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