Algebraic cycles and Lehn-Lehn-Sorger-van Straten eightfolds

This article is about Lehn-Lehn-Sorger-van Straten eightfolds $Z$, and their anti-symplectic involution $\iota$. When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $\iota$ on the Chow group of $0$-cycles of $Z$. The formula is in agreement with the Bloch-Beilinson conjectures, and has some non-trivial consequences for the Chow ring of the quotient.