---
tag: G9AW6R1D2F
kind: concept
name: Kirillov orbit method
msc: ["22E27", "22E45", 22D10]
sources:
  - url: https://en.wikipedia.org/wiki/Orbit_method
same_as:
  wikidata: Q7100049
  wikipedia: Orbit method
---

## Summary

The orbit method relates the irreducible unitary representations of a Lie group $G$ to the coadjoint orbits.
These are the orbits of $G$ in the dual $\mathfrak g^*$ of its Lie algebra.
Kirillov showed that for connected simply connected nilpotent Lie groups this relation is an exact bijection, built by inducing characters from polarizing subgroups.
For other groups it is a guiding principle rather than a theorem.
It works for exponential solvable groups and gives good heuristics for compact and reductive groups.
It ties representation theory to symplectic geometry, since each coadjoint orbit carries a natural symplectic form.

## Explanations

### L1 human

The dual of the Lie algebra is a vector space on which the group acts, and it breaks into orbits.
The orbit method says: each orbit should give one irreducible representation, and each irreducible representation should come from one orbit.
For the Heisenberg group, the orbits are parallel planes and single points.
The planes give the representations of quantum mechanics, one for each value of Planck's constant, and the points give the one-dimensional characters.
In this way the method is a mathematical version of passing from classical to quantum mechanics.