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.