Tutorial

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 ?
Overloaded instances of '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 ?
Overloaded instances of '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.