# Composition Series and Character formulasΒΆ

Let us review some basic examples

atlas> set G=SL(2,R)
Variable G: RealForm
atlas> p=trivial(G)
atlas> set p=trivial(G)
Variable p: Param
atlas> p
Value: final parameter(x=2,lambda=[1]/1,nu=[1]/1)
atlas> set x=x(p)
Variable x: KGBElt
atlas> involution (x)
Value:
| -1 |

atlas> infinitesimal_character (p)
Value: [ 1 ]/1
atlas> rho(G)
Value: [ 1 ]/1
atlas>


This is the minimal principal series containing the trivial representation as unique irreducible quotient.

On the other end we can talk about the discrete series

atlas> whattype discrete_series ?
(KGBElt,ratvec)->Param
(RealForm,ratvec)->Param
atlas> set q=discrete_series (KGB (G,0), rho(G))
Variable q: Param
atlas> q
Value: final parameter(x=0,lambda=[1]/1,nu=[0]/1)
atlas>


We can look at the block of the trivial representation to find other representations of this group

atlas> print_block (p)
Parameter defines element 2 of the following block:
0:  0  [i1]  1   (2,*)  *(x=0,lam_rho= [0], nu= [0]/1)  e
1:  0  [i1]  0   (2,*)  *(x=1,lam_rho= [0], nu= [0]/1)  e
2:  1  [r1]  2   (0,1)  *(x=2,lam_rho= [0], nu= [1]/1)  1^e
atlas>


Here the trivial representation is #2 and the other two are discrete series

atlas> set r=discrete_series (KGB(G,1), rho(G))
Variable r: Param
atlas> r
Value: final parameter(x=1,lambda=[1]/1,nu=[0]/1)
atlas> set x_b=KGB(G,0)
Variable x_b: KGBElt
atlas> hc_parameter (q,x_b)
Value: [ 1 ]/1
atlas>
atlas> hc_parameter (r,x_b)
Value: [ -1 ]/1
atlas>


So, the Harish-Chandra parameter of q is 1 and that of r is -1; the holomorphic and antiholomorphic one respectively.

Now let us look at another group

atlas> G:=PGL(2,R)
Value: disconnected split real group with Lie algebra 'sl(2,R)'
atlas> set p=trivial(G)
Variable p: Param (overriding previous instance, which had type Param)
atlas> print_block (p)
Parameter defines element 1 of the following block:
0:  0  [i2]  0   (1,2)  *(x=0,lam_rho= [0], nu= [0]/1)  e
1:  1  [r2]  2   (0,*)  *(x=1,lam_rho= [0], nu= [1]/2)  1^e
2:  1  [r2]  1   (0,*)  *(x=1,lam_rho= [1], nu= [1]/2)  1^e


In this case we only have one discrete series and two minimal principal series

atlas> set q=discrete_series (KGB(G,0), rho(G))
Variable q: Param (overriding previous instance, which had type Param)
atlas> q
Value: final parameter(x=0,lambda=[1]/2,nu=[0]/1)
atlas>  rho(G)
Value: [ 1 ]/2
atlas>


Note that $$\rho=1/2$$ in this case. So $$X^* +\rho \cong \mathbb Z +1/2$$

Also there are only two KGB elements in this group

atlas> print_KGB(G)
kgbsize: 2
0:  0  [n]   0    1  (0)#0 e
1:  1  [r]   1    *  (0)#1 1^e
atlas>


Equivalently, note that the simple reflection $$s_\alpha$$ is in the Weyl group of $$K$$, which is disconnected in this case.

On the other hand, we have two principal series in this block associated to the KGB element x=1. They both have infinitesimal character rho. But they differ in the disconnectedness of $$G$$.

Now to know about more representations we look at other blocks

atlas> block_sizes (G)
Value:
| 0, 1 |
| 1, 3 |

atlas>


So we have three representations at infinitesimal character rho and we have one extra at different infinitesimal character.