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Jean-Louis Milhorat

Publications and source records attributed to Jean-Louis Milhorat.

5 recordsLinked to original sources
arXiv2023

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

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.

math.DG↗
arXiv2019

The First Eigenvalue of the Dirac Operator on Compact Outer Spin Symmetric Spaces

In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit expression of the first eigenvalue for "outer" compact spin symmetric spaces.

math.DG↗
arXiv2014

On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a Kähler or Quaternion-Kähler structure

It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a ''{lowest}'' action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for the symmetric space $\mathrm{F}_4/\mathrm{Spin}_9$, proving that it is valid for all the ''{possible}'' holonomies in the Berger's list occurring in that context. The proof requires a characterization of the first eigenvalue of the Dirac operator which was given in Commun. Math. Phys. \textbf{259} (2005), n.~1, 71-78. By the way, we review an incorrect statement in a proof of that article.

math.DG↗
arXiv2005

A formula for the First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

Let $G/K$ be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of $G$ sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of $G$. As an example of application, we give the list of the first eigenvalues for the spin compact irreducible symmetric spaces endowed with a quaternion-Kähler structure.

math.DG↗