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

TL;DR

This paper demonstrates the representability of certain moduli spaces related to local shtukas and zip stacks, establishing a homeomorphism between their coarse moduli stacks in positive characteristic, which advances the understanding of Shimura varieties.

Contribution

It proves the representability of Viehmann's double coset spaces by Lusztig varieties and relates moduli stacks of truncated local shtukas to zip stacks, including in mixed characteristic.

Findings

Viehmann's double coset spaces are representable by Lusztig varieties.

A homeomorphism is established between moduli stacks of truncated local shtukas and G-zips.

Results enhance understanding of zip period maps in Shimura varieties.

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.