The Casimir operator of a metric connection with skew-symmetric torsion

For any triple $(M^n,g,\mathrm{\nabla })$ consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second order operator $\mathrm{\Omega }$ acting on spinor fields. In case of a reductive space and its canonical connection our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly K"ahler, cocalibrated ${\mathrm{G}}_2$-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of $\mathrm{\nabla }$-parallel spinors.