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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
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

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.

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