Langlands classification

<h2 id="notation">Notation</h2>
<ul><li>g is the Lie algebra of a real reductive Lie group G in the <a href="/facts/Harish-Chandra_class/QAyGTk89">Harish-Chandra class</a>.</li>
<li>K is a maximal compact subgroup of G, with Lie algebra k.</li>
<li>ω is a <a href="/facts/Cartan_involution/G8g40WGg">Cartan involution</a> of G, fixing K.</li>
<li>p is the −1 eigenspace of a Cartan involution of g.</li>
<li>a is a maximal abelian subspace of p.</li>
<li>Σ is the <a href="/facts/Root_system/TFaEdwMU">root system</a> of a in g.</li>
<li>Δ is a set of <a href="/facts/Simple_root_(root_system)/TFaEdwMU">simple roots</a> of Σ.</li></ul>
<h2 id="classification">Classification</h2>
The Langlands classification states that the irreducible <a href="/facts/Admissible_representation/KjgWLM3s">admissible representations</a> of (g, K) are parameterized by triples

(F, σ, λ)
where

<ul><li>F is a subset of Δ</li>
<li>Q is the standard <a href="/facts/Borel_subgroup/F80yLMqA">parabolic subgroup</a> of F, with <a href="/facts/Langlands_decomposition/ffJ1aql1">Langlands decomposition</a> Q = MAN</li>
<li>σ is an irreducible tempered representation of the semisimple Lie group M (up to isomorphism)</li>
<li>λ is an element of Hom(aF, C) with α(Re(λ)) > 0 for all simple roots α not in F.</li></ul>
More precisely, the irreducible admissible representation given by the data above is the irreducible quotient of a parabolically induced representation.
For an example of the Langlands classification, see the <a href="/facts/Representation_theory_of_SL2(R)/33K7XTrU">representation theory of SL2(R)</a>.

<h2 id="variations">Variations</h2>
There are several minor variations of the Langlands classification. For example: 

<ul><li>Instead of taking an irreducible quotient, one can take an irreducible submodule.</li>
<li>Since tempered representations are in turn given as certain representations induced from discrete series or limit of discrete series representations, one can do both inductions at once and get a Langlands classification parameterized by discrete series or limit of discrete series representations instead of tempered representations. The problem with doing this is that it is tricky to decide when two irreducible representations are the same.</li></ul>

<ul><li>Adams, Jeffrey; Barbasch, Dan; Vogan, David A. (1992), <a href="https://books.google.com/books?id=T37ryFaTWm4C">The Langlands classification and irreducible characters for real reductive groups</a>, Progress in Mathematics, vol. 104, Boston, MA: Birkhäuser Boston, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-3634-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1162533">1162533</a></li>
<li>E. P. van den Ban, Induced representations and the Langlands classification, in <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0609-2 (T. Bailey and A. W. Knapp, eds.).</li>
<li><a href="/facts/Armand_Borel/QkRUS4ej">Borel, A.</a> and <a href="/facts/Nolan_Wallach/hpTW63TC">Wallach, N.</a> Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition. Mathematical Surveys and Monographs, 67. American Mathematical Society, Providence, RI, 2000. xviii+260 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0851-6</li>
<li>Langlands, Robert P. (1989) [1973], <a href="http://publications.ias.edu/rpl/paper/16">"On the classification of irreducible representations of real algebraic groups"</a>, in Sally, Paul J.; Vogan, David A. (eds.), Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, pp. 101–170, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-1526-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1011897">1011897</a></li>
<li>Vogan, David A. (2000), <a href="http://atlas.math.umd.edu/papers/kyoto.pdf">"A Langlands classification for unitary representations"</a> (PDF), in Kobayashi, Toshiyuki; <a href="/facts/Masaki_Kashiwara/29L71Lgt">Kashiwara, Masaki</a>; Matsuki, Toshihiko; Nishiyama, Kyo; Oshima, Toshio (eds.), Analysis on homogeneous spaces and representation theory of Lie groups, Okayama--Kyoto (1997), Adv. Stud. Pure Math., vol. 26, Tokyo: Math. Soc. Japan, pp. 299–324, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-4-314-10138-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1770725">1770725</a></li>
<li>D. Vogan, Representations of real reductive Lie groups, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-7643-3037-6</li></ul>

Langlands classification open-in-new

Langlands classification