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
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
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.
|||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|
|||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|
|[a1]||V.S. Varadarajan, "Harmonic analysis on real reductive groups" , Springer (1977)|
Discrete series (of representations). A.I. Shtern (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Discrete_series_(of_representations)&oldid=13646