These are the unitary reps of Sp(8,R) of infinitesimal character
[4,2,2,0].

U4220:

atlas> for p in U4220 do prints(p) od

final parameter(x=33,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) 
final parameter(x=32,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1)
final parameter(x=30,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1)
final parameter(x=28,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1)
final parameter(x=12,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=10,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=9,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=7,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=5,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=4,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=2,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=0,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)


cohom ind from unitary on Sp(6), parabolic ([1,2,3],KGB(G,0):

atlas> for p in u0 do prints(theta_induce_irreducible(p,G)) od

1*parameter(x=30,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=28,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=7,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=4,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=2,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=0,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]


cohom ind from unitary on Sp(6), parabolic ([1,2,3],KGB(G,5):

atlas> for p in u5 do prints(theta_induce_irreducible(p,G)) od

1*parameter(x=32,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=33,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=5,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=9,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=10,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
1*parameter(x=12,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]


Remove cohom ind ones from list of all unitary: lists of cohom ind
are DISJOINT as theta-stable Q varies, and cover U4220.


TWO ORBITS of W(L) on fundamental fiber, represented by KGBElts 0 and
5. Orbit of 0: 0, 2, 3, 4, 7...

Now the atlas code that made these lists:

atlas> set G=Sp(8,R)
Variable G: RealForm (overriding previous instance, which had type RealForm)

atlas> set P=all_parameters_gamma(G,[4,2,2,0])
Variable P: [Param]
atlas> #P
Value: 159

{here's code to pick the unitary ones out of a list of parameters}

atlas> set U([Param] P) = ## for p in P do if is_unitary(p) then [p] else [] fi od
Added definition [2] of U: ([Param]->[Param])
atlas> set U4220 = U(P)
Variable U4220: [Param]
atlas> #U4220
Value: 12

atlas> set Q0=([1,2,3],KGB(G,0))
Variable Q0: ([int],KGBElt) (overriding previous instance, which had type ([int],KGBElt))
atlas> set L0=Q0.Levi
Variable L0: RealForm (overriding previous instance, which had type RealForm)
atlas> L0
Value: connected quasisplit real group with Lie algebra 'sp(6,R).u(1)'
atlas> rho_u(Q0)  
Value: [ 4, 0, 0, 0 ]/1

atlas> set p0=all_parameters_gamma(L0,[4,2,2,0]-[4,0,0,0])
Variable p0: [Param] (overriding previous instance, which had type [Param])
atlas> set u0=U(p0)
Variable u0: [Param] (overriding previous instance, which had type [Param])
atlas> #u0
Value: 6

atlas> simple_coroots (G)        
Value: 
|  1,  0,  0, 0 |
| -1,  1,  0, 0 |
|  0, -1,  1, 0 |
|  0,  0, -1, 1 |

{0th column [1,-1,0,0] takes value 2 on lambda. Conjecture suggests
{ all reps in U4220 should be good range cohom ind from an
{omit-root-zero parabolic}
{Need such parabolics:}

atlas> set Q0=([1,2,3],KGB(G,0))

{[1,2,3] is all of the simple roots but "alpha"; 0 puts us in
{ fundamental fiber, makes  Q0 theta-stable}

Variable Q0: ([int],KGBElt) (overriding previous instance, which had type ([int],KGBElt))

atlas> set Q5=([1,2,3],KGB(G,5))
Variable Q5: ([int],KGBElt) (overriding previous instance, which had type ([int],KGBElt))
atlas> set L0=Q0.Levi
Variable L0: RealForm (overriding previous instance, which had type
RealForm)

atlas> set L5=Q5.Levi
Variable L5: RealForm (overriding previous instance, which had type RealForm)

atlas> L0
Value: connected quasisplit real group with Lie algebra 'sp(6,R).u(1)'

{for cohom ind, get rho-shift in infl char by rho_u(Q)}

atlas> set p0=all_parameters_gamma(L0,lambda-rho_u(Q0))
Variable p0: [Param] (overriding previous instance, which had type [Param])
atlas> #p0
Value: 31

atlas> {pick out unitary}

atlas> set u0=U(p0)
Variable u0: [Param] (overriding previous instance, which had type [Param])
atlas> #u0
Value: 6

atlas> set p5=all_parameters_gamma(L0,lambda-rho_u(Q5))
Variable p5: [Param] (overriding previous instance, which had type [Param])

atlas> set u5=U(p5)
Variable u5: [Param] (overriding previous instance, which had type [Param])
atlas> #u5
Value: 6

atlas> for p in u0 do prints(theta_induce_irreducible(p,G)) od

1*parameter(x=30,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=28,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=7,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=4,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=2,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=0,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
Value: [(),(),(),(),(),()]

atlas> for p in u5 do prints(theta_induce_irreducible(p,G)) od

1*parameter(x=30,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=28,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=7,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=4,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=2,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=0,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]
Value: [(),(),(),(),(),()]

atlas> for p in U4220 do prints(p) od
final parameter(x=33,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1)
final parameter(x=32,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1)
final parameter(x=30,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1)
final parameter(x=28,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1)
final parameter(x=12,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=10,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=9,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=7,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=5,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=4,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=2,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)
final parameter(x=0,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1)

atlas> Q5
Value: ([1,2,3],KGB element #5)
atlas> Q0
Value: ([1,2,3],KGB element #0)
atlas> set p5=all_parameters_gamma(L5,lambda-rho_u(Q5))
Variable p5: [Param] (overriding previous instance, which had type [Param])
atlas> set u5=U(p5)
Variable u5: [Param] (overriding previous instance, which had type [Param])
atlas> for p in u5 do prints(theta_induce_irreducible(p,G)) od

1*parameter(x=32,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=33,lambda=[4,2,2,1]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=5,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=9,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=10,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

1*parameter(x=12,lambda=[4,2,2,0]/1,nu=[0,0,0,0]/1) [44]

{Jeff suggested an easy way to get a list of all the theta-stable
parabolics with simple root set {1,2,3}:}

atlas> ## for P in theta_stable_parabolics(G) do let (S,)=P in if S=[1,2,3] then [P] else [] fi od
Value: [([1,2,3],KGB element #149),([1,2,3],KGB element #150)]

{This is our Q0 and Q5, except represented by their _longest_ KGB
elements rather than choices of shortest}

{Now try the conjecture on a different infinitesimal character:}

atlas> set lambda1=[4,2,1,1,0]
Variable lambda1: [int]
atlas> set lambda1=[4,2,1,1]  
Variable lambda1: [int] (overriding previous instance, which had type [int])
atlas> set P=all_parameters_gamma(G,lambda1)
Variable P: [Param] (overriding previous instance, which had type [Param])

{Here are the unitary reps of G at infinitesimal character [4,2,2,1]}

atlas> set U4211 = U(P)
Variable U4211: [Param]
atlas> #U4211
Value: 26


atlas> set p5=all_parameters_gamma(L5,lambda1-rho_u(Q5))
Variable p5: [Param] (overriding previous instance, which had type [Param])
atlas> set p0=all_parameters_gamma(L0,lambda1-rho_u(Q5))
Variable p0: [Param] (overriding previous instance, which had type [Param])
atlas> set p0=all_parameters_gamma(L0,lambda1-rho_u(Q0))
Variable p0: [Param] (overriding previous instance, which had type [Param])
atlas> set u5=U(p5)
Variable u5: [Param] (overriding previous instance, which had type [Param])
atlas> #u5
Value: 13

atlas> set u0=U(p0)
Variable u0: [Param] (overriding previous instance, which had type [Param])
atlas> #u0
Value: 13

atlas> U4211[0]
Value: final parameter(x=101,lambda=[4,4,1,1]/1,nu=[0,2,0,1]/1)
atlas> set p=$
Variable p: Param

{the next command lists all the theta-stable parabolics from which p
is good-range-cohomologically-induced}

atlas> set list = is_good_range_induced_from (p)
Variable list: [Param]
atlas> list
Value: [final parameter(x=35,lambda=[0,4,1,1]/1,nu=[0,2,0,1]/1),
        final parameter(x=101,lambda=[4,4,1,1]/1,nu=[0,2,0,1]/1)]

atlas> list[0].real_form
Value: connected quasisplit real group with Lie algebra 'sp(6,R).u(1)'
atlas> list[1].real_form
Value: connected split real group with Lie algebra 'sp(8,R)'

{any p is good-range-induced from G}

atlas> #P
Value: 230

{DON'T claim that nonunitary have to be cohom induced}

atlas> #p0
Value: 44
atlas> #p5
Value: 44

atlas> #P          

{says 88 of 230 G reps are cohom ind in good range from Sp(6); 26 unitary}