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 the Berger’s list occurring in that context. The proof is based on a characterization of the first eigenvalue of the Dirac operator given in [Mil05] and [Mil06]. By the way, we review an incorrect statement in the proof of the first lemma in [Mil05].