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Note on Weight Spaces of Irreducible Linear Representations

Published online by Cambridge University Press:  20 November 2018

F. W. Lemire
F. W. Lemire*
Affiliation:
University of British Columbia
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Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 3 , August 1968 , pp. 399 - 403
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Chandra, Harish, Some applications of the universal enveloping algebra of a semi-simple Lie algebra. Trans. Amer. Math. Soc. 70 (1951) 28-99.Google Scholar
2. Lemire, F. W., Irreducible representations of a simple Lie algebra admitting a one dimensional weight space. Proc. Amer. Math. Soc. (to appear).Google Scholar
3. Serre, Jean-Pierre, Algèbres de Lie semi-simples complexes. Benjamin, New York, 1966.Google Scholar