Example \(G=PSL(2,\mathbb R)\)ΒΆ

Another group we can look at is:

atlas> G:PSL(2,R)
Variable G: RealForm (overriding previous instance, which had type RealForm)
atlas> G
Value: disconnected split real group with Lie algebra 'sl(2,R)'
atlas>

Here the complex Lie group is \(G(\mathbb C )=PSL(2,\mathbb C )=SL(2,\mathbb C)/{\pm 1}\). Its real points are disconnected, and they are the group \(PSL(2, \mathbb R ) \cong SO(2,1)\):

atlas> rho(G)
Value: [ 1 ]/2
atlas> set parameters=all_parameters_gamma (G,rho(G))
Variable parameters: [Param] (overriding previous instance, which had type [Param])
atlas>

Note we can use rho(G) instead of the vector value for \(\rho\ \). The parameters for this group are almost like those for \(SL(2,\mathbb R)\), except that the Weyl group of the compact Cartan has changed and the number of parameters is now just three:

atlas> #parameters
Value: 3
atlas> void: for p in parameters do prints(p) od
final parameter (x=0,lambda=[1]/2,nu=[0]/1)
final parameter (x=1,lambda=[1]/2,nu=[1]/2)
final parameter (x=1,lambda=[3]/2,nu=[1]/2)
atlas>

We still have two principal series with infinitesimal character \(\rho\). But we now only have one discrete series representation associated to the compact Cartan, namely the sum of the two discrete series for \(SL(2,\mathbb R)\) are now a single irreducible representation of \(PSL(2, \mathbb R )\).

Now let us look at the trivial representation

atlas> p:trivial(G)
Variable p: Param
atlas> p
atlas> dimension (p)
Value: 1
atlas>
One thing to have in mind is that the trivial representation is
always given by the maximal number x and lambda=nu=rho

This parameter has composition series:

Value: final parameter (x=1,lambda=[1]/2,nu=[1]/2)
atlas> composition_series(I(p))
Value: (
1*final parameter (x=0,lambda=[1]/2,nu=[0]/1)
1*final parameter (x=1,lambda=[1]/2,nu=[1]/2),"irr")
atlas>

Actually it is best to use the command show(composition_series(I(p))))

atlas> show(composition_series(I(p)))
1*J(x=0,lambda=[1/2],nu=[0/1])
1*J(x=1,lambda=[1/2],nu=[1/2])
atlas>

So, this induced representation for \(PSL(2,\mathbb R )\) has two factors: the trivial representation (with x=1 and \(\lambda=\nu=\rho\) ) and a discrete series (with x=0).

What is the other principal series attached to the split Cartan? For \(SL(2,\mathbb R )\) the other representation attached to the split Cartan was an infinite demensional irreducible principal series. However here we have:

atlas> q:parameters[2]
Variable q: Param
atlas> q
Value: final parameter (x=1,lambda=[3]/2,nu=[1]/2)
atlas> is_finite_dimensional (q)
Value: true
atlas> p=q
Value: false
atlas>
atlas> p
Value: final parameter (x=1,lambda=[1]/2,nu=[1]/2)
atlas> q
Value: final parameter (x=1,lambda=[3]/2,nu=[1]/2)
atlas>

This is another one dimensional representation of G not equivalent to the trivial representation. Recall that \(PSL (2,\mathbb R )\) is disconnected, so q is the parameter for the sign representation. In other words the standard module attached to this parameter is a principal series which has as its unique irreducible quotient the sign representation of \(PSL (2,\mathbb R )\).

Now for another example:

atlas> set p=parameter(KGB(G,1),[1]/2,[1])
Variable p: Param
atlas> p
Value: final parameter (x=1,lambda=[1]/2,nu=[1]/1)
atlas> show(composition_series (I(p)))
1*J(x=1,lambda=[1/2],nu=[1/1])
atlas>

So, the composition series gives an irreducible. Even though nu is an integer this is not an irreducibility point.