Introduction
=============

So far the focus has been on Cartan subgroups, whose information is
encoded on the element ``x`` as the Cartan involution of the complex
abstract group :math:`H`, that determines the real group :math:`H(\mathbb
R)`. Now ``x`` is really a :math:`K`-orbit on :math:`G/B`. So, it is
the support of the corresponding :math:`D`-module. In order to explain this in
detail we will look at some easy cases. In particular we will be
talking about the principal series of split groups.

So, we start with a group :math:`G` and a parameter ``p=(x, lambda,
nu)`` where ``x`` encodes the above information and :math:`\lambda \in
X^* /(1-\theta )X^*` and :math:`\nu \in {X^* \otimes \mathbb Q
}^{-\theta}`. With these data we obtain a character of
:math:`H(\mathbb R)` with differential :math:`{(1+\theta )\over
2}\lambda + \nu`.

From this character we get a representation of the group G.

In this Chapter we will focus on the minimal principal series for split groups