Pappas-Zhu Local Models

• Pappas-Zhu local models are flat projective schemes defined over discrete valuation rings that encode the étale-local structure of Shimura varieties and moduli of shtukas.
• They are constructed as the scheme-theoretic closure of generic Schubert loci in Beilinson–Drinfeld Grassmannians, with special fibres organized by admissible subsets of the extended Iwahori–Weyl group.
• Recent advances have established their normality and Cohen–Macaulayness, with explicit moduli-theoretic realizations enhancing applications in arithmetic geometry.

Pappas–Zhu local models are flat projective schemes over a discrete valuation ring that encode the étale-local structure of integral models of Shimura varieties and of moduli of shtukas with parahoric level. In the mixed-characteristic formulation, they are defined as closures of generic Schubert loci inside Beilinson–Drinfeld Grassmannians attached to parahoric group schemes; their special fibres are governed by admissible subsets of the Iwahori–Weyl group and decompose into unions of affine Schubert varieties (Haines et al., 2019). Subsequent work has clarified their scheme-theoretic special fibres, extended the construction to Weil-restricted groups, supplied v-sheaf and canonical-scheme formulations, and established Cohen–Macaulayness in full generality, including residue characteristic $2$ and non-reduced root systems (He et al., 6 Mar 2026).

1. Construction and foundational formulations

Let $F$ be a non-archimedean local field with ring of integers $\mathcal O_F$, let $G/F$ be a connected reductive group, let $\mathcal G_f$ be a parahoric group scheme attached to a facet $f$ in the Bruhat–Tits building, and let $\{\mu\}$ be a geometric conjugacy class of cocharacters with reflex field $E$. In the Pappas–Zhu construction, the local model $M^{\mathrm{loc}}_{\mathcal G,\mu}$ is the scheme-theoretic closure of the generic Schubert locus $X_\mu$ in a Beilinson–Drinfeld Grassmannian $F$0 over $F$1. The generic fibre is the Schubert variety $F$2, with

$F$3

and the special fibre embeds into a twisted affine flag variety $F$4 attached to an equal-characteristic model $F$5 and a corresponding facet $F$6 (Haines et al., 2019).

Zhu’s equal-characteristic theory of global Schubert varieties provides the parallel deformation picture over a curve. For a Bruhat–Tits group scheme $F$7 over $F$8, the global Schubert variety $F$9 is flat over a ramified cover $\mathcal O_F$0, has generic fibre the partial flag variety $\mathcal O_F$1, and special fibre supported on a Schubert union in the affine flag variety. The coherence theorem identifies Hilbert functions of these Schubert unions with those of the generic Schubert varieties, a mechanism that underlies scheme-theoretic descriptions of special fibres in several local-model problems (Zhu, 2010).

Levin extended the Pappas–Zhu construction to groups of the form $\mathcal O_F$2, including possibly wildly ramified extensions $\mathcal O_F$3. In that setting one replaces the usual Beilinson–Drinfeld interpolation by a mixed-characteristic Grassmannian $\mathcal O_F$4 built from the minimal polynomial $\mathcal O_F$5 of a uniformizer, with generic fibre identified with $\mathcal O_F$6 and special fibre identified with the affine flag variety of a parahoric $\mathcal O_F$7 (Levin, 2014).

A later v-sheaf formulation places the same objects in the geometry of diamonds and integral local Shimura varieties. For minuscule $\mathcal O_F$8, the v-sheaf local model $\mathcal O_F$9 is defined as the closure of the Schubert cell $G/F$0 inside the Beilinson–Drinfeld affine Grassmannian over $G/F$1, and by results of Anschütz–Gleason–Lourenço–Richarz there is a unique absolutely weakly normal proper flat $G/F$2-scheme $G/F$3 whose associated v-sheaf is $G/F$4 (Daniels et al., 2024).

2. Special fibres and admissible-set combinatorics

The combinatorics of Pappas–Zhu local models is organized by the extended Iwahori–Weyl group

$G/F$5

equipped with its Bruhat order and length function. For a dominant cocharacter $G/F$6, the admissible set is

$G/F$7

At parahoric level $G/F$8, with $G/F$9 the subgroup generated by the corresponding simple affine reflections and $\mathcal G_f$0 the minimal left coset representatives, one has

$\mathcal G_f$1

For Pappas–Zhu local models, the reduced special fibre is the union of $\mathcal G_f$2-Schubert varieties indexed by $\mathcal G_f$3 (He et al., 6 Mar 2026).

The closure relation among the Schubert strata is governed by the Bruhat order. If $\mathcal G_f$4 denotes the corresponding stratum, then

$\mathcal G_f$5

The top-dimensional components correspond to maximal elements of the admissible set and have dimension $\mathcal G_f$6 (He et al., 6 Mar 2026).

In the Haines–Richarz framework, under minimal normality hypotheses on the generic Schubert variety and on maximal Schubert varieties in the admissible locus, the special fibre is not merely a reduced union set-theoretically: it is geometrically reduced and equals the admissible union as a closed subscheme,

$\mathcal G_f$7

where $\mathcal G_f$8 denotes the admissible locus in the twisted affine flag variety (Haines et al., 2019).

Zhu’s proof of the coherence conjecture gives a complementary route to special-fibre identification in equal characteristic and in ramified unitary cases. The equality of Hilbert functions for generic Schubert varieties and admissible Schubert unions forces the special fibre of the corresponding global Schubert variety to coincide scheme-theoretically with the expected admissible union (Zhu, 2010).

3. Normality, reducedness, and Cohen–Macaulayness

The singularity theory of Pappas–Zhu local models developed in stages. Haines and Richarz proved that, under normality assumptions on the generic Schubert variety and on maximal Schubert varieties in the admissible locus, the total space $\mathcal G_f$9 is normal, the special fibre is geometrically reduced, and each irreducible component of the special fibre is normal, Cohen–Macaulay, has rational singularities, and is Frobenius split. They further showed that if $f$0, then the entire local model is Cohen–Macaulay; the proof uses Beilinson–Drinfeld Grassmannians, the coherence theorem, Frobenius splitting, and a homological criterion of Blickle–Schwede (Haines et al., 2019).

The remaining gap in residue characteristic $f$1 and in non-reduced root systems was closed by the shellability theorem of 2026. For any dominant $f$2 and any spherical parahoric $f$3, the augmented admissible set

$f$4

is dual EL-shellable. This resolves a conjecture of Görtz and yields a characteristic-free proof that special fibres of local models satisfying the He–Pappas–Rapoport axiomatic description are Cohen–Macaulay. In particular, Pappas–Zhu local models have Cohen–Macaulay special fibres for all parahoric $f$5, and by flatness the entire local model is Cohen–Macaulay (He et al., 6 Mar 2026).

That argument is intrinsic to admissible sets rather than dependent on Frobenius-splitting techniques. It therefore applies uniformly to the previously open cases of residue characteristic $f$6 and non-reduced root systems, and it gives a new proof that the integral models constructed by Kisin–Pappas–Zhou are Cohen–Macaulay whenever the usual local model diagram is available (He et al., 6 Mar 2026).

The shellability method has a narrower output than a full singularity package. It proves Cohen–Macaulayness but does not imply normality or rational singularities in small characteristics, and in small-prime equal-characteristic situations one may need seminormalizations of Schubert varieties; a plausible implication is that the Cohen–Macaulay and normality problems should be regarded as partially decoupled in these regimes (He et al., 6 Mar 2026).

4. Shellability, augmented admissible sets, and inductive geometry

The need for augmentation comes from the fact that $f$7 has a unique minimal element but typically several maximal elements $f$8. Introducing a formal top element $f$9 produces a graded poset with unique maximal element, which is the appropriate setting for dual EL-shellability. The interior cover relations are labelled by affine roots $\{\mu\}$0, while the augmented edges $\{\mu\}$1 receive labels $\{\mu\}$2 inserted between type I and type II affine roots in a total order refining Bruhat order on $\{\mu\}$3 (He et al., 6 Mar 2026).

A dual EL-labelling has the property that every interval possesses a unique label-increasing maximal chain, and that chain is lexicographically minimal. The shellability theorem constructs such labellings using acute presentations and the quantum Bruhat graph. The associated order complex is Cohen–Macaulay, and in the recursive form used by Görtz this implies that $\{\mu\}$4 is $\{\mu\}$5-Cohen–Macaulay for $\{\mu\}$6 (He et al., 6 Mar 2026).

This combinatorics translates directly into geometry. The explicit shelling order yields a component-by-component construction of the special fibre: first one attaches maximal Schubert components in an order determined by the augmented labels $\{\mu\}$7, then one fills in lower-dimensional strata according to the reflection order on affine roots. At each step the intersection with the previously built part is Cohen–Macaulay of codimension one in the newly added maximal stratum, so Cohen–Macaulayness is preserved inductively (He et al., 6 Mar 2026).

One concrete consequence is that if $\{\mu\}$8 is a lower set in Bruhat order, then the union

$\{\mu\}$9

is Cohen–Macaulay. This gives a controlled partial-construction theorem for unions of top-dimensional components, not only for the complete special fibre (He et al., 6 Mar 2026).

5. Moduli-theoretic realizations and explicit families

Although the Pappas–Zhu construction is group-theoretic, several important families admit explicit moduli descriptions. For ramified unitary groups of signature $E$0 with arbitrary parahoric level and residue characteristic $E$1, Luo proved that the strengthened spin condition $E$2 gives a moduli description of the Pappas–Zhu local model. More precisely, the functor $E$3, defined using $E$4–$E$5 together with $E$6, represents $E$7. In the strongly non-special maximal parahoric case, Luo also obtained explicit equations for the special fibre and proved that it is normal and Cohen–Macaulay (Luo, 2024).

An earlier result of Smithling handled a special maximal parahoric case for ramified quasi-split $E$8 with odd $E$9 and signature $M^{\mathrm{loc}}_{\mathcal G,\mu}$0. There a refined condition $M^{\mathrm{loc}}_{\mathcal G,\mu}$1 defines a closed subscheme $M^{\mathrm{loc}}_{\mathcal G,\mu}$2 satisfying

$M^{\mathrm{loc}}_{\mathcal G,\mu}$3

thus characterizing the flat Pappas–Zhu local model in that case. The analysis proceeds through an explicit affine chart near the worst point and a detailed study of wedge and spin conditions in terms of lattice-linear algebra (Smithling, 2014).

For quasi-split but non-split even orthogonal similitude groups of type $M^{\mathrm{loc}}_{\mathcal G,\mu}$4, the canonical Pappas–Zhu local model was identified with the Pappas–Rapoport spin local model. For arbitrary parahoric level and residue characteristic $M^{\mathrm{loc}}_{\mathcal G,\mu}$5, the spin local model is flat, normal, Cohen–Macaulay, and has reduced special fibre; equivalently, it agrees with the canonical local model. In a maximal parahoric case, an explicit regular semi-stable model is obtained by blowing up the spin local model along the unique closed Schubert cell in its special fibre (Yang et al., 27 Feb 2026).

For orthogonal local models of Hodge type attached to a single lattice, the Pappas–Zhu model admits an explicit affine chart around the worst point given as a hypersurface in a determinantal scheme. The equations show that the special fibre is reduced and Cohen–Macaulay, and they permit an explicit analysis of irreducible components in several parity configurations (Zachos, 2020).

These families illustrate a recurrent pattern: the abstract Schubert-closure construction often coincides with a more concrete lattice-theoretic, wedge, or spin moduli problem, but the identification is highly sensitive to the group, ramification, signature, and parahoric type.

6. Local model diagrams, Shimura varieties, and current directions

The principal arithmetic use of Pappas–Zhu local models is through the local model diagram

$M^{\mathrm{loc}}_{\mathcal G,\mu}$6

with $M^{\mathrm{loc}}_{\mathcal G,\mu}$7 smooth and $M^{\mathrm{loc}}_{\mathcal G,\mu}$8 étale or fppf. Whenever such a diagram is available, geometric properties of the local model transfer to the integral model. In particular, Cohen–Macaulayness of the special fibre of $M^{\mathrm{loc}}_{\mathcal G,\mu}$9 implies the same property for the integral model, and this mechanism underlies the new proofs of Cohen–Macaulayness for integral models considered by Kisin–Pappas and by Kisin–Pappas–Zhou (He et al., 6 Mar 2026).

For Hodge-type Shimura varieties with quasi-parahoric level at $X_\mu$0, canonical integral models are now known to exist and to be controlled by Pappas–Zhu local models. There are v-sheaf local model diagrams and, under the hypotheses that $X_\mu$1 is coprime to $X_\mu$2 and that $X_\mu$3 is acceptable and $X_\mu$4-smooth, scheme-theoretic local model diagrams

$X_\mu$5

with $X_\mu$6 a $X_\mu$7-torsor and $X_\mu$8 smooth and $X_\mu$9-equivariant. The same work proves uniformization of isogeny classes by integral local Shimura varieties and a factorization statement predicted by Kisin–Pappas (Daniels et al., 2024).

Equal-characteristic analogues remain integral to the theory. Global Schubert varieties supply local models for moduli stacks of global $F$00-shtukas with parahoric level, and the normality, reducedness, and Cohen–Macaulayness results for these equal-characteristic local models strengthen the geometric foundation for shtuka moduli (Haines et al., 2019).

Among the current directions, the shellability approach suggests extensions beyond admissible sets to subsets indexing EKOR strata in basic loci of Coxeter type. The conjectural dual EL-shellability of $F$01 would imply Cohen–Macaulayness for unions of EKOR strata. This suggests that the combinatorics of affine Weyl groups may continue to govern singularity questions beyond the traditional local-model setting (He et al., 6 Mar 2026).