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 ?
Overloaded instances of '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
Base grading: [1].
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.