View source for Discrete series (of representations)

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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

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

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

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

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

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

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|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====
<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====
Especially for a semi-simple [[Lie group|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|harmonic analysis]] on the group.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.S. Varadarajan,   "Harmonic analysis on real reductive groups" , Springer  (1977)</TD></TR></table>