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<title>Admissible representation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Admissible representation</span></span>

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<p>In mathematics, <b>admissible representations</b> are a well-behaved class of <a href="Group_representation" title="Group representation">representations</a> used in the <a href="Representation_theory" title="Representation theory">representation theory</a> of <a href="Reductive_group" title="Reductive group">reductive</a> <a href="Lie_group" title="Lie group">Lie groups</a> and <a href="Locally_compact_group" title="Locally compact group">locally compact</a> <a href="Totally_disconnected_group" title="Totally disconnected group">totally disconnected groups</a>. They were introduced by <a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a>.

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<div class="mw-heading mw-heading2"><h2 id="Real_or_complex_reductive_Lie_groups">Real or complex reductive Lie groups</h2></div>

<p>Let <i>G</i> be a connected reductive (real or complex) Lie group. Let <i>K</i> be a maximal compact subgroup. A continuous representation (π, <i>V</i>) of <i>G</i> on a complex <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> <i>V</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is called <b>admissible</b> if π restricted to <i>K</i> is <a href="Unitary_representation" title="Unitary representation">unitary</a> and each <a href="Irreducible_representation" title="Irreducible representation">irreducible</a> unitary representation of <i>K</i> occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of <i>G</i>.

</p><p>An admissible representation π induces a <a href="(g%2CK)-module" title="(g,K)-module"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathfrak {g}},K)}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathfrak {g}},K)}</annotation>

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</math></span><img src="./fc413e0c6483c4923b796846b98ef2e3bbdb8efa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle ({\mathfrak {g}},K)}" loading="lazy"></span>-module</a> which is easier to deal with as it is an algebraic object. Two admissible representations are said to be <b>infinitesimally equivalent</b> if their associated <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathfrak {g}},K)}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathfrak {g}},K)}</annotation>

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</math></span><img src="./fc413e0c6483c4923b796846b98ef2e3bbdb8efa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle ({\mathfrak {g}},K)}" loading="lazy"></span>-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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathfrak {g}},K)}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathfrak {g}},K)}</annotation>

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</math></span><img src="./fc413e0c6483c4923b796846b98ef2e3bbdb8efa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle ({\mathfrak {g}},K)}" loading="lazy"></span>-modules. This reduces the study of the equivalence classes of irreducible unitary representations of <i>G</i> 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 <a href="Robert_Langlands" title="Robert Langlands">Robert Langlands</a> and is called the <a href="Langlands_classification" title="Langlands classification">Langlands classification</a>.

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<div class="mw-heading mw-heading2"><h2 id="Totally_disconnected_groups">Totally disconnected groups</h2></div>

<p>Let <i>G</i> be a <a href="Locally_profinite_group" title="Locally profinite group">locally compact totally disconnected group</a> (such as a reductive algebraic group over a nonarchimedean <a href="Local_field" title="Local field">local field</a> or over the finite <a href="Adele_ring" title="Adele ring">adeles</a> of a <a href="Global_field" title="Global field">global field</a>). A representation (π, <i>V</i>) of <i>G</i> on a complex vector space <i>V</i> is called <b>smooth</b> if the subgroup of <i>G</i> fixing any vector of <i>V</i> is <a href="Open_set" title="Open set">open</a>. If, in addition, the space of vectors fixed by any <a href="Compact_space" title="Compact space">compact</a> open subgroup is finite dimensional then π is called <b>admissible</b>. Admissible representations of <i>p</i>-adic groups admit more algebraic description through the action of the <a href="Hecke_algebra_of_a_locally_compact_group" class="mw-redirect" title="Hecke algebra of a locally compact group">Hecke algebra</a> of locally constant functions on <i>G</i>.

</p><p>Deep studies of admissible representations of <i>p</i>-adic reductive groups were undertaken by <a href="Bill_Casselman_(mathematician)" class="mw-redirect" title="Bill Casselman (mathematician)">Casselman</a> and by <a href="Joseph_Bernstein" title="Joseph Bernstein">Bernstein</a> and <a href="Andrey_Zelevinsky" class="mw-redirect" title="Andrey Zelevinsky">Zelevinsky</a> in the 1970s. Progress was made more recently by <a href="Roger_Evans_Howe" title="Roger Evans Howe">Howe</a>, Moy, <a href="Gopal_Prasad" title="Gopal Prasad">Gopal Prasad</a> and Bushnell and Kutzko, who developed a <i>theory of types</i> and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.

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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">I.e. a homomorphism <span class="nowrap">π : <i>G</i> → GL(<i>V</i>)</span> (where GL(<i>V</i>) is the group of <a href="Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded linear operators</a> on <i>V</i> whose inverse is also bounded and linear) such that the associated map <span class="nowrap"><i>G</i> × <i>V</i> → <i>V</i></span> is continuous.</span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBushnellHenniart2006" class="citation cs2"><a href="Colin_J._Bushnell" title="Colin J. Bushnell">Bushnell, Colin J.</a>; Henniart, Guy (2006), <i>The local Langlands conjecture for GL(2)</i>, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-31511-X">10.1007/3-540-31511-X</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-31486-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2234120">2234120</a></cite></li>

<li><cite id="CITEREFBushnellPhilip_C._Kutzko1993" class="citation book cs1">Bushnell, Colin J.; Philip C. Kutzko (1993). <i>The admissible dual of GL(N) via compact open subgroups</i>. Annals of Mathematics Studies 129. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-02114-7</bdi>.</cite></li>

<li>Chapter VIII of <cite id="CITEREFKnapp2001" class="citation book cs1">Knapp, Anthony W. (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QCcW1h835pwC"><i>Representation Theory of Semisimple Groups: An Overview Based on Examples</i></a>. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-09089-0</bdi>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2022-01-09" href="https://en.wikipedia.org/wiki/?title=Admissible_representation&oldid=1064595430">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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