---
tag: 9AB7HYGPEK
kind: concept
name: Discrete series representation
msc: ["22E46", 22D10]
specializes: [74BPYJNZTM]
sources:
  - url: https://en.wikipedia.org/wiki/Discrete_series_representation
same_as:
  wikidata: Q5282059
  wikipedia: Discrete series representation
---

## Summary

A discrete series representation of a locally compact group $G$ is an irreducible unitary representation.
It occurs as a direct summand of the regular representation on $L^2(G)$.
These are the representations that contribute point masses to the Plancherel measure.
Let $G$ be a connected semisimple Lie group with finite center.
Harish-Chandra showed that $G$ has discrete series exactly when it has a compact Cartan subgroup, and he classified them.
They are the building blocks from which the [[Langlands classification|YHFP9HGBTT]] constructs all irreducible admissible representations.

## Explanations

### L1 human

The functions on a group can be broken into pieces on which the group acts irreducibly, much as a sound breaks into pure tones.
For the circle or the real line, all pieces come in continuous families, like frequencies.
Some groups, such as $\mathrm{SL}_2(\R)$, also have isolated pieces that appear on their own, with positive "weight".
These isolated pieces are the discrete series.
For $\mathrm{SL}_2(\R)$, they are realised on spaces of holomorphic functions on the upper half-plane and are closely linked to modular forms.