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<title>Unipotent representation</title>

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

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In mathematics, a <b>unipotent representation</b> of a <a href="Reductive_group" title="Reductive group">reductive group</a> is a <a href="Group_representation" title="Group representation">representation</a> that has some similarities with <a href="Unipotent" title="Unipotent">unipotent</a> <a href="Conjugacy_class" title="Conjugacy class">conjugacy classes</a> of groups.

</p><p>Informally, <a href="Langlands_philosophy" class="mw-redirect" title="Langlands philosophy">Langlands philosophy</a> suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a <a href="Langlands_dual_group" title="Langlands dual group">Langlands dual group</a>, and the unipotent representations should be roughly the ones corresponding to unipotent classes in the dual group.

</p><p>Unipotent representations are supposed to be the basic "building blocks" out of which one can construct all other representations in the following sense.

Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that all irreducible representations can be obtained from unipotent representations of possibly smaller groups by some sort of systematic process, such as (cohomological or parabolic) induction.

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<div class="mw-heading mw-heading2"><h2 id="Finite_fields">Finite fields</h2></div>

<p>Over finite fields, the unipotent representations are those that occur in the decomposition of the <a href="Deligne%E2%80%93Lusztig_theory" title="Deligne–Lusztig theory">Deligne–Lusztig characters</a> <i>R</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sup><br><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>T</i></sub></span></span> of the trivial representation 1 of a torus <i>T</i> . They were classified by Lusztig (<a href="#CITEREFLusztig1978">1978</a>, <a href="#CITEREFLusztig1979">1979</a>).

Some examples of unipotent representations over finite fields are the trivial 1-dimensional representation, the <a href="Steinberg_representation" title="Steinberg representation">Steinberg representation</a>, and <a href="%CE%9810" title="Θ10">θ<sub>10</sub></a>.

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<div class="mw-heading mw-heading2"><h2 id="Non-archimedean_local_fields">Non-archimedean local fields</h2></div>

<p><a href="#CITEREFLusztig1995">Lusztig (1995)</a> classified the unipotent characters over non-archimedean local fields.

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<div class="mw-heading mw-heading2"><h2 id="Archimedean_local_fields">Archimedean local fields</h2></div>

<p><a href="#CITEREFVogan1987">Vogan (1987)</a> discusses several different possible definitions of unipotent representations of real <a href="Lie_group" title="Lie group">Lie groups</a>.

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<ul><li><a href="Deligne%E2%80%93Lusztig_theory" title="Deligne–Lusztig theory">Deligne–Lusztig theory</a></li></ul>

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

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</style><cite id="CITEREFBarbasch1991" class="citation cs2">Barbasch, Dan (1991), <a rel="nofollow" class="external text" href="http://mathunion.org/ICM/ICM1990.2/">"Unipotent representations for real reductive groups"</a>, in Satake, Ichirô (ed.), <i>Proceedings of the International Congress of Mathematicians, Vol. II (Kyoto, 1990)</i>, Tokyo: Math. Soc. Japan, pp. <span class="nowrap">769–</span>777, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-4-431-70047-0</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=1159263">1159263</a></cite></li>

<li><cite id="CITEREFLusztig1979" class="citation cs2">Lusztig, George (1979), "Unipotent representations of a finite Chevalley group of type E<sub>8</sub>", <i>The Quarterly Journal of Mathematics</i>, Second Series, <b>30</b> (3): <span class="nowrap">315–</span>338, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fqmath%2F30.3.315">10.1093/qmath/30.3.315</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-5606">0033-5606</a>, <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=0545068">0545068</a></cite></li>

<li><cite id="CITEREFLusztig1978" class="citation cs2">Lusztig, George (1978), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wn27F59-SwAC"><i>Representations of finite Chevalley groups</i></a>, CBMS Regional Conference Series in Mathematics, vol. 39, Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-1689-9</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=0518617">0518617</a></cite></li>

<li><cite id="CITEREFLusztig1995" class="citation cs2">Lusztig, George (1995), "Classification of unipotent representations of simple p-adic groups", <i>International Mathematics Research Notices</i>, <b>1995</b> (11): <span class="nowrap">517–</span>589, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0111248">math/0111248</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FS1073792895000353">10.1155/S1073792895000353</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1073-7928">1073-7928</a>, <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=1369407">1369407</a></cite></li>

<li><cite id="CITEREFVogan1987" class="citation cs2">Vogan, David A. (1987), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0O-9c_kImJYC"><i>Unitary representations of reductive Lie groups</i></a>, Annals of Mathematics Studies, vol. 118, <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-08482-4</bdi></cite></li></ul></div><div><div class="zim-footer">

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