Orbit method: Difference between revisions

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{{Short description|Construction in representation theory}}

{{Use dmy dates|date=September 2015}}

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 [[groupGroup action (mathematics)|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.<ref>{{Citation [[David| Vogan]]last=Howe proposed| thatfirst=Roger the| orbittitle=Kirillov methodtheory shouldfor servecompact asp-adic agroups unifying| principleyear=1977 in| thejournal=[[Pacific descriptionJournal of theMathematics]] unitary| dualsvolume=73 of| realissue=2 reductive| Liepages=365–381 groups| doi=10.2140/pjm.1977.73.365 | doi-access=free }}</ref>

[[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.<ref>{{Citation | last=Vogan | first=David | title=Representations of reductive Lie groups | year=1986 | journal=Proceedings of the International Congress of Mathematicians (Berkeley, California)| pages=245–266 }}</ref>

== Relation with symplectic geometry ==

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, PukanszkyPukánszky condition). This point of view has been significantly advanced by Kostant in his theory of [[geometric quantization]] of coadjoint orbits.

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 ==

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== See also ==

*[[Dixmier mapping]]

*[[Polarization (Lie algebra)#Pukanszky condition|Pukánszky condition]]

== References ==

{{reflist}}

*{{Citation | last1=Dulfo | last2=Pederson| last3=Vergne| title=The Orbit Method in Representation Theory: Proceedings of a Conference Held in Copenhagen, August to September 1988 (Progress in Mathematics)|year=1990 | publisher= Birkhäuser}}

*{{Citation | last1=Kirillov | first1=A. A. | title=Unitary representations of nilpotent Lie groups |mr=0125908 | year=1961 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=138 | pages=283–284}}

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

*{{Citation | last1=Kirillov | first1=A. A. | title=Elements of the theory of representations | origyearorig-year=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|issue=4|year=1999|pages=433–488|url=httphttps://www.ams.org/bull/1999-36-04/S0273-0979-99-00849-6/home.html|doi=10.1090/s0273-0979-99-00849-6|doi-access=free}}.

*{{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___location=Providence, RI|year=2004|isbn=978-0-8218-3530-02}}.

[[Category:Representation theory of Lie groups]]