Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387. MR 0142696 (26:265)
  • [2] B. Kostant, Eigenvalues of a Laplacian and commutative Lie subalgebras, Topology 3 (1965), suppl. 2, 147-159. MR 0167567 (29:4839)
  • [3] B. Kostant, Clifford algebra analogue of the Hopf-Koszul-Samelson Theorem, the $ \rho $-decomposition $ C(\mathfrak{g})=End\ V_\rho \otimes C(P)$, and the $ \mathfrak{g}$-module structure of $ \wedge \mathfrak{g}$, Adv. Math. 125 (1997), 275-350. MR 1434113 (98k:17009)
  • [4] B. Kostant, The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations. Internat. Math. Res. Notices 5 (1998), 225-252. MR 1616913 (99c:17010)
  • [5] P. Cellini, P. Papi, ad-nilpotent ideals of a Borel subalgebra, J. Algebra 225 (2000), no. 1, 130-141. MR 1743654 (2001g:17017)
  • [6] R. Suter, Abelian ideals in a Borel subalgebra of a complex simple Lie algebra, Invent. Math. 156 (2004), no. 1, 175-221. MR 2047661 (2005b:17020)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 17B10, 17B20

Retrieve articles in all journals with MSC (2010): 17B10, 17B20


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.

American Mathematical Society