G-Zips Stack: Structure & Applications

• G-Zips Stack is a smooth Artin quotient stack parametrizing generalized F-zip objects with a reductive group structure in characteristic p.
• The stack features a stratification by E-orbits indexed via Weyl group minimal representatives, key to understanding moduli spaces like Shimura varieties.
• Its structure supports automorphic vector bundles and a finitely generated Cox ring, establishing it as a Mori dream space ideal for GIT and related studies.

A G-Zips Stack is a smooth Artin quotient stack, introduced by Pink–Wedhorn–Ziegler, that parametrizes certain generalized $F$-zip objects with reductive group structure over a field of characteristic $p>0$. It serves as a model for stratifications, automorphic vector bundles, and invariants in the geometry of Shimura varieties and related moduli spaces in positive characteristic.

1. Zip Data, Definition, and Stack Structure

Given a connected reductive group $G$ over $\mathbb{F}_p$ and a cocharacter $\mu: \mathbb{G}_{m,k} \to G_k$ (with $k$ an algebraically closed field of characteristic $p$), the cocharacter $\mu$ determines:

• Opposite parabolic subgroups $P_-=P$, $P_+$ in $p>0$0
• Common Levi $p>0$1
• $p>0$2, $p>0$3 with Frobenius map $p>0$4

The zip group is defined as

$p>0$5

where $p>0$6, $p>0$7 are projections to Levi factors.

$p>0$8 acts on $p>0$9 by $G$0. The stack of $G$1-zips of type $G$2 is the quotient stack

$G$3

which is a smooth Artin stack of dimension zero, and consists of finitely many points, each corresponding to an $G$4-orbit in $G$5 (Koskivirta, 2024, Yatsyshyn, 2012, Pink et al., 2012).

2. Stratification, Weyl Group, and Closure Relations

The $G$6-orbit stratification of $G$7 is indexed by the set $G$8 of minimal-length representatives for $G$9, with $\mathbb{F}_p$0 the subset of simple roots determined by $\mathbb{F}_p$1. For $\mathbb{F}_p$2, the corresponding stratum $\mathbb{F}_p$3 is locally closed, smooth, and

$\mathbb{F}_p$4

with closure

$\mathbb{F}_p$5

where $\mathbb{F}_p$6 is a refinement of the Bruhat order. Thus, $\mathbb{F}_p$7–Zip$\mathbb{F}_p$8 is naturally stratified, with a unique open dense stratum ("$\mathbb{F}_p$9-ordinary" locus) and boundary strata of codimension $\mu: \mathbb{G}_{m,k} \to G_k$0 (Koskivirta, 2024, Lang, 19 May 2025, Pink et al., 2012, Goldring et al., 2016).

This stratification underlies, for example, the Ekedahl–Oort stratification of moduli spaces of $\mu: \mathbb{G}_{m,k} \to G_k$1-divisible groups and Shimura varieties (Lopuhaä-Zwakenberg, 2017, Yatsyshyn, 2012).

3. Line Bundles, Global Sections, Hasse Invariants

Every character $\mu: \mathbb{G}_{m,k} \to G_k$2 defines an $\mu: \mathbb{G}_{m,k} \to G_k$3-equivariant line bundle $\mu: \mathbb{G}_{m,k} \to G_k$4 on $\mu: \mathbb{G}_{m,k} \to G_k$5–Zip$\mu: \mathbb{G}_{m,k} \to G_k$6. The Picard group fits into an exact sequence

$\mu: \mathbb{G}_{m,k} \to G_k$7

and, as $\mu: \mathbb{G}_{m,k} \to G_k$8 is finite, rationally

$\mu: \mathbb{G}_{m,k} \to G_k$9

A global section exists for $k$0 on $k$1–Zip$k$2 if and only if:

• $k$3 is trivial ($k$4 the Frobenius-fixed subgroup)
• $k$5 for all $k$6 in the set of simple roots $k$7, where $k$8 solves $k$9

If these hold, $p$0 is $p$1-dimensional. Strict positivity $p$2 for all $p$3 yields a $p$4-ordinary Hasse invariant: a nonvanishing section whose zero locus is the complement of the open stratum (Koskivirta, 2024).

4. The Cox Ring and the Mori Dream Space Property

The Cox ring of $p$5–Zip$p$6 is

$p$7

graded by the effective cone

$p$8

A major result is the finite generation of $p$9: $\mu$0–Zip$\mu$1 is a Mori dream space in the sense that its Cox ring is finitely generated, the effective monoid is a rational polyhedral cone defined by finitely many linear inequalities, and chamber decompositions recover the natural stratification of the stack. These structural properties enable "variation of GIT" arguments and deep control of automorphic forms (Koskivirta, 2024).

5. Automorphic Vector Bundles and Global Sections

Automorphic vector bundles arise from $\mu$2-representations, especially from $\mu$3- and $\mu$4-representations inflated via the projections. For $\mu$5, $\mu$6 gives a bundle $\mu$7 and its global sections form the ring

$\mu$8

with multiplication via tensor product of representations. There is a conjecture of finite generation for $\mu$9; it is established in several groups (e.g., $P_-=P$0, and certain unitary groups) (Koskivirta, 2024).

Explicit criteria for the nonvanishing of sections are given in terms of intersection with weight cones and the action of the Brylinski–Kostant filtration in the presence of additional monodromy operators. These control the existence and dimension of spaces of global sections and connect automorphic forms on $P_-=P$1–zip stacks to those on Shimura varieties (Imai et al., 2020, Koskivirta, 2018, Goldring et al., 2017).

6. Cohomological and Motivic Properties

$P_-=P$2–Zip$P_-=P$3 is zero-dimensional, with motive and compactly supported cohomology closely related to the combinatorics of its $P_-=P$4-orbits. For stacks of local $P_-=P$5-shtukas, the compactly supported motive decomposes as a direct sum over $P_-=P$6–zip stacks indexed by dominant cocharacters, with each motive being Tate and cohomology concentrated in even degrees (Yaylali, 29 Oct 2025).

Perverse sheaves on $P_-=P$7–Zip$P_-=P$8 are classified by the $P_-=P$9-orbit combinatorics together with the representation theory of finite groups of Lie type arising as stabilizers of the orbits, with simple perverse sheaves explicitly described as intersection complexes $P_+$0 for $P_+$1 an orbit and $P_+$2 an irreducible representation of the finite stabilizer (Lang, 15 May 2025).

7. K-Theory, Chow Rings, and Zeta Functions

The equivariant $P_+$3-theory and Chow rings of $P_+$4–Zip$P_+$5 are computed as quotients of the representation or character rings of the Levi $P_+$6, subject to relations imposed by Frobenius. For $P_+$7 with simply connected derived group,

$P_+$8

(Cooper, 2024).

The Chow ring admits presentations as an invariant subring modulo Frobenius-twisted relations, rationally generated by the closures of $P_+$9-orbits. The zeta function of the stack is a rational function determined by the combinatorics of the Weyl group and orbit enumeration (Lopuhaä-Zwakenberg, 2017, Brokemper, 2016).

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• (Koskivirta, 2024) The stack of $p>0$00-zips is a Mori dream space
• (Yatsyshyn, 2012) Purity of G-zips
• (Pink et al., 2012) $p>0$01-zips with additional structure
• (Cooper, 2024) Grothendieck group of the stack of G-Zips
• (Imai et al., 2020) Automorphic vector bundles on the stack of $p>0$02-zips
• (Lang, 15 May 2025) Perverse sheaves on the stack of $p>0$03-zips
• (Goldring et al., 2017) Automorphic vector bundles with global sections on $p>0$04-$p>0$05-schemes
• (Yaylali, 29 Oct 2025) Truncations and the Motive of the Stack of Local $p>0$06-Shtukas
• (Koskivirta, 2018) Automorphic forms on the stack of G-Zips
• (Brokemper, 2016) On the Chow Ring of the Stack of truncated Barsotti-Tate Groups
• (Lopuhaä-Zwakenberg, 2017) The zeta function of stacks of $p>0$07-zips and truncated Barsotti-Tate groups
• (Lang, 19 May 2025) Abstract zip data