# zbMATH — the first resource for mathematics

Minimal K-types and classification of irreducible representations of reductive Lie groups. (English. Russian original) Zbl 0581.22016
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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.