A Remark on the First Eigenvalue of the Laplace Operator on 1-forms for Compact Inner Symmetric Spaces

TL;DR

This paper investigates the first non-zero eigenvalue of the Laplace operator on 1-forms for compact inner symmetric spaces, linking it to the Casimir eigenvalue of roots of the Lie group, with corrections to previous results on functions.

Contribution

It establishes a precise relationship between the first eigenvalue on 1-forms and the Casimir eigenvalues of roots, correcting earlier inaccuracies on the spectrum for functions.

Findings

Eigenvalue on 1-forms equals the Casimir eigenvalue of roots.

Results connect spectral properties to Lie algebra root data.

Previous incorrect statements on functions' spectrum have been revised.

Abstract

We remark that on a compact inner symmetric space $G/K$, indowed with the Riemmannian metric given by the Killing form of $G$ signed-changed, the first (non-zero) eigenvalue of the Laplace operator on $1$-forms is the Casimir eigenvalue of the highest either long or short root of $G$, according as the highest weight of the isotropy representation is long or short. Some results for the first (non-zero) eigenvalue on functions are derived. This is a revision of the first version of the preprint: a non correct statement about the spectrum on functions has been reviewed.