197:-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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233:-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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314:-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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466:. Annals of Mathematics Studies 129. Princeton University Press.
262:(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
302:. If, in addition, the space of vectors fixed by any
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341:in the 1970s. Progress was made more recently by
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325:Deep studies of admissible representations of
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462:Bushnell, Colin J.; Philip C. Kutzko (1993).
329:-adic reductive groups were undertaken by
260:locally compact totally disconnected group
349:and Bushnell and Kutzko, who developed a
118:An admissible representation π induces a
426:The local Langlands conjecture for GL(2)
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64:Real or complex reductive Lie groups
226:{\displaystyle ({\mathfrak {g}},K)}
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190:{\displaystyle ({\mathfrak {g}},K)}
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148:{\displaystyle ({\mathfrak {g}},K)}
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318:of locally constant functions on
310:. Admissible representations of
488:. Princeton University Press.
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1:
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274:). A representation (π,
34:are a well-behaved class of
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250:Totally disconnected groups
54:totally disconnected groups
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482:Knapp, Anthony W. (2001).
282:on a complex vector space
159:infinitesimally equivalent
107:unitary representation of
56:. They were introduced by
32:admissible representations
424:; Henniart, Guy (2006),
387:bounded linear operators
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244:Langlands classification
27:Class of representations
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513:Representation theory
438:10.1007/3-540-31511-X
294:fixing any vector of
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40:representation theory
18:Smooth representation
369:I.e. a homomorphism
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161:if their associated
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290:if the subgroup of
266:or over the finite
95:if π restricted to
422:Bushnell, Colin J.
385:) is the group of
242:and is called the
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447:978-3-540-31486-8
16:(Redirected from
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480:Chapter VIII of
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84:on a complex
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347:Gopal Prasad
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38:used in the
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264:local field
105:irreducible
415:References
381:(where GL(
339:Zelevinsky
308:admissible
286:is called
93:admissible
91:is called
47:Lie groups
371:π :
335:Bernstein
331:Casselman
103:and each
44:reductive
507:Category
345:, Moy,
456:2234120
333:and by
304:compact
155:-module
101:unitary
492:
470:
454:
444:
288:smooth
268:adeles
375:→ GL(
357:Notes
278:) of
270:of a
258:be a
80:) of
490:ISBN
468:ISBN
442:ISBN
343:Howe
337:and
300:open
254:Let
68:Let
49:and
434:doi
389:on
298:is
99:is
42:of
509::
452:MR
450:,
440:,
432:,
402:→
398:×
322:.
246:.
115:.
60:.
498:.
476:.
436::
404:V
400:V
396:G
391:V
383:V
379:)
377:V
373:G
327:p
320:G
312:p
296:V
292:G
284:V
280:G
276:V
256:G
235:G
221:)
218:K
215:,
210:g
205:(
185:)
182:K
179:,
174:g
169:(
143:)
140:K
137:,
132:g
127:(
113:G
109:K
97:K
89:V
82:G
78:V
74:K
70:G
20:)
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