Minimal K-types and classification of irreducible representations of reductive Lie groups. (English. Russian original) Zbl 0581.22016

Zhelobenko, D. P.

Funct. Anal. Appl. 18, 333-335 (1984); translation from Funkts. Anal. Prilozh. 18, No. 4, 79-80 (1984).

Let G be a reductive Lie group and let K be its maximal compact subgroup. A new, simpler approach, based on the theory of S-algebras [see the author, Dokl. Akad. Nauk SSSR 273, 785-788 (1983; Zbl 0568.17005)], to the minimal K-types [see D. A. Vogan jun., Ann. Math., II. Ser. 109, 1-60 (1979; Zbl 0424.22010)] is given. The proof of the main result on the single appearance of minimal K-types is very elegant and short. A classification scheme of \(\hat G\) based on K-types is sketched.

Reviewer: E.Ljubenova

MSC:

22E47	Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E60	Lie algebras of Lie groups
22E46	Semisimple Lie groups and their representations
17B20	Simple, semisimple, reductive (super)algebras

Keywords:

irreducible representations; reductive Lie group; S-algebras; minimal K- types

References:

[1]	D. A. Vogan, ”The algebraic structure of the representation of semisimple Lie groups. I,” Ann. Math.,109, No. 1, 1-60 (1979). · Zbl 0424.22010 · doi:10.2307/1971266
[2]	D. P. Zhelobenko, ”S-Algebras and Verma modules over reductive Lie algebras,” Dokl. Akad. Nauk SSSR,273, No. 4, 785-788 (1983). · Zbl 0568.17005
[3]	D. P. Zhelobenko, ”Z-Algebras over reductive Lie algebras,” Dokl. Akad. Nauk SSSR,273, No. 6, 1301-1304 (1983). · Zbl 0568.17006
[4]	D. P. Zhelobenko, Harmonic Analysis on Semisimple Complex Lie Groups [in Russian], Nauka, Moscow (1974).
[5]	H. Hecht and W. Schmid, ”A proof of Blattner’s conjecture,” Invent. Math.,31, No. 2, 129-154 (1975). · Zbl 0319.22012 · doi:10.1007/BF01404112