# More parameter commands¶

## Principal series commands¶

There is another command which we will use here to look at more examples of minimal principal series. Namely, in addition to the command all_minimal_principal_series, the command minimal_principal_series helps us identify a particular representation in the series. Let us compare their use with some examples:

atlas> set G=Sp(4,R)
Variable G: RealForm
atlas> whattype minimal_principal_series ?
(RealForm,ratvec,ratvec)->Param
RealForm->Param


We will use the first syntax above:

atlas> minimal_principal_series(G,rho(G),rho(G))
Value: final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1)
atlas>
atlas> minimal_principal_series(G,rho(G),[0,0])
Value: final parameter (x=10,lambda=[2,1]/1,nu=[0,0]/1)


So we get the single trivial or the representation with nu=0. Now, recall that for the first command, we need to provide a real form and a rational vector:

atlas> whattype all_minimal_principal_series ?
(RealForm,ratvec)->[Param]
atlas>

atlas> set ps= all_minimal_principal_series (G,rho(G))
Variable ps: [Param]
atlas>

atlas> void: for p in ps do prints(p) od
final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[3,1]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[2,2]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[3,2]/1,nu=[2,1]/1)
atlas>


So, in this case we obtain again the four principal series of $$Sp(4,R)$$ at infinitesimal character rho.

The nus all equal rho and the lambdas are all the possible lambdas in $$X^*/2X^*$$.

Note that the group does not have to be semisimple:

atlas> G:=GL(2,R)
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas> set ps= all_minimal_principal_series (G,rho(G))
Variable ps: [Param] (overriding previous instance, which had type [Param])
atlas> void: for p in ps do prints(p) od
final parameter(x=1,lambda=[1,-1]/2,nu=[1,-1]/2)
final parameter(x=1,lambda=[3,-1]/2,nu=[1,-1]/2)
final parameter(x=1,lambda=[1,1]/2,nu=[1,-1]/2)
final parameter(x=1,lambda=[3,1]/2,nu=[1,-1]/2)
atlas>


WARNING: This command does not work for non-split groups:

atlas> G:=U(2,2)
Value: connected quasisplit real group with Lie algebra 'su(2,2).u(1)'
atlas> set ps= all_minimal_principal_series (G,rho(G))
group is not split
(in call at atlas-scripts/basic.at:8:57-71 of error@string, built-in)
[b=false, message="group is not split"]
(in call at atlas-scripts/all_parameters.at:109:4-44 of assert@(bool,string),
defined at atlas-scripts/basic.at:8:4-74)
[G=connected quasisplit real group with Lie algebra 'su(2,2).u(1)', gamma=
[ 3,  1, -1, -3 ]/2]
(in call at <standard input>:5:7-45 of all_minimal_principal_series@(RealForm,
ratvec), defined at atlas-scripts/all_parameters.at:108:4--110:63)
Command 'set ps' interrupted, nothing defined.
atlas>


## all_parameters_gamma¶

For this group we need to use the command that lists all representations with a given parameter for $$G$$

atlas> G:=U(2,2)
Value: connected quasisplit real group with Lie algebra 'su(2,2).u(1)'
atlas> set params=all_parameters_gamma (G,rho(G))
Variable params: [Param] (overriding previous instance, which had type [Param])
atlas> #params
Value: 21
atlas>
atlas> void: for p in params do prints(p) od
final parameter(x=20,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/2)
final parameter(x=19,lambda=[3,1,-1,-3]/2,nu=[3,0,0,-3]/2)
final parameter(x=18,lambda=[3,1,-1,-3]/2,nu=[3,0,0,-3]/2)
final parameter(x=17,lambda=[3,1,-1,-3]/2,nu=[1,1,-1,-1]/1)
final parameter(x=16,lambda=[3,1,-1,-3]/2,nu=[1,0,-1,0]/1)
final parameter(x=15,lambda=[3,1,-1,-3]/2,nu=[1,0,-1,0]/1)
final parameter(x=14,lambda=[3,1,-1,-3]/2,nu=[0,1,0,-1]/1)
final parameter(x=13,lambda=[3,1,-1,-3]/2,nu=[0,1,0,-1]/1)
final parameter(x=12,lambda=[3,1,-1,-3]/2,nu=[1,-1,1,-1]/2)
final parameter(x=11,lambda=[3,1,-1,-3]/2,nu=[1,-1,0,0]/2)
final parameter(x=10,lambda=[3,1,-1,-3]/2,nu=[1,-1,0,0]/2)
final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[0,1,-1,0]/2)
final parameter(x=8,lambda=[3,1,-1,-3]/2,nu=[0,1,-1,0]/2)
final parameter(x=7,lambda=[3,1,-1,-3]/2,nu=[0,0,1,-1]/2)
final parameter(x=6,lambda=[3,1,-1,-3]/2,nu=[0,0,1,-1]/2)
final parameter(x=5,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1)
final parameter(x=4,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1)
final parameter(x=3,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1)
final parameter(x=2,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1)
final parameter(x=1,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1)
final parameter(x=0,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1)
atlas>


Recall that all Cartan subgroups of $$U(2,2)$$ are connected. And we can find the information on the Cartan subgroup associated to each parameter as follows

atlas> p:=trivial(G)
Value: final parameter(x=20,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/2)
atlas>
atlas> H:=Cartan_class(p)
Value: Cartan class #2, occurring for 1 real form and for 2 dual real forms
atlas>
atlas> print_Cartan_info (H)
compact: 0, complex: 2, split: 0
canonical twisted involution: 2,1,3,2
twisted involution orbit size: 3; fiber size: 1; strong inv: 3
imaginary root system: empty
real root system: A1.A1
complex factor: A1
atlas>


This is the most split Cartan subgroup in $$U(2,2)$$. It is just two copies of $${\mathbb C}^x$$. So it is connected. In fact this group has three minimal principal series (with x=17 and x=12) not comming from the disconnectedness of the Cartan subgroup but from the Weyl group. We will address this later.

## all_parameters¶

This command helps us find representations with the same differential

atlas> G:=Sp(4,R)
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> set params=all_parameters_gamma (G,rho(G))
Variable params: [Param] (overriding previous instance, which had type [Param])
atlas> void: for p in params do prints(p) od
final parameter(x=10,lambda=[2,1]/1,nu=[2,1]/1)
final parameter(x=10,lambda=[3,1]/1,nu=[2,1]/1)
final parameter(x=10,lambda=[2,2]/1,nu=[2,1]/1)
final parameter(x=10,lambda=[3,2]/1,nu=[2,1]/1)
final parameter(x=9,lambda=[2,1]/1,nu=[3,3]/2)
final parameter(x=8,lambda=[2,1]/1,nu=[2,0]/1)
final parameter(x=8,lambda=[3,1]/1,nu=[2,0]/1)
final parameter(x=7,lambda=[2,1]/1,nu=[2,0]/1)
final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1)
final parameter(x=6,lambda=[2,1]/1,nu=[0,1]/1)
final parameter(x=6,lambda=[2,2]/1,nu=[0,1]/1)
final parameter(x=5,lambda=[2,1]/1,nu=[0,1]/1)
final parameter(x=5,lambda=[2,2]/1,nu=[0,1]/1)
final parameter(x=4,lambda=[2,1]/1,nu=[1,-1]/2)
final parameter(x=3,lambda=[2,1]/1,nu=[0,0]/1)
final parameter(x=2,lambda=[2,1]/1,nu=[0,0]/1)
final parameter(x=1,lambda=[2,1]/1,nu=[0,0]/1)
final parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1)
atlas>
atlas> p:=params[8]
Value: final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1)
atlas> set others=all_parameters (p)
Variable others: [Param] (overriding previous instance, which had type [Param])
atlas> void: for p in others do prints(p) od
final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1)
final parameter(x=7,lambda=[2,1]/1,nu=[2,0]/1)
atlas> void: for q in others do prints(q) od
final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1)
final parameter(x=7,lambda=[2,1]/1,nu=[2,0]/1)
atlas>


This Cartan subgroup has two connected components. So if you hand in a parameter for this subgroup, the total number of parameters with the same differential is two and this command gives the list of all of them.