Maximal eigenvalues of a Casimir operator and multiplicity-free modules

Abstract

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