Orbit method: Difference between revisions

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One of the key observations of Kirillov was that coadjoint orbits of a Lie group ''G'' have natural structure of [[symplectic manifold]]s whose symplectic structure is invariant under ''G''. If an orbit is the [[phase space]] of a ''G''-invariant [[Hamiltonian mechanics|classical mechanical system]] then the corresponding quantum mechanical system ought to be described via an irreducible unitary representation of ''G''. Geometric invariants of the orbit translate into algebraic invariants of the corresponding representation. In this way the orbit method may be viewed as a precise mathematical manifestation of a vague physical principle of quantization. In the case of a nilpotent group ''G'' the correspondence involves all orbits, but for a general ''G'' additional restrictions on the orbit are necessary (polarizability, integrality, Pukanszky condition). This point of view has been significantly advanced by Kostant in his theory of [[geometric quantization]] of coadjoint orbits.

== Kirillov character formula ==

{{main|Kirillov character formula}}

For a [[Lie group]] <math>G</math>, the [[Kirillov orbit method]] gives a heuristic method in [[representation theory]]. It connects the [[Fourier transform]]s of [[coadjoint orbit]]s, which lie in the [[dual space]] of the [[Lie algebra]] of ''G'', to the [[infinitesimal character]]s of the [[irreducible representation]]s. The method got its name after the [[Russia]]n mathematician [[Alexandre Kirillov]].

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At its simplest, it states that a character of a Lie group may be given by the [[Fourier transform]] of the [[Dirac delta function]] [[support (mathematics)|support]]ed on the coadjoint orbits, weighted by the square-root of the [[Jacobian matrix and determinant|Jacobian]] of the [[exponential map (Lie theory)|exponential map]], denoted by <math>j</math>. It does not apply to all Lie groups, but works for a number of classes of [[connected space|connected]] Lie groups, including [[nilpotent]], some [[Semisimple Lie group|semisimple]] groups, and [[compact group]]s.

== Special cases ==

=== Nilpotent group case ===

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Complex irreducible representations of [[compact Lie group]]s have been completely classified. They are always finite-dimensional, unitarizable (i.e. admit an invariant positive definite [[Hermitian form]]) and are parametrized by their [[highest weight]]s, which are precisely the dominant integral weights for the group. If ''G'' is a compact [[semisimple Lie group]] with a [[Cartan subalgebra]] ''h'' then its coadjoint orbits are [[closed set|closed]] and each of them intersects the positive Weyl chamber ''h''^*₊ in a single point. An orbit is '''integral''' if this point belongs to the weight lattice of ''G''.

The highest weight theory can be restated in the form of a bijection between the set of integral coadjoint orbits and the set of equivalence classes of irreducible unitary representations of ''G'': the highest weight representation ''L''(''&lambda;λ'') with highest weight ''&lambda;λ''&isin;∈''h''^*₊ corresponds to the integral coadjoint orbit ''G''&middot;·''&lambda;λ''. The [[Kirillov character formula]] amounts to the character formula earlier proved by [[Harish-Chandra]].

== See also ==

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*{{Citation | last1=Kirillov | first1=A. A. | title=Unitary representations of nilpotent Lie groups | doi=10.1070/RM1962v017n04ABEH004118 |mr=0142001 | year=1962 | journal=Russian mathematical surveys | issn=0042-1316 | volume=17 | issue=4 | pages=53–104}}

*{{Citation | last1=Kirillov | first1=A. A. | title=Elements of the theory of representations | origyear=1972 | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series= Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-07476-4 |mr=0412321 | year=1976 | volume=220}}

* {{citation|title=Merits and demerits of the orbit method|first=A. A.|last=Kirillov|journal=Bull. Amer. Math. Soc.|volume=36|year=1999|pages=433&ndash;488433–488|url=http://www.ams.org/bull/1999-36-04/S0273-0979-99-00849-6/home.html|doi=10.1090/s0273-0979-99-00849-6}}.

*{{eom|id=O/o070020|first=A. A.|last=Kirillov}}

* {{citation|last=Kirillov|first=A. A.|title=Lectures on the orbit method|series=[[Graduate Studies in Mathematics]]|volume=64|publisher=American Mathematical Society|publication-place=Providence, RI|year=2004|isbn=0-8218-3530-0}}.

[[Category:Representation theory of Lie groups]]