# 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 $$H$$, that determines the real group $$H(\mathbb R)$$. Now x is really a $$K$$-orbit on $$G/B$$. So, it is the support of the corresponding $$D$$-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 $$G$$ and a parameter p=(x, lambda, nu) where x encodes the above information and $$\lambda \in X^* /(1-\theta )X^*$$ and $$\nu \in {X^* \otimes \mathbb Q }^{-\theta}$$. With these data we obtain a character of $$H(\mathbb R)$$ with differential $${(1+\theta )\over 2}\lambda + \nu$$.

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