Maximal eigenvalues of a Casimir operator and multiplicity-free modules

TL;DR

This paper investigates the eigenvalues of the Casimir operator on the exterior algebra of a semisimple Lie algebra, revealing their maximal values, monotonicity, and the structure of associated multiplicity-free modules.

Contribution

It establishes explicit maximal eigenvalues of the Casimir operator on exterior algebra components and characterizes the structure of the eigenspaces as multiplicity-free modules with specific highest weights.

Findings

Maximal eigenvalue of Casimir on  g is one third of dim g

Eigenvalues on ^i g increase with i up to r

Eigenspaces are multiplicity-free modules with specific highest weights

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 $\w^i\g$ 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 $\g$-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 $\rho$, where $\rho$ is one half the sum of positive roots.