Orbit method: Difference between revisions

Revision as of 02:21, 23 September 2020

Revision as of 06:39, 22 April 2021

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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 [[Group 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 | last=Howe | first=Roger | title=Kirillov theory for compact p-adic groups | year=1977 | journal=[[Pacific Journal of Mathematics]] | volume=73 | issue=2 | pages=365-381365–381 | doi=10.2140/pjm.1977.73.365 }}</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=245-266.245–266 }}</ref>

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*{{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 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 | 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=http://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}}