Motivation of the definition of principal series.

Question

I am reading the book representation theory of semisimple groups. On page 33, the principal series representation $\mathcal{P}^{k,iv}$ is defined as follows.

What are motivations of the above formula? Any help will be greatly appreciated!

Answer 1

This is an instance of a "formula" giving no clues, indeed!

In fact, it may well be that this formula and certain similar arose historically through experimentation, but I would argue that _by_now_, with some decades' hindsight, it's harder to understand as a formula than as an artifact or side-effect of something else, as follows.

A more structural characterization/definition of (smooth, not-necessarily unitary) principal series representations $I(s,k)$ of $G=SL_2(\mathbb C)$ is as (smooth) $\mathbb C$-valued functions $f$ on $G$ itself which are left $P$-equivariant by a character $\pmatrix{a & * \cr 0 & a^{-1}}\to |a|^s\cdot (a/|a|)^k$ of the upper-triangular (=standard parabolic) $P$ for arbitrary complex $s$ and integer $k$. In this guise, the action of $G$ itself is simply right translation. Unitariness is easier to discuss here, too, in my opinion.

Traditionally, this can be made more tangible by using the fact that $PwN$ (the larger Bruhat cell) is dense in $G$, where $w=\pmatrix{0&-1\cr 1&0}$ and $N$ is the upper-triangular unipotent matrices (=unipotent radical of $P$). That is, functions in $I(s,k)$ are completely determined by their behavior on the copy of $N$ in $PwN$. But now the right translation action of $G$ has to be rewritten in terms of the Bruhat decomposition $PwN$, producing the formulas you've written.