Principal series representation

Remarks

The shift by $\rho$ makes $I(\sigma, \nu)$ unitary when $\nu \in i \mathfrak a^*$; these are the unitary principal series. Every function in the space is determined by its restriction to $K$, because $G = KP$ by the Iwasawa decomposition.

Example: for $G = \mathrm{SL}_2(\R)$, let $P$ be the upper triangular matrices, so $M = \set{\pm 1}$, $A$ is the positive diagonal matrices and $N$ the unipotent upper triangular ones. Here $\rho(\log \operatorname{diag}(t, t^{-1})) = \log t$, so $\operatorname{diag}(t, t^{-1})^\rho = t$. The restriction to $K = \mathrm{SO}(2)$ identifies the space with the functions on the circle that are even or odd according to $\sigma(-1) = \pm 1$. So each character $e^{in\theta}$ of $K$ with $(-1)^n = \sigma(-1)$ occurs once, and $I(\sigma, \nu)$ is admissible.

Non-example: inducing the trivial representation from the compact subgroup $K$ instead of from $P$ gives the representation of $G$ on $L^2(G/K)$. It is not a principal series representation: $K$ is not parabolic, and the representation is not even admissible.

Some authors call representations induced from any parabolic subgroup, with discrete series or tempered data on $M$, generalised principal series.