Maximal eigenvalues of a Casimir operator and multiplicityfree modules
Author:
Gang Han
Journal:
Proc. Amer. Math. Soc. 141 (2013), 377382
MSC (2010):
Primary 17B10, 17B20
Published electronically:
May 29, 2012
MathSciNet review:
2996942
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Abstract: Let be a finitedimensional 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 multiplicityfree module whose highest weight vectors correspond to certain adnilpotent 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:
http://dx.doi.org/10.1090/S000299392012113176
Keywords:
Casimir operator,
exterior algebra,
multiplicityfree 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.
