# Discrete series (of representations)

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).

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