Cuspidal Data for RepresentationsΒΆ

Given a parameter (x, lambda, nu) we can obtain information about the cuspidal data used to construct the representation. Let us review the parameters of all the representations of \(G=SL(2,\mathbb R)\) with infinitesimal character rho

atlas> set G=SL(2,R)
Variable G: RealForm
atlas> G
Value: connected split real group with Lie algebra 'sl(2,R)'
atlas> rho(G)
Value: [ 1 ]/1
atlas> set parameters=all_parameters_gamma(G,rho(G))
Variable parameters: [Param]
atlas> void: for p in parameters do prints(p) od
final parameter (x=0,lambda=[1]/1,nu=[0]/1)
final parameter (x=1,lambda=[1]/1,nu=[0]/1)
final parameter (x=2,lambda=[1]/1,nu=[1]/1)
final parameter (x=2,lambda=[2]/1,nu=[1]/1)
atlas>
atlas> set t=trivial(G)
Variable t: Param
atlas> t
Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas>

Now, let us find the cuspidal data for``t``

atlas> set (P,q)=cuspidal_data(t)
Variable P: ([int],KGBElt)
Variable q: Param
atlas> q
Value: final parameter (x=0,lambda=[0]/1,nu=[1]/1)
atlas> Levi(P)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas> real_form(q)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas> induce_irreducible (q,P,G)
Value:
1*final parameter (x=0,lambda=[1]/1,nu=[0]/1)
1*final parameter (x=1,lambda=[1]/1,nu=[0]/1)
1*final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas>

Recall that the Cartan subgroup for this parameter is the split Cartan subgroup:

atlas> set x=x(t)
Variable x: KGBElt
atlas> x
Value: KGB element #2
atlas> set H=Cartan_class(x)
Variable H: CartanClass (overriding previous instance, which had type string (constant))
atlas> H
Value: Cartan class #1, occurring for 1 real form and for 2 dual real forms
atlas> print_Cartan_info(H)
compact: 0, complex: 0, split: 1
canonical twisted involution: 1
twisted involution orbit size: 1; fiber size: 1; strong inv: 1
imaginary root system: empty
real root system: A1
complex factor: empty

So, we can extract the character of the Cartan subgroup by finding the Cuspidal data for the representation with parameter t.

The standard representation containing the trivial is induced from a parabolic subgroup P with Levi factor equal to \(GL(1,R)\) and a character q of \(GL(1,R)\) with lambda=0 and nu=1.

Moreover, we can see above that when we induce we obtain the composition series of the spherical principal series that contains the trivial representation and the two discrete series:

atlas> t
Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas>

Similarly we can do the same for the non-spherical principal series:

atlas> set p=parameters[3]
Variable p: Param
atlas> p
Value: final parameter (x=2,lambda=[2]/1,nu=[1]/1)
atlas> set (P,q)=cuspidal_data(p)
Variable P: ([int],KGBElt) (overriding previous instance, which had type ([int],KGBElt))
Variable q: Param (overriding previous instance, which had type Param)
atlas> real_form(q)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas> q
Value: final parameter (x=0,lambda=[1]/1,nu=[1]/1)
atlas> induce_irreducible (q,P,G)
Value:
1*final parameter (x=2,lambda=[2]/1,nu=[1]/1)
atlas>
atlas> p
Value: final parameter (x=2,lambda=[2]/1,nu=[1]/1)
atlas>

So, we get the irreducible, non-spherical principal series by inducing the character on \(GL(1,R)\) with lambda and nu both equal to 1 and from the same parabolic subgroup as in the previous case.

Similarly, just to look at another example with non-integral infinitesimal character:

atlas> set u=parameter(x, [2], [3/2])
Variable u: Param
atlas> u
Value: final parameter (x=2,lambda=[2]/1,nu=[3]/2)
atlas>
atlas> set (P,q)=cuspidal_data(u)
Variable P: ([int],KGBElt) (overriding previous instance, which had type ([int],KGBElt))
Variable q: Param (overriding previous instance, which had type Param)
atlas> q
Value: final parameter (x=0,lambda=[1]/1,nu=[3]/2)
atlas> Levi(P)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas> induce_irreducible(q,P,G)
Value:
1*final parameter (x=2,lambda=[2]/1,nu=[3]/2)
atlas> u
Value: final parameter (x=2,lambda=[2]/1,nu=[3]/2)
atlas>

So the induced representation is also irreducible as was expected.