## Maximal eigenvalues of a Casimir operator and multiplicity-free modules

HTML articles powered by AMS MathViewer

- by Gang Han
- Proc. Amer. Math. Soc.
**141**(2013), 377-382 - DOI: https://doi.org/10.1090/S0002-9939-2012-11317-6
- Published electronically: May 29, 2012
- PDF | 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.## References

- Bertram Kostant,
*Lie algebra cohomology and the generalized Borel-Weil theorem*, Ann. of Math. (2)**74**(1961), 329โ387. MR**142696**, DOI 10.2307/1970237 - Bertram Kostant,
*Eigenvalues of the Laplacian and commutative Lie subalgebras*, Topology**3**(1965), no.ย suppl, suppl. 2, 147โ159 (German). MR**167567**, DOI 10.1016/0040-9383(65)90073-X - Bertram Kostant,
*Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the $\rho$-decomposition $C(\mathfrak {g})=\textrm {End}\, V_\rho \otimes C(P)$, and the $\mathfrak {g}$-module structure of $\bigwedge \mathfrak {g}$*, Adv. Math.**125**(1997), no.ย 2, 275โ350. MR**1434113**, DOI 10.1006/aima.1997.1608 - Bertram Kostant,
*The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations*, Internat. Math. Res. Notices**5**(1998), 225โ252. MR**1616913**, DOI 10.1155/S107379289800018X - Paola Cellini and Paolo Papi,
*ad-nilpotent ideals of a Borel subalgebra*, J. Algebra**225**(2000), no.ย 1, 130โ141. MR**1743654**, DOI 10.1006/jabr.1999.8099 - Ruedi Suter,
*Abelian ideals in a Borel subalgebra of a complex simple Lie algebra*, Invent. Math.**156**(2004), no.ย 1, 175โ221. MR**2047661**, DOI 10.1007/s00222-003-0337-0

## Bibliographic Information

**Gang Han**- Affiliation: Department of Mathematics, College of Science, Zhejiang University, Hangzhou 310027, Peopleโs 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