Principal Series and Discrete Series revisited

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 |

So the Cartan involution acts by -1 which means the Cartan subgroup is split. This is a minimal spherical principal series with infinitesimal character rho.

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 also talked 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>

To find more representations we 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.

But, recall there is another representation with infinitesimal character rho which is not in the trivial block

atlas> set params=all_parameters_gamma (G, rho(G))
Variable params: [Param] (overriding previous instance, which had type [Param])
atlas> void: for p in params do prints(p) od
final parameter(x=2,lambda=[1]/1,nu=[1]/1)
final parameter(x=2,lambda=[2]/1,nu=[1]/1)
final parameter(x=1,lambda=[1]/1,nu=[0]/1)
final parameter(x=0,lambda=[1]/1,nu=[0]/1)
atlas>

And recall that the second representation in this list corresponds to the irreducible non-spherical principal series of \(SL(2,R)\)

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

So, this representation is a singleton block.

In any case, other than principal series or discrete series, there is nothing else for this group at fixed infinitesimal character.

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, namely number 0; and the others are 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>
atlas> hc_parameter(q)
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>

So there is only one \(KGB\) element to use for the compact Cartan subgroup and this means we only have one discrete series.

Equivalently, note that the simple reflection \(s_\alpha\) is in the Weyl group of \(K\), which is disconnected in this case. So \(s_\alpha\) flips the positive and negative \(K\) types.

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>

This says that we have three representations for \(PGL(2,R)\) at infinitesimal character rho and we have one extra at a different infinitesimal character.

More on the block_sizes command later.