# 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

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

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

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); 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, "$C^\star$ 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, "$L^2$-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