128.78.162.220: small corrections in a formula

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small corrections in a formula

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In mathematics, '''admissible representations''' are a well-behaved class of [[Group representation|representations]] used in the [[representation theory]] of [[reductive group|reductive]] [[Lie group]]s and [[locally compact group|locally compact]] [[totally disconnected group]]s. They were introduced by [[Harish-Chandra]].

==Real or complex reductive Lie groups==
Let ''G'' be a connected reductive (real or complex) Lie group. Let ''K'' be a maximal compact subgroup. A continuous representation (π, ''V'') of ''G'' on a complex [[Hilbert space]] ''V''<ref>I.e. a homomorphism {{nowrap|π : ''G'' → GL(''V'')}} (where GL(''V'') is the group of [[bounded linear operator]]s on ''V'' whose inverse is also bounded and linear) such that the associated map {{nowrap|''G'' × ''V'' → ''V''}} is continuous.</ref> is called '''admissible''' if π restricted to ''K'' is unitary and each irreducible unitary representation of ''K'' occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of ''G''.

An admissible representation π induces a [[(g,K)-module|<math>(\mathfrak{g},K)</math>-module]] which is easier to deal with as it is an algebraic object. Two admissible representations are said to be '''infinitesimally equivalent''' if their associated <math>(\mathfrak{g},K)</math>-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 <math>(\mathfrak{g},K)</math>-modules. This reduces the study of the equivalence classes of irreducible unitary representations of ''G'' 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 [[Robert Langlands]] and is called the [[Langlands classification]].

== Totally disconnected groups ==

Let ''G'' be a [[locally profinite group|locally compact totally disconnected group]] (such as a reductive algebraic group over a [[local field]] or over the finite [[adele ring|adeles]] of a [[global field]]). A representation (π, ''V'') of ''G'' on a complex vector space ''V'' is called '''smooth''' if the subgroup of ''G'' fixing any vector of ''V'' is [[open set|open]]. If, in addition, the space of vectors fixed by any [[compact space|compact]] open subgroup is finite dimensional then π is called '''admissible'''. Admissible representations of ''p''-adic groups admit more algebraic description through the action of the [[Hecke algebra of a locally compact group|Hecke algebra]] of locally constant functions on ''G''.

Deep studies of admissible representations of ''p''-adic reductive groups were undertaken by [[Bill Casselman (mathematician)|Casselman]] and by [[Joseph Bernstein|Bernstein]] and [[Andrey Zelevinsky|Zelevinsky]] in the 1970s. Much progress has been made more recently by [[Roger Evans Howe|Howe]] and Moy and Bushnell and Kutzko, who developed a ''theory of types'' and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.

==Notes==
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==References==
*{{Citation | last1=Bushnell | first1=Colin J. |authorlink1=Colin J. Bushnell | last2=Henniart | first2=Guy | title=The local Langlands conjecture for GL(2) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-31486-8 | doi=10.1007/3-540-31511-X | mr=2234120 | year=2006 | volume=335}}

*{{cite book|last = Bushnell|first = Colin J.|author2 = Philip C. Kutzko|title = The admissible dual of GL(N) via compact open subgroups|series = Annals of Mathematics Studies 129|publisher=Princeton University Press|year = 1993|isbn=0-691-02114-7}}

*Chapter VIII of {{cite book|last = Knapp|first = Anthony W.|title = Representation Theory of Semisimple Groups: An Overview Based on Examples|publisher = Princeton University Press|year = 2001|isbn = 0-691-09089-0|url = http://books.google.com/books?id=QCcW1h835pwC}}

[[Category:Representation theory]]

Carl Johan Malmsten - Revision history

128.78.162.220: small corrections in a formula