146:. 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
444:. 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
154:
additional restrictions on the orbit are necessary (polarizability, integrality, Pukánszky condition). This point of view has been significantly advanced by
Kostant in his theory of
380:
279:
193:
741:
722:
647:
563:
The Orbit Method in
Representation Theory: Proceedings of a Conference Held in Copenhagen, August to September 1988 (Progress in Mathematics)
110:
proposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.
714:
254:
519:
402:
have been completely classified. They are always finite-dimensional, unitarizable (i.e. admit an invariant positive definite
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258:
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then the corresponding quantum mechanical system ought to be described via an irreducible unitary representation of
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55:
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178:
95:
735:
294:
17:
611:
682:
638:, Grundlehren der Mathematischen Wissenschaften, vol. 220, Berlin, New York:
549:
Proceedings of the
International Congress of Mathematicians (Berkeley, California)
118:
One of the key observations of
Kirillov was that coadjoint orbits of a Lie group
216:
131:
107:
63:
31:
242:
At its simplest, it states that a character of a Lie group may be given by the
426:
212:
619:
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281:. It does not apply to all Lie groups, but works for a number of classes of
46:
and by a few similar names) establishes a correspondence between irreducible
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325:
286:
173:
51:
598:
Kirillov, A. A. (1962), "Unitary representations of nilpotent Lie groups",
570:
Kirillov, A. A. (1961), "Unitary representations of nilpotent Lie groups",
390:
of the representation as a certain integral over the corresponding orbit.
410:, which are precisely the dominant integral weights for the group. If
232:
517:
Howe, Roger (1977), "Kirillov theory for compact p-adic groups",
547:
Vogan, David (1986), "Representations of reductive Lie groups",
717:, vol. 64, Providence, RI: American Mathematical Society,
253:
on the coadjoint orbits, weighted by the square-root of the
150:
the correspondence involves all orbits, but for a general
429:
and each of them intersects the positive Weyl chamber
357:
267:
181:
102:
found a version of the orbit method that applies to
479:amounts to the character formula earlier proved by
374:
328:. Kirillov proved that the equivalence classes of
273:
187:
440:if this point belongs to the weight lattice of
126:whose symplectic structure is invariant under
8:
467:corresponds to the integral coadjoint orbit
681:
666:"Merits and demerits of the orbit method"
636:Elements of the theory of representations
532:
366:
360:
359:
356:
266:
180:
75:
71:
509:
398:Complex irreducible representations of
27:Construction in representation theory
7:
448:: the highest weight representation
231:. The method got its name after the
742:Representation theory of Lie groups
375:{\displaystyle {\mathfrak {g}}^{*}}
361:
347:, that is the orbits of the action
25:
114:Relation with symplectic geometry
561:Dulfo; Pederson; Vergne (1990),
406:) and are parametrized by their
715:Graduate Studies in Mathematics
612:10.1070/RM1962v017n04ABEH004118
436:in a single point. An orbit is
66:. The theory was introduced by
520:Pacific Journal of Mathematics
425:then its coadjoint orbits are
44:the method of coadjoint orbits
1:
683:10.1090/s0273-0979-99-00849-6
711:Lectures on the orbit method
600:Russian Mathematical Surveys
199:gives a heuristic method in
698:Encyclopedia of Mathematics
229:irreducible representations
140:classical mechanical system
758:
572:Doklady Akademii Nauk SSSR
477:Kirillov character formula
384:Kirillov character formula
168:Kirillov character formula
165:
162:Kirillov character formula
122:have natural structure of
94:and others to the case of
691:Kirillov, A. A. (2001) ,
634:Kirillov, A. A. (1976) ,
339:are parametrized by the
62:on the dual space of its
709:Kirillov, A. A. (2004),
664:Kirillov, A. A. (1999),
388:Harish-Chandra character
382:of its Lie algebra. The
225:infinitesimal characters
534:10.2140/pjm.1977.73.365
333:unitary representations
48:unitary representations
670:Bull. Amer. Math. Soc.
456:) with highest weight
394:Compact Lie group case
376:
285:Lie groups, including
275:
189:
156:geometric quantization
82:and later extended by
377:
276:
201:representation theory
197:Kirillov orbit method
190:
158:of coadjoint orbits.
130:. If an orbit is the
18:Kirillov orbit theory
416:semisimple Lie group
355:
306:Nilpotent group case
265:
248:Dirac delta function
179:
124:symplectic manifolds
498:Pukánszky condition
211:, which lie in the
60:action of the group
38:(also known as the
400:compact Lie groups
372:
351:on the dual space
271:
237:Alexandre Kirillov
205:Fourier transforms
203:. It connects the
185:
106:-adic Lie groups.
724:978-0-8218-3530-2
649:978-0-387-07476-4
420:Cartan subalgebra
274:{\displaystyle j}
244:Fourier transform
188:{\displaystyle G}
16:(Redirected from
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341:coadjoint orbits
320:simply connected
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209:coadjoint orbits
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80:nilpotent groups
58:: orbits of the
56:coadjoint orbits
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493:Dixmier mapping
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408:highest weights
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259:exponential map
177:
176:
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96:solvable groups
92:Lajos Pukánszky
88:Louis Auslander
84:Bertram Kostant
40:Kirillov theory
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15:
12:
11:
5:
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693:"Orbit method"
688:
676:(4): 433–488,
661:
648:
631:
595:
567:
555:
554:
539:
527:(2): 365–381,
508:
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502:
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488:
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481:Harish-Chandra
464:
433:
404:Hermitian form
395:
392:
386:expresses the
369:
363:
307:
304:
302:
299:
295:compact groups
270:
235:mathematician
184:
166:Main article:
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14:
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9:
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606:(4): 53–104,
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414:is a compact
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301:Special cases
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261:, denoted by
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565:, Birkhäuser
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293:groups, and
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220:
171:
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147:
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135:
127:
119:
117:
103:
43:
39:
36:orbit method
35:
29:
578:: 283–284,
330:irreducible
217:Lie algebra
138:-invariant
132:phase space
108:David Vogan
64:Lie algebra
32:mathematics
504:References
291:semisimple
213:dual space
100:Roger Howe
703:EMS Press
620:0042-1316
584:0002-3264
551:: 245–266
368:∗
326:Lie group
323:nilpotent
316:connected
287:nilpotent
283:connected
251:supported
223:, to the
174:Lie group
52:Lie group
736:Category
487:See also
438:integral
255:Jacobian
68:Kirillov
54:and its
658:0412321
628:0142001
592:0125908
418:with a
289:, some
257:of the
246:of the
233:Russian
227:of the
215:of the
70: (
721:
656:
646:
626:
618:
590:
582:
475:. The
427:closed
195:, the
172:For a
78:) for
34:, the
314:be a
134:of a
50:of a
719:ISBN
644:ISBN
616:ISSN
580:ISSN
310:Let
76:1962
72:1961
678:doi
608:doi
576:138
529:doi
343:of
335:of
219:of
207:of
30:In
738::
713:,
701:,
695:,
674:36
672:,
668:,
654:MR
652:,
642:,
624:MR
622:,
614:,
604:17
602:,
588:MR
586:,
574:,
525:73
523:,
483:.
318:,
297:.
239:.
98:.
90:,
86:,
74:,
42:,
728:.
687:.
680::
610::
531::
473:λ
471:·
469:G
465:+
462:h
460:∈
458:λ
454:λ
452:(
450:L
446:G
442:G
434:+
431:h
423:h
412:G
362:g
349:G
345:G
337:G
312:G
269:j
221:G
183:G
152:G
148:G
144:G
136:G
128:G
120:G
104:p
20:)
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