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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Maximal eigenvalues of a Casimir operator and multiplicity-free modules
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by Gang Han PDF
Proc. Amer. Math. Soc. 141 (2013), 377-382 Request permission

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, Peopleโ€™s Republic of China
  • Email: mathhg@hotmail.com
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2996942