A relation between zip stacks and moduli stacks of truncated local shtukas

Abstract

Let $G$ be a reductive group over a field $k$, and let $\mu$ be a cocharacter of $G$. We prove that Viehmann’s double coset spaces associated with $(G,\mu )$ are represented by certain Lusztig varieties, and establish a similar result for the mixed characteristic case. This representability enables a comparison between moduli stacks of truncated local shtukas and zip stacks. Over a perfect field of positive characteristic, we establish a homeomorphism between the coarse moduli stack of $1$-$1$-truncated local $G$-shtukas and that of $G$-zips, enriching our understanding of zip period maps in the context of Shimura varieties.