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
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>
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
atlas>
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)
atlas>
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)
atlas>
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
atlas>
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
atlas>
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'
(KGBElt,ratvec)->Param
(RealForm,ratvec)->Param
atlas>
atlas>
atlas> whattype all_discrete_series_gamma ?
No overloads for 'all_discrete_series_gamma'
atlas>