Background¶

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.