186:-modules are isomorphic. Though for general admissible representations, this notion is different than the usual equivalence, it is an important result that the two notions of equivalence agree for unitary (admissible) representations. Additionally, there is a notion of unitarity of
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to the study of infinitesimal equivalence classes of admissible representations and the determination of which of these classes are infinitesimally unitary. The problem of parameterizing the infinitesimal equivalence classes of admissible representations was fully solved by
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and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.
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occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of
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which is easier to deal with as it is an algebraic object. Two admissible representations are said to be
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222:-modules. This reduces the study of the equivalence classes of irreducible unitary representations of
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Representation Theory of
Semisimple Groups: An Overview Based on Examples
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303:-adic groups admit more algebraic description through the action of the
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whose inverse is also bounded and linear) such that the associated map
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be a maximal compact subgroup. A continuous representation (π,
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455:. Annals of Mathematics Studies 129. Princeton University Press.
251:(such as a reductive algebraic group over a nonarchimedean
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be a connected reductive (real or complex) Lie group. Let
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The admissible dual of GL(N) via compact open subgroups
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open subgroup is finite dimensional then π is called
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330:in the 1970s. Progress was made more recently by
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314:Deep studies of admissible representations of
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451:Bushnell, Colin J.; Philip C. Kutzko (1993).
318:-adic reductive groups were undertaken by
249:locally compact totally disconnected group
338:and Bushnell and Kutzko, who developed a
107:An admissible representation π induces a
415:The local Langlands conjecture for GL(2)
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53:Real or complex reductive Lie groups
215:{\displaystyle ({\mathfrak {g}},K)}
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179:{\displaystyle ({\mathfrak {g}},K)}
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137:{\displaystyle ({\mathfrak {g}},K)}
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307:of locally constant functions on
299:. Admissible representations of
477:. Princeton University Press.
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1:
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263:). A representation (π,
23:are a well-behaved class of
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239:Totally disconnected groups
43:totally disconnected groups
10:
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471:Knapp, Anthony W. (2001).
271:on a complex vector space
148:infinitesimally equivalent
96:unitary representation of
45:. They were introduced by
21:admissible representations
413:; Henniart, Guy (2006),
376:bounded linear operators
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233:Langlands classification
16:Class of representations
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502:Representation theory
427:10.1007/3-540-31511-X
283:fixing any vector of
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29:representation theory
358:I.e. a homomorphism
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150:if their associated
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279:if the subgroup of
255:or over the finite
84:if π restricted to
411:Bushnell, Colin J.
374:) is the group of
231:and is called the
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436:978-3-540-31486-8
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469:Chapter VIII of
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19:In mathematics,
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75:Hilbert space
73:on a complex
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336:Gopal Prasad
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261:global field
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27:used in the
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253:local field
94:irreducible
404:References
370:(where GL(
328:Zelevinsky
297:admissible
275:is called
82:admissible
80:is called
36:Lie groups
360:π :
324:Bernstein
320:Casselman
92:and each
33:reductive
496:Category
334:, Moy,
445:2234120
322:and by
293:compact
144:-module
90:unitary
481:
459:
443:
433:
277:smooth
257:adeles
364:→ GL(
346:Notes
267:) of
259:of a
247:be a
69:) of
479:ISBN
457:ISBN
431:ISBN
332:Howe
326:and
289:open
243:Let
57:Let
38:and
423:doi
378:on
287:is
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31:of
498::
441:MR
439:,
429:,
421:,
391:→
387:×
311:.
235:.
104:.
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487:.
465:.
425::
393:V
389:V
385:G
380:V
372:V
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366:V
362:G
316:p
309:G
301:p
285:V
281:G
273:V
269:G
265:V
245:G
224:G
210:)
207:K
204:,
199:g
194:(
174:)
171:K
168:,
163:g
158:(
132:)
129:K
126:,
121:g
116:(
102:G
98:K
86:K
78:V
71:G
67:V
63:K
59:G
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