Orbit method - Imported Knowlpedia Content

Definition

The orbit method is the representation-theoretic program that associates irreducible unitary representations of a Lie group $G$ with its coadjoint orbits in $\mathfrak g^*$ , interpreting each orbit as a classical symplectic phase space and its representation as a quantization. For a connected, simply connected nilpotent Lie group, the precise Kirillov correspondence sends the orbit of $\ell$ to the equivalence class of the representation induced from the character $e^{i\ell}$ of a subgroup integrating a polarization at $\ell$ . In this setting it is a bijection $\mathfrak g^*/G\to\widehat G$ , independent of the polarization chosen.

The nilpotent correspondence

For connected simply connected nilpotent $G$ , the exponential map is a diffeomorphism and every $\ell\in\mathfrak g^*$ admits a polarization. If $P=\exp(\mathfrak p)$ , Kirillov constructs

\pi_\ell=\operatorname{Ind}_P^G\chi_\ell,\qquad \chi_\ell(\exp X)=e^{i\ell(X)}.

The representation is irreducible; changing $\ell$ within its coadjoint orbit or changing the polarization does not change its unitary-equivalence class; and every irreducible unitary representation arises this way Kirillov, Chapter 3. With the quotient topology on $\mathfrak g^*/G$ and the Fell topology on $\widehat G$ , the correspondence is a homeomorphism.

Geometry encoded by an orbit

Each orbit carries the Kirillov–Kostant–Souriau symplectic form. Orbit dimension predicts the number of variables in an induced model, stabilizers control the inducing subgroup, and invariant functions on $\mathfrak g^*$ reflect central or infinitesimal-character data. For nilpotent groups, the orbit method also yields character and Plancherel formulas. These are not merely analogies: they are compatible parts of the classification developed in Corwin–Greenleaf, Chapters 1–4.

Examples and limitations

For an abelian Lie group, every coadjoint orbit is a point and the method reduces to the classification by unitary characters. For the Heisenberg group, nonzero central values give the familiar infinite-dimensional Schrödinger representations. For compact or noncompact semisimple groups, however, raw coadjoint orbits do not stand in a simple bijection with the unitary dual: integrality, admissible orbit data, coverings, and choices of quantization intervene. Kirillov’s survey explicitly separates the nilpotent success from these later obstacles Kirillov, §§2–4.

Conventions and scope

References

A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics 64, American Mathematical Society, 2004. AMS record. Relevant: Chapters 1–5, especially Chapter 3.
A. A. Kirillov, “Merits and Demerits of the Orbit Method,” Bulletin of the American Mathematical Society 36 (1999), 433–488. DOI record. Relevant: §§2–4.
Lawrence J. Corwin and Frederick P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications, Part I: Basic Theory and Examples, Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, 1990. Publisher front matter. Relevant: Chapters 1–4.