---
tag: CWDF4KGHS3
kind: concept
name: Principal series representation
msc: ["22E46", 22D30]
specializes: [6Q7AHAZ6Q4]
sources:
  - url: https://en.wikipedia.org/wiki/Principal_series_representation
same_as:
  wikidata: Q7245090
  wikipedia: Principal series representation
---

## Summary

A principal series representation of a real reductive group is induced from a minimal parabolic subgroup.
The inducing data are a character of its split part and an irreducible representation of its compact part.
For $\mathrm{SL}_2(\R)$, it acts on homogeneous functions on the plane, or equivalently on functions on the circle.
The unitary principal series supply the continuous part of the Plancherel decomposition of $L^2(G)$.
By a theorem of Casselman, every irreducible admissible representation embeds into some principal series, which makes these representations a universal source.

## Explanations

### L1 human

Pick a large subgroup $P$ of $G$, such as the upper triangular matrices in $\mathrm{SL}_2(\R)$.
A character of $P$ is a simple one-dimensional rule.
Inducing spreads this rule over the whole group: the space consists of functions on $G$ that transform under $P$ by the chosen character.
The resulting family depends on a complex parameter $\nu$, and almost all its members are irreducible.