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| In [[mathematics]], the '''orbit method''' (also known as the '''Kirillov theory''', '''the method of coadjoint orbits''' and by a few similar names) establishes a correspondence between irreducible [[unitary representation]]s of a [[Lie group]] and its [[coadjoint orbit]]s: orbits of the [[group action|action of the group]] on the dual space of its [[Lie algebra]]. The theory was introduced by {{harvs|txt|last=Kirillov|authorlink=Alexandre Kirillov|year1=1961|year2=1962}} for [[nilpotent group]]s and later extended by [[Bertram Kostant]], [[Louis Auslander]], [[Lajos Pukánszky]] and others to the case of [[solvable group]]s. [[Roger Evans Howe|Roger Howe]] found a version of the orbit method that applies to ''p''-adic Lie groups. [[David Vogan]] proposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.
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| == Relation with symplectic geometry ==
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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.
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| == Nilpotent group case ==
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| Let ''G'' be a [[connected space|connected]], [[simply connected]] [[nilpotent group|nilpotent]] [[Lie group]]. Kirillov proved that the equivalence classes of [[irreducible representation|irreducible]] [[unitary representation]]s of ''G'' are parametrized by the ''coadjoint orbits'' of ''G'', that is the orbits of the action ''G'' on the dual space <math> \mathfrak{g}^*</math> of its Lie algebra. The [[Kirillov character formula]] expresses the [[Harish-Chandra character]] of the representation as a certain integral over the corresponding orbit.
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| == Compact Lie 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''<sup>*</sup><sub>+</sub> in a single point. An orbit is '''integral''' if this point belongs to the weight lattice of ''G''.
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| 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''(''λ'') with highest weight ''λ''∈''h''<sup>*</sup><sub>+</sub> corresponds to the integral coadjoint orbit ''G''·''λ''. The [[Kirillov character formula]] amounts to the character formula earlier proved by [[Harish-Chandra]].
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| == See also ==
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| *[[Dixmier mapping]]
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| *[[Pukánszky condition]]
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| == References ==
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| *{{Citation | last1=Kirillov | first1=A. A. | title=Unitary representations of nilpotent Lie groups | id={{MR|0125908}} | year=1961 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=138 | pages=283–284}}
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| *{{Citation | last1=Kirillov | first1=A. A. | title=Unitary representations of nilpotent Lie groups | doi=10.1070/RM1962v017n04ABEH004118 | id={{MR|0142001}} | year=1962 | journal=Russian mathematical surveys | issn=0042-1316 | volume=17 | issue=4 | pages=53–104}}
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| *{{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 | id={{MR|0412321}} | year=1976 | volume=220}}
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| * {{citation|title=Merits and demerits of the orbit method|first=A. A.|last=Kirillov|journal=Bull. Amer. Math. Soc.|volume=36|year=1999|pages=433–488|url=http://www.ams.org/bull/1999-36-04/S0273-0979-99-00849-6/home.html}}.
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| *{{eom|id=O/o070020|first=A. A.|last=Kirillov}}
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| * {{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}}.
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| [[Category:Representation theory of Lie groups]]
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