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

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.