# Example $$G=PSL(2,\mathbb R)$$¶

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

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 subgroup? For $$SL(2,\mathbb R )$$ the other representation attached to the split Cartan subgroup 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.