Background
===========

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

:math:`{\mathcal I}_W /W\leftrightarrow \text{conjugacy classes of
Cartan subgroups in quasisplit group.}`

Now let us fix :math:`x_b` and define the set

.. math:: \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 :math:`{\mathcal I}_W` this just
means that the elements in this preimage are in the Cartan subgroup :math:`H`. 

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

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

So when talking about representations associated to a non split
Cartan subgroup, the element :math:`x` not only gives you the Cartan subgroup but also a
:math:`K`-conjugacy class of Borel subgroups for that Cartan subgroup.

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

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