gl(1,C)'
6 connected quasisplit real group with Lie algebra 'sl(3,C).gl(1,C).gl(1,C)'
7 connected quasisplit real group with Lie algebra 'sl(4,C).gl(1,C)'
Value: [(),(),(),(),(),(),(),()]

{The winner is i=5, so we call that parabolic P and its Levi factor
L. You might wish that you could construct the Langlands decomposition
P=MAN, but the groups M and A are NOT real points of complex algebraic
groups, so Atlas can't talk about them separately.}

atlas> set P=PP[5]
Variable P: KGPElt
atlas> set L = P.Levi
Variable L: RealForm
atlas> L
Value: connected quasisplit real group with Lie algebra
'sl(2,C).sl(2,C).gl(1,C).gl(1,C)'

{Something I didn't do in the seminar but should have is}

atlas> simple_roots(L)
Value: 
|  1,  0,  0,  0 |
| -1,  0,  0,  0 |
|  0,  1,  0,  0 |
|  0, -1,  0,  0 |
|  0,  0,  1,  0 |
|  0,  0, -1,  0 |
|  0,  0,  0,  1 |
|  0,  0,  0, -1 |

{The columns are the simple roots:

[1,-1,0,0,0,0,0,0],  [0,0,1,-1,0,0,0,0]

and minus theta applied to these (same thing in the LAST four coords
instead of the first two). In case you don't believe the "-theta"
claim,}

atlas> (-KGB(G,23).involution)*simple_roots(L)
Value: 
|  0,  0,  0,  1 |
|  0,  0,  0, -1 |
|  0,  0,  1,  0 |
|  0,  0, -1,  0 |
|  0,  1,  0,  0 |
|  0, -1,  0,  0 |
|  1,  0,  0,  0 |
| -1,  0,  0,  0 |

{Normalized real parabolic induction is accomplished by the atlas
function real_induce_irreducible(parameter for L,G).}
atlas> set pL=L.trivial
Variable pL: Param
atlas> pL
Value: final parameter(x=3,lambda=[1,-1,1,-1,1,-1,1,-1]/2,nu=[1,-1,1,-1,1,-1,1,-1]/2)

atlas> set q=real_induce_irreducible(pL,G)
Variable q: ParamPol
atlas> q
Value: 
1*parameter(x=16,lambda=[3,3,-1,-1,1,1,-3,-3]/2,nu=[1,1,-1,-1,1,1,-1,-1]/2) [0]
atlas> is_unitary(first_param (q))
Value: true
atlas> pL
Value: final parameter(x=3,lambda=[1,-1,1,-1,1,-1,1,-1]/2,nu=[1,-1,1,-1,1,-1,1,-1]/2)

atlas> set pL1=parameter(KGB(L,3),[1,-1,1,-1,1,-1,1,-1]/2,[1,-1,1,-1,1,-1,1,-1]/2+[1,1,-1,-1,0,0,0,0])
Variable p1: Param
atlas> set q1=real_induce_irreducible(pL1,G) 
Variable q1: ParamPol

atlas> set r = first_param (q1)
Variable r: Param
atlas> is_unitary(r)
Value: true

atlas> set pL2=parameter(KGB(L,3),[1,-1,1,-1,1,-1,1,-1]/2,[1,-1,1,-1,1,-1,1,-1]/2+[1,1,-1,-1,0,0,0,0]*(3/4))
Variable pL2: Param (overriding previous instance, which had type Param)
atlas> set q2=real_induce_irreducible(pL1,G)
Variable q2: ParamPol
atlas> q2
Value: 
1*parameter(x=23,lambda=[3,1,-1,-3,3,1,-1,-3]/2,nu=[7,1,-1,-7,7,1,-1,-7]/8) [0]
atlas> is_unitary(first_param (q2))
Value: true
atlas> set pL3=parameter(KGB(L,3),[1,-1,1,-1,1,-1,1,-1]/2,[1,-1,1,-1,1,-1,1,-1]/2+[1,1,-1,-1,0,0,0,0]*(3/2))  
Variable pL3: Param
atlas> set q3=real_induce_irreducible(pL3,G)
Variable q3: ParamPol

atlas> is_unitary(first_param (q3))
Value: false
atlas> set pL4=parameter(KGB(L,3),[1,-1,1,-1,1,-1,1,-1]/2,[1,-1,1,-1,1,-1,1,-1]/2+[1,1,-1,-1,0,0,0,0]*2)  
Variable pL4: Param
atlas> set q4=real_induce_irreducible(p3,G)
Variable q4: ParamPol

atlas> is_unitary(first_param (q4))
Value: true
atlas> print_branch_irr(q4,20)
(1+0s)*(KGB element #0,[ -1,  0,  1,  2,  1,  0, -1, -2 ])
(1+0s)*(KGB element #0,[ -1,  0,  1,  1,  2,  0, -1, -2 ])
(1+0s)*(KGB element #0,[ -1,  0,  0,  1,  2,  1, -1, -2 ])
(1+0s)*(KGB element #0,[  0,  0,  1,  1,  2,  0, -1, -3 ])
(1+0s)*(KGB element #0,[  0,  0,  0,  1,  2,  1, -1, -3 ])
(1+0s)*(KGB element #0,[  0,  1,  0,  1,  2,  1, -2, -3 ])
(1+0s)*(KGB element #0,[  0,  0,  1,  0,  3,  0, -1, -3 ])
(1+0s)*(KGB element #0,[  0,  0,  0,  0,  3,  1, -1, -3 ])

atlas> print_branch_irr(first_param(q4),20)
(1+0s)*(KGB element #0,[ -1,  0,  1,  2,  1,  0, -1, -2 ])

atlas> infinitesimal_character (q4)
Value: [  3,  1, -1, -3,  3,  1, -1, -3 ]/2