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 F4/Spin9, proving that it is valid for all the “possible” holonomies in Berger's list occurring in that context. The proof is based on a characterization of the first eigenvalue of the Dirac operator given in Milhorat (2005) and Milhorat (2006). By the way, we review an incorrect statement in the proof of the first lemma in Milhorat (2005).