Discrete series (of representations)

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

 (1)
 (2)

are satisfied for all vectors of the space of the representation . If and are two non-equivalent representations of in the spaces and , respectively, which belong to the discrete series, then the relations

 (3)
 (4)

are valid for all , . 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 is compact if and only if all continuous irreducible unitary representations of belong to the discrete series, and if is compact and the Haar measure satisfies the condition , then the number coincides with the dimension of the representation . Simply-connected nilpotent real Lie groups and complex semi-simple Lie groups have no discrete series.

The equivalence class of a representation forming part of the discrete series is a closed point in the dual space of the group , and the Plancherel measure of this point coincides with the formal degree ; if, in addition, some non-zero matrix entry of the representation is summable, the representation is an open point in the support of the regular representation of , but open points in 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