Maximal eigenvalues of a Casimir operator and multiplicity-free modules

2011/03/18 by Gang Han, Han, Gang
Mathematics · #17B10 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10

paper · pdf · doi:10.48550/arxiv.1103.3545

arxiv created 2011/03/18 · openalex publication_date 2011/03/18 · arxiv updated 2011/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \g be a finite-dimensional complex semisimple Lie algebra and \b a Borel subalgebra. Then \g acts on its exterior algebra \w\g naturally. We prove that the maximal eigenvalue of the Casimir operator on \w\g is one third of the dimension of \g, that the maximal eigenvalue m_i of the Casimir operator on \wⁱ\g is increasing for 0≤ i≤ r, where r is the number of positive roots, and that the corresponding eigenspace M_i is a multiplicity-free \g-module whose highest weight vectors corresponding to certain ad-nilpotent ideals of \b. We also obtain a result describing the set of weights of the irreducible representation of \g with highest weight a multiple of ρ, where ρ is one half the sum of positive roots.

Citations

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra

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