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 representable by certain Lusztig varieties, and establish a similar result for the mixed characteristic case. This representability enables a comparison between the 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\text{-}1)-truncated local (G)-shtukas and that of (G)-zips, thereby enriching our understanding of zip period maps in the context of Shimura varieties.