Descent of tautological sheaves from Hilbert schemes to Enriques manifolds

Let $X$ be a K3 surface which doubly covers an Enriques surface $S$. If $n\in \mathbb{N}$ is an odd number, then the Hilbert scheme of $n$-points $X^{[n]}$ admits a natural quotient $S_{[n]}$. This quotient is an Enriques manifold in the sense of Oguiso and Schröer. In this paper we construct slope stable sheaves on $S_{[n]}$ and study some of their properties.