Geodesic
One-Dimensional Representations of the Cycle Subalgebra of a Semi-Simple Lie Algebra
Canadian mathematical bulletin, Tome 13 (1970) no. 4, pp. 463-467

Voir la notice de l'article provenant de la source Cambridge University Press

Let L denote a semi-simple, finite dimensional Lie algebra over an algebraically closed field K of characteristic zero. If denotes a Cartan subalgebra of L and denotes the centralizer of in the universal enveloping algebra U of L, then it has been shown that each algebra homomorphism (called a "mass-function" on ) uniquely determines a linear irreducible representation of L. The technique involved in this construction is analogous to the Harish-Chandra construction [2] of dominated irreducible representations of L starting from a linear functional . The difference between the two results lies in the fact that all linear functionals on are readily obtained, whereas since is in general a noncommutative algebra the construction of mass-functions is decidedly nontrivial. 
Lemire, F. W. One-Dimensional Representations of the Cycle Subalgebra of a Semi-Simple Lie Algebra. Canadian mathematical bulletin, Tome 13 (1970) no. 4, pp. 463-467. doi: 10.4153/CMB-1970-085-9
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[1] 1. Bouwer, I. Z., Standard representations of simple Lie algebras, Canad. J. Math. 20 (1968), 344-361. Google Scholar

[2] 2. Harish-Chandra, , On some applications of the universal enveloping algebra of a semi-simple Lie algebra, Trans. Amer. Math. Soc. 70 (1951), 28-99. Google Scholar

[3] 3. Lemire, F. W., Irreducible representations of a simple Lie algebra admitting a one-dimensional weight space, Proc. Amer. Math. Soc. 19 (1968), 1161-1164. Google Scholar

[4] 4. Jacobson, N., Lie algebras, Interscience, New York, 1962. Google Scholar

[5] 5. de Siebenthal, J., Sur certains modules dans une algèbre de Lie semi-simple, Comment. Math. Helv. 44(1969), 1-44. Google Scholar

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