Descent of tautological sheaves from Hilbert schemes to Enriques manifolds

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