BackgroundΒΆ

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.