2.1 Quadratic spaces and GSpin groups

Let $p$ be an odd prime. Let $(V,q)$ be a non-degenerate self-dual quadratic space over $\mathbb{Z}_{p}$ of rank $n\geqslant 3$ . By definition, the Clifford algebra $C(V)$ is the quotient of the tensor algebra $V^{\otimes }$ by the two-sided ideal generated by elements of the form $v\otimes v-q(v)$ . It is free of rank $2^{n}$ over $\mathbb{Z}_{p}$ . The linear map $v\mapsto -v$ preserves the quadratic form $q$ on $V$ and induces an involution on $C(V)$ . This involution decomposes $C(V)=C^{+}(V)\,\oplus \,C^{-}(V)$ into even and odd parts. The image of the injection $V{\hookrightarrow}C^{-}(V)$ generates $C(V)$ as a $\mathbb{Z}_{p}$ -algebra.

We also have a canonical involution $\ast :C(V)\rightarrow C(V)$ , which is a $\mathbb{Z}_{p}$ -linear endomorphism characterized by $(v_{1}v_{2}\cdots v_{k})^{\ast }=v_{k}\cdots v_{2}v_{1}$ for $v_{i}\in V$ . The spinor similitude group $G=\operatorname{GSpin}(V)$ is the reductive group over $\mathbb{Z}_{p}$ such that for a $\mathbb{Z}_{p}$ -algebra $R$ ,

$$\begin{eqnarray}G(R)=\{g\in C^{+}(V)^{\times }:gV_{R}g^{-1}=V_{R},\;g^{\ast }g\in R^{\times }\}.\end{eqnarray}$$

The character $\unicode[STIX]{x1D702}_{G}:G\rightarrow \mathbb{G}_{m}$ given by $g\mapsto g^{\ast }g$ is called the spinor similitude.

Let $G$ act on $C(V)$ by the conjugation action $g\cdot v:=gvg^{-1}$ . This action stabilizes $V$ and preserves the quadratic form $q$ . Thus, we obtain a homomorphism

$$\begin{eqnarray}G\rightarrow \operatorname{SO}(V).\end{eqnarray}$$

The kernel of the above morphism is the central $\mathbb{G}_{m}$ inside $G$ given by the natural inclusion $R^{\times }\subset G(R)$ for any $\mathbb{Z}_{p}$ -algebra $R$ . The restriction of $\unicode[STIX]{x1D702}_{G}$ on the central $\mathbb{G}_{m}$ is given by $g\mapsto g^{2}$ . Note that the central $\mathbb{G}_{m}$ in $G$ is equal to the identity component of the center of $G$ and it is equal to the center of $G$ precisely when $n$ is odd.

2.2 Basic elements in GSpin groups

Let $k=\overline{\mathbb{F}}_{p}$ , $W=W(k)$ and $K=W[1/p]$ . Let $\unicode[STIX]{x1D70E}\in \operatorname{Aut}(W)$ be the lift of the absolute $p$ -Frobenius on $k$ . Let $D=\operatorname{Hom}_{\mathbb{Z}_{p}}(C(V),\mathbb{Z}_{p})$ be the contragredient $G$ -representation of $C(V)$ .

Any $b\in G(K)$ determines two isocrystals

$$\begin{eqnarray}(V_{K},\unicode[STIX]{x1D6F7}=b\circ \unicode[STIX]{x1D70E}),\quad (D_{K},F=b\circ \unicode[STIX]{x1D70E}).\end{eqnarray}$$

Denote by $\mathbb{T}$ the pro-torus over $\mathbb{Q}_{p}$ of character group $\mathbb{Q}$ . Recall that $b\in G(K)$ is basic if its slope morphism $\unicode[STIX]{x1D708}_{b}:\mathbb{T}_{K}\rightarrow G_{K}$ factors through (the identity component of) $Z(G_{K})$ , i.e. factors through the central $\mathbb{G}_{m}$ . By [Reference Howard and PappasHP17, Lemma 4.2.4], $b$ is basic if and only if $(V_{K},\unicode[STIX]{x1D6F7})$ is isoclinic of slope 0, if and only if $(D_{K},F)$ is isoclinic of slope $-\unicode[STIX]{x1D708}_{b}\in \operatorname{Hom}(\mathbb{T}_{K},\mathbb{G}_{m})\cong \mathbb{Q}.$ The map $b\mapsto \unicode[STIX]{x1D708}_{b}$ gives a bijection between the set of basic $\unicode[STIX]{x1D70E}$ -conjugacy classes and the set $\frac{1}{2}\mathbb{Z}$ . Moreover, the $\mathbb{Q}_{p}$ -quadratic space

$$\begin{eqnarray}V_{K}^{\unicode[STIX]{x1D6F7}}=\{x\in V_{K}:\unicode[STIX]{x1D6F7}x=x\}\end{eqnarray}$$

has the same dimension and determinant as $V_{\mathbb{Q}_{p}}$ and has Hasse invariant $(-1)^{2\unicode[STIX]{x1D708}_{b}}$  [Reference Howard and PappasHP17, Proposition 4.2.5].

2.3 Local unramified Shimura–Hodge data

Since $V$ is self-dual, we know that $V_{\mathbb{Q}_{p}}$ has Hasse invariant $+1$ . In particular, $V$ contains at least one hyperbolic plane and we can pick a $\mathbb{Z}_{p}$ -basis $x_{1},\ldots ,x_{n}$ of $V$ such that the Gram matrix of the quadratic form $q$ under this basisFootnote ² is of the form

$$\begin{eqnarray}\left(\begin{array}{@{}cccccc@{}}0 & 1 & & & \\ 1 & 0 & & & \\ & & \ast & & & \\ & & & \ast & & \\ & & & & \ddots & \\ & & & & & \ast \end{array}\right).\end{eqnarray}$$

We will fix $x_{1},\ldots ,x_{n}$ once and for all. Define a cocharacter

$$\begin{eqnarray}\unicode[STIX]{x1D707}:\mathbb{G}_{m}\rightarrow G,\quad t\mapsto t^{-1}x_{1}x_{2}+x_{2}x_{1}.\end{eqnarray}$$

Pick an explicit element $b=x_{3}(p^{-1}x_{1}+x_{2})\in G(\mathbb{Q}_{p})$ . Then one can show that $b$ is basic with $\unicode[STIX]{x1D708}_{b}=\frac{1}{2}$ . Thus, $V_{K}^{\unicode[STIX]{x1D6F7}}$ has the opposite Hasse invariant $-1$ (cf. § 2.2).

Fix any $\unicode[STIX]{x1D6FF}\in C(V)^{\times }$ such that $\unicode[STIX]{x1D6FF}^{\ast }=-\unicode[STIX]{x1D6FF}$ . Then $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FF}}(c_{1},c_{2})=\text{Trd}(c_{1}\unicode[STIX]{x1D6FF}c_{2}^{\ast })$ defines a non-degenerate symplectic form on $C(V)$ , where $\text{Trd}:C(V)\rightarrow \mathbb{Z}_{p}$ is the reduced trace (see [Reference Howard and PappasHP17, § 4.1.2]). We have a closed immersion into the symplectic similitude group

$$\begin{eqnarray}G{\hookrightarrow}\operatorname{GSp}(C(V),\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FF}}).\end{eqnarray}$$

By [Reference Howard and PappasHP17, Proposition 4.2.6], the tuple $(G,b,\unicode[STIX]{x1D707},C(V))$ defines a local unramified Shimura–Hodge datum in the sense of [Reference Howard and PappasHP17, Definition 2.2.4]. In fact, for the fixed $G$ and $\unicode[STIX]{x1D707}$ , the $\unicode[STIX]{x1D70E}$ -conjugacy class of $b$ is the unique basic $\unicode[STIX]{x1D70E}$ -conjugacy class for which $(G,b,\unicode[STIX]{x1D707})$ is a local unramified Shimura–Hodge datum (see [Reference Howard and PappasHP17, Remark 4.2.7]).

2.4 GSpin Rapoport–Zink spaces

There is a unique (up to isomorphism) $p$ -divisible group $\mathbb{X}_{0}/k$ such that its (contravariant) Dieudonné module $\mathbb{D}(\mathbb{X}_{0})$ is given by the $W$ -lattice $D_{W}$ in the isocrystal $D_{K}$ . The non-degenerate symplectic form $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FF}}$ induces a principal polarization $\unicode[STIX]{x1D706}_{0}$ of $\mathbb{X}_{0}$ . Fix a collection of tensors $(s_{\unicode[STIX]{x1D6FC}})$ on $C(V)$ , including the symplectic form $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FF}}$ as a member, that cut out $G$ from $\operatorname{GL}(C(V))$ . By [Reference Howard and PappasHP17, § 4.3], we have a GSpin Rapoport–Zink space

$$\begin{eqnarray}\operatorname{RZ}:=\operatorname{RZ}(G,b,\unicode[STIX]{x1D707},C(V),(s_{\unicode[STIX]{x1D6FC}})).\end{eqnarray}$$

It is a formal scheme over $W$ , together with a closed immersion into the symplectic Rapoport–Zink space $\operatorname{RZ}(\mathbb{X}_{0},\unicode[STIX]{x1D706}_{0})$ . Moreover, the formal scheme $\operatorname{RZ}$ itself depends only on the local unramified Shimura–Hodge datum $(G,b,\unicode[STIX]{x1D707},C(V))$ and not on the choice of the tensors $(s_{\unicode[STIX]{x1D6FC}})$ .

Denote by $(X,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D706})$ the universal triple over $\operatorname{RZ}(\mathbb{X}_{0},\unicode[STIX]{x1D706}_{0})$ , where $X$ is the universal $p$ -divisible group, $\unicode[STIX]{x1D70C}$ is the universal quasi-isogeny and $\unicode[STIX]{x1D706}$ is the universal polarization. Consider the restriction of this triple to the closed formal subscheme $\operatorname{RZ}$ of $\operatorname{RZ}(\mathbb{X}_{0},\unicode[STIX]{x1D706}_{0})$ . We denote this restricted triple also by $(X,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D706})$ and call it the universal triple over $\operatorname{RZ}$ .

Over $\operatorname{RZ}$ , the universal quasi-isogeny $\unicode[STIX]{x1D70C}$ respects the polarizations $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D706}_{0}$ up to a scalar $c(\unicode[STIX]{x1D70C})\in \mathbb{Q}_{p}^{\times }$ , i.e.  $\unicode[STIX]{x1D70C}^{\vee }\circ \unicode[STIX]{x1D706}\circ \unicode[STIX]{x1D70C}=c^{-1}(\unicode[STIX]{x1D70C})\cdot \unicode[STIX]{x1D706}_{0}$ (Zariski locally on $\operatorname{RZ}_{k}$ ). Let $\operatorname{RZ}^{(\ell )}\subseteq \operatorname{RZ}$ be the closed and open formal subscheme where $\operatorname{ord}_{p}(c(\unicode[STIX]{x1D70C}))=\ell$ . We have the decomposition into a disjoint union

$$\begin{eqnarray}\operatorname{RZ}=\coprod _{\ell \in \mathbb{Z}}\operatorname{RZ}^{(\ell )}.\end{eqnarray}$$

In fact, each $\operatorname{RZ}^{(\ell )}$ is connected and they are mutually (non-canonically) isomorphic; see [Reference Howard and PappasHP17, §§ 4.3.3 and 4.3.4].

2.5 The group $J_{b}$

The algebraic group $J_{b}=\operatorname{GSpin}(V_{K}^{\unicode[STIX]{x1D6F7}})$ has $\mathbb{Q}_{p}$ -points

$$\begin{eqnarray}J_{b}(\mathbb{Q}_{p})=\{g\in G(K):gb=b\unicode[STIX]{x1D70E}(g)\}\end{eqnarray}$$

and $J_{b}(\mathbb{Q}_{p})$ acts on $\operatorname{RZ}$ via its action on $\mathbb{X}_{0}$ as quasi-endomorphisms. The action of $g\in J_{b}(\mathbb{Q}_{p})$ on $\operatorname{RZ}$ restricts to isomorphisms

(2.5.0.1) $$\begin{eqnarray}\operatorname{RZ}^{(\ell )}\overset{{\sim}}{\longrightarrow }\operatorname{RZ}^{(\ell +\operatorname{ord}_{p}(\unicode[STIX]{x1D702}_{b}(g)))},\quad \ell \in \mathbb{Z},\end{eqnarray}$$

where $\unicode[STIX]{x1D702}_{b}:J_{b}(\mathbb{Q}_{p})\rightarrow \mathbb{Q}_{p}^{\times }$ is the spinor similitude. In particular, $p^{\mathbb{Z}}\subseteq J_{b}(\mathbb{Q}_{p})$ acts on $\operatorname{RZ}$ and, since $\unicode[STIX]{x1D702}_{b}(p)=p^{2}$ , we have an isomorphism

$$\begin{eqnarray}p^{\mathbb{Z}}\backslash \operatorname{RZ}\cong \operatorname{RZ}^{(0)}\coprod \operatorname{RZ}^{(1)}.\end{eqnarray}$$

Remark 2.5.1. In this article we are interested in studying the fixed locus $\operatorname{RZ}^{g}$ of $\operatorname{RZ}$ under $g\in J_{b}(\mathbb{Q}_{p})$ . By (2.5.0.1), this is non-empty only when $\operatorname{ord}_{p}(\unicode[STIX]{x1D702}_{b}(g))=0$ . Since $p^{\mathbb{Z}}$ is central in $J_{b}(\mathbb{Q}_{p})$ , one could also study $(p^{\mathbb{Z}}\backslash \operatorname{RZ})^{g}$ for $g\in J_{b}(\mathbb{Q}_{p})$ . However, by (2.5.0.1), we know that $(p^{\mathbb{Z}}\backslash \operatorname{RZ})^{g}\neq \varnothing$ only if $\operatorname{ord}_{p}(\unicode[STIX]{x1D702}_{b}(g))$ is even and in this case

$$\begin{eqnarray}(p^{\mathbb{Z}}\backslash \operatorname{RZ})^{g}\cong p^{\mathbb{Z}}\backslash \operatorname{RZ}^{g_{0}},\end{eqnarray}$$

where $g_{0}=p^{-\text{ord}_{p}(\unicode[STIX]{x1D702}_{b}(g))/2}g$ . Hence, the study of $(p^{\mathbb{Z}}\backslash \operatorname{RZ})^{g}$ for general $g$ reduces to the study of $\operatorname{RZ}^{g}$ for $g$ satisfying $\operatorname{ord}_{p}(\unicode[STIX]{x1D702}_{b}(g))=0$ .

2.6 Special endomorphisms

Using the injection $V{\hookrightarrow}C(V)^{\text{op}}$ , we can view

$$\begin{eqnarray}V\subseteq \operatorname{End}_{\mathbb{Z}_{p}}(D)\end{eqnarray}$$

as special endomorphisms of $D$ : the action of $v\in V$ on $D$ is explicitly given by

$$\begin{eqnarray}(vd)(c)=d(vc),\quad d\in D,\quad c\in C(V).\end{eqnarray}$$

Base changing to $K$ gives $V_{K}\subseteq \operatorname{End}_{K}(D_{K})$ . Since the space $\operatorname{End}_{K,F}(D_{K})$ of $F$ -equivariant endomorphisms of $D_{K}$ can be identified with the space $\operatorname{End}^{0}(\mathbb{X}_{0})$ of quasi-endomorphisms of $\mathbb{X}_{0}$ , we obtain an embedding of $\mathbb{Q}_{p}$ -vector spaces

$$\begin{eqnarray}V_{K}^{\unicode[STIX]{x1D6F7}}{\hookrightarrow}\operatorname{End}^{0}(\mathbb{X}_{0}).\end{eqnarray}$$

Elements of $V_{K}^{\unicode[STIX]{x1D6F7}}$ are thus viewed as quasi-endomorphisms of $\mathbb{X}_{0}$ and we call them special quasi-endomorphisms.

2.7 Vertex lattices

Definition 2.7.1. A vertex lattice is a $\mathbb{Z}_{p}$ -lattice $\unicode[STIX]{x1D6EC}\subseteq V_{K}^{\unicode[STIX]{x1D6F7}}$ such that

$$\begin{eqnarray}p\unicode[STIX]{x1D6EC}\subseteq \unicode[STIX]{x1D6EC}^{\vee }\subseteq \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

We define

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{0}=\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D6EC}^{\vee }.\end{eqnarray}$$

Then the quadratic form $v\mapsto p\cdot q(v)$ makes $\unicode[STIX]{x1D6FA}_{0}$ a non-degenerate quadratic space over $\mathbb{F}_{p}$ . The type of $\unicode[STIX]{x1D6EC}$ is defined to be $t_{\unicode[STIX]{x1D6EC}}:=\dim _{\mathbb{F}_{p}}\unicode[STIX]{x1D6FA}_{0}.$

By [Reference Howard and PappasHP17, 5.1.2], the type of a vertex lattice is always an even integer $t$ such that $2\leqslant t\leqslant t_{\text{max}}$ , where

$$\begin{eqnarray}t_{\text{max}}=\left\{\begin{array}{@{}ll@{}}n-2\quad & \text{if }n\text{ is even and }\det (V_{\mathbb{Q}_{p}})=(-1)^{n/2}\in \mathbb{Q}_{p}^{\times }/(\mathbb{Q}_{p}^{\times })^{2},\\ n-1\quad & \text{if }n\text{ is odd},\\ n\quad & \text{if }n\text{ is even and }\det (V_{\mathbb{Q}_{p}})\neq (-1)^{n/2}\in \mathbb{Q}_{p}^{\times }/(\mathbb{Q}_{p}^{\times })^{2}.\end{array}\right.\end{eqnarray}$$

It follows that the quadratic space $\unicode[STIX]{x1D6FA}_{0}$ is always non-split, because otherwise a Lagrangian subspace ${\mathcal{L}}\subseteq \unicode[STIX]{x1D6FA}_{0}$ would provide a vertex lattice $\unicode[STIX]{x1D6EC}^{\vee }+{\mathcal{L}}\subseteq V_{K}^{\unicode[STIX]{x1D6F7}}$ of type 0 (cf. [Reference Howard and PappasHP17, 5.3.1]).

2.8 The variety $S_{\unicode[STIX]{x1D6EC}}$

Definition 2.8.1. Define

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D6FA}_{0}\otimes _{\mathbb{F}_{p}}k\cong \unicode[STIX]{x1D6EC}_{W}/\unicode[STIX]{x1D6EC}_{W}^{\vee }.\end{eqnarray}$$

Let $d=t_{\unicode[STIX]{x1D6EC}}/2$ . Let $\operatorname{OGr}(\unicode[STIX]{x1D6FA})$ be the moduli space of Lagrangian subspaces ${\mathcal{L}}\subseteq \unicode[STIX]{x1D6FA}$ . We define $S_{\unicode[STIX]{x1D6EC}}\subseteq \operatorname{OGr}(\unicode[STIX]{x1D6FA})$ to be the reduced closed subscheme of $\operatorname{OGr}(\unicode[STIX]{x1D6FA})$ with $k$ -points given as follows:

$$\begin{eqnarray}\displaystyle S_{\unicode[STIX]{x1D6EC}}(k) & = & \displaystyle \{\text{Lagrangian subspaces }{\mathcal{L}}\subseteq \unicode[STIX]{x1D6FA}:\dim ({\mathcal{L}}+\unicode[STIX]{x1D6F7}({\mathcal{L}}))=d+1\}\nonumber\\ \displaystyle & \cong & \displaystyle \{({\mathcal{L}}_{d-1},{\mathcal{L}}_{d}):{\mathcal{L}}_{d}\subseteq \unicode[STIX]{x1D6FA}\text{Lagrangian},{\mathcal{L}}_{d-1}\subseteq {\mathcal{L}}_{d}\cap \unicode[STIX]{x1D6F7}{\mathcal{L}}_{d},~\dim {\mathcal{L}}_{d-1}=d-1\},\nonumber\end{eqnarray}$$

where the last bijection is given by ${\mathcal{L}}\mapsto ({\mathcal{L}}\cap \unicode[STIX]{x1D6F7}{\mathcal{L}},{\mathcal{L}})$ .

More precisely, for any $k$ -algebra $R$ , the $R$ -points $S_{\unicode[STIX]{x1D6EC}}(R)$ is the set of pairs $({\mathcal{L}}_{d-1},{\mathcal{L}}_{d})$ such that:

• ∙ ${\mathcal{L}}_{d}$ is a totally isotropic $R$ -module local direct summand of $\unicode[STIX]{x1D6FA}\,\otimes _{k}\,R$ of local rank $d$ ;

• ∙ ${\mathcal{L}}_{d-1}$ is an $R$ -module local direct summand of $\unicode[STIX]{x1D6FA}\,\otimes _{k}R$ of local rank $d-1$ ;

• ∙ ${\mathcal{L}}_{d-1}\subset {\mathcal{L}}_{d}\cap \unicode[STIX]{x1D6F7}{\mathcal{L}}_{d}$ , where $\unicode[STIX]{x1D6F7}$ acts on $\unicode[STIX]{x1D6FA}\,\otimes _{k}\,R=\unicode[STIX]{x1D6FA}_{0}\,\otimes _{\mathbb{F}_{p}}\,R$ via the identity on $\unicode[STIX]{x1D6FA}_{0}$ and the $p$ -Frobenius on $R$ . In particular, ${\mathcal{L}}_{d-1}$ is totally isotropic and is a local direct summand of ${\mathcal{L}}_{d}$ and of $\unicode[STIX]{x1D6F7}{\mathcal{L}}_{d}$ . (For the last statement, see Remark 2.8.2 below.)

By [Reference Howard and PappasHP17, 5.3.2], $S_{\unicode[STIX]{x1D6EC}}$ is a $k$ -variety with two isomorphic connected components $S_{\unicode[STIX]{x1D6EC}}^{\pm }$ , each being projective and smooth of dimension $t_{\unicode[STIX]{x1D6EC}}/2-1$ . For more details, see [Reference Howard and PappasHP17, § 5.3] and [Reference Howard and PappasHP14, § 3.2].

2.9 Structure of the reduced scheme $\operatorname{RZ}^{\text{red}}$

Consider the reduced subscheme $\operatorname{RZ}^{(\ell ),\text{red}}$ of $\operatorname{RZ}^{(\ell )}$ . By the result [Reference Howard and PappasHP17, Theorem 6.4.1], the irreducible components of $\operatorname{RZ}^{(\ell ),\text{red}}$ are precisely $\{\operatorname{RZ}_{\unicode[STIX]{x1D6EC}}^{(\ell ),\text{red}}\}$ , where $\unicode[STIX]{x1D6EC}$ runs through the vertex lattices of the maximal type $t_{\unicode[STIX]{x1D6EC}}=t_{\text{max}}$ . Moreover, there is an isomorphism of $k$ -schemes [Reference Howard and PappasHP17, Theorem 6.3.1]

(2.9.1.1) $$\begin{eqnarray}p^{\mathbb{Z}}\backslash \operatorname{RZ}_{\unicode[STIX]{x1D6EC}}^{\text{red}}\overset{{\sim}}{\longrightarrow }S_{\unicode[STIX]{x1D6EC}},\end{eqnarray}$$

which also induces an isomorphism between $\operatorname{RZ}_{\unicode[STIX]{x1D6EC}}^{(\ell ),\text{red}}$ and $S_{\unicode[STIX]{x1D6EC}}^{\pm }$ for each $\ell \in \mathbb{Z}$ . In particular, $\operatorname{RZ}^{\text{red}}$ is equidimensional of dimension $t_{\text{max}}/2-1$ .

2.10 The Bruhat–Tits stratification

For any vertex lattices $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ , the intersection $\operatorname{RZ}_{\unicode[STIX]{x1D6EC}_{1}}^{\text{red}}\cap \operatorname{RZ}_{\unicode[STIX]{x1D6EC}_{2}}^{\text{red}}$ is non-empty if and only if $\unicode[STIX]{x1D6EC}_{1}\cap \unicode[STIX]{x1D6EC}_{2}$ is also a vertex lattice, in which case it is equal to $\operatorname{RZ}_{\unicode[STIX]{x1D6EC}_{1}\cap \unicode[STIX]{x1D6EC}_{2}}^{\text{red}}$  [Reference Howard and PappasHP17, Corollary 6.2.4]. In this way we obtain a Bruhat–Tits stratification on $\operatorname{RZ}^{\text{red}}$ . Associated to a vertex lattice $\unicode[STIX]{x1D6EC}$ , we define an open subscheme of $\operatorname{RZ}_{\unicode[STIX]{x1D6EC}}^{\text{red}}$ given by

$$\begin{eqnarray}\operatorname{BT}_{\unicode[STIX]{x1D6EC}}=\operatorname{RZ}_{\unicode[STIX]{x1D6EC}}^{\text{red}}-\mathop{\bigcup }_{\unicode[STIX]{x1D6EC}^{\prime }\subsetneq \unicode[STIX]{x1D6EC}}\operatorname{RZ}_{\unicode[STIX]{x1D6EC}^{\prime }}^{\text{red}}.\end{eqnarray}$$

Then

$$\begin{eqnarray}\operatorname{RZ}^{\text{red}}=\coprod _{\unicode[STIX]{x1D6EC}}\operatorname{BT}_{\unicode[STIX]{x1D6EC}}\end{eqnarray}$$

is a disjoint union of locally closed subschemes, indexed by all vertex lattices.

2.11 Special lattices

One can further parametrize the $k$ -points in each $\operatorname{RZ}_{\unicode[STIX]{x1D6EC}}$ using special lattices.

We have a bijection [Reference Howard and PappasHP17, Proposition 6.2.2]

(2.11.1.1) $$\begin{eqnarray}p^{\mathbb{Z}}\backslash \operatorname{RZ}(k)\overset{{\sim}}{\longrightarrow }\{\text{special lattices }L\subseteq V_{K}\}.\end{eqnarray}$$

To construct this bijection, one uses the fact [Reference Howard and PappasHP17, Corollary 3.2.3] that $p^{\mathbb{Z}}\backslash \operatorname{RZ}(k)$ can be identified with the affine Deligne–Lusztig set

(2.11.1.2) $$\begin{eqnarray}X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}}(k)=\{g\in G(K):g^{-1}b\unicode[STIX]{x1D70E}(g)\in G(W)\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}(p)G(W)\}/G(W).\end{eqnarray}$$

The special lattice associated to $g\in G(K)$ is then given by $g\unicode[STIX]{x1D707}(p^{-1}).V_{W}\subseteq V_{K}$ . Conversely, given a special lattice $L\subseteq V_{K}$ , there exists some $g\in G(K)$ such that $g\unicode[STIX]{x1D707}(p^{-1}).V_{W}=L$ and $g.V_{W}=\unicode[STIX]{x1D6F7}(L)$ . The point in $\operatorname{RZ}(k)$ then corresponds to the image of $g$ in $X_{G,b,\unicode[STIX]{x1D707}^{\unicode[STIX]{x1D70E}}}(k)$ . The Dieudonné module of the $p$ -divisible group at this point is given by $M=gD_{W}\subseteq D_{K}$ and the image of Verschiebung is $(F^{-1}p)M=g\cdot p\unicode[STIX]{x1D707}(p^{-1})D_{W}$ .

Lemma 2.11.2. Suppose that $x_{0}\in \operatorname{RZ}(k)$ corresponds to a special lattice $L$ under (2.11.1.1). Let $M=\mathbb{D}(X_{0})\subset D_{K}$ be the Dieudonné module of the $p$ -divisible group $X_{0}$ corresponding to $x_{0}$ . Then we have

$$\begin{eqnarray}L=\{v\in V_{K}\mid v(F^{-1}p)M\subset (F^{-1}p)M\},\quad \unicode[STIX]{x1D6F7}L=\{v\in V_{K}\mid vM\subset M\}.\end{eqnarray}$$

Here we view $V_{K}\subset \operatorname{End}_{K}(D_{K})$ as in § 2.6.

2.12 Special lattices and vertex lattices

For any vertex lattice $\unicode[STIX]{x1D6EC}$ , the bijection (2.11.1.1) induces a bijection

(2.12.0.1) $$\begin{eqnarray}p^{\mathbb{Z}}\backslash \operatorname{RZ}_{\unicode[STIX]{x1D6EC}}(k)\overset{{\sim}}{\longrightarrow }\{\text{special lattices }L\subseteq V_{K}:\unicode[STIX]{x1D6EC}_{W}^{\vee }\subseteq L\subseteq \unicode[STIX]{x1D6EC}_{W}\}=\{\text{special lattices }L\subseteq V_{K}:\unicode[STIX]{x1D6EC}_{W}^{\vee }\subseteq L\}.\end{eqnarray}$$

Sending a special lattice $L$ to ${\mathcal{L}}:=L/\unicode[STIX]{x1D6EC}_{W}^{\vee }\subseteq \unicode[STIX]{x1D6FA}$ gives a bijection between the right-hand side of (2.12.0.1) and $S_{\unicode[STIX]{x1D6EC}}(k)$ , which is the effect of the isomorphism (2.9.1.1) on $k$ -points.

Definition 2.12.1. For each special lattice $L\subseteq V_{K}$ , there is a unique minimal vertex lattice $\unicode[STIX]{x1D6EC}(L)\subseteq V_{K}^{\unicode[STIX]{x1D6F7}}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(L)_{W}^{\vee }\subseteq L\subseteq \unicode[STIX]{x1D6EC}(L)_{W}.\end{eqnarray}$$

In fact, let $L^{(r)}=L+\unicode[STIX]{x1D6F7}(L)+\cdots +\unicode[STIX]{x1D6F7}^{r}(L)$ . Then there exists a unique integer $1\leqslant d\leqslant t_{\text{max}}/2$ such that $L^{(i)}\subsetneq L^{(i+1)}$ for $i<d$ , and $L^{(d)}=L^{(d+1)}$ . Then $L^{(i+1)}/L^{(i)}$ all have $W$ -length 1 for $i<d$ and

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(L):=(L^{(d)})^{\unicode[STIX]{x1D6F7}}\subseteq V_{K}^{\unicode[STIX]{x1D6F7}}\end{eqnarray}$$

is a vertex lattice of type $2d$ and $\unicode[STIX]{x1D6EC}(L)^{\vee }=L^{\unicode[STIX]{x1D6F7}}$ .

Notice that $\unicode[STIX]{x1D6EC}(L)_{W}$ is the smallest $\unicode[STIX]{x1D6F7}$ -invariant lattice containing $L$ and $\unicode[STIX]{x1D6EC}(L)_{W}^{\vee }$ is the largest $\unicode[STIX]{x1D6F7}$ -invariant lattice contained in $L$ . It follows that the element of $\operatorname{RZ}(k)$ corresponding to a special lattice $L$ lies in $\operatorname{RZ}_{\unicode[STIX]{x1D6EC}}$ if and only if $\unicode[STIX]{x1D6EC}(L)\subseteq \unicode[STIX]{x1D6EC}$ , and it lies in $\operatorname{BT}_{\unicode[STIX]{x1D6EC}}$ if and only if $\unicode[STIX]{x1D6EC}(L)=\unicode[STIX]{x1D6EC}$ . Thus, we have the bijection

(2.12.1.1) $$\begin{eqnarray}p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}(k)\overset{{\sim}}{\longrightarrow }\{L\text{ special lattices}:\unicode[STIX]{x1D6EC}(L)=\unicode[STIX]{x1D6EC}\}.\end{eqnarray}$$

2.13 Deligne–Lusztig varieties

For any vertex lattice $\unicode[STIX]{x1D6EC}$ , by [Reference Howard and PappasHP17, Theorem 6.5.6], $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}$ is a smooth quasi-projective variety of dimension $t_{\unicode[STIX]{x1D6EC}}/2-1$ , isomorphic to a disjoint union of two Deligne–Lusztig varieties $X_{B}(w^{\pm })$ associated to two Coxeter elements $w^{\pm }$ in the Weyl group of $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ . Here $\unicode[STIX]{x1D6FA}_{0}:=\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D6EC}^{\vee }$ is the quadratic space over $\mathbb{F}_{p}$ defined in Definition 2.7.1. In particular, the $k$ -variety $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}$ only depends on the quadratic space $\unicode[STIX]{x1D6FA}_{0}$ .

Let us recall the definition of $X_{B}(w^{\pm })$ . Let $d=t_{\unicode[STIX]{x1D6EC}}/2$ . Let $\langle \cdot ,\cdot \rangle$ be the bilinear pairing on $\unicode[STIX]{x1D6FA}_{0}$ . Since $\unicode[STIX]{x1D6FA}_{0}$ is a non-degenerate non-split quadratic space over $\mathbb{F}_{p}$ (§ 2.7), one can choose a basis $e_{1},\ldots ,e_{d},f_{d},\ldots ,f_{1}$ of $\unicode[STIX]{x1D6FA}$ such that $\langle e_{i},f_{i}\rangle =1~\forall i$ and all other pairings between the basis vectors are 0, and $\unicode[STIX]{x1D6F7}$ fixes $e_{i},f_{i}$ for $i=1,\ldots ,d-1$ and interchanges $e_{d}$ with $f_{d}$ . This choice of basis gives a maximal $\unicode[STIX]{x1D6F7}$ -stable torus $T\subseteq \operatorname{SO}(\unicode[STIX]{x1D6FA})$ (diagonal under this basis) and a $\unicode[STIX]{x1D6F7}$ -stable Borel subgroup $B\supseteq T$ as the common stabilizer of the two complete isotropic flags

$$\begin{eqnarray}{\mathcal{F}}^{\pm }:\langle e_{1}\rangle \subseteq \langle e_{1},e_{2}\rangle \subseteq \cdots \subseteq \langle e_{1},\ldots ,e_{d-1},e_{d}^{\pm }\rangle ,\end{eqnarray}$$

where $e_{d}^{+}:=e_{d}$ and $e_{d}^{-}:=f_{d}$ . Let $s_{i}$ ( $i=1,\ldots ,d-2$ ) be the reflection $e_{i}\leftrightarrow e_{i+1}$ , $f_{i}\leftrightarrow f_{i+1}$ and let $t^{\pm }$ be the reflection $e_{d-1}\leftrightarrow e_{d}^{\pm }$ , $f_{d-1}\leftrightarrow e_{d}^{\mp }$ . Then the Weyl group $W(T)=N(T)/T$ is generated by $s_{1},\ldots ,s_{d-2},t^{+},t^{-}$ . We also know that $W(T)$ sits in a split exact sequence

$$\begin{eqnarray}0\rightarrow (\mathbb{Z}/2\mathbb{Z})^{d-1}\rightarrow W(T)\rightarrow S_{d}\rightarrow 0.\end{eqnarray}$$

Since $\unicode[STIX]{x1D6F7}$ fixes $s_{i}$ and swaps $t^{+}$ and $t^{-}$ , we know that the $d-1$ elements $s_{1},\ldots ,s_{d-2},t^{+}$ (respectively $s_{1},\ldots ,s_{d-2},t^{-}$ ) form a set of representatives of $\unicode[STIX]{x1D6F7}$ -orbits of the simple reflections. Therefore,

$$\begin{eqnarray}w^{\pm }:=t^{\mp }s_{d-2}\cdots s_{2}s_{1}\in W(T)\end{eqnarray}$$

are Coxeter elements of minimal length. The Deligne–Lusztig variety associated to $B$ and the Coxeter element $w^{\pm }$ is defined to be

$$\begin{eqnarray}X_{B}(w^{\pm }):=\{g\in \operatorname{SO}(\unicode[STIX]{x1D6FA})/B:\operatorname{inv}(g,\unicode[STIX]{x1D6F7}(g))=w^{\pm }\},\end{eqnarray}$$

where $\operatorname{inv}(g,h)\in B\backslash \operatorname{SO}(\unicode[STIX]{x1D6FA})/B\cong W(T)$ is the relative position between the two Borels $gBg^{-1}$ and $hBh^{-1}$ . The variety $X_{B}(w^{\pm })$ has dimension $d-1$ . Under the map $g\mapsto g{\mathcal{F}}^{\pm }$ , the disjoint union $X_{B}(w^{+})\coprod X_{B}(w^{-})$ can be identified with the variety of complete isotropic flags

$$\begin{eqnarray}{\mathcal{F}}:{\mathcal{F}}_{1}\subseteq {\mathcal{F}}_{2}\subseteq \cdots \subseteq {\mathcal{F}}_{d}\end{eqnarray}$$

such that ${\mathcal{F}}_{i}={\mathcal{F}}_{i-1}+\unicode[STIX]{x1D6F7}({\mathcal{F}}_{i-1})$ and $\dim _{k}({\mathcal{F}}_{d}+\unicode[STIX]{x1D6F7}({\mathcal{F}}_{d}))=d+1$ . The two components are interchanged by an orthogonal transformation of determinant $-1$ . Notice that such ${\mathcal{F}}$ is determined by the isotropic line ${\mathcal{F}}_{1}$ by

$$\begin{eqnarray}{\mathcal{F}}_{i}={\mathcal{F}}_{1}+\unicode[STIX]{x1D6F7}(F_{1})+\cdots +\unicode[STIX]{x1D6F7}^{i-1}({\mathcal{F}}_{1})\end{eqnarray}$$

and is also determined by the Lagrangian ${\mathcal{F}}_{d}$ by

$$\begin{eqnarray}{\mathcal{F}}_{i}={\mathcal{F}}_{d}\cap \unicode[STIX]{x1D6F7}({\mathcal{F}}_{d})\cap \cdots \cap \unicode[STIX]{x1D6F7}^{d-i}({\mathcal{F}}_{d}).\end{eqnarray}$$

The bijection (2.12.1.1) induces a bijection

(2.13.0.2) $$\begin{eqnarray}p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}(k)\overset{{\sim}}{\longrightarrow }X_{B}(w^{+})(k)\coprod X_{B}(w^{-})(k)\end{eqnarray}$$

by sending a special lattice $L$ with $\unicode[STIX]{x1D6EC}(L)=\unicode[STIX]{x1D6EC}$ to the flag determined by the Lagrangian ${\mathcal{F}}_{d}=L/\unicode[STIX]{x1D6EC}_{W}^{\vee }$ . This bijection is the restriction of the isomorphism (2.9.1.1) to $k$ -points and we obtain the desired isomorphism

(2.13.0.3) $$\begin{eqnarray}p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}\cong X_{B}(w^{+})\coprod X_{B}(w^{-}).\end{eqnarray}$$

2.14 Special cycles

Definition 2.14.3. When $m=n$ and $T(\mathbf{v})$ is non-singular, we obtain a lattice

$$\begin{eqnarray}L(\mathbf{v})=\mathbb{Z}_{p}v_{1}+\cdots +\mathbb{Z}_{p}v_{n}\subseteq V_{K}^{\unicode[STIX]{x1D6F7}}.\end{eqnarray}$$

By the Cartan decomposition, $T(\mathbf{v})\in \operatorname{GL}_{n}(\mathbb{Z}_{p})\operatorname{diag}(p^{r_{1}},p^{r_{2}},\ldots ,p^{r_{n}})\operatorname{GL}_{n}(\mathbb{Z}_{p})$ for a unique non-increasing sequence of integers $r_{1}\geqslant \cdots \geqslant r_{n}$ . Note that if we view the matrix $T(\mathbf{v})^{-1}$ as a linear operator $V_{K}^{\unicode[STIX]{x1D6F7}}\rightarrow V_{K}^{\unicode[STIX]{x1D6F7}}$ using the basis $\mathbf{v}$ , it sends $\mathbf{v}$ to the dual basis of $\mathbf{v}$ and in particular it sends any $\mathbb{Z}_{p}$ -basis of $L(\mathbf{v})$ to a $\mathbb{Z}_{p}$ -basis of $L(\mathbf{v})^{\vee }$ . Therefore, the tuple $(r_{1},\ldots ,r_{n})$ is characterized by the condition that there is a basis $e_{1},\ldots ,e_{n}$ of $L(\mathbf{v})$ such that $p^{-r_{1}}e_{1},\ldots ,p^{-r_{n}}e_{n}$ form a basis of $L(\mathbf{v})^{\vee }$ . From this characterization, we also see that the tuple $(r_{1},\ldots ,r_{n})$ is an invariant only depending on the lattice $L(\mathbf{v})$ . We say that $\mathbf{v}$ is minuscule if $T(\mathbf{v})$ is non-singular and $r_{1}=1,~r_{n}\geqslant 0$ .

3.1 The GSpin Rapoport–Zink subspace

From now on we assume that $n\geqslant 4$ . Suppose that the last basis vector $x_{n}\in V$ has norm 1. Then the quadratic subspace of rank $n-1$

$$\begin{eqnarray}V^{\flat }=\mathbb{Z}_{p}x_{1}+\cdots +\mathbb{Z}_{p}x_{n-1}\end{eqnarray}$$

is also self-dual. Let $G^{\flat }=\operatorname{GSpin}(V^{\flat })$ . Analogously, we define the element

$$\begin{eqnarray}b^{\flat }=x_{3}(p^{-1}x_{1}+x_{2})\in G^{\flat }(\mathbb{Q}_{p})\end{eqnarray}$$

and the cocharacter

$$\begin{eqnarray}\unicode[STIX]{x1D707}^{\flat }:\mathbb{G}_{m}\rightarrow G^{\flat },\quad t\mapsto t^{-1}x_{1}x_{2}+x_{2}x_{1}.\end{eqnarray}$$

As in § 2.4, we have an associated GSpin Rapoport–Zink space

$$\begin{eqnarray}\operatorname{RZ}^{\flat }=\operatorname{RZ}(G^{\flat },b^{\flat },\unicode[STIX]{x1D707}^{\flat },C(V^{\flat })).\end{eqnarray}$$

The embedding $V^{\flat }{\hookrightarrow}V$ induces an embedding of Clifford algebras $C(V^{\flat }){\hookrightarrow}C(V)$ and a closed embedding of group schemes $G^{\flat }{\hookrightarrow}G$ over $\mathbb{Z}_{p}$ , which maps $b^{\flat }$ to $b$ and $\unicode[STIX]{x1D707}^{\flat }$ to $\unicode[STIX]{x1D707}$ . Thus, by the functoriality of Rapoport–Zink spaces [Reference KimKim13, 4.9.6], we have a closed immersion

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}:\operatorname{RZ}^{\flat }{\hookrightarrow}\operatorname{RZ}\end{eqnarray}$$

of formal schemes over $W$ .

3.2 Relation with the special divisor ${\mathcal{Z}}(x_{n})$

For compatible choices of symplectic forms $\unicode[STIX]{x1D713}^{\flat }$ on $C(V^{\flat })$ and $\unicode[STIX]{x1D713}$ on $C(V)$ , the closed embedding of group schemes $\operatorname{GSp}(C(V^{\flat }),\unicode[STIX]{x1D713}^{\flat }){\hookrightarrow}\operatorname{GSp}(C(V),\unicode[STIX]{x1D713})$ induces a closed immersion of symplectic Rapoport–Zink spaces (§ 2.4)

$$\begin{eqnarray}\unicode[STIX]{x1D719}:\operatorname{RZ}(\mathbb{X}_{0}^{\flat },\unicode[STIX]{x1D706}_{0}^{\flat }){\hookrightarrow}\operatorname{RZ}(\mathbb{X}_{0},\unicode[STIX]{x1D706}_{0}).\end{eqnarray}$$

Since we have a decomposition of $\operatorname{GSp}(C(V^{\flat }),\unicode[STIX]{x1D713}^{\flat })$ -representations

$$\begin{eqnarray}C(V)\cong C(V^{\flat })\oplus C(V^{\flat })x_{n},\end{eqnarray}$$

we know that the moduli interpretation of $\unicode[STIX]{x1D719}$ is given by sending a triple $(X^{\flat },\unicode[STIX]{x1D70C}^{\flat },\unicode[STIX]{x1D706}^{\flat })$ to the $p$ -divisible group $X=X^{\flat }\,\oplus \,X^{\flat }$ with the quasi-isogeny $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}^{\flat }\,\oplus \,\unicode[STIX]{x1D70C}^{\flat }$ and polarization $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}^{\flat }\,\oplus \,\unicode[STIX]{x1D706}^{\flat }$ .

By the functoriality of Rapoport–Zink spaces [Reference KimKim13, 4.9.6], we have a commutative diagram of closed immersions

(3.2.0.1)

Here the two vertical arrows are induced by the closed immersions $\operatorname{GSpin}(V^{\flat }){\hookrightarrow}\operatorname{GSp}(C(V^{\flat }),\unicode[STIX]{x1D713}^{\flat })$ and $\operatorname{GSpin}(V){\hookrightarrow}\operatorname{GSp}(C(V),\unicode[STIX]{x1D713})$ (§ 2.4).

Lemma 3.2.1. Diagram (3.2.0.1) is Cartesian, i.e. we have

(3.2.1.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })=\unicode[STIX]{x1D719}(\operatorname{RZ}(\mathbb{X}_{0}^{\flat },\unicode[STIX]{x1D706}_{0}^{\flat }))\cap \operatorname{RZ}\end{eqnarray}$$

inside $\operatorname{RZ}(\mathbb{X}_{0},\unicode[STIX]{x1D706}_{0})$ .

Proof. Let $X^{\flat }$ be the universal $p$ -divisible group over $\operatorname{RZ}^{\flat }$ and $\unicode[STIX]{x1D70C}^{\flat }$ be the universal quasi-isogeny. Then it follows from the commutative diagram (3.2.0.1) that the image of $(X^{\flat },\unicode[STIX]{x1D70C})$ under $\unicode[STIX]{x1D6FF}$ is given by the $p$ -divisible group $(X^{\flat }\,\oplus \,X^{\flat },\unicode[STIX]{x1D70C}^{\flat }\,\oplus \,\unicode[STIX]{x1D70C}^{\flat })$ . Since $x_{n}$ has norm 1, right multiplication by $x_{n}$ swaps the two factors $C(V^{\flat })$ and $C(V^{\flat })x_{n}$ . It follows that the quasi-endomorphism

$$\begin{eqnarray}(\unicode[STIX]{x1D70C}^{\flat }\oplus \unicode[STIX]{x1D70C}^{\flat })\circ x_{n}\circ (\unicode[STIX]{x1D70C}^{\flat }\oplus \unicode[STIX]{x1D70C}^{\flat })^{-1}:(X^{\flat }\oplus X^{\flat })\rightarrow (X^{\flat }\oplus X^{\flat })\end{eqnarray}$$

(uniquely determined by the rigidity of quasi-isogenies) simply swaps the two factors, which is an actual endomorphism (i.e. the swapping) of $X^{\flat }\,\oplus \,X^{\flat }$ . By Definition 2.14.1 of ${\mathcal{Z}}(x_{n})$ , we have $\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\subseteq {\mathcal{Z}}(x_{n})$ .

Conversely, over ${\mathcal{Z}}(x_{n})$ the universal $p$ -divisible group $X$ admits an action of $C(x_{n})^{\text{op}}\otimes C(V)$ , where $C(x_{n})$ is the Clifford algebra of the rank-one quadratic space $\mathbb{Z}_{p}x_{n}$ . Notice that

$$\begin{eqnarray}C(x_{n})^{\text{op}}\otimes C(V)\cong (C(x_{n})^{\text{op}}\otimes C(x_{n}))\oplus (C(x_{n})^{\text{op}}\otimes C(V^{\flat })).\end{eqnarray}$$

It follows that over ${\mathcal{Z}}(x_{n})$ the universal $p$ -divisible group $X$ admits an action of $C(x_{n})^{\text{op}}\otimes C(x_{n})$ , which is isomorphic to the matrix algebra $M_{2}(\mathbb{Z}_{p})$ . The two natural idempotents of $M_{2}(\mathbb{Z}_{p})$ then decompose $X$ into a direct sum of the form $X^{\flat }\,\oplus \,X^{\flat }$ . Hence, ${\mathcal{Z}}(x_{n})\subseteq \unicode[STIX]{x1D719}(\operatorname{RZ}(\mathbb{X}_{0}^{\flat },\unicode[STIX]{x1D706}_{0}^{\flat }))\cap \operatorname{RZ}$ . The latter is equal to $\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })$ by (3.2.1.1) and hence ${\mathcal{Z}}(x_{n})\subseteq \unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })$ .◻

3.3 Arithmetic intersection of GGP cycles

Definition 3.3.1. The closed immersion $\unicode[STIX]{x1D6FF}$ induces a closed immersion of formal schemes

$$\begin{eqnarray}(\operatorname{id},\unicode[STIX]{x1D6FF}):\operatorname{RZ}^{\flat }\rightarrow \operatorname{RZ}^{\flat }\times _{W}\operatorname{RZ}.\end{eqnarray}$$

Denote by $\unicode[STIX]{x1D6E5}$ the image of $(\operatorname{id},\unicode[STIX]{x1D6FF})$ , which we call the GGP cycle.

The embedding $V^{\flat }{\hookrightarrow}V$ also induces an embedding of quadratic spaces $V_{K}^{\flat ,\unicode[STIX]{x1D6F7}}{\hookrightarrow}V_{K}^{\unicode[STIX]{x1D6F7}}$ and hence we can view

$$\begin{eqnarray}J_{b^{\flat }}=\operatorname{GSpin}(V_{K}^{\flat ,\unicode[STIX]{x1D6F7}}){\hookrightarrow}J_{b}\end{eqnarray}$$

as an algebraic subgroup over $\mathbb{Q}_{p}$ .

For any $g\in J_{b}(\mathbb{Q}_{p})$ , we obtain a formal subscheme

$$\begin{eqnarray}g\unicode[STIX]{x1D6E5}:=(\operatorname{id}\times g)\unicode[STIX]{x1D6E5}\subseteq \operatorname{RZ}^{\flat }\times _{W}\operatorname{RZ},\end{eqnarray}$$

via the action of $g$ on $\operatorname{RZ}$ . Our goal is to compute the arithmetic intersection number

$$\begin{eqnarray}\langle \unicode[STIX]{x1D6E5},g\unicode[STIX]{x1D6E5}\rangle\end{eqnarray}$$

when $g$ is regular semisimple and minuscule.

Definition 3.3.2. We say that $g\in J_{b}(\mathbb{Q}_{p})$ is regular semisimple if the

$$\begin{eqnarray}\mathbf{v}(g):=(x_{n},gx_{n},\ldots ,g^{n-1}x_{n})\end{eqnarray}$$

forms a $\mathbb{Q}_{p}$ -basis of $V_{K}^{\unicode[STIX]{x1D6F7}}$ . Equivalently, the fundamental matrix $T(g):=T(\mathbf{v}(g))$ is non-singular (Definition 2.14.1). We say that $g$ is minuscule if $\mathbf{v}(g)$ is minuscule (Definition 2.14.3).

3.4 Fixed points

Let $g\in J_{b}(\mathbb{Q}_{p})$ and let $\operatorname{RZ}^{g}\subseteq \operatorname{RZ}$ be the fixed locus of $g$ . Then by definition we have

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}\cap g\unicode[STIX]{x1D6E5}\cong \unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\cap \operatorname{RZ}^{g}.\end{eqnarray}$$

Definition 3.4.1. Let $g\in J_{b}(\mathbb{Q}_{p})$ be regular semisimple. We define the lattice

$$\begin{eqnarray}L(g):=\mathbb{Z}_{p}x_{n}+\cdots +\mathbb{Z}_{p}g^{n-1}x_{n}\subseteq V_{K}^{\unicode[STIX]{x1D6F7}}.\end{eqnarray}$$

Proposition 3.4.4.

$$\begin{eqnarray}p^{\mathbb{Z}}\backslash (\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\cap \operatorname{RZ}^{g})(k)=\coprod _{\unicode[STIX]{x1D6EC}\in \operatorname{VL}(g)}p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{\bar{g}}(k).\end{eqnarray}$$

3.5 Fixed points in a Bruhat–Tits stratum

Let $\unicode[STIX]{x1D6EC}$ be a vertex lattice and $\unicode[STIX]{x1D6FA}_{0}=\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D6EC}^{\vee }$ (§ 2.7). By the isomorphism (2.13.0.3), $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}$ is the disjoint union of two isomorphic Deligne–Lusztig varieties $X_{B}(w^{\pm })$ associated to the Coxeter elements $w^{\pm }$ for $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ . Write $X:=X_{B}(w^{\pm })$ . To compute $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{\bar{g}}$ , it suffices to compute the $\bar{g}$ -fixed points $X^{\bar{g}}$ .

Remark 3.5.3. Recall that a maximal torus $T^{\prime }$ is of Coxeter type if $T^{\prime }=hTh^{-1}$ for some $h\in \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ such that $h^{-1}\unicode[STIX]{x1D6F7}(h)$ lifts to a Coxeter element $w$ in the Weyl group $W(T)=N(T)/T$ . In other words, $T^{\prime }$ is conjugate to $T$ over $k$ but its Frobenius structure is given by $w\cdot \unicode[STIX]{x1D6F7}$ . For the Coxeter element $w=w^{\pm }$ constructed in § 2.13, we know that an element $(\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{d},\unicode[STIX]{x1D706}_{d}^{-1},\ldots ,\unicode[STIX]{x1D706}_{1}^{-1})$ of $T(k)$ is fixed by $w\cdot \unicode[STIX]{x1D6F7}$ if and only if

$$\begin{eqnarray}(\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2},\ldots ,\unicode[STIX]{x1D706}_{d-1},\unicode[STIX]{x1D706}_{d})=(\unicode[STIX]{x1D706}_{d}^{\mp p},\unicode[STIX]{x1D706}_{1}^{p},\ldots ,\unicode[STIX]{x1D706}_{d-2}^{p},\unicode[STIX]{x1D706}_{d-1}^{\pm p}).\end{eqnarray}$$

It follows that a semisimple element $\bar{g}\in \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})(\mathbb{F}_{p})$ is contained in a maximal torus of Coxeter type if and only if the eigenvalues of $\bar{g}$ on $\unicode[STIX]{x1D6FA}_{0}\otimes k$ belong to a single Galois orbit.

Proof. (1) Suppose that $X^{\bar{g}}$ is non-empty. Then it is a general fact about Deligne–Lusztig varieties that $\bar{g}$ must be semisimple [Reference LusztigLus11, 5.9(a)]. Let $T(w)\subseteq \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ be a torus of Coxeter type (associated to $w=w^{+}$ or $w^{-}$ ) and $B(w)\supseteq T(w)$ be a Borel. Assume that $\bar{g}$ is semisimple. Then we know from [Reference Deligne and LusztigDL76, Proposition 4.7] that $X^{\bar{g}}$ is a disjoint union of Deligne–Lusztig varieties $X_{T^{\prime }\subseteq B^{\prime }}$ for the group $G^{\prime }=Z^{\circ }(\bar{g})$ and the pairs

$$\begin{eqnarray}(T^{\prime },B^{\prime })=(hT(w)h^{-1},hB(w)h^{-1}\cap G^{\prime }),\end{eqnarray}$$

where $h$ runs over classes $G^{\prime }(\mathbb{F}_{p})\backslash \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})(\mathbb{F}_{p})$ such that $\bar{g}\in hT(w)h^{-1}$ . Therefore, $X^{\bar{g}}$ is non-empty if and only if there exists $h\in \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})(\mathbb{F}_{p})$ such that $\bar{g}\in hT(w)h^{-1}$ , if and only if $\bar{g}$ is contained in a maximal torus of Coxeter type (as so is $T(w)$ ).

(2) By part (1), we know that $X^{\bar{g}}$ is further finite if and only if all $X_{T^{\prime }\subseteq B^{\prime }}$ are zero dimensional, if and only if all $B^{\prime }=hBh^{-1}\cap G^{\prime }$ are tori. This happens exactly when $G^{\prime }=Z^{\circ }(\bar{g})$ itself is a torus, i.e. when $\bar{g}$ is regular. In this case, $G^{\prime }$ is a maximal torus of Coxeter type in $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ and the cardinality of $X^{\bar{g}}$ is equal to the cardinality of $N(T(w))(\mathbb{F}_{p})/T(w)(\mathbb{F}_{p})$ . The latter group is isomorphic to $(N(T(w))/T(w))^{\unicode[STIX]{x1D6F7}}$ by Lang’s theorem and hence is isomorphic to the $\unicode[STIX]{x1D6F7}$ -twisted centralizer of $w$ in the Weyl group $W(T)=N(T)/T$ :

$$\begin{eqnarray}Z_{\unicode[STIX]{x1D6F7}}(w):=\{x\in W(T):xw=w\unicode[STIX]{x1D6F7}(x)\}.\end{eqnarray}$$

The cardinality of $Z_{\unicode[STIX]{x1D6F7}}(w)$ is known as the Coxeter number of the group $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ , which is equal to $d=t_{\unicode[STIX]{x1D6EC}}/2$ since $\operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})$ is a non-split even orthogonal group [Reference LusztigLus76/77, 1.15].◻

3.6 Point counting in the minuscule case

Let $g\in J_{b}(\mathbb{Q}_{p})$ be regular semisimple and minuscule. Then $\unicode[STIX]{x1D6FA}_{0}(g):=L(g)^{\vee }/L(g)$ is a $\mathbb{F}_{p}$ -vector space (see Definition 2.14.3) and hence $L(g)^{\vee }$ is a vertex lattice.

From now on we assume that $\operatorname{RZ}^{g}$ is non-empty. Let $\bar{g}\in \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0}(g))(\mathbb{F}_{p})$ be as in Remark 3.6.1.

Definition 3.6.2. For any polynomial $R(T)$ , we define its reciprocal to be

$$\begin{eqnarray}R^{\ast }(T):=T^{\deg R(T)}\cdot R(1/T).\end{eqnarray}$$

We say that $R(T)$ is self-reciprocal if $R(T)=R^{\ast }(T)$ .

Theorem 3.6.4. Assume that $\operatorname{RZ}^{g}$ is non-empty. Then $p^{\mathbb{Z}}\backslash (\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\cap \operatorname{RZ}^{g})(k)$ is non-empty if and only if $P(T)$ has a unique self-reciprocal monic irreducible factor $Q(T)$ such that $m(Q(T))$ is odd. In this case, $p^{\mathbb{Z}}\backslash (\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\cap \operatorname{RZ}^{g})(k)$ is finite and has cardinality

$$\begin{eqnarray}\deg Q(T)\cdot \mathop{\prod }_{R(T)}(1+m(R(T))),\end{eqnarray}$$

where $R(T)$ runs over all non-self-reciprocal monic irreducible factors of $P(T)$ .

Proof. By Proposition 3.4.4, we know that $p^{\mathbb{Z}}\backslash (\unicode[STIX]{x1D6FF}(\operatorname{RZ}^{\flat })\cap \operatorname{RZ}^{g})(k)$ is non-empty if and only if $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{g}$ is non-empty for some $\unicode[STIX]{x1D6EC}\in \operatorname{VL}(g)$ . For any $\unicode[STIX]{x1D6EC}\in \operatorname{VL}(g)$ , by definition we have a chain of inclusions of lattices

$$\begin{eqnarray}L(g)\subseteq \unicode[STIX]{x1D6EC}^{\vee }\subseteq \unicode[STIX]{x1D6EC}\subseteq L(g)^{\vee },\end{eqnarray}$$

which induces a filtration of $\mathbb{F}_{p}$ -vector spaces

$$\begin{eqnarray}0\subseteq \unicode[STIX]{x1D6EC}^{\vee }/L(g)\subseteq \unicode[STIX]{x1D6EC}/L(g)\subseteq \unicode[STIX]{x1D6FA}_{0}(g).\end{eqnarray}$$

It follows that the map $\unicode[STIX]{x1D6EC}\mapsto \unicode[STIX]{x1D6EC}^{\vee }/L(g)$ gives a bijection

(3.6.4.1) $$\begin{eqnarray}\operatorname{VL}(g)\cong \{\text{totally isotropic }\bar{g}\text{-invariant subspaces }U\subseteq \unicode[STIX]{x1D6FA}_{0}(g)\}.\end{eqnarray}$$

By the bijection (3.6.4.1), $\operatorname{VL}(g)$ is non-empty if and only if there is a totally isotropic $\bar{g}$ -invariant subspace $U$ of $\unicode[STIX]{x1D6FA}_{0}(g)$ . Such a subspace $U$ induces a filtration

(3.6.4.2) $$\begin{eqnarray}0\subseteq U\subseteq U^{\bot }\subseteq \unicode[STIX]{x1D6FA}_{0}(g).\end{eqnarray}$$

Since $U$ and $U^{\bot }$ are $\bar{g}$ -invariant, we obtain a decomposition of the characteristic polynomial

(3.6.4.3) $$\begin{eqnarray}P(T)=P_{1}(T)Q(T)P_{2}(T),\end{eqnarray}$$

where $P_{1}(T),Q(T)$ and $P_{2}(T)$ are respectively the characteristic polynomials of $\bar{g}$ acting on the associated graded $U$ , $U^{\bot }/U$ and $\unicode[STIX]{x1D6FA}_{0}(g)/U^{\bot }$ . Notice that the non-degenerate quadratic form on $\unicode[STIX]{x1D6FA}_{0}(g)$ identifies $\unicode[STIX]{x1D6FA}_{0}(g)/U^{\bot }$ with the linear dual of $U$ , from which we know that $P_{2}(T)=P_{1}^{\ast }(T)$ . Similarly, we know that $Q(T)=Q^{\ast }(T)$ , i.e.  $Q(T)$ is self-reciprocal.

Let $\unicode[STIX]{x1D6EC}=L(g)+U^{\bot }$ be the $g$ -vertex lattice corresponding to $U$ under the bijection (3.6.4.1) and let $\unicode[STIX]{x1D6FA}_{0}=\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D6EC}^{\vee }$ and $\bar{g}_{0}\in \operatorname{SO}(\unicode[STIX]{x1D6FA}_{0})(\mathbb{F}_{p})$ be the induced action of $\bar{g}$ on $\unicode[STIX]{x1D6FA}_{0}$ . Since the minimal polynomial of $\bar{g}$ is equal to its characteristic polynomial $P(T)$ (Remark 3.6.1), we know that the minimal polynomial of $\bar{g}_{0}$ is equal to its characteristic polynomial $Q(T)$ under the decomposition (3.6.4.3). If $\bar{g}_{0}$ is semisimple, then its eigenvalues are distinct. If $\bar{g}$ is further contained in a torus of Coxeter type, then we know that its eigenvalues belong to a single Galois orbit (Remark 3.5.3), so $Q(T)$ is irreducible. Conversely, if $Q(T)$ is irreducible, then clearly $\bar{g}_{0}$ is semisimple and contained in a torus of Coxeter type. Hence, we know that $\bar{g}_{0}$ is semisimple and contained in a torus of Coxeter type if and only if $Q(T)$ is irreducible.

Therefore, by Proposition 3.5.2(1), $\operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{\bar{g}_{0}}$ is non-empty if and only if $Q(T)$ is irreducible. In this case, $\bar{g}_{0}$ is indeed regular semisimple and the cardinality of $p^{\mathbb{Z}}\backslash \operatorname{BT}_{\unicode[STIX]{x1D6EC}}^{\bar{g}_{0}}$ is equal to $2\cdot \#X^{\bar{g}_{0}}$ (due to two connected components), which is equal to $\dim _{\mathbb{F}_{p}}\unicode[STIX]{x1D6FA}_{0}=\deg Q(T)$ by Proposition 3.5.2(2).

Since $P_{2}(T)=P_{1}^{\ast }(T)$ , we know that the multiplicity of $R(T)$ in $P_{1}(T)P_{2}(T)$ is even for any self-reciprocal factor $R(T)$ . Hence, $Q(T)$ is the unique self-reciprocal monic irreducible factor of $P(T)$ such that $m(Q(T))$ is odd. Finally, the factorizations (3.6.4.3) with $P_{2}(T)=P_{1}^{\ast }(T)$ correspond bijectively to the filtrations (3.6.4.2). Notice such a factorization is given by a choice of the polynomial

$$\begin{eqnarray}P_{1}(T)=\mathop{\prod }_{R(T)\neq R^{\ast }(T)}R(T)^{e(R(T))}\cdot \mathop{\prod }_{R(T)=R^{\ast }(T)}R(T)^{\lfloor m(R(T))/2\rfloor },\end{eqnarray}$$

where $R(T)$ runs over all monic irreducible factors of $P(T)$ and $0\leqslant e(R(T))\leqslant m(R(T))$ . So, the total number of such factorizations is exactly given by

$$\begin{eqnarray}\mathop{\prod }_{R(T)\neq R^{\ast }(T)}(1+m(R(T))).\end{eqnarray}$$

The proof is finished. ◻

4.1 The analogue of a result of Madapusi Pera on special cycles

We start with a general lemma on Clifford algebras.

Proof. Assume that $v$ is orthogonal to $\unicode[STIX]{x1D709}$ . Then $v\unicode[STIX]{x1D709}=-\unicode[STIX]{x1D709}v$ , so $v$ preserves $\ker (\unicode[STIX]{x1D709})$ .

Conversely, assume that $v$ preserves $\ker (\unicode[STIX]{x1D709})$ . Write $q$ for the quadratic form and write $\langle \,,\rangle$ for the corresponding bilinear pairing. Since ${\mathcal{O}}\unicode[STIX]{x1D709}$ is a direct summand of $\mathbf{L}$ , there exists an ${\mathcal{O}}$ -module homomorphism $\mathbf{L}\rightarrow {\mathcal{O}}$ sending $\unicode[STIX]{x1D709}$ to $1$ . Since $\mathbf{L}$ is self-dual, we know that there exists $\unicode[STIX]{x1D701}\in \mathbf{L}$ representing such a homomorphism. Namely, we have

$$\begin{eqnarray}\langle \unicode[STIX]{x1D701},\unicode[STIX]{x1D709}\rangle =1.\end{eqnarray}$$

It immediately follows that we have an ${\mathcal{O}}$ -module direct sum $\mathbf{L}=\unicode[STIX]{x1D709}^{\bot }\,\oplus \,{\mathcal{O}}\unicode[STIX]{x1D701}$ . Replacing $\unicode[STIX]{x1D701}$ by $\unicode[STIX]{x1D701}-(q(\unicode[STIX]{x1D701})/2)\unicode[STIX]{x1D709}$ , we may arrange that $\unicode[STIX]{x1D701}$ is isotropic. We have

$$\begin{eqnarray}q(\unicode[STIX]{x1D701}+\unicode[STIX]{x1D709})=2\langle \unicode[STIX]{x1D701},\unicode[STIX]{x1D709}\rangle =2\end{eqnarray}$$

and in $C(\mathbf{L})$ we have

$$\begin{eqnarray}q(\unicode[STIX]{x1D701}+\unicode[STIX]{x1D709})=\unicode[STIX]{x1D701}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}\unicode[STIX]{x1D701}.\end{eqnarray}$$

Hence, in $C(\mathbf{L})$ , we have

(4.1.2.1) $$\begin{eqnarray}\unicode[STIX]{x1D709}\unicode[STIX]{x1D701}+\unicode[STIX]{x1D701}\unicode[STIX]{x1D709}=2.\end{eqnarray}$$

Write

$$\begin{eqnarray}v=v_{1}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D701},\end{eqnarray}$$

with $v_{1}\in \unicode[STIX]{x1D709}^{\bot }$ and $\unicode[STIX]{x1D706}\in {\mathcal{O}}$ . By the first part of the proof we know that $v_{1}$ preserves $\ker (\unicode[STIX]{x1D709})$ . Therefore, $\unicode[STIX]{x1D706}\unicode[STIX]{x1D701}$ preserves $\ker (\unicode[STIX]{x1D709})$ . Note that $\unicode[STIX]{x1D709}\in \ker (\unicode[STIX]{x1D709})$ as $\unicode[STIX]{x1D709}$ is isotropic. It follows that, in $C(\mathbf{L})$ ,

This is possible only when $\unicode[STIX]{x1D706}=0$ and hence we have $v=v_{1}\in \unicode[STIX]{x1D709}^{\bot }$ .◻

The next result is a Rapoport–Zink space analogue of [Reference Madapusi PeraMad16, Proposition 5.16], which is in the context of special cycles on $\operatorname{GSpin}$ Shimura varieties. We only state a weaker analogue as it is sufficient for our need. The proof builds on [Reference Madapusi PeraMad16, Proposition 5.16], too. We first introduce some definitions.

Let $y_{0}\in \operatorname{RZ}(k)$ be an arbitrary element. Let $\mathbf{v}$ be as in Definition 2.14.1 such that the special cycle ${\mathcal{Z}}:={\mathcal{Z}}(\mathbf{v})$ contains $y_{0}$ . In particular, $\mathbf{v}\subset L\cap \unicode[STIX]{x1D6F7}L$ by Lemma 2.14.2. Let $\widehat{\operatorname{RZ}}_{y_{0}}$ and $\widehat{{\mathcal{Z}}}_{y_{0}}$ be the formal completions of $\operatorname{RZ}$ and ${\mathcal{Z}}$ at $y_{0}$ , respectively.

Theorem 4.1.7. For any ${\mathcal{O}}\in \mathscr{C}$ , there is a bijection

$$\begin{eqnarray}f_{{\mathcal{O}}}:\widehat{\operatorname{RZ}}_{y_{0}}({\mathcal{O}})\overset{{\sim}}{\longrightarrow }\{\text{isotropic lines in }(\unicode[STIX]{x1D6F7}L)_{{\mathcal{O}}}:=\unicode[STIX]{x1D6F7}L\otimes _{W}{\mathcal{O}}\text{ lifting }\operatorname{Fil}^{1}(\unicode[STIX]{x1D6F7}L)_{k}\}\end{eqnarray}$$

such that the following properties hold. Here we equip $(\unicode[STIX]{x1D6F7}L)_{{\mathcal{O}}}$ with the ${\mathcal{O}}$ -bilinear form obtained by extension of scalars of the $W$ -bilinear form on $\unicode[STIX]{x1D6F7}L$ .

1. (1) $f_{{\mathcal{O}}}$ is functorial in ${\mathcal{O}}\in \mathscr{C}$ in the following sense. Let ${\mathcal{O}}^{\prime }\in \mathscr{C}$ be another object of $\mathscr{C}$ and let $\unicode[STIX]{x1D719}:{\mathcal{O}}\rightarrow {\mathcal{O}}^{\prime }$ be a morphism in $\mathscr{C}$ . Then we have a commuting diagram

   
   Here the top horizontal map is the natural map induced by $\unicode[STIX]{x1D719}$ and the bottom horizontal map is given by base change along $\unicode[STIX]{x1D719}$ .

2. (2) $f_{{\mathcal{O}}}$ restricts to a bijection

   $$\begin{eqnarray}\displaystyle & \displaystyle f_{{\mathcal{O}},\mathbf{v}}:\widehat{{\mathcal{Z}}}_{y_{0}}({\mathcal{O}})\overset{{\sim}}{\longrightarrow } & \displaystyle \nonumber\\ \displaystyle & \displaystyle \{\text{isotropic lines in }(\unicode[STIX]{x1D6F7}L)_{{\mathcal{O}}}\text{ lifting }\operatorname{Fil}^{1}(\unicode[STIX]{x1D6F7}L)_{k}\text{ and orthogonal to the image of }\mathbf{v}\text{ in }(\unicode[STIX]{x1D6F7}L)_{{\mathcal{O}}}\}. & \displaystyle \nonumber\end{eqnarray}$$

Proof. The existence and construction of the bijection $f_{{\mathcal{O}}}$ and the property (1) are consequences of [Reference Madapusi PeraMad16, Proposition 5.16] and the global construction of $\operatorname{RZ}$ in [Reference Howard and PappasHP17] using the integral model of the $\operatorname{GSpin}$ Shimura variety. We explain this more precisely below.

Consider

$$\begin{eqnarray}\mathscr{S}=\mathscr{S}_{U_{p}U^{p}},\end{eqnarray}$$

the canonical integral model over $\mathbb{Z}_{(p)}$ of the Shimura variety associated to the $\operatorname{GSpin}$ Shimura datum associated to a quadratic space $V_{\mathbb{Q}}$ over $\mathbb{Q}$ , at a suitable level $U^{p}$ away from $p$ and a hyperspecial level $U_{p}$ at $p$ . See [Reference Howard and PappasHP17, § 7] or [Reference Madapusi PeraMad16] for more details on this notion. By [Reference Howard and PappasHP17, 7.2.3], we may assume that the following package of data:

• ∙ the Shimura datum associated to $V_{\mathbb{Q}}$ ;

• ∙ the Kuga–Satake Hodge embedding (see [Reference Howard and PappasHP17, 4.14]) of the Shimura datum into a $\operatorname{GSp}$ Shimura datum;

• ∙ the chosen hyperspecial level at $p$ ;

• ∙ an element $x_{00}\in \mathop{\varprojlim }\nolimits_{U^{p}}\mathscr{S}_{U^{p}U_{p}}(k)$

induces, in the fashion of [Reference Howard and PappasHP17, 3.1.4], the local unramified Shimura–Hodge datum that we used to define $\operatorname{RZ}$ . Let $\widehat{\mathscr{S}}$ be the formal scheme over $\mathbb{Z}_{p}$ obtained from $p$ -adic completion of $\mathscr{S}$ and let $\widehat{\mathscr{S}}_{W}$ be the base change to $W$ of $\widehat{\mathscr{S}}$ . Then, as in [Reference Howard and PappasHP17, 3.2.14], we have a morphism of formal schemes over $W$ :

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}:\operatorname{RZ}\rightarrow \widehat{\mathscr{S}}_{W}.\end{eqnarray}$$

We know that $\unicode[STIX]{x1D6E9}$ maps $y_{00}$ to the $k$ -point of $\widehat{\mathscr{S}}_{W}$ induced by $x_{00}$ . Moreover, let

$$\begin{eqnarray}x_{0}:=\unicode[STIX]{x1D6E9}(y_{0})\in \widehat{\mathscr{S}}_{W}(k)=\mathscr{S}(k)\end{eqnarray}$$

and let $\widehat{U}$ be the formal completion of $\mathscr{S}$ at $x_{0}$ (or, what amounts to the same thing, the formal completion of $\widehat{\mathscr{S}}_{W}$ at $x_{0}$ ). By the construction of $\operatorname{RZ}$ in [Reference Howard and PappasHP17, § 3], we know that $\unicode[STIX]{x1D6E9}$ induces an isomorphism $\widehat{\operatorname{RZ}}_{y_{0}}\overset{{\sim}}{\longrightarrow }\widehat{U}$ .

In [Reference Madapusi PeraMad16], two crystals $\mathbf{H}_{\operatorname{cris}},\mathbf{L}_{\operatorname{cris}}$ are constructed on $\mathscr{S}_{k}$ . (In fact, [Reference Madapusi PeraMad16] works over $\mathbb{F}_{p}$ , but we always base change from $\mathbb{F}_{p}$ to $k$ .) Here $\mathbf{H}_{\operatorname{cris}}$ is by definition the first relative crystalline cohomology of the Kuga–Satake abelian scheme over $\mathscr{S}_{k}$ in the sense of [Reference Madapusi PeraMad16].Footnote ⁴ The specialization of $\mathbf{H}_{\operatorname{cris}}$ over $\operatorname{Spec}k$ via $x_{00}$ is identified with the Dieudonné module $C_{W}$ , which is the covariant Diedonné module of the $p$ -divisible group $\mathbb{X}_{0}$ considered in this article (and [Reference Howard and PappasHP17]) and the contravariant Diedonné module of the Kuga–Satake abelian variety at $x_{00}$ considered in [Reference Madapusi PeraMad16].Footnote ⁵ Moreover, the embedding $V{\hookrightarrow}\operatorname{End}_{\mathbb{Z}_{p}}(C)$ has a crystalline realization, which is a sub-crystal $\mathbf{L}_{\operatorname{cris}}$ of $\operatorname{End}(\mathbf{H}_{\operatorname{cris}})$ . For details, see [Reference Madapusi PeraMad16, § 4]. Among others, $\mathbf{L}_{\operatorname{cris}}$ has the following structures:

• ∙ its specialization $\mathbf{L}_{\operatorname{cris},x_{0}}$ to any $x_{0}\in \mathscr{S}(k)$ , viewed as a $W$ -module, has the structure of a $W$ -quadratic space;

• ∙ $\mathbf{L}_{\operatorname{cris},x_{0}}\,\otimes _{W}\,k$ contains a canonical isotropic line $\operatorname{Fil}^{1}(\mathbf{L}_{\operatorname{cris},x_{0}}\,\otimes _{W}\,k)$ .

By the definition of $\unicode[STIX]{x1D6E9}$ and the definition of the parametrization of $\operatorname{RZ}(k)$ by the affine Deligne–Lusztig set (cf. [Reference Howard and PappasHP17, § 2.4]), we know that when $y_{0}\in \operatorname{RZ}(k)$ corresponds to the special lattice $L$ under (2.11.1.1), the following statements are true.

1. (a) There is an isomorphism of Dieudonné modules $(gC)_{W}\overset{{\sim}}{\longrightarrow }\mathbf{H}_{\operatorname{cris},x_{0}}$ .

2. (b) There is a $W$ -linear isometry $(\unicode[STIX]{x1D6F7}L)_{W}\overset{{\sim}}{\longrightarrow }\mathbf{L}_{\operatorname{cris},x_{0}}$ under which $\operatorname{Fil}^{1}(\unicode[STIX]{x1D6F7}L)_{k}$ is identified with

   $$\begin{eqnarray}\operatorname{Fil}^{1}(\mathbf{L}_{\operatorname{ cris},x_{0}}\otimes _{W}k).\end{eqnarray}$$

3. (c) We have a commutative diagram

   
   where:

   • ∙ the right vertical map is induced by the map in (a);

   • ∙ the left vertical map is the map in (b);

   • ∙ the bottom horizontal map arises from the fact that $\mathbf{L}_{\operatorname{cris}}$ is a sub-crystal of $\operatorname{End}(\mathbf{H}_{\operatorname{cris}})$ .

In the rest of the proof we make the identifications in (a) and (b) above and omit them from the notation. Abbreviate $\mathbf{H}:=\mathbf{H}_{\operatorname{cris},x_{0}}$ and $\mathbf{L}:=\mathbf{L}_{\operatorname{cris},x_{0}}$ .

Now, in [Reference Madapusi PeraMad16, Proposition 5.16], Madapusi Pera constructed a bijection

$$\begin{eqnarray}\widehat{U}({\mathcal{O}})\overset{{\sim}}{\longrightarrow }\{\text{isotropic lines in }\mathbf{L}_{{\mathcal{O}}}:=\mathbf{L}\otimes _{W}{\mathcal{O}}\text{ lifting }\operatorname{Fil}^{1}\mathbf{L}_{k}\}.\end{eqnarray}$$

Moreover, by the construction given in [Reference Madapusi PeraMad16, Proposition 5.16], the above bijection is functorial in ${\mathcal{O}}\in \mathscr{C}$ . We define $f_{{\mathcal{O}}}$ as the above bijection precomposed with the isomorphism $\unicode[STIX]{x1D6E9}:\widehat{\operatorname{RZ}}_{y_{0}}\overset{{\sim}}{\longrightarrow }\widehat{U}$ .

It remains to prove property (2). Note that $\mathbf{H}=gC_{W}$ is the covariant Dieudonné module of the $p$ -divisible group $X_{y_{0}}$ over $k$ determined by $y_{0}\in \operatorname{RZ}(k)$ . Given $y\in \widehat{\operatorname{RZ}}_{y_{0}}({\mathcal{O}})$ lifting $y_{0}$ , by Grothendieck–Messing theory (for covariant Dieudonné modules) we know that $y\in \widehat{{\mathcal{Z}}}_{y_{0}}$ if and only if the image of $\mathbf{v}$ in $\operatorname{End}_{{\mathcal{O}}}(\mathbf{H}_{{\mathcal{O}}})$ stabilizes $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}\subset \mathbf{H}_{{\mathcal{O}}}$ , where $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}$ is the Hodge filtration corresponding to the deformation from $k$ to ${\mathcal{O}}$ of the $X_{y_{0}}$ determined by $y$ . Now, as is stated in the proof of [Reference Madapusi PeraMad16, Proposition 5.16],Footnote ⁶ we know that $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}$ is the kernel in $\mathbf{H}_{{\mathcal{O}}}$ of any ${\mathcal{O}}$ -generator $\unicode[STIX]{x1D709}$ of the isotropic line $f_{{\mathcal{O}}}(y)$ . Here $\unicode[STIX]{x1D709}\in \mathbf{L}_{{\mathcal{O}}}$ is viewed as an element of $\operatorname{End}_{{\mathcal{O}}}(\mathbf{H}_{{\mathcal{O}}})$ . By Lemma 4.1.2, $\mathbf{v}$ preserves $\operatorname{Fil}^{1}\mathbf{H}_{{\mathcal{O}}}=\ker \unicode[STIX]{x1D709}$ if and only if $\mathbf{v}$ is orthogonal to $\unicode[STIX]{x1D709}$ (inside $\mathbf{L}_{{\mathcal{O}}}$ ). Thus, $y\in \widehat{{\mathcal{Z}}}_{y_{0}}$ if and only if $f_{{\mathcal{O}}}(y)$ is orthogonal to the image of $\mathbf{v}$ in $\mathbf{L}_{{\mathcal{O}}}=(\unicode[STIX]{x1D6F7}L)_{{\mathcal{O}}}$ .◻

4.2 Reducedness of minuscule special cycles