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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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 $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra and $\mathfrak b$ a Borel subalgebra. Then $\mathfrak g$ acts on its exterior algebra $\wedge \mathfrak g$ naturally. We prove that the maximal eigenvalue of the Casimir operator on $\wedge \mathfrak g$ is one third of the dimension of $\mathfrak g$, that the maximal eigenvalue $m_i$ of the Casimir operator on $\wedge ^i\mathfrak g$ is increasing for $0\le i\le r$, where $r$ is the number of positive roots, and that the corresponding eigenspace $M_i$ is a multiplicity-free $\mathfrak g$-module whose highest weight vectors correspond to certain ad-nilpotent ideals of $\mathfrak {b}$. We also obtain a result describing the set of weights of the irreducible representation of $\mathfrak g$ with highest weight a multiple of $\rho$, where $\rho$ 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

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.