Maximal eigenvalues of a Casimir operator and multiplicity-free modules

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.