Maximal eigenvalues of a Casimir operator and multiplicity-free modules

Let $\text{g}$ be a finite-dimensional complex semisimple Lie algebra and $\text{b}$ a Borel subalgebra. Then $\text{g}$ acts on its exterior algebra $\text{w}\text{g}$ naturally. We prove that the maximal eigenvalue of the Casimir operator on $\text{w}\text{g}$ is one third of the dimension of $\text{g}$, that the maximal eigenvalue $m_i$ of the Casimir operator on ${\text{w}}^i\text{g}$ 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 $\text{g}$-module whose highest weight vectors corresponding to certain ad-nilpotent ideals of $\text{b}$. We also obtain a result describing the set of weights of the irreducible representation of $\text{g}$ with highest weight a multiple of $\rho$, where $\rho$ is one half the sum of positive roots.