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The family of continuous irreducible unitary representations of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331201.png" /> which are equivalent to the subrepresentations of the regular representation of this group. If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331202.png" /> is unimodular, then a continuous irreducible unitary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331203.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331204.png" /> belongs to the discrete series if and only if the matrix entries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331205.png" /> lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331206.png" />. In such a case there exists a positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331207.png" />, known as the formal degree of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331208.png" />, such that the relations
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d0331209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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The family of continuous irreducible unitary representations of a locally compact group  $  G $
 +
which are equivalent to the subrepresentations of the regular representation of this group. If the group  $  G $
 +
is unimodular, then a continuous irreducible unitary representation  $  \pi $
 +
of  $  G $
 +
belongs to the discrete series if and only if the matrix entries of  $  \pi $
 +
lie in  $  L _ {2} ( G) $.  
 +
In such a case there exists a positive number  $  d _  \pi  $,
 +
known as the formal degree of the representation  $  \pi $,
 +
such that the relations
  
are satisfied for all vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312011.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312012.png" /> of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312015.png" /> are two non-equivalent representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312016.png" /> in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312018.png" />, respectively, which belong to the discrete series, then the relations
+
$$ \tag{1 }
 +
\int\limits _ { G } ( \pi ( g) \xi , \eta ) ( {\pi ( g) \xi  ^  \prime  , \eta  ^  \prime
 +
} bar )  dg  = d _  \pi  ^ {-} 1 ( \xi , \xi  ^  \prime  ) ( {\eta , \eta
 +
^  \prime  } bar );
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{2 }
 +
( \pi ( g) \xi , \eta ) * ( \pi ( g) \xi  ^  \prime  , \eta  ^  \prime  )  = d _  \pi  ^ {-} 1 ( \xi , \eta  ^  \prime  ) ( \pi ( g) \xi  ^  \prime  , \eta ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
are satisfied for all vectors  $  \xi , \eta , \xi  ^  \prime  , \eta  ^  \prime  $
 +
of the space  $  H _  \pi  $
 +
of the representation  $  \pi $.  
 +
If  $  \pi _ {1} $
 +
and  $  \pi _ {2} $
 +
are two non-equivalent representations of  $  G $
 +
in the spaces  $  H _ {1} $
 +
and  $  H _ {2} $,
 +
respectively, which belong to the discrete series, then the relations
  
are valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312022.png" />. The relations (1)(4) are generalizations of the orthogonality relations for the matrix entries of representations of compact topological groups (cf. [[Representation of a compact group|Representation of a compact group]]); the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312023.png" /> is compact if and only if all continuous irreducible unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312024.png" /> belong to the discrete series, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312025.png" /> is compact and the Haar measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312026.png" /> satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312027.png" />, then the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312028.png" /> coincides with the dimension of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312029.png" />. Simply-connected nilpotent real Lie groups and complex semi-simple Lie groups have no discrete series.
+
$$ \tag{3 }
 +
\int\limits _ { G } ( \pi _ {1} ( g) \xi _ {1} \eta _ {1} ) ( {\pi _ {2} ( g)
 +
\xi _ {2} \eta _ {2} } bar ) dg  = 0 ,
 +
$$
  
The equivalence class of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312030.png" /> forming part of the discrete series is a closed point in the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312031.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312032.png" />, and the Plancherel measure of this point coincides with the formal degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312033.png" />; if, in addition, some non-zero matrix entry of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312034.png" /> is summable, the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312035.png" /> is an open point in the support of the regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312036.png" />, but open points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312037.png" /> need not correspond to representations of the discrete series. The properties of discrete series representations may be partly extended to the case of non-unimodular locally compact groups.
+
$$ \tag{4 }
 +
( \pi _ {1} ( g) \xi _ {1} , \eta _ {1} ) * ( \pi _ {2} ( g) \xi _ {2} , \eta _ {2} )  =  0 ,
 +
$$
 +
 
 +
are valid for all  $  \xi _ {1} , \eta _ {1} \in H _ {1} $,
 +
$  \xi _ {2} , \eta _ {2} \in H _ {2} $.
 +
The relations (1)–(4) are generalizations of the orthogonality relations for the matrix entries of representations of compact topological groups (cf. [[Representation of a compact group|Representation of a compact group]]); the group  $  G $
 +
is compact if and only if all continuous irreducible unitary representations of  $  G $
 +
belong to the discrete series, and if  $  G $
 +
is compact and the Haar measure  $  dg $
 +
satisfies the condition  $  \int _ {G} dg = 1 $,
 +
then the number  $  d _  \pi  $
 +
coincides with the dimension of the representation  $  \pi $.
 +
Simply-connected nilpotent real Lie groups and complex semi-simple Lie groups have no discrete series.
 +
 
 +
The equivalence class of a representation $  \pi $
 +
forming part of the discrete series is a closed point in the dual space $  \widehat{G}  $
 +
of the group $  G $,  
 +
and the Plancherel measure of this point coincides with the formal degree d _  \pi  $;  
 +
if, in addition, some non-zero matrix entry of the representation $  \pi $
 +
is summable, the representation $  \pi $
 +
is an open point in the support of the regular representation of $  G $,  
 +
but open points in $  \widehat{G}  $
 +
need not correspond to representations of the discrete series. The properties of discrete series representations may be partly extended to the case of non-unimodular locally compact groups.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312038.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  Harish-Chandra,  "Discrete series for semisimple Lie groups I"  ''Acta Math.'' , '''113'''  (1965)  pp. 241–318</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  Harish-Chandra,  "Discrete series for semisimple Lie groups II"  ''Acta Math.'' , '''116'''  (1966)  pp. 1–111</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Schmid,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312039.png" />-cohomology and the discrete series"  ''Ann. of Math.'' , '''103'''  (1976)  pp. 375–394</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  A. Kleppner,  R. Lipsman,  "The Plancherel formula for group extensions"  ''Ann. Sci. Ecole Norm. Sup.'' , '''5'''  (1972)  pp. 459–516</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  A. Kleppner,  R. Lipsman,  "The Plancherel formula for group extensions II"  ''Ann. Sci. Ecole Norm. Sup.'' , '''6'''  (1973)  pp. 103–132</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312038.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  Harish-Chandra,  "Discrete series for semisimple Lie groups I"  ''Acta Math.'' , '''113'''  (1965)  pp. 241–318</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  Harish-Chandra,  "Discrete series for semisimple Lie groups II"  ''Acta Math.'' , '''116'''  (1966)  pp. 1–111</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Schmid,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033120/d03312039.png" />-cohomology and the discrete series"  ''Ann. of Math.'' , '''103'''  (1976)  pp. 375–394</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  A. Kleppner,  R. Lipsman,  "The Plancherel formula for group extensions"  ''Ann. Sci. Ecole Norm. Sup.'' , '''5'''  (1972)  pp. 459–516</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  A. Kleppner,  R. Lipsman,  "The Plancherel formula for group extensions II"  ''Ann. Sci. Ecole Norm. Sup.'' , '''6'''  (1973)  pp. 103–132</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:36, 5 June 2020


The family of continuous irreducible unitary representations of a locally compact group $ G $ which are equivalent to the subrepresentations of the regular representation of this group. If the group $ G $ is unimodular, then a continuous irreducible unitary representation $ \pi $ of $ G $ belongs to the discrete series if and only if the matrix entries of $ \pi $ lie in $ L _ {2} ( G) $. In such a case there exists a positive number $ d _ \pi $, known as the formal degree of the representation $ \pi $, such that the relations

$$ \tag{1 } \int\limits _ { G } ( \pi ( g) \xi , \eta ) ( {\pi ( g) \xi ^ \prime , \eta ^ \prime } bar ) dg = d _ \pi ^ {-} 1 ( \xi , \xi ^ \prime ) ( {\eta , \eta ^ \prime } bar ); $$

$$ \tag{2 } ( \pi ( g) \xi , \eta ) * ( \pi ( g) \xi ^ \prime , \eta ^ \prime ) = d _ \pi ^ {-} 1 ( \xi , \eta ^ \prime ) ( \pi ( g) \xi ^ \prime , \eta ) , $$

are satisfied for all vectors $ \xi , \eta , \xi ^ \prime , \eta ^ \prime $ of the space $ H _ \pi $ of the representation $ \pi $. If $ \pi _ {1} $ and $ \pi _ {2} $ are two non-equivalent representations of $ G $ in the spaces $ H _ {1} $ and $ H _ {2} $, respectively, which belong to the discrete series, then the relations

$$ \tag{3 } \int\limits _ { G } ( \pi _ {1} ( g) \xi _ {1} \eta _ {1} ) ( {\pi _ {2} ( g) \xi _ {2} \eta _ {2} } bar ) dg = 0 , $$

$$ \tag{4 } ( \pi _ {1} ( g) \xi _ {1} , \eta _ {1} ) * ( \pi _ {2} ( g) \xi _ {2} , \eta _ {2} ) = 0 , $$

are valid for all $ \xi _ {1} , \eta _ {1} \in H _ {1} $, $ \xi _ {2} , \eta _ {2} \in H _ {2} $. The relations (1)–(4) are generalizations of the orthogonality relations for the matrix entries of representations of compact topological groups (cf. Representation of a compact group); the group $ G $ is compact if and only if all continuous irreducible unitary representations of $ G $ belong to the discrete series, and if $ G $ is compact and the Haar measure $ dg $ satisfies the condition $ \int _ {G} dg = 1 $, then the number $ d _ \pi $ coincides with the dimension of the representation $ \pi $. Simply-connected nilpotent real Lie groups and complex semi-simple Lie groups have no discrete series.

The equivalence class of a representation $ \pi $ forming part of the discrete series is a closed point in the dual space $ \widehat{G} $ of the group $ G $, and the Plancherel measure of this point coincides with the formal degree $ d _ \pi $; if, in addition, some non-zero matrix entry of the representation $ \pi $ is summable, the representation $ \pi $ is an open point in the support of the regular representation of $ G $, but open points in $ \widehat{G} $ need not correspond to representations of the discrete series. The properties of discrete series representations may be partly extended to the case of non-unimodular locally compact groups.

References

[1] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
[2a] Harish-Chandra, "Discrete series for semisimple Lie groups I" Acta Math. , 113 (1965) pp. 241–318
[2b] Harish-Chandra, "Discrete series for semisimple Lie groups II" Acta Math. , 116 (1966) pp. 1–111
[3] W. Schmid, "-cohomology and the discrete series" Ann. of Math. , 103 (1976) pp. 375–394
[4a] A. Kleppner, R. Lipsman, "The Plancherel formula for group extensions" Ann. Sci. Ecole Norm. Sup. , 5 (1972) pp. 459–516
[4b] A. Kleppner, R. Lipsman, "The Plancherel formula for group extensions II" Ann. Sci. Ecole Norm. Sup. , 6 (1973) pp. 103–132

Comments

Especially for a semi-simple Lie group the representations belonging to the discrete series of the group or of some of its subgroups play an essential role in the harmonic analysis on the group.

References

[a1] V.S. Varadarajan, "Harmonic analysis on real reductive groups" , Springer (1977)
How to Cite This Entry:
Discrete series (of representations). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_series_(of_representations)&oldid=46736
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article