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

Qijun Yan

Abstract

Let \(G\) be a reductive group over a field \(k\), and let \(μ\) be a cocharacter of \(G\). We prove that Viehmann's double coset spaces associated with \((G, μ)\) 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.

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

Abstract

Let be a reductive group over a field , and let be a cocharacter of . We prove that Viehmann's double coset spaces associated with \((G, μ)\) 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 -truncated local -shtukas and that of -zips, thereby enriching our understanding of zip period maps in the context of Shimura varieties.
Paper Structure (28 sections, 26 theorems, 126 equations)

This paper contains 28 sections, 26 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. An open problem
  5. Organization of the Paper
  6. Notation and conventions
  7. Acknowledgements
  8. Preliminaries on Loop Groups and group action data
  9. Loop Groups
  10. Frobenii on the Loop Group
  11. Group action data
  12. Unipotent Groups versus Pro-Unipotent Groups
  13. Representability of $\prescript{}{1}{\mathcal{C}_1^\mu}$
  14. The Sheaf $\prescript{}{1}{\mathcal{C}_1^\mu}$ is Representable
  15. Connections with affine Schubert Varieties
  16. ...and 13 more sections

Key Result

Theorem A

The fpqc sheaf $\prescript{}{1}{\mathcal{C}_1^\mu} = \mathcal{K}_1 \backslash \mathcal{K}\mu(t) \mathcal{K}/\mathcal{K}_1$ is represented by a geometrically connected, smooth, separated scheme of dimension $\dim G$ and of finite type over $k$. It is strongly quasi-affine and a $G^2 := G \times_k G$- where $\mathsf{E}_{\mu} \subseteq P_- \times P_+$ is the zip group of $\mu$ (see Definition Def:Emu

Theorems & Definitions (65)

  • Definition 1.1
  • Theorem A: Representability of $\prescript{}{1}{\mathcal{C}_1^\mu}$
  • Theorem $\text{A}'$
  • Corollary 1.2
  • Corollary 1.3
  • Theorem B
  • Theorem C
  • Conjecture 1.4
  • Theorem 2.1
  • proof
  • ...and 55 more