On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a K\"ahler or Quaternion-K\"ahler structure

TL;DR

This paper investigates the first eigenvalue of the Dirac operator on compact spin symmetric spaces with Kähler or Quaternion-Kähler structures, linking it to holonomy actions and verifying the result on specific symmetric spaces.

Contribution

It establishes a connection between the first Dirac eigenvalue and holonomy actions on spinors, extending the result to all relevant holonomies in Berger's list.

Findings

The first eigenvalue relates to the lowest holonomy action on spinors.

Verification of the result on the symmetric space F4/Spin9.

Correction of an earlier incorrect statement in related literature.

Abstract

It is shown that on a compact spin symmetric space with a K\"ahler or Quaternion-K\"ahler 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.