Example :math:`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 :math:`G(\mathbb C )=PSL(2,\mathbb C )=SL(2,\mathbb C)/{\pm 1}`. Its real points are disconnected, and they are the group :math:`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 :math:`\rho\ `. The parameters for this group are almost like those for :math:`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 :math:`\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 :math:`SL(2,\mathbb R)` are now a single irreducible representation of :math:`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 :math:`PSL(2,\mathbb R )` has two factors: the trivial representation (with ``x=1`` and :math:`\lambda=\nu=\rho` ) and a discrete series (with ``x=0``). What is the other principal series attached to the split Cartan subgroup? For :math:`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 :math:`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 :math:`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.