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 [[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 [[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.}}</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 ==