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Perverse sheaves on the stack of $G$-zips

Christopher Lang

TL;DR

The paper addresses the problem of computing simple perverse sheaves on the stack of $G$-zips, which serves as a tractable proxy for Ekedahl-Oort stratifications arising from Shimura varieties. It develops a general framework in which simple perverse sheaves on $G\\text{-Zip}^\\mu$ over $\\overline{\\mathbb{F}}_q$ are classified by pairs $(w,\\theta)$ with $w$ running over the $E_\\mu$-orbits on $G$ (encoded as $^I W$) and $\\theta$ an irreducible representation of the stabilizer’s component group $\\Pi_w$, with $\\Pi_w$ described via Levi subgroups and twisted Frobenius actions. The authors validate and illustrate the framework through explicit group-theoretic computations in several families: GL$_n$ (central and regular cocharacters), Sp$_4$, and GU$_n$ with signature $(1,n-1)$, deriving concrete descriptions of $\\Pi_w$ and its representations and highlighting the role of Deligne-Lusztig theory. They also connect these results to truncated displays by showing that the quotient model $G\\text{-Disp}_\\mu^{W_n}$ preserves simple objects under truncation and that the pullback to displays coherently extends the $G$-zip picture. Overall, the work provides a concrete, computable approach to perverse sheaves on $G$-zips with applications to Shimura varieties and their EO strata, and it establishes a robust bridge to the display-theoretic framework for arithmetic geometry.

Abstract

We explain how to compute simple perverse sheaves on the stack of $G$-zips and do these computations in several examples.

Perverse sheaves on the stack of $G$-zips

TL;DR

The paper addresses the problem of computing simple perverse sheaves on the stack of -zips, which serves as a tractable proxy for Ekedahl-Oort stratifications arising from Shimura varieties. It develops a general framework in which simple perverse sheaves on over are classified by pairs with running over the -orbits on (encoded as ) and an irreducible representation of the stabilizer’s component group , with described via Levi subgroups and twisted Frobenius actions. The authors validate and illustrate the framework through explicit group-theoretic computations in several families: GL (central and regular cocharacters), Sp, and GU with signature , deriving concrete descriptions of and its representations and highlighting the role of Deligne-Lusztig theory. They also connect these results to truncated displays by showing that the quotient model preserves simple objects under truncation and that the pullback to displays coherently extends the -zip picture. Overall, the work provides a concrete, computable approach to perverse sheaves on -zips with applications to Shimura varieties and their EO strata, and it establishes a robust bridge to the display-theoretic framework for arithmetic geometry.

Abstract

We explain how to compute simple perverse sheaves on the stack of -zips and do these computations in several examples.
Paper Structure (6 sections, 8 theorems, 72 equations)

This paper contains 6 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Perverse Sheaves on G-Zips
  3. Examples for GLn
  4. Symplectic Group
  5. Unitary Group with Signature (1,n-1)
  6. Pullback to (Truncated) Displays

Key Result

Theorem 1

Given a connected reductive group $G$ over $\mathbb{F}_q$ and a cocharacter $\mu:\mathbb{G}_{m,\overline{\mathbb{F}}_q}\to G_{\overline{\mathbb{F}}_q}$, the simple perverse sheaves on the stack of $G$-zips of type $\mu$ are given by pairs $(w,\theta)$, where $w$ is an orbit of the $E_\mu$ action on

Theorems & Definitions (21)

  • Theorem 1: Theorem \ref{['thm_simp_perv']}
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4: Pink_2015
  • Theorem 2.5: Pink_2015
  • Example 2.6: $P=G$
  • Example 2.7
  • Example 2.8: $\mathop{\mathrm{GL}}\nolimits_n, Q=B$
  • ...and 11 more