---
tag: 95Q8WZMN0W
kind: concept
name: Unitary dual
msc: ["22D10", "43A40"]
sources:
  - url: https://en.wikipedia.org/wiki/Unitary_representation
---

## Summary

The unitary dual $\widehat{G}$ of a [[locally compact group|AHV05ZBCYW]] $G$ is the set of equivalence classes of irreducible [[unitary representations|74BPYJNZTM]] of $G$.
It usually carries the Fell topology.
It is the space on which a "Fourier transform" of functions on $G$ lives.
For abelian groups it is the Pontryagin dual, a group again.
For nonabelian groups it is only a topological space, often non-Hausdorff.
Determining it for a given group is a central problem of representation theory.

## Explanations

### L1 human

For the circle, the irreducible unitary representations are the characters $z \mapsto z^n$, one for each integer $n$, so the unitary dual of the circle is $\Z$.
This is why functions on the circle have Fourier series indexed by integers.
For the real line the characters are $x \mapsto e^{ixy}$, one for each real $y$, which gives the Fourier transform.
For a general group the unitary dual plays the same role: it lists the basic frequencies in which functions on the group can be decomposed.