Discrete Series

Recall that there is a map \(\rho :\mathcal X\rightarrow {\mathcal I}_W\) (involutions in \(W\)). And the conjugacy classes of involutions in W give a map:

\({\mathcal I}_W /W\leftrightarrow \text{conjugacy classes of Cartans in quasisplit group.}\)

Now let us fix \(x_b\) and define the set

\[\mathcal F := {\rho }^{-1}(Id)=\{x\in \mathcal X |x\in H \}\]

This is the distinguished fiber above the identity element in the Weyl group or the identity involution in \({\mathcal I}_W\) this just means that the elements in this preimage are in the Cartan \(H\).

So, this \(\mathcal F\) parametrizes the Borel subgroups containing a compact Cartan up to conjugation by \(K\). And these in turn parametrize the discrete series with fixed infinitesimal character.

Explicitly, if we fix infinitesimal character \(\rho\), \(x=wx_b\), corresponds to the discrete series with Harish Chandra parameter \(w\rho\).

So when talking about representations associated to a non split Cartan, the element \(x\) not only gives you the Cartan but also a \(K\)-conjugacy class of Borels for that Cartan.

Now we can focus on the case when \(\theta _x\) is acting by \(Id\) which corresponds to the discrete series representations.

In other words, assuming that \(G=G(\mathbb C)\) has discrete series representations is equivalent to having a distinguished involution equal to the Identity.

Example \(SL(2,R)\)

Recall the \(K\backslash G/B\) elements of \(SL(2,R)\):

atlas> set G=SL(2,R)
Variable G: RealForm
atlas> G
Value: connected split real group with Lie algebra 'sl(2,R)'
atlas> print_KGB (G)
kgbsize: 3
Base grading: [1].
0:  0  [n]   1    2  (0)#0 e
1:  0  [n]   0    2  (1)#0 e
2:  1  [r]   2    *  (0)#1 1^e

If we again look at the block of the trivial

atlas> set B=block_of (trivial (G))
Variable B: [Param]
atlas> set show([Param] params)= void: for p in params do prints(p) od
Added definition [6] of show: ([Param]->)
atlas> show(B)
final parameter (x=0,lambda=[1]/1,nu=[0]/1)
final parameter (x=1,lambda=[1]/1,nu=[0]/1)
final parameter (x=2,lambda=[1]/1,nu=[1]/1)

We focus on the first two elements:

atlas> B[0]
Value: final parameter (x=0,lambda=[1]/1,nu=[0]/1)
atlas> B[1]
Value: final parameter (x=1,lambda=[1]/1,nu=[0]/1)

Recall that these are the (discrete series) representations associated to the compact Cartan. Note that they both have parameter lambda=rho. This is because the software is using a different x. In order to understand which representation is which, we need to conjugate x=1 to x=0. This will conjugate lambda to -lambda. Remember that we have to fix a KGB element x_b to fix a real group \(K\). Let us fix it to be x=0:

atlas> set x_b=KGB(G,0)
Variable x_b: KGBElt
atlas> x_b
Value: KGB element #0

Then the harish chandra parameters of the discrete series with respect to the fixed element x=0 will be:

atlas> hc_parameter(B[0],x_b)
Value: [ 1 ]/1
atlas> hc_parameter(B[1],x_b)
Value: [ -1 ]/1

But by choosing x_b =1 we get the opposite situation for the Harish Chandra parameters.

Example \(Sp(4,R)\)

There are a couple of commands that will give you discrete series:

atlas> whattype discrete_series ?
Overloaded instances of 'discrete_series'
atlas> whattype all_discrete_series_gamma ?
No overloads for 'all_discrete_series_gamma'