On the stability of Einstein metrics carrying a special twisted spinor

We prove linear semi-stability for a large class of Einstein metrics of non-positive scalar curvature. More precisely, we show that any Einstein $n$-manifold with non-positive scalar curvature carrying a parallel twisted pure spin$^r$ spinor is linearly semi-stable, under mild restrictions on $n$ and $r$. We thus extend the parallel spin and spin$^c$ stability results of Dai--Wang--Wei. As an application, our result implies linear semi-stability for all negative quaternion-K{\"a}hler manifolds of dimension greater than $8$.