Maximal eigenvalues of a Casimir operator and multiplicity-free modules

Author:
Gang Han

Journal:
Proc. Amer. Math. Soc. **141** (2013), 377-382

MSC (2010):
Primary 17B10, 17B20

DOI:
https://doi.org/10.1090/S0002-9939-2012-11317-6

Published electronically:
May 29, 2012

MathSciNet review:
2996942

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Abstract: Let be a finite-dimensional complex semisimple Lie algebra and a Borel subalgebra. Then acts on its exterior algebra naturally. We prove that the maximal eigenvalue of the Casimir operator on is one third of the dimension of , that the maximal eigenvalue of the Casimir operator on is increasing for , where is the number of positive roots, and that the corresponding eigenspace is a multiplicity-free -module whose highest weight vectors correspond to certain ad-nilpotent ideals of . We also obtain a result describing the set of weights of the irreducible representation of with highest weight a multiple of , where is one half the sum of positive roots.

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Additional Information

**Gang Han**

Affiliation:
Department of Mathematics, College of Science, Zhejiang University, Hangzhou 310027, People’s Republic of China

Email:
mathhg@hotmail.com

DOI:
https://doi.org/10.1090/S0002-9939-2012-11317-6

Keywords:
Casimir operator,
exterior algebra,
multiplicity-free module

Received by editor(s):
January 15, 2011

Received by editor(s) in revised form:
June 21, 2011, and June 24, 2011

Published electronically:
May 29, 2012

Additional Notes:
The author was supported by NSFC Grant No. 10801116 and by the Fundamental Research Funds for the Central Universities

Communicated by:
Gail R. Letzter

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.