Cuspidal Data for Representations

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>

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
atlas>

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

For more on induction, see the Section Parabolic Induction in this tutorial.

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.

We can 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.