2.1 The moduli space ${\mathcal N}_{E/F}^h(r,n-r)$

We fix a prime $p>2$ . Let F be a finite extension of ${\mathbb Q}_p$ , with ring of integers $O_F$ and residue field ${\mathbb F}_q$ . We fix a uniformizer $\pi $ . Let E be a quadratic unramified extension of F, with ring of integers $O_E$ and residue field ${\mathbb F}_{q^2}$ . Let $\breve {E}$ be the completion of a maximal unramified extension of E. Denote by $^*$ the nontrivial Galois automorphism of E over F. We recall the definition of a strict formal $O_F$ -module from [Reference Rapoport and ZinkRZ17].

Definition 2.1. Let S be a scheme such that p is locally nilpotent in $O_S$ . A formal $O_F$ -module over a scheme S is a formal p-divisible group X over S with an $O_F$ -action

$$ \begin{align*} i_X:O_F \rightarrow \operatorname{\mathrm{End}} X. \end{align*} $$

Let X be a formal $O_F$ -module over an $O_F$ -scheme S. We call X a strict formal $O_F$ -module if $O_F$ acts on $\operatorname {\mathrm {Lie}} X$ via the structure morphism $O_F \rightarrow O_S$ . For $S=\operatorname {\mathrm {Spec}} k$ , where k is a field of characteristic p, a strict formal $O_F$ -module X over S is called supersingular if all slopes of X as a strict $O_F$ -module are $1/2$ .

Let h be an integer with $0 \leq h \leq n$ . We fix a triple $(\mathbb {X}, i_{{\mathbb X}}, \lambda _{{\mathbb X}})$ consisting of the following data:

(1) $\mathbb {X}$ is a supersingular strict formal $O_F$ -module of F-height $2n$ over ${\mathbb F}_{q^2}$ ;

(2) $i_{{\mathbb X}}:O_E \rightarrow \operatorname {\mathrm {End}} {\mathbb X}$ is an $O_E$ -action on ${\mathbb X}$ that extends the $O_F$ -action on ${\mathbb X}$ ;

(3) $\lambda _{{\mathbb X}}$ is a polarization

$$ \begin{align*} \lambda_{{\mathbb X}} :{\mathbb X} \rightarrow {\mathbb X}^{\vee}, \end{align*} $$

such that the corresponding Rosati involution induces the involution $^*$ on $O_E$ .

We also assume that $({\mathbb X},i_{{\mathbb X}},\lambda _{{\mathbb X}})$ satisfies the following conditions.

(a) For all $a \in O_E$ , the action $i_{{\mathbb X}}$ satisfies

$$ \begin{align*} \operatorname{\mathrm{Charpol}}(i_{{\mathbb X}}(a) \vert \operatorname{\mathrm{Lie}} {\mathbb X}) = (T-a)^r(T-a^*)^{n-r}. \end{align*} $$

Here, we view $(T-a)^r(T-a^*)^{n-r}$ as an element of ${\mathbb F}_{q^2}[T]$ via the structure morphism. We call this condition the determinant condition of signature $(r,n-r)$ .

(b) We assume that $\operatorname {\mathrm {Ker}} \lambda _{{\mathbb X}} \subset {\mathbb X}[\pi ]$ and its order is $q^{2h}$ .

Now, we can define our moduli problem.

Let (Nilp) be the category of $O_E$ -schemes S such that $\pi $ is locally nilpotent on S. Let ${\mathcal N}_{E/F}^h(r,n-r)$ be the set-valued functor on (Nilp) which sends a scheme $S \in $ (Nilp) to the set of isomorphism classes of tuples $(X,i_X,\lambda _X,\rho _X)$ .

Here, X is a formal $O_F$ -module of F-height $2n$ over S and $i_X$ is an $O_E$ -action on X satisfying the determinant condition of signature $(r,n-r)$

$$ \begin{align*} \operatorname{\mathrm{Charpol}}(i_X(a)\vert \operatorname{\mathrm{Lie}} X)=(T-a)^r(T-a^*)^{n-r}, \quad \forall a \in E. \end{align*} $$

Here, we view $(T-a)^r(T-a^*)^{n-r}$ as an element of $O_S[T]$ via the structure morphism $O_E \rightarrow O_S$ .

Furthermore, $\rho _X$ is an $O_E$ -linear quasi-isogeny

$$ \begin{align*} \rho_X:X_{\overline{S}} \rightarrow \mathbb{X}\times_{\operatorname{\mathrm{Spec}}{\mathbb F}_{q^2}}\overline{S}, \end{align*} $$

of height 0, where $\overline {S}=S\otimes _{O_E}{\mathbb F}_{q^2}$ and $X_{\overline {S}}$ is the base change $X\times _S\overline {S}$ .

Finally, $\lambda _X:X \rightarrow X^{\vee } $ is a polarization such that its Rosati involution induces the involution $^*$ on $O_E$ , and locally on S the following diagram commutes up to a constant in $O_F^{\times }$ :

Two quadruples $(X,i_X,\lambda _X,\rho _X)$ and $(X',i_{X'},\lambda _{X'},\rho _{X'})$ are isomorphic if there exists an $O_E$ -linear isomorphism $\alpha :X \rightarrow X'$ such that $(\alpha \times _S \overline {S})\circ \rho _{X'}=\rho _X$ and $\alpha ^{\vee } \circ \lambda _{X'} \circ \alpha $ is an $O_F^{\times }$ -multiple of $\lambda _{X}$ .

The functor ${\mathcal N}_{E/F}^h(r,n-r) \otimes O_{\breve {E}}$ is representable by a formal scheme over $\operatorname {\mathrm {Spf}} O_{\breve {E}}$ which is locally formally of finite type. This is explained in [Reference MihatschMih22]. Indeed, we can use [Reference Rapoport and ZinkRZ, Theorem 2.16], and the fact that the condition that the $O_F$ -action on X lifts from ${\mathbb X}$ and the condition that the lifted action is strict are closed conditions.

Furthermore, when F is an unramified extension of ${\mathbb Q}_p$ , we will fix a triple $({\mathbb X},i_{{\mathbb X}},\lambda _{{\mathbb X}})$ in Remark 2.13. Then, ${\mathcal N}_{E/F}^h(r,n-r)$ is representable by a formal scheme over $\operatorname {\mathrm {Spf}} O_E$ which is locally formally of finite type.

From now on, we will restrict ourselves to the case $r=1$ . Note that the case $(r=1, h=0, F={\mathbb Q}_p)$ is studied in [Reference Vollaard and WedhornVW11]. For simplicity, denote by ${\mathcal N}$ the moduli problem ${\mathcal N}_{E/F}^h(1,n-1)$ .

2.2 The moduli space ${\mathcal N}_{E/K}^h(r,n-r)$

Let K be a finite extension of ${\mathbb Q}_p$ contained in F, with ring of integers $O_K$ and residue field ${\mathbb F}_s$ . In this subsection, we will define the moduli space ${\mathcal N}_{E/K}^h(r,n-r)$ . For this, we imitate the construction in [Reference MihatschMih22]. We will use the notation in Section 2. Also, we will use the theory of O-displays in [Reference Ahsendorf, Cheng and ZinkACZ16].

Let $[F:K]=f$ . We denote by $\breve {K}$ the completion of a maximal unramified extension of K, and $^F:\breve {K} \rightarrow \breve {K}$ the Frobenius automorphism. We choose a decomposition $\Psi :=\operatorname {\mathrm {Hom}}_K(E,\breve {K})=\Psi _0 \sqcup \Psi _1$ such that $(\Psi _0)^{*}=\Psi _1$ , where $^{*}$ is the nontrivial Galois automorphism of E over F. We fix an element $\psi _0 \in \Psi _0$ , and $\breve {E}:=E \otimes _{E,\psi _0} \breve {K}$ .

Definition 2.3 [Reference MihatschMih22, Definition 2.7]

For $a \in E$ , we define the following polynomials:

$$ \begin{align*} P^{E/K}_{(0,1)}(a;t)&=\prod_{\psi \in \Psi_1}\psi((t-a)) \in E[t]; \\ P^{E/K}_{(1,0)}(a;t)&= P^{E/K}_{(0,1)}(a;t)(t-a)(t-a^{*})^{-1} \in E[t]; \\ P^{E/K}_{(r,n-r)}(a;t)&=(P^{E/K}_{(1,0)}(a;t))^r(P^{E/K}_{(0,1)}(a;t))^{n-r} \in E[t]. \end{align*} $$

Let X be a hermitian $O_E \text {-} O_K \text {-} h$ -module over a $\operatorname {\mathrm {Spf}}{O_E}$ -scheme S. Then, by the $O_E$ -action, we have the grading

$$ \begin{align*} \operatorname{\mathrm{Lie}}(X)=\bigoplus_{\psi \in \Psi} \operatorname{\mathrm{Lie}}_{\psi}(X). \end{align*} $$

Here, $\operatorname {\mathrm {Lie}}_{\psi }(X)$ is the direct summand on which $O_{E}$ acts via $\psi $ . We define the following determinant condition.

Definition 2.5 (cf. [Reference MihatschMih22, Definition 2.8])

Let S be a scheme over $\operatorname {\mathrm {Spf}}{O_E}$ . A hermitian $O_E \text {-} O_K \text {-} h$ -module $(X,i_X,\lambda _X)$ of rank n over S is of signature $(r,n-r)$ if for all $a \in O_E$ ,

(2.2.1) $$ \begin{align} \operatorname{\mathrm{Charpol}}(i_X(a) \vert \operatorname{\mathrm{Lie}} X)&= P^{E/K}_{(r,n-r)}(a;t), \end{align} $$

(2.2.2) $$ \begin{align} (i_X(a)-a) \vert_{\operatorname{\mathrm{Lie}}_{\psi_0}(X)}&=0. \end{align} $$

Here, we view $P^{E/K}_{(r,n-r)}(a;t)$ as an element of $O_S[t]$ via the structure morphism. The second equation means that $O_E$ acts on $\operatorname {\mathrm {Lie}}_{\psi _0}(X)$ via the structure morphism. Note that (2.2.1) implies (2.2.2) since E is unramified over ${\mathbb Q}_p$ .

Let $({\mathbb X},i_{{\mathbb X}},\lambda _{{\mathbb X}})$ be a hermitian $O_E \text {-} O_K \text {-} h$ -module of signature $(r,n-r)$ over ${\mathbb F}_{q^2}$ . Let ${\mathcal N}_{E/K}^h(r,n-r)$ be the set-valued functor on (Nilp) which sends a scheme $S \in $ (Nilp) to the set of isomorphism classes of tuples $(X,i_X,\lambda _X,\rho _X)$ . Here, $(X,i_X,\lambda _X)$ is a hermitian $O_E \text {-} O_K \text {-} h$ -module of signature $(r,n-r)$ over S and $\rho _X$ is an $O_E$ -linear quasi-isogeny

$$ \begin{align*} \rho_X: X\times_S \overline{S} \rightarrow {\mathbb X} \times_{{\mathbb F}^{q^2}}\overline{S} \end{align*} $$

of height 0.

Furthermore, we require that locally on S the following diagram commutes up to a constant in $O_K^{\times }$ :

Two quadruples $(X,i_X,\lambda _X,\rho _X)$ and $(Y,i_Y,\lambda _Y,\rho _Y)$ are isomorphic if there exists an $O_E$ -linear isomorphism $\alpha :X \rightarrow Y$ with $\rho _Y \circ (\alpha \times _S \overline {S})=\rho _X$ .

The functor ${\mathcal N}_{E/K}^h(r,n-r) \otimes O_{\breve {E}}$ is representable by a formal scheme which is locally formally of finite type over $\operatorname {\mathrm {Spf}}{O_{\breve {E}}}$ (see [Reference MihatschMih22]).

Now, let $S=\operatorname {\mathrm {Spec}}{R}$ be an affine scheme over $\operatorname {\mathrm {Spf}}{O_E}$ and $(X,i_X,\lambda _X)$ be a hermitian $O_E \text {-} O_K \text {-} h$ -module of signature $(r,n-r)$ over S. Let $(P,Q,F,F_1)$ be the $O_K$ -display (i.e., an $O_K$ -window over $W_{O_K}(R)$ ) of $(X,i_X,\lambda _X)$ . We denote by $\langle \cdot , \cdot \rangle : P \times P \rightarrow W_{O_K}(R)$ the $W_{O_K}(R)$ -bilinear alternating form induced by $\lambda _X$ . From the $O_E$ -action, we have the decomposition

$$ \begin{align*} O_E \otimes_{O_K} W_{O_K}(R) \simeq \prod_{\psi \in \Psi}O_E \otimes_{O_{E},\psi} W_{O_K}(R). \end{align*} $$

This decomposition gives gradings

$$ \begin{align*} P & =\prod_{\psi \in \Psi}P_{\psi}=\prod_{\psi \in \Psi_0}P_{\psi} \oplus P_{\psi^*},\\ Q & =\prod_{\psi \in \Psi}Q_{\psi}=\prod_{\psi \in \Psi_0}Q_{\psi} \oplus Q_{\psi^*}. \end{align*} $$

Note that our pairing satisfies

$$ \begin{align*} \langle \cdot, \cdot \rangle|_{P_{\psi} \times P_{\psi'}}\equiv 0 \qquad \text{if} \quad \psi' \neq \psi^*. \end{align*} $$

Therefore, we can define

$$ \begin{align*} \langle \cdot, \cdot \rangle|_{\psi}:=\langle \cdot, \cdot \rangle|_{P_{\psi} \times P_{\psi^*}}. \end{align*} $$

Let $P_{\psi }=L_{\psi } \oplus T_{\psi }$ , $Q_{\psi }=L_{\psi }+I_{O_K}(R)T_{\psi }$ be a normal decomposition. From the normal decomposition, we get a $^F$ -linear isomorphism

$$ \begin{align*} \begin{array}{ccc} \Phi_{\psi}:P_{\psi}=L_{\psi}\oplus T_{\psi} & \rightarrow &P_{^F\psi} \\ \quad\qquad\qquad (l,t)& \mapsto & (F_1(l)+F(t)). \end{array} \end{align*} $$

Now, we have the following lemma.

Proof. This is essentially from the proof of [Reference MihatschMih22, Proposition 3.4]. By the signature condition, we have

$$ \begin{align*} \begin{aligned} L_{\psi}=P_{\psi}, T_{\psi}=0,\quad\text{if } \psi \in \Psi_0\backslash \lbrace \psi_0\rbrace,\\ L_{\psi}=0, T_{\psi}=P_{\psi},\quad\text{if } \psi \in \Psi_1\backslash \lbrace \psi_0^*\rbrace. \end{aligned} \end{align*} $$

Therefore, we have

$$ \begin{align*} &\begin{array}{cccc} \Phi_{\psi}:P_{\psi} & \rightarrow &P_{^F\psi}& \\ \qquad x& \mapsto & F_1(x),& \text{ if } \psi \in \Psi_0 \backslash \lbrace \psi_0 \rbrace; \end{array} \\&\begin{array}{cccc} \Phi_{\psi}:P_{\psi} & \rightarrow &P_{^F\psi}& \\ \qquad x& \mapsto & F(x),& \text{ if } \psi \in \Psi_1 \backslash \lbrace \psi_0^* \rbrace. \end{array} \end{align*} $$

Note that $\Phi _{\psi }$ is a $^F$ -linear isomorphism; hence,

$$ \begin{align*} \begin{aligned} \Phi(P_{^{F^i}\psi_0})=P_{^{F^{i+1}}\psi_0},\\ \Phi(P_{^{F^i}\psi_0^*})=P_{^{F^{i+1}}\psi_0^*}. \end{aligned} \end{align*} $$

First, assume that $^{F^i}\psi _0^* \in \Psi _0 \backslash \lbrace \psi _0 \rbrace $ . Then, $^{F^i}\psi _0 \in \Psi _1 \backslash \lbrace \psi _0^* \rbrace $ . This implies that $\Phi =F_1$ on $P_{^{F^{i}}\psi _0^*}$ and $\Phi =F$ on $P_{^{F^{i}}\psi _0}$ . Therefore, for $x \in P_{^{F^{i}}\psi _0}$ , $y \in P_{^{F^{i}}\psi _0^*}$ , we have

$$ \begin{align*} \langle \Phi(x), \Phi(y) \rangle=\langle Fx,F_1y \rangle=^F\langle x,y \rangle. \end{align*} $$

Here, we used the fact that $\langle F \cdot , F_1 \cdot \rangle = \langle F_1 \cdot , F \cdot \rangle =^F\langle \cdot , \cdot \rangle $ .

In the case that $^{F^i}\psi _0^* \in \Psi _1 \backslash \lbrace \psi _0^* \rbrace $ , we can prove the lemma in the same way.

Since $\Phi _{\psi }$ is a $^F$ -linear isomorphism and there is a unique way to extend this pairing to the whole of $P_{^F\psi } \times P_{^F\psi ^*}$ , one could get all the information about the pairing on $P_{^F\psi } \times P_{^F\psi ^*}$ from the pairing on $P_{\psi } \times P_{\psi ^*}$ . With this lemma, we can follow all the steps in [Reference MihatschMih22, Chapter 3]. Indeed, the only difference is the polarization. The polarization $\lambda _X$ induces our alternating form $\langle \cdot , \cdot \rangle :P \times P \rightarrow W_{O_K}(R)$ , and by taking $\langle \cdot , \cdot \rangle _{\psi _0}$ , we get a polarization of the corresponding $O_E \text {-} O_{F} \text {-} h\text {-}$ module. The only thing we need to check is if the order of the kernel of this polarization is $s^{2h}$ . Note that the order of the kernel of $\lambda _X$ is Zariski locally constant; therefore, we can check this at the geometric points of S. Since the order of the kernel of $\lambda _X$ is $q^{2h}=s^{2fh}$ , we can check the above claim by using Lemma 2.10 at geometric points. This proves the following analogue of [Reference MihatschMih22, Proposition 3.4]. Let $Sch / \operatorname {\mathrm {Spf}} O_E$ (resp. $Sch'/ \operatorname {\mathrm {Spf}} O_E$ ) be the category of schemes (resp. locally noetherian schemes) over $\operatorname {\mathrm {Spf}} O_E$ together with the Zariski topology.

Proposition 2.11 (cf. [Reference MihatschMih22, Proposition 3.4])

There is an isomorphism of stacks over $\operatorname {\mathrm {Sch}}/\operatorname {\mathrm {Spf}} O_E$

$$ \begin{align*} {\mathcal C}_{K,F}: O_E \text{-} O_K \text{-} h \text{-} \text{Herm} \overset{\simeq}{\rightarrow} O_E \text{-} O_{F} \text{-} h\text{-} \text{Herm} \end{align*} $$

that is equivariant for the Rosati involutions and sends objects of signature $(r,n-r)$ to objects of signature $(r,n-r)$ .

The following proposition is an analogue of [Reference MihatschMih22, Theorem 3.1].

Proposition 2.12 (cf. [Reference MihatschMih22, Theorem 3.1])

The isomorphism ${\mathcal C}_{K,F}$ induces an isomorphism

$$ \begin{align*} c_{K,F}:{\mathcal N}_{E/F}^h(r,n-r) \simeq {\mathcal N}_{E/K}^h(r,n-r). \end{align*} $$

Proof. When $h=0$ , it is proved in [Reference MihatschMih22, Proposition 2.17]. Since ${\mathcal N}_{E/F}^0(1,n-1)_{O_{\breve {E}}}$ and ${\mathcal N}_{E/F}^n(1,n-1)_{O_{\breve {E}}}$ are isomorphic (see Definition 5.2), ${\mathcal N}_{E/F}^n(1,n-1)_{O_{\breve {E}}}$ is also formally smooth over $\operatorname {\mathrm {Spf}} O_{\breve {E}}$ . Now, assume that $1 \leq h \leq n-1$ . By Proposition 2.12, it suffices to show that ${\mathcal N}_{E/{\mathbb Q}_p}^h(1,n-1)_{O_{\breve {E}}}$ has semistable reduction. Since this moduli problem is of PEL type, it suffices to show that its local model has semistable reduction ([Reference Rapoport and ZinkRZ, Proposition 3.33]). To define the local model ${\mathcal N}^{loc}$ in our case, we need to use the notation in [Reference Rapoport, Smithling and ZhangRSZ20, Appendix B]. Let $l (\cdot , \cdot )$ be an $E/F$ -hermitian form on $E^n$ given by the matrix

$$ \begin{align*} \left(\begin{array}{@{}cc@{}} p I_h &\\ & I_{n-h} \end{array} \right). \end{align*} $$

Fix an element $\delta \in O_E^{\times }$ such that $\delta ^*=-\delta $ . Let $\theta ^{-1}_{F/{\mathbb Q}_p}$ be a generator of the inverse different of $F/{\mathbb Q}_p$ . Let $(\cdot , \cdot )$ be the ${\mathbb Q}_p$ -bilinear alternating form

$$ \begin{align*} (x,y)=Tr_{E/{\mathbb Q}_p}(\theta^{-1}_{F/{\mathbb Q}_p} \delta l(x,y)), \quad x,y \in E^n. \end{align*} $$

Let $\Lambda _0=O_E^n$ and $\Lambda _1=p^{-1}O_E^h \oplus O_E^{n-h}$ . Then, the dual $\Lambda _1^{\vee }$ of the lattice $\Lambda _1$ with respect to $(\cdot , \cdot )$ is $\Lambda _0$ . Now, let ${\mathcal L}$ be the self-dual lattice chain

$$ \begin{align*} \lbrace \dots \subset p \Lambda_1 \subset \Lambda_0 \subset \Lambda_1= \Lambda_0^{\vee} \subset p^{-1} \Lambda_0 \subset \cdots \rbrace. \end{align*} $$

Then, ${\mathcal N}^{loc}$ is the functor which sends each $O_{\breve {E}}$ -scheme S to the set of isomorphism classes of families $(\Lambda \otimes _{{\mathbb Z}_p} O_S \twoheadrightarrow {\mathcal P}_{\Lambda })_{\Lambda \in {\mathcal L}}$ such that

$\bullet $ For each $\Lambda $ , ${\mathcal P}_{\Lambda }$ is an $O_E \otimes _{{\mathbb Z}_p} O_S$ -linear quotient of $\Lambda \otimes _{{\mathbb Z}_p} O_S$ , locally free on S as an $O_S$ -module.

$\bullet $ For each inclusion $\Lambda \subset \Lambda '$ in ${\mathcal L}$ , the arrow $\Lambda \otimes _{{\mathbb Z}_p} O_S \rightarrow \Lambda ' \otimes _{{\mathbb Z}_p} O_S$ induces an arrow ${\mathcal P}_{\Lambda } \rightarrow {\mathcal P}_{\Lambda '}$ .

$\bullet $ For each $\Lambda $ , the isomorphism $\Lambda \otimes _{{\mathbb Z}_p} O_S \xrightarrow {p \otimes 1} (p \Lambda ) \otimes _{{\mathbb Z}_p} O_S$ identifies ${\mathcal P}_{\Lambda } \overset {~}{\rightarrow } {\mathcal P}_{p\Lambda }$ .

$\bullet $ For each $\Lambda $ , the perfect pairing $(\Lambda \otimes _{{\mathbb Z}_p} O_S) \times (\Lambda ^{\vee } \otimes _{{\mathbb Z}_p} O_S) \xrightarrow {(\cdot , \cdot ) \otimes O_S} O_S$ identifies $(\operatorname {\mathrm {Ker}}(\Lambda \otimes _{{\mathbb Z}_p} O_S \twoheadrightarrow {\mathcal P}_{\Lambda }))^{\perp }$ with $\operatorname {\mathrm {Ker}}(\Lambda ^{\vee } \otimes _{{\mathbb Z}_p} O_S \twoheadrightarrow P_{\Lambda ^{\vee }})$ .

We need to impose one more condition.

By the $O_E$ -action on S, there is a natural identification

$$ \begin{align*} O_{E} \otimes_{{\mathbb Z}_p} O_S \xrightarrow{\simeq} \prod_{\psi \in \Psi} O_S. \end{align*} $$

This induces a decomposition

$$ \begin{align*} {\mathcal P}_{\Lambda} \xrightarrow{\simeq} \bigoplus_{\psi \in \Psi} {\mathcal P}_{\Lambda, \psi}. \end{align*} $$

$\bullet $ For each $\Lambda $ , ${\mathcal P}_{\Lambda }$ satisfies

(2.2.3) $$ \begin{align} \operatorname{\mathrm{Charpol}}_{O_S}(a\otimes 1 \vert {\mathcal P}_{\Lambda}) &= P^{E/{\mathbb Q}_p}_{(1,n-1)}(a;t), \end{align} $$

(2.2.4) $$ \begin{align} (a \otimes 1 - 1 \otimes a) \vert P_{\Lambda, \psi_0}&=0. \end{align} $$

Here, $P_{\Lambda , \psi _0}$ is the direct summand on which $O_{E}$ acts via $\psi _0$ . These two conditions follow from the conditions (2.2.1) and (2.2.2).

Now, fix a scheme S over $O_{\breve {E}}$ , and let $(\Lambda \otimes _{{\mathbb Z}_p} O_S \twoheadrightarrow {\mathcal P}_{\Lambda })_{\Lambda \in {\mathcal L}} \in {\mathcal N}^{loc}(S)$ . By the signature condition (2.2.3), we have

$$ \begin{align*} \left\{ \begin{array}{@{}l} {\mathcal P}_{\Lambda, \psi_0} \text{ is locally free of rank 1 over } O_S,\\ {\mathcal P}_{\Lambda, \psi_0^*}={\mathcal P}_{\Lambda^{\vee}, \psi_0}^{\perp} \subset (\Lambda \otimes_{{\mathbb Z}_p} O_S)_{\psi_0^*},\\ {\mathcal P}_{\Lambda, \psi}=0 \qquad\text{ if } \psi \in \Psi_0 \backslash \lbrace \psi_0 \rbrace,\\ {\mathcal P}_{\Lambda, \psi^*}=(\Lambda \otimes_{{\mathbb Z}_p} O_S)_{\psi^*} \qquad \text{if } \psi \in \Psi_1 \backslash \lbrace \psi_0^* \rbrace.\\ \end{array} \right. \end{align*} $$

Therefore, $(\Lambda \otimes _{{\mathbb Z}_p} O_S \twoheadrightarrow {\mathcal P}_{\Lambda })_{\Lambda \in {\mathcal L}}$ is determined by $({\mathcal P}_{\Lambda ,\psi _0})_{\Lambda \in {\mathcal L}}$ .

Also, by the condition (2.2.4), $O_E$ acts on ${\mathcal P}_{\Lambda , \psi _0}$ via the structure morphism; therefore, ${\mathcal P}_{\Lambda , \psi _0}$ is a quotient of

$$ \begin{align*} A_{\Lambda} :=(\Lambda \otimes_{{\mathbb Z}_p} O_S) \otimes_{O_E \otimes_{{\mathbb Z}_p} O_S} O_S, \end{align*} $$

which is locally free of rank n over $O_S$ .

It follows that the map $(\Lambda \otimes _{{\mathbb Z}_p} O_S \twoheadrightarrow {\mathcal P}_{\Lambda })_{\Lambda \in {\mathcal L}} \mapsto (A_{\Lambda } \twoheadrightarrow {\mathcal P}_{\Lambda , \psi _0})_{\Lambda \in {\mathcal L}}$ is an isomorphism from ${\mathcal N}^{loc}$ to the standard local model over $\operatorname {\mathrm {Spec}} O_{\breve {E}}$ in [Reference GörtzGör01] (for the group $GL_n$ , the cocharacter $\mu =(1^{(n-1)},0)$ and the lattice chain ${\mathcal L}$ ). Therefore, by [Reference GörtzGör01, 4.4.5] (in the case $k=h, r=1$ ), this local model has semistable reduction.

2.3 Description of the points of $\mathcal {N}$

Let k be a fixed algebraic closure of $O_E/\pi O_E={\mathbb F}_{q^2}$ . In this subsection, we will study the set ${\mathcal N}(k)$ . For this, we need to use relative Dieudonné theory in the sense of [Reference Rapoport and ZinkRZ, Proposition 3.56]. We use the following notation.

Let $\breve {F}$ be the completion of a maximal unramified extension of F containing E and $O_{\breve {F}}$ its ring of integers. Let $F^u$ be the maximal unramified extension of ${\mathbb Q}_p$ in F and $O_{F^u}$ its ring of integers. Let L be a perfect field with an ${\mathbb F}_q$ -algebra structure $\alpha _0: {\mathbb F}_q \rightarrow L$ . Then, we get a map $O_{F^u} \rightarrow W(L)$ induced from $\alpha _0: {\mathbb F}_q \hookrightarrow L$ . We define $W_{O_F}(L)=O_F \otimes _{O_{F^u},\alpha _0}W(L)$ . This is the ring of relative Witt vectors of L. In particular, $W_{O_F}(k)=O_{\breve {F}}$ .

Let $\sigma $ be the Frobenius element in $\operatorname {\mathrm {Gal}}(\breve {F}/F)$ .

We recall from [Reference Rapoport and ZinkRZ, Proposition 3.56] (or [Reference Kudla and RapoportKR14b, Notation]) the definition of the relative Dieudonné module. Let X be a formal $O_F$ -module of F-height $2n$ over L. Let $(\tilde {M}, \tilde {{\mathcal V}})$ be the (absolute) Dieudonné module of X. Consider the decomposition

$$ \begin{align*} O_F \otimes_{{\mathbb Z}_p}W(L)=\prod_{\alpha:{\mathbb F}_{q} \rightarrow k} O_F \otimes_{O_{F^u},\alpha} W(L). \end{align*} $$

Here, $\alpha $ runs over the set of ${\mathbb F}_p$ -embeddings $\alpha :{\mathbb F}_{q} \rightarrow L$ . Via this decomposition, the action of $O_F$ on $\tilde {M}$ induces the decomposition

$$ \begin{align*} \tilde{M}=\bigoplus_{\alpha:{\mathbb F}_{q} \rightarrow L} \tilde{M}^{\alpha}. \end{align*} $$

We define the relative Dieudonné module of X as

$$ \begin{align*} (M^{\alpha_0},{\mathcal V}=\tilde{{\mathcal V}}^f), \end{align*} $$

where $f=[ F^u : {\mathbb Q}_p ]=[ {\mathbb F}_{q}:{\mathbb F}_p ]$ .

Now, let $(\mathbb {M},{\mathcal V})$ be the relative Dieudonné module of $\mathbb {X}$ , and let $N=\mathbb {M} \otimes _{{\mathbb Z}}{\mathbb Q}$ be its relative Dieudonné isocrystal. Denote by $N_k=\mathbb {M} \otimes _{O_E}\breve {F}$ its base change. The $O_E$ -action $i_{{\mathbb X}}$ on ${\mathbb X}$ induces an E-action on $N_k$ . Let ${\mathcal F}$ be the Frobenius of ${\mathbb M}$ . The polarization $\lambda _{{\mathbb X}}$ of ${\mathbb X}$ induces a nondegenerate $\breve {F}$ -bilinear alternating form on $N_k$

$$ \begin{align*} \langle\cdot, \cdot\rangle:N_k \times N_k \rightarrow \breve{F}, \end{align*} $$

such that for all $x,y \in N_k, a \in E$ , it satisfies

(2.3.1) $$ \begin{align} \langle {\mathcal F} x,y \rangle=\langle x,{\mathcal V} y \rangle^{\sigma} ,\\[-36pt]\nonumber \end{align} $$

(2.3.2) $$ \begin{align} \langle i_{{\mathbb X}}(a)x,y \rangle =\langle x,i_{{\mathbb X}}(a^{*})y \rangle. \end{align} $$

Since we have the decomposition $E \otimes _F \breve {F} \simeq \breve {F} \times \breve {F}$ , the E-action $i_{{\mathbb X}}$ on $N_k$ induces a ${\mathbb Z}/2{\mathbb Z}$ -grading

$$ \begin{align*} N_k=N_{k,0} \oplus N_{k,1}. \end{align*} $$

Note that by (2.3.2), each $N_{k,i}$ is totally isotropic with respect to $\langle \cdot , \cdot \rangle $ . Also, for $i=0,1$ , we have that ${\mathcal F}:N_{k,i} \rightarrow N_{k,i+1}$ and ${\mathcal V}:N_{k,i} \rightarrow N_{k,i+1}$ are homogeneous of degree 1 with respect to the decomposition.

For an $O_{\breve {F}}$ -lattice $M=M_0 \oplus M_1$ , we define the dual lattice $M_i^{\perp }$ of $M_i$ as

$$ \begin{align*} M_i^{\perp}=\lbrace x \in N_{k,i+1} \vert \langle x, M_i \rangle \subset O_{\breve{F}} \rbrace. \end{align*} $$

For $O_{\breve {F}}$ -lattices $M_i \subset M^{\prime }_i \subset N_{k,i}$ , we denote by $[M^{\prime }_i:M_i]$ the index of $M_i$ in $M^{\prime }_i$ , i.e., the length of the $O_{\breve {F}}$ -module $M^{\prime }_i/M_i$ . If $[M^{\prime }_i:M_i]=t$ , we write $M_i \overset {t}{\subset} M^{\prime }_i$ .

By the relative Dieudonné theory, we have the following proposition.

Proof. The first condition is from the relative Dieudonné theory. The condition on the characteristic polynomial follows from the determinant condition of signature $(1,n-1)$ . Also, the polarization condition implies that

$$ \begin{align*}\begin{array}{ll} &M \overset{2h}{\subset} M^{\perp}\overset{2n-2h}{\subset} \pi^{-1}M \\ \Longleftrightarrow& M_0 \oplus M_1 \overset{2h}{\subset} M_1^{\perp} \oplus M_0^{\perp} \overset{2n-2h}{\subset} \pi^{-1}M_0 \oplus \pi^{-1}M_1\\ \Longleftrightarrow & M_0 \overset{h}{\subset} M_1^{\perp} \overset{n-h}{\subset}\pi^{-1}M_0. \end{array}\\[-36pt] \end{align*} $$

We will use the following lemma in the next subsection.

Lemma 2.17 [Reference VollaardVol10, Lemma 1.5]

Let $M=M_0 \oplus M_1$ be an $O_E$ -invariant lattice in $N_k$ . Assume that M is invariant under ${\mathcal F}$ and ${\mathcal V}$ . Then, M satisfies the determinant condition of signature $(1,n-1)$ if and only if

$$ \begin{align*} \begin{aligned} \pi M_0 \overset{n-1}{\subset} {\mathcal F} M_1 \overset{1}{\subset} M_0,\\ \pi M_1 \overset{1}{\subset} {\mathcal F} M_0 \overset{n-1}{\subset} M_1. \end{aligned} \end{align*} $$

2.4 Description of the points of $\mathcal {N}$ II

In this subsection, we will describe the set $\mathcal {N}(k)$ as the set of lattices in $N_{k,0}$ . We use the following notation.

Let $\tau $ be the $\sigma ^2$ -linear operator ${\mathcal V}^{-1}{\mathcal F}$ on $N_k$ , and let $N_{k,0}^{\tau }$ be the set of $\tau $ -invariant elements in $N_{k,0}$ . Then, $N_{k,0}^{\tau }$ is an E-vector space. Note that $\tau $ is isoclinic of slope $0$ and hence there exists a $\tau $ -invariant lattice in $N_{k,0}$ . For every $\tau $ -invariant lattice A in $N_{k,0}$ , there exists a $\tau $ -invariant basis of A (see [Reference VollaardVol10, 1.10]). Therefore, we have $N_{k,0}=N_{k,0}^{\tau } \otimes _E \breve {F}$ .

We define $\lbrace x, y \rbrace :=\langle x, {\mathcal F} y \rangle $ . This is a nondegenerate form on $N_{k,0}$ which is linear in the first variable and $\sigma $ -linear in the second variable.

Also, this form $\lbrace \cdot , \cdot \rbrace $ satisfies the following properties (see [Reference VollaardVol10, 1.11]):

$$ \begin{align*} \lbrace x,y \rbrace&=-\lbrace y, \tau^{-1}(x)\rbrace^{\sigma}, \\\lbrace \tau(x),\tau(y) \rbrace&=\lbrace x, y\rbrace^{\sigma^2}. \end{align*} $$

For an $O_{\breve {F}}$ -lattice A in $N_{k,0}$ , we define $A^{\vee }$ the dual lattice of A with respect to the form $\lbrace \cdot , \cdot \rbrace $ as

$$ \begin{align*} A^{\vee}=\lbrace x \in N_{k,0} \vert \lbrace x, A \rbrace \subset O_{\breve{F}} \rbrace. \end{align*} $$

For an $O_{\breve {F}}$ -lattice $A \subset N_{k,0}$ , we have

$$ \begin{align*} (A^{\vee})^{\vee}&=\tau(A),\\ \tau(A^{\vee})&=\tau(A)^{\vee}. \end{align*} $$

We can now state the following description of $\mathcal {N}(k)$ .

Proposition 2.18. There is a bijection between $\mathcal {N}(k)$ and the set

$$ \begin{align*} \left\{\begin{array}{@{}c@{}} O_{\breve{F}}-lattices A \overset{h}{\subset} B \subset N_{k,0} \end{array} \middle| \begin{array}{c} \pi B^{\vee} \overset{1}{\subset} A \overset{n-1}{\subset} B^{\vee},\\ \pi B \subset A \subset B \end{array}\!\!\! \right\}. \end{align*} $$

Proof. For $M=M_0 \oplus M_1 \in {\mathcal N}(k)$ , let $A=M_0$ , $B=M_1^{\perp }$ . Then, by Proposition 2.16, we have $\pi B \subset A \overset {h}{\subset} B$ . Now, we will show the following equality:

(2.4.1) $$ \begin{align} \pi(M_1^{\perp})^{\vee}={\mathcal F} M_1. \end{align} $$

Indeed, we have

$$ \begin{align*} (M_1^{\perp})^{\vee}&=\lbrace y \in N_{k,0} \vert \lbrace y, M_1^{\perp} \rbrace \subset O_{\breve{F}} \rbrace \\& =\lbrace y \in N_{k,0} \vert \langle y, {\mathcal F} M_1^{\perp} \rangle \subset O_{\breve{F}} \rbrace\\&=\lbrace y \in N_{k,0} \vert \langle {\mathcal F} M_1^{\perp}, y \rangle \subset O_{\breve{F}} \rbrace \\& =\lbrace y \in N_{k,0} \vert \langle M_1^{\perp}, {\mathcal V} y \rangle \subset O_{\breve{F}} \rbrace\\&={\mathcal V}^{-1}((M_1^{\perp})^{\perp})={\mathcal V}^{-1}M_1. \end{align*} $$

Therefore, by multiplying by $\pi $ , we get the equality (2.4.1).

By Lemma 2.17 and (2.4.1), we have $\pi B^{\vee } \overset {1}{\subset} A \overset {n-1}{\subset} B^{\vee }$ .

Similarly, we have ${\mathcal V} M_1 \overset {1}{\subset} M_0 \Longleftrightarrow M_1 \overset {1}{\subset} {\mathcal V}^{-1}M_0 \Longleftrightarrow {\mathcal F} M_1 \subset {\mathcal V}^{-1}{\mathcal F}(M_0) \Longleftrightarrow \pi (M_1^{\perp })^{\vee } \overset {1}{\subset} \tau (M_0) \Longleftrightarrow \pi M_0^{\vee } \overset {1}{\subset} M_1^{\perp }$ . Therefore, we have $\pi A^{\vee } \overset {1}{\subset} B \overset {n-1}{\subset} A^{\vee }$ .

Conversely, if we have $O_{\breve {F}}$ -lattices $A,B$ satisfying the above conditions, then one can easily show that $A \oplus B^{\perp }$ is an element in ${\mathcal N}(k)$ . Indeed, for $M_0=A$ and $M_1=B^{\perp }$ , it satisfies

$$ \begin{align*} \pi M_0=\pi A \overset{n-1}{\subset} {\mathcal F} M_1=\pi (M_1^{\perp})^{\vee}=\pi B^{\vee} \overset{1}{\subset} A=M_0, \end{align*} $$

and

$$ \begin{align*} M_0=A \overset{h}{\subset} B=(M_1)^{\perp} \overset{n-h}{\subset} \pi^{-1}A=\pi^{-1}M_0. \end{align*} $$

From now on, we identify ${\mathcal N}(k)$ with the set defined in Proposition 2.18.

2.5 The sets $R_{\Lambda }$ , $S_{\Lambda }$ indexed by vertex lattices $\Lambda $

In this section, we will define the sets $R_{\Lambda }$ and $S_{\Lambda }$ indexed by the lattices $\Lambda $ which are called vertex lattices. First, we start with the definition of the vertex lattices.

Definition 2.20. For $(A \overset {h}{\subset} B)$ a pair in ${\mathcal N}(k)$ (via Proposition 2.18), we define

$$ \begin{align*} T_iA&:=A+\tau(A)+\cdots+\tau^{i-1}(A), \\ T_iB&:=B+\tau(B)+\cdots+\tau^{i-1}(B). \end{align*} $$

By [Reference Rapoport and ZinkRZ, Proposition 2.17], there exist positive integers $c,d$ such that $T_c(A)$ and $T_d(B)$ are $\tau $ -invariant. Now, we will show the following lemma.

Lemma 2.21. Let $(A \overset {h}{\subset} B)$ be a pair in ${\mathcal N}(k)$ (via Proposition 2.18). Let $c, d$ be the smallest positive integers such that $T_cA, T_dB$ are $\tau $ -invariant, and write $\Lambda _A:=T_c(A)$ , $\Lambda _B:=T_d(B)$ . Then, at least one of the following assertions holds.

(1) $\Lambda _B$ is a vertex lattice in ${\mathcal L}_0$ , and

$$ \begin{align*} \begin{array}{cccccccc} &\pi A^{\vee} &\overset{1}{\subset} &B &\subset& \Lambda_B &\subset &\Lambda_B^{\vee} \\ &\cup & &\cup & &&&\\ \pi \Lambda_B^{\vee} \subset&\pi B^{\vee}&\overset{1}{\subset}&A. \end{array} \end{align*} $$

(2) $\Lambda _A$ is a vertex lattice in ${\mathcal L}_1$ , and

$$ \begin{align*} \begin{array}{cccccccc} &\pi B^{\vee} &\overset{1}{\subset} &A &\subset& \Lambda_A &\subset &\pi \Lambda_A^{\vee} \\ &\cup & &\cup & &&&\\ \pi^2 \Lambda_A^{\vee} \subset&\pi^2 A^{\vee}&\overset{1}{\subset}& \pi B. \end{array} \end{align*} $$

To prove Lemma 2.21, we need the following lemma.

Lemma 2.22. For $(A \overset {h}{\subset} B)$ in Lemma 2.21, we have the following properties. For $1 \leq i <c$ , $1 \leq j<d$ ,

(2.5.1) $$ \begin{align} T_{i}A \cap \tau(T_{i}A)=\tau(T_{i-1}A),\\[-36pt]\nonumber \end{align} $$

(2.5.2) $$ \begin{align} T_{i-1}A \overset{1}{\subset} T_{i}A,\\[-36pt]\nonumber \end{align} $$

(2.5.3) $$ \begin{align} T_{j}B \cap \tau(T_{j}B)=\tau(T_{j-1}B),\\[-36pt]\nonumber \end{align} $$

(2.5.4) $$ \begin{align} T_{j-1}B \overset{1}{\subset} T_{j}B. \end{align} $$

Proof. We will show (2.5.1) and (2.5.2). The proof of (2.5.3) and (2.5.4) is similar.

Note that we have

(2.5.5) $$ \begin{align} \pi B^{\vee} \overset{1}{\subset} A \overset{n-1}{\subset} B^{\vee},\\[-36pt]\nonumber \end{align} $$

(2.5.6) $$ \begin{align} \pi A^{\vee} \overset{1}{\subset} B \overset{n-1}{\subset} A^{\vee}. \end{align} $$

Therefore, we have $\pi B^{\vee } \overset {1}{\subset} A$ and $\pi B^{\vee } \overset {1}{\subset} \tau (A)$ by taking the dual of (2.5.6). If A is $\tau $ -invariant, then $c=0$ , and hence there is nothing to prove. Now, assume that A is not $\tau $ -invariant. Since $\pi B^{\vee } \subset A \cap \tau (A) \subsetneq A$ and $\pi B^{\vee }$ is of index 1 in A, $A \cap \tau (A)$ must be $\pi B^{\vee }$ . Since $A \neq \tau (A)$ , $A \cap \tau (A) \overset {1}{\subset} A$ and $A \cap \tau (A) \overset {1}{\subset} \tau (A)$ , we have that A and $\tau (A)$ should have index 1 in $T_1A=A+\tau (A)$ . This shows (2.5.2) when $i=1$ .

For (2.5.1), note that $\tau (A) \overset {1}{\subset} T_1A$ and $\tau (A) \overset {1}{\subset} \tau (T_1A)$ . If $T_1A$ is $\tau $ -invariant, then $c=1$ . Therefore, there is nothing to show. Assume that $T_1A$ is not $\tau $ -invariant. Then, $T_1A \cap \tau (T_1A)=\tau (A)$ . This shows (2.5.1) for $i=1$ .

For arbitrary i, we use induction on i. Assume that (2.5.1) and (2.5.2) are true for $1 \leq i \leq k-1 <c$ . If $k=c$ , then there is nothing to show. Therefore, assume that $k <c$ .

Since $T_{k-2}A \overset {1}{\subset} T_{k-1}A$ , we have $\tau (T_{k-2}A) \overset {1}{\subset}\tau (T_{k-1}A)$ . Also, $T_{k-1}A \cap \tau (T_{k-1}A)=\tau (T_{k-2}A)$ implies that $\tau (T_{k-2}A) \overset {1}{\subset} T_{k-1}A$ . Since $k-1 <c$ , we have $T_{k-1}A\neq \tau (T_{k-1}A)$ . Therefore, $T_{k-1}A$ and $\tau (T_{k-1}A)$ are both of index $1$ in $T_kA=T_{k-1}A+\tau (T_{k-1}A)$ . This proves (2.5.1) for $i=k$ .

Also, note that $\tau (T_{k-1}A) \overset {1}{\subset} T_kA$ and $\tau (T_{k-1}A) \overset {1}{\subset} \tau (T_kA)$ . Since $k<c$ , $T_kA \neq \tau (T_kA)$ , and hence $T_kA \cap \tau (T_kA)=\tau (T_{k-1}A)$ . This proves (2.5.2) for $i=k$ .

We now go back to the proof of Lemma 2.21.

Proof of Lemma 2.21

We will prove this lemma by dividing the proof into six cases and their subcases.

Case 1. If $B \in {\mathcal L}_0$ , then (1) holds.

Case 2. If $A \in {\mathcal L}_1$ , then (2) holds.

Case 3. Assume that A is $\tau $ -invariant, but not a vertex lattice in ${\mathcal L}_1$ . Then, $A \nsubseteq \pi A^{\vee }$ . Since $\pi A^{\vee }$ is of index 1 in B, and $A \subset B$ , we have $B=A+\pi A^{\vee }$ . Since A is $\tau $ -invariant, B is also $\tau $ -invariant. Therefore, if $B \subset B^{\vee }$ , then $B \in {\mathcal L}_0$ , and hence (1) holds. Therefore, it suffices to show that $B \subset B^{\vee }$ . Assume that $B \nsubseteq B^{\vee }$ . Since $\pi B^{\vee }$ is of index 1 in A and $\pi B \subset A$ , we have $A=\pi B + \pi B^{\vee }$ . However, $\pi B^{\vee } \subset \pi A^{\vee }$ and $\pi B \subset \pi A^{\vee }$ imply that $A=\pi B + \pi B^{\vee } \subset \pi A^{\vee }$ , which contradicts our assumption that A is not a vertex lattice.

Case 4. This is almost identical to Case 3. Assume that B is $\tau $ -invariant, but not a vertex lattice in ${\mathcal L}_0$ . Then, $B \nsubseteq B^{\vee }$ . Since $\pi B^{\vee }$ is of index 1 in A and $\pi B \subset A$ , we have that $A=\pi B+ \pi B^{\vee }$ . In particular, A is also $\tau $ -invariant. Also, $\pi B^{\vee } \subset \pi A^{\vee }$ and $\pi B \subset \pi A^{\vee }$ imply that $A \subset \pi A^{\vee }$ . Therefore, A is a vertex lattice in ${\mathcal L}_1$ and (2) holds in this case.

Case 5. Assume that $A,B$ are not $\tau $ -invariant and $B \subset B^{\vee }$ . In this case, we have

(2.5.7) $$ \begin{align} A \cap \tau(A)= \pi B^{\vee}, \\[-36pt]\nonumber\end{align} $$

(2.5.8) $$ \begin{align} B \cap \tau(B)=\pi A^{\vee}. \end{align} $$

Also, note that

$$ \begin{align*} B+\tau(B) &\subset B^{\vee} \subset \pi^{-1} \tau(A), \\\tau(B)+\tau^2(B)& \subset \tau(B^{\vee}) \subset \pi^{-1} \tau(A). \end{align*} $$

Therefore, we have

$$ \begin{align*} T_2B \subset \pi^{-1} \tau(A) \subset \pi^{-1} T_1A, \end{align*} $$

and

(2.5.9) $$ \begin{align} T_dB \subset \pi^{-1} T_{d-1}A. \end{align} $$

Case 5-1. Assume that $d-1 < c$ . Since $T_dB$ is $\tau $ -invariant, (2.5.9) implies that

$$ \begin{align*} T_dB \subset \bigcap_{l \in {\mathbb Z}}\pi^{-1} \tau^l(T_{d-1}A)\overset{(2.5.1)}{=}\bigcap_{l \in {\mathbb Z}}\pi^{-1} \tau^l(A)\overset{(2.5.7)}{=}\bigcap_{l \in {\mathbb Z}}\pi^{-1} \tau^l(\pi B^{\vee})=(T_dB)^{\vee}. \end{align*} $$

The last equality is induced by

$$ \begin{align*} (T_dB)^{\vee}=B^{\vee} \cap \tau(B^{\vee}) \cap \cdots \cap \tau^{d-1}(B^{\vee}), \end{align*} $$

and the fact that $(T_dB)^{\vee }$ is $\tau $ -invariant. Therefore, (1) holds in this case.

Case 5-2. Assume that $d-1 \geq c$ . Then, $T_cA \subset T_cB$ and $T_cA$ is $\tau $ -invariant. Therefore, we have

$$ \begin{align*} T_cA \subset \bigcap_{l \in {\mathbb Z}} \tau^l(T_{c}B)\overset{(2.5.3)}{=}\bigcap_{l \in {\mathbb Z}} \tau^l(B)\overset{(2.5.8)}{=}\bigcap_{l \in {\mathbb Z}} \tau^l(\pi A^{\vee})=\pi (T_cA)^{\vee}. \end{align*} $$

The last equality is induced by

$$ \begin{align*} (T_cA)^{\vee}=A^{\vee} \cap \tau(A^{\vee}) \cap \cdots \cap \tau^{c-1}(A^{\vee}), \end{align*} $$

and the fact that $(T_cA)^{\vee }$ is $\tau $ -invariant. Therefore, (2) holds in this case.

Case 6. Assume that $A,B$ are not $\tau $ -invariant and $B \nsubseteq B^{\vee }$ . In this case, (2.5.7) and (2.5.8) hold and we have $A=\pi B + \pi B^{\vee } \subset \pi A^{\vee }$ (see Case 4). By (2.5.8), we have $A \subset B$ and $A \subset \tau (B)$ . Therefore, $T_1A \subset \tau (B)$ and

$$ \begin{align*} T_cA \subset \tau(T_{c-1}B). \end{align*} $$

Case 6-1. Assume that $c \leq d$ . Then, we have

$$ \begin{align*} T_cA \subset \bigcap_{l \in {\mathbb Z}} \tau^l(T_{c-1}B)\overset{(2.5.3)}{=}\bigcap_{l \in {\mathbb Z}} \tau^l(B)\overset{(2.5.8)}{=}\bigcap_{l \in {\mathbb Z}} \tau^l(\pi A^{\vee})=\pi (T_cA)^{\vee}. \end{align*} $$

Therefore, (2) holds in this case.

Case 6-2. Assume that $d<c$ . Then, $B \subset \pi ^{-1}A$ implies that $T_dB \subset \pi ^{-1}T_dA$ . Therefore, we have

$$ \begin{align*} T_dB \subset \bigcap_{l \in {\mathbb Z}}\pi^{-1} \tau^l(T_{d}A)\overset{(2.5.1)}{=}\bigcap_{l \in {\mathbb Z}}\pi^{-1} \tau^l(A)\overset{(2.5.7)}{=}\bigcap_{l \in {\mathbb Z}}\pi^{-1} \tau^l(\pi B^{\vee})=(T_dB)^{\vee}. \end{align*} $$

This is a contradiction, since $B \nsubseteq B^{\vee }$ and $B \subset T_dB \subset (T_dB)^{\vee } \subset B^{\vee }$ .

This completes the proof of Lemma 2.21.

Now, let us give the definition of the sets $R_{\Lambda }(k), S_{\Lambda }(k)$ .

Definition 2.23. (1) For a vertex lattice $\Lambda \in {\mathcal L}_1$ , we define the subset $R_{\Lambda }(k)$ of ${\mathcal N}(k)$ as

$$ \begin{align*} R_{\Lambda}(k):=\left\{\begin{array}{@{}c@{}} (A, B) \in {\mathcal N}(k) \end{array} \middle| \begin{array}{c} A \subset \Lambda \end{array}\!\!\! \right\}. \end{align*} $$

(2) For a vertex lattice $\Lambda \in {\mathcal L}_0$ , we define the subset $S_{\Lambda }(k)$ of ${\mathcal N}(k)$ as

$$ \begin{align*} S_{\Lambda}(k):=\left\{\begin{array}{@{}c@{}} (A, B) \in {\mathcal N}(k) \end{array} \middle| \begin{array}{c} B \subset \Lambda \end{array}\!\!\! \right\}. \end{align*} $$

Proof. (1) is clear from the definition.

For (2), we will show that $S_{\Lambda _1}(k) \cap S_{\Lambda _2}(k) \subset S_{\Lambda _1 \cap \Lambda _2}(k)$ . Let $(A,B)$ be an element in $S_{\Lambda _1}(k) \cap S_{\Lambda _2}(k)$ . Note that $(A,B)$ satisfies the following diagrams:

$$ \begin{align*} \begin{array}{cccccccc} &\pi A^{\vee} &\overset{1}{\subset} &B &\subset& \Lambda_1 & \subset&\Lambda_1^{\vee}\\ &\cup & &\cup & &&&\\ \pi \Lambda_1^{\vee} \subset&\pi B^{\vee}&\overset{1}{\subset}&A, \end{array} \end{align*} $$

and

$$ \begin{align*} \begin{array}{cccccccc} &\pi A^{\vee} &\overset{1}{\subset} &B &\subset& \Lambda_2 & \subset &\Lambda_2^{\vee}\\ &\cup & &\cup & &&&\\ \pi \Lambda_2^{\vee} \subset&\pi B^{\vee}&\overset{1}{\subset}&A. \end{array} \end{align*} $$

These two diagrams imply that

$$ \begin{align*} \begin{array}{cccccccc} &&\pi A^{\vee} &\overset{1}{\subset} &B &\subset& \Lambda_1 \cap\Lambda_2\subset \Lambda_1^{\vee}\subset (\Lambda_1 \cap\Lambda_2)^{\vee} \\ &&\cup & &\cup & &\\ \pi (\Lambda_1 \cap\Lambda_2)^{\vee}=&\pi \Lambda_1^{\vee}+\pi \Lambda_2^{\vee} \subset&\pi B^{\vee}&\overset{1}{\subset}&A. \end{array} \end{align*} $$

Therefore, $\Lambda _1 \cap \Lambda _2$ is in ${\mathcal L}_0^+$ , and $(A,B)$ should be contained in $S_{\Lambda _1 \cap \Lambda _2}(k)$ .

Conversely, $ S_{\Lambda _1 \cap \Lambda _2}(k)\subset S_{\Lambda _1}(k) \cap S_{\Lambda _2}(k)$ is obvious from (1). This completes the proof of the proposition.

Now, let us consider the intersection $R_{\Lambda _1}(k) \cap S_{\Lambda _0}(k)$ .

Proposition 2.33. Let $\Lambda _1 \in {\mathcal L}_1^+, \Lambda _0 \in {\mathcal L}_0^+$ .

(1) If $\pi \Lambda _1^{\vee } \nsubseteq \Lambda _0$ , then $R_{\Lambda _1}(k) \cap S_{\Lambda _0}(k)=\emptyset $ .

(2) If $\pi \Lambda _1^{\vee } \subset \Lambda _0$ , then

$$ \begin{equation*} R_{\Lambda_1}(k) \cap S_{\Lambda_0}(k)=\left\{\begin{array}{@{}c@{}}O_{\breve{F}}-lattices\\A \overset{h}{\subset} B \subset N_{k,0}\end{array}\middle|\begin{array}{@{}cccccccc@{}}& \pi \Lambda_1^{\vee}&\subset&\pi A^{\vee}&\overset{1}{\subset} & B & \subset&\Lambda_0\\&\cup&&&&&&\\& \Lambda_1&\supset&A&\overset{1}{\supset} & \pi B^{\vee}& \supset&\pi \Lambda_0^{\vee} \\\end{array}\right\}. \end{equation*} $$

Remark 2.34. Let $h=1$ , $\Lambda _1 \in {\mathcal L}_1^+, \Lambda _0 \in {\mathcal L}_0^+$ and $\pi \Lambda _1^{\vee } \subset \Lambda _0$ . For any $(A,B) \in R_{\Lambda _1}(k)$ , we have $A=\Lambda _1$ by Remark 2.30. Therefore,

$$ \begin{align*} R_{\Lambda_1}(k) \cap S_{\Lambda_0}(k)=\left\{\begin{array}{@{}c@{}} O_{\breve{F}}-lattices\\ B \subset N_{k,0} \end{array} \middle| \begin{array}{c} \pi \Lambda_1^{\vee} \overset{1}{\subset} B \subset \Lambda_0 \end{array} \right\}. \end{align*} $$

This is isomorphic to ${\mathbb P}^{m-1}(k)$ , where $m=[\Lambda _0 : \pi \Lambda _1^{\vee }]$ .

Remark 2.35. We can apply our method to ${\mathcal N}_{E/F}^0(2,2)$ , which has been studied in [Reference Howard and PappasHP14]. We should note that all of the following descriptions of k-points are already obtained in loc. cit. with a different method.

By using the relative Dieudonné theory and steps similar to those in Section 2, we can show that there is a bijection between $\mathcal {N}(k)$ and the set

$$ \begin{align*} \left\{\begin{array}{@{}c@{}} O_{\breve{F}}\text{-lattice } B \subset N_{k,0} \end{array} \middle| \begin{array}{c} \pi B^{\vee} \overset{2}{\subset} B \overset{2}{\subset} B^{\vee} \end{array} \right\}. \end{align*} $$

We can divide the set into three cases.

Case 1. $B \cap \tau (B) \overset {1}{\subset} B$ .

Case 2. $B \cap \tau (B)=\pi B^{\vee }$ and $B \overset {1}{\subset} T_1B$ .

Case 3. $B \cap \tau (B)=\pi B^{\vee }$ and $T_1B=B^{\vee }$ .

In Case 1, let $\pi A^{\vee }=B \cap \tau (B)$ . Then, the pair $(A,B)$ satisfies

$$ \begin{align*} \begin{array}{c} \pi A^{\vee} \overset{1}{\subset} B \overset{3}{\subset} A^{\vee};\\ \pi B^{\vee} \overset{1}{\subset} A \overset{3}{\subset} B^{\vee};\\ \pi B \overset{3}{\subset} A \overset{1}{\subset} B. \end{array} \end{align*} $$

Therefore, in a similar way to the proof of Lemma 2.21, we can show that at least one of the following is true.

1. (1) A is $\tau $ -invariant and $A=\pi A^{\vee }$ .

2. (2) $\Lambda _B \subset \Lambda _B^{\vee }$ .

In Case 2, one can prove that $\Lambda _B \subset \Lambda _B^{\vee }$ .

In Case 3, since $B \cap \tau (B) =\pi B^{\vee }$ , we have $B^{\vee } + \tau (B^{\vee }) =\pi ^{-1} \tau (B)$ by taking the dual. Since $B^{\vee }=B+\tau (B)$ , we have

$$ \begin{align*} B+\tau(B)+\tau^2(B)=\pi^{-1} \tau(B). \end{align*} $$

Let d be the smallest integer such that $T_dB$ is $\tau $ -invariant. Then, $T_dB=\pi ^{-1}\tau (T_{d-2}B)$ is $\tau $ -invariant, and this means that $T_{d-2}B$ is also $\tau $ -invariant. This is possible only when B is $\tau $ -invariant.

In summary, $B \cap \tau (B)$ is a vertex lattice of type $0$ , or $\Lambda _B \subset \Lambda _B^{\vee }$ (hence, $\Lambda _B$ is a vertex lattice). This is the analogue of Lemma 2.21.

Therefore, for each vertex lattice $\Lambda $ , we can attach the following set.

(1) If $\Lambda =\pi \Lambda ^{\vee }$ , then we attach the set

$$ \begin{align*} \left\{\begin{array}{@{}c@{}} O_{\breve{F}}-lattices\\ B \subset N_{k,0} \end{array} \middle| \begin{array}{c} \Lambda \overset{1}{\subset} B \overset{2}{\subset} B^{\vee} \overset{1}{\subset} \Lambda^{\vee} \end{array}\!\!\! \right\}. \end{align*} $$

This is the set of k-points of a Fermat hypersurface.

(2) If $\pi \Lambda ^{\vee } \overset {2}{\subset} \Lambda $ , then we attach the set

$$ \begin{align*} \left\{\begin{array}{@{}c@{}} O_{\breve{F}}-lattices\\ B \subset N_{k,0} \end{array} \middle| \begin{array}{c} B=\Lambda \end{array}\!\!\! \right\}. \end{align*} $$

This is one k-point.

(3) If $\pi \Lambda ^{\vee } \overset {4}{\subset} \Lambda $ , then we attach the set

$$ \begin{align*} \left\{\begin{array}{@{}c@{}} O_{\breve{F}}-lattices\\ B \subset N_{k,0} \end{array} \middle| \begin{array}{c} \pi \Lambda^{\vee} \overset{1}{\subset} \pi B^{\vee} \overset{2}{\subset} B \overset{1}{\subset} \Lambda = \Lambda^{\vee} \end{array}\!\!\! \right\}. \end{align*} $$

This is the set of k-points of a Fermat hypersurface.

${\mathcal N}(k)$ is the union of the above sets and this is the same result as in [Reference Howard and PappasHP14].

3.1 Strict formal $O_F$ -modules $X_{\Lambda ^+}$ and $X_{\Lambda ^-}$

In this subsection, we fix a vertex lattice $\Lambda \in {\mathcal L}_i^+$ , for $i=0,1$ . We will define the strict formal $O_F$ -modules $X_{\Lambda ^+}$ , $X_{\Lambda ^-}$ over ${\mathbb F}_{q^2}$ with $O_E$ -action, polarizations $\lambda _{\Lambda ^{\pm }}$ and quasi-isogenies $\rho _{\Lambda ^{\pm }}:X_{\Lambda ^{\pm }} \rightarrow {\mathbb X}$ . For this, we will construct the following two Dieudonné submodules of N.

First, if $\Lambda \in {\mathcal L}_0^+$ , we define the lattices $\Lambda ^+$ and $\Lambda ^-$ by

$$ \begin{align*} \Lambda_0^+&=\Lambda,\\\Lambda_1^+&={\mathcal V}^{-1}(\Lambda), \\ \Lambda_0^-&=p \Lambda^{\vee}\\ \Lambda_1^-&={\mathcal V}(\Lambda^{\vee}), \\ \Lambda^+&=\Lambda_0^+ \oplus \Lambda_1^+,\\ \Lambda^-&=\Lambda_0^- \oplus \Lambda_1^-. \end{align*} $$

Then, one can easily show that $\Lambda ^-=\Lambda _0^- \oplus \Lambda _1^-=(\Lambda _1^+)^{\perp } \oplus (\Lambda _0^+)^{\perp }=(\Lambda ^+)^{\perp }$ . Since ${\mathcal F}={\mathcal V}$ on $\Lambda ^+$ and $\Lambda ^-$ , we have that $\Lambda ^+$ and $\Lambda ^-$ are Dieudonné submodules of N.

In the case $\Lambda \in {\mathcal L}_1^+$ , we define the lattices $\Lambda ^+$ and $\Lambda ^-$ by

$$ \begin{align*} \Lambda_0^+&=\Lambda, \\ \Lambda_1^+&={\mathcal V}^{-1}(\Lambda), \\\Lambda_0^-&=p^2 \Lambda^{\vee}, \\\Lambda_1^-&=p {\mathcal V}(\Lambda^{\vee}), \\\Lambda^+&=\Lambda_0^+ \oplus \Lambda_1^+, \\\Lambda^-&=\Lambda_0^- \oplus \Lambda_1^-. \end{align*} $$

Then, we have $\Lambda ^-=\Lambda _0^- \oplus \Lambda _1^-=p(\Lambda _1^+)^{\perp } \oplus p(\Lambda _0^+)^{\perp }=p(\Lambda ^+)^{\perp }$ . Again, these $\Lambda ^+$ and $\Lambda ^-$ are Dieudonné submodules of N.

For $\Lambda \in {\mathcal L}_i^+$ , we have $\Lambda \subset p^i \Lambda ^{\vee }$ . Therefore, the pairing $p^{-i+1} \langle \cdot , \cdot \rangle $ on N induces a $W_{O_F}({\mathbb F}_{q^2})$ -pairing on $\Lambda ^+$ and $\Lambda ^-$ , respectively.

Now, let $X_{\Lambda ^+}$ and $X_{\Lambda ^-}$ be the strict formal $O_F$ -modules over ${\mathbb F}_{q^2}$ associated to $\Lambda ^+$ and $\Lambda ^-$ with quasi-isogenies $\rho _{\Lambda ^{\pm }}:X_{\Lambda ^{\pm }} \rightarrow {\mathbb X}$ .

We will use these two strict formal $O_F$ -modules to define the subschemes ${\mathcal N}_{\Lambda }$ of ${\mathcal N}$ .

3.2 Subschemes $\widetilde {{\mathcal N}}_{\Lambda }$ attached to vertex lattices $\Lambda $

We fix $\Lambda \in {\mathcal L}_i^+$ , for $i=0,1$ . Let S be an ${\mathbb F}_{q^2}$ -scheme. For $(X,i_X,\lambda _X,\rho _X) \in {\mathcal N}(S)$ , consider

$$ \begin{align*} \rho_{X,\Lambda^+}&:X \xrightarrow{\rho_X} {\mathbb X}_S \xrightarrow{(\rho_{\Lambda^+})^{-1}_S} (X_{\Lambda^+})_S, \\ \rho_{X,\Lambda^-}&:(X_{\Lambda^-})_S \xrightarrow{(\rho_{\Lambda^-})_S} {\mathbb X}_S \xrightarrow{\rho_X^{-1}} X. \end{align*} $$

For $\Lambda $ in ${\mathcal L}_0^+$ (resp. ${\mathcal L}_1^+$ ), we define $\widetilde {{\mathcal N}}_{\Lambda }$ as the subfunctor of ${\mathcal N} \otimes _{O_E}{\mathbb F}_{q^2}$ consisting of tuples $(X,i_X,\lambda _X,\rho _X) \in {\mathcal N}(S)$ such that $\rho _{X,\Lambda ^-}$ (resp. $\rho _{X,\Lambda ^+}$ ) is an isogeny.

We have the following lemma.

Proof. This is clear from the definition of $\widetilde {{\mathcal N}}_{\Lambda }$ . For $\Lambda \in {\mathcal L}_1^+$ , a point $(A,B^{\perp }) \in \widetilde {{\mathcal N}}_{\Lambda }(k)$ satisfies $A \subset \Lambda _0^+$ and $B^{\perp } \subset \Lambda _1^+$ . This is contained in $R_{\Lambda }(k)$ if and only if $ A \subset \Lambda $ . The condition $A \subset \Lambda $ is $A \subset \Lambda _0^{+}$ . The condition $B^{\perp } \subset \Lambda _1^+={\mathcal V}^{-1}\Lambda $ is equivalent to $p\Lambda ^{\vee }\subset B$ , which is a redundant condition.

For $\Lambda \in {\mathcal L}_0^+$ , a point $(A,B^{\perp }) \in \widetilde {{\mathcal N}}_{\Lambda }(k)$ satisfies $\Lambda _0^- \subset A$ and $\Lambda _1^- \subset B^{\perp }$ . This is contained in $S_{\Lambda }(k)$ if and only if $ B \subset \Lambda $ . Note that

$$ \begin{align*}\begin{array}{lllll} &B \subset \Lambda \Longleftrightarrow& \Lambda^{\perp}\subset B^{\perp} \Longleftrightarrow & {\mathcal V}(\Lambda^{\vee}) \subset B^{\perp} \Longleftrightarrow& \Lambda_1^{-} \subset B^{\perp}. \end{array} \end{align*} $$

In this case, the condition $p\Lambda ^{\vee }=\Lambda _0^- \subset A$ is a redundant condition.

3.3 Deligne–Lusztig varieties

In this subsection, we will recall some results about Deligne–Lusztig varieties.

Let G be a connected reductive group over a finite field ${\mathfrak K}$ . Denote by $G_{\overline {{\mathfrak K}}}$ the base change of G over $\overline {{\mathfrak K}}$ , where $\overline {{\mathfrak K}}$ is a fixed algebraic closure of ${\mathfrak K}$ . Let ${\mathcal F}:G \rightarrow G$ be the Frobenius morphism with respect to ${\mathfrak K}$ , and let $(W,S)$ be the Weyl system of $G_{\overline {{\mathfrak K}}}$ . Then, ${\mathcal F}$ gives an automorphism of W. By Lang’s theorem, G is quasi-split, and hence ${\mathcal F}(S)=S$ .

For $I \subset S$ , let $W_I$ be the subgroup of W generated by I, and let $P_I=BW_IB$ be the corresponding standard parabolic subgroup of G.

For I, $J \subset S$ , we denote by $^IW^J$ the set of minimal length representatives $w\in W$ in the double coset $W_I \backslash W / W_J$ .

Now, we define the generalized Deligne–Lusztig varieties as follows.

Definition 3.3. Let $I \subset S$ . For each $w \in W$ , we define the generalized Deligne–Lusztig variety $X_I(w)$ by

$$ \begin{align*} X_I(w):=\{g \in G/P_I : g^{-1}{\mathcal F}(g) \in P_IwP_{{\mathcal F}(I)} \}. \end{align*} $$

We will need the following two results later.

3.4 The Deligne–Lusztig variety $Y_{\Lambda }$

In this subsection, we will define the Deligne–Lusztig variety $Y_{\Lambda }$ . For $i=0,1$ , we fix a vertex lattice $\Lambda \in {\mathcal L}_i^+$ . We use the following notation.

$\bullet $ Let $V_{\Lambda }$ be $\Lambda _0^+/\Lambda _0^-$ and let $(\cdot , \cdot )$ be the skew-hermitian form on $V_{\Lambda }$ induced by $p^{-i}\lbrace \cdot , \cdot \rbrace $ . Note that $V_{\Lambda }$ is an ${\mathbb F}_{q^2}$ -vector space of dimension $d:=t(\Lambda )$ . Here, we fix $\overline {\delta } \in {\mathbb F}_{q^2}^{\times }$ such that $\overline {\delta }^*=-\overline {\delta }$ . Then, we can choose a basis of $V_{\Lambda }$ such that the form $(\cdot , \cdot )$ with respect to this basis is given by the matrix

$$ \begin{align*} \overline{\delta}\left(\begin{array}{ccc} & & 1\\ &\unicode{x22F0} & \\ 1& & \end{array}\right). \end{align*} $$

$\bullet $ Let $J_{\Lambda }$ be the special unitary group associated to $(V_{\Lambda }, (\cdot , \cdot ))$ . This is a connected reductive group over ${\mathbb F}_q$ .

$\bullet $ Let ${\mathcal F}:J_{\Lambda } \rightarrow J_{\Lambda }$ be the Frobenius morphism over ${\mathbb F}_q$ and $(W,S)$ be the Weyl system of $J_{\Lambda }$ .

Note that

$$ \begin{align*} J_{\Lambda} \otimes_{{\mathbb F}_q} {\mathbb F}_{q^2} \simeq SL(V_{\Lambda})=SL_{d,{\mathbb F}_{q^2}}. \end{align*} $$

Therefore, we can identify W with the symmetric group $S_d$ , and S with $\lbrace s_1, \dots , s_d \rbrace $ , where $s_i$ is the transposition of i and $i+1$ .

The Frobenius ${\mathcal F}$ induces an automorphism of W, and this is given by the conjugation with $w_0 \in S_d$ , where $w_0(i)=d+1-i$ for all i.

$\bullet $ For an ${\mathbb F}_{q^2}$ -algebra R, we denote by $V_{\Lambda , R}$ the base change $V_{\Lambda } \otimes _{{\mathbb F}_{q^2}}R$ . Let $\sigma $ be the Frobenius of R. For an R-module M, denote by $M^{(\sigma )}=M \otimes _{R,\sigma }R$ the Frobenius twist, and denote by $M^*=Hom_R(M,R)$ . Let U be a locally direct summand of $V_{\Lambda ,R}$ of rank m. We define its dual module $U^{\curlyvee }$ as follows. Since $(\cdot ,\cdot )$ induces an R-linear isomorphism

$$ \begin{align*} \psi:(V_{\Lambda,R})^{(\sigma)} \simeq (V_{\Lambda,R})^*, \end{align*} $$

$\psi (U^{(\sigma )})$ is a locally direct summand of $(V_{\Lambda ,R})^*$ of rank m. Let $U^{\curlyvee }$ be the kernel of the composition

$$ \begin{align*} V_{\Lambda,R} \simeq (V_{\Lambda,R})^{**} \twoheadrightarrow \psi(U^{(\sigma)})^*. \end{align*} $$

This is a locally direct summand of $V_{\Lambda ,R}$ of rank $d-m$ .

In particular, if $R=k$ , then

$$ \begin{align*} U^{\curlyvee}=\lbrace x \in V_{\Lambda,k}:(x,U)=0\rbrace. \end{align*} $$

Remark 3.6. Let $R=k$ . For a lattice A such that $p^{i+1}\Lambda ^{\vee } \subset A \subset \Lambda $ , the quotient $A/p^{i+1}\Lambda ^{\vee }$ is a subspace of $V_{\Lambda ,k}$ . Then, by definition, we have

$$ \begin{align*} p^{i+1}A^{\vee}/p^{i+1}\Lambda^{\vee}=(A/p^{i+1}\Lambda^{\vee})^{\curlyvee}. \end{align*} $$

We will need the following lemma.

Then, we have the following proposition.

Proof. (1) For any $(A \subset B) \in S_{\Lambda }(k)$ , it satisfies

$$ \begin{align*} \begin{array}{cccccccc} &p A^{\vee} &\overset{1}{\subset} &B &\subset& \Lambda &\subset &\Lambda^{\vee} \\ &\cup & &\cup & &&&\\ p \Lambda^{\vee} \subset&p B^{\vee}&\overset{1}{\subset}&A.\\ \end{array} \end{align*} $$

Therefore, the map $(A,B) \rightarrow (A/p \Lambda ^{\vee } \subset B/ p \Lambda ^{\vee })=:(\overline {A},\overline {B})$ is a bijection from $S_{\Lambda }(k)$ to the set of flags ${\mathfrak F}=(\overline {A},\overline {B})$ satisfying

$$ \begin{align*} \begin{array}{cccccccc} &\overline{A}^{\curlyvee} &\overset{1}{\subset} &\overline{B} &\overset{l}{\subset}& V_{\Lambda,k} &\\ &\cup & &\cup & &&&\text{(see Remark}\ {3.6)}\\ 0 \overset{l}{\subset}&\overline{B}^{\curlyvee}&\overset{1}{\subset}&\overline{A}.\\ \end{array} \end{align*} $$

Now, it suffices to show that $X_{I_{\Lambda }}(id)(k) \sqcup X_{I_{\Lambda }}(w_{\Lambda })(k)$ is the set of all flags ${\mathfrak F}=(\overline {A},\overline {B})$ satisfying the above condition.

We fix a standard basis $\lbrace e_1,\dots , e_d \rbrace $ of $V_{\Lambda ,k}$ such that the hermitian form $(\cdot ,\cdot )$ on $V_{\Lambda ,k}$ is given by the matrix

$$ \begin{align*} \overline{\delta}\left(\begin{array}{ccc} & & 1\\ &\unicode{x22F0} & \\ 1& & \end{array}\right). \end{align*} $$

Let $\overline {A}_0=\text {span}\lbrace e_1,\dots ,e_{l+1}\rbrace $ , $\overline {B}_0=\text {span}\lbrace e_1,\dots , e_{l+h+1}\rbrace $ , and let ${\mathfrak F}_0$ be the flag $(\overline {A}_0,\overline {B}_0)$ . Then, $P_{I_{\Lambda }}$ is the stabilizer of ${\mathfrak F}_0$ , and we have

$$ \begin{align*}\overline{B_0}^{\curlyvee}\overset{1}{\subset} \overline{A}_0 \overset{h-1}{\subset}\overline{A}_0^{\curlyvee}\overset{1}{\subset}\overline{B}_0.\end{align*} $$

First, if $A \subset pA^{\vee } \Longleftrightarrow \overline {A} \subset \overline {A}^{\curlyvee }$ , then we have that

(3.4.1) $$ \begin{align}\overline{B}^{\curlyvee}\overset{1}{\subset} \overline{A} \overset{h-1}{\subset}\overline{A}^{\curlyvee}\overset{1}{\subset}\overline{B}. \end{align} $$

By the transitivity of the $J_{\Lambda ,k}$ -action on flags of the same type, we can choose an element g such that

(3.4.2) $$ \begin{align}(\overline{B}^{\curlyvee}, \overline{A} ,\overline{A}^{\curlyvee},\overline{B})=g(\overline{B}_0^{\curlyvee}, \overline{A}_0 ,\overline{A}_0^{\curlyvee},\overline{B}_0). \end{align} $$

In particular, we have that

$$ \begin{align*} (g{\mathfrak F}_0)^{\curlyvee}&=({\mathfrak F})^{\curlyvee}=(\overline{B}^{\curlyvee},\overline{A}^{\curlyvee}),\\ g{\mathfrak F}_0^{\curlyvee}&=g(\overline{B}_0^{\curlyvee},\overline{A}_0^{\curlyvee})=(\overline{B}^{\curlyvee},\overline{A}^{\curlyvee}). \end{align*} $$

By Lemma 3.7, we have that

$$ \begin{align*} {\mathcal F}(g){\mathfrak F}_0^{\curlyvee}=(g{\mathfrak F}_0)^{\curlyvee}=g{\mathfrak F}_0^{\curlyvee}. \end{align*} $$

Therefore, we have ${\mathcal F}(g)P_{{\mathcal F}(I_{\Lambda })}=gP_{{\mathcal F}(I_{\Lambda })}$ , and hence $g^{-1}{\mathcal F}(g) \in P_{I_{\Lambda }} id P_{{\mathcal F}(I_{\Lambda })}$ , which is equivalent to $gP_{I_{\Lambda }} \in X_{I_{\Lambda }}(id)(k)$ .

Conversely, if $g P_{I_{\Lambda }} \in X_{I_{\Lambda }}(id)(k)$ , then we have ${\mathcal F}(g){\mathcal F}(P_{I_{\Lambda }})=g{\mathcal F}(P_{I_{\Lambda }})$ . We define ${\mathfrak F}=(\overline {A},\overline {B})=g(\overline {A}_0,\overline {B}_0)$ . Then, by Lemma 3.7 again, we have

$$ \begin{align*} {\mathfrak F}^{\curlyvee}=(g{\mathfrak F}_0)^{\curlyvee}={\mathcal F}(g){\mathfrak F}_0^{\curlyvee}=g{\mathfrak F}_0^{\curlyvee}. \end{align*} $$

This implies (3.4.2) and (3.4.1), and hence ${\mathfrak F}=(\overline {A}, \overline {B})$ gives a point in $S_{\Lambda }(k)$ .

Now, assume that $A \nsubseteq pA^{\vee } \Longleftrightarrow \overline {A} \nsubseteq \overline {A}^{\curlyvee }$ . Then, the flag ${\mathfrak F}=(\overline {A} \subset \overline {B})$ satisfies

Since $\overline {A}^{\curlyvee }+\overline {A}=\overline {B}$ , we can choose a basis $\lbrace e_1',e_2',\dots ,e_{l+h+1}'\rbrace $ of $\overline {B}$ satisfying the following properties:

1. (1) $\lbrace e_1',\dots ,e_l'\rbrace $ is a basis of $\overline {B}^{\curlyvee }$ .

2. (2) $e_{l+1}' \in \overline {A} \backslash \overline {A}^{\curlyvee }$ .

3. (3) $\lbrace e_1',\dots ,e_l',e_{l+2}',\dots ,e_{l+h+1}'\rbrace $ is a basis of $\overline {A}^{\curlyvee }$ .

Choose an element $g \in J_{\Lambda ,k}$ such that

$$ \begin{align*} (e_1',\dots,e_{l+h+1}')=g(e_1,\dots,e_{l+h+1}). \end{align*} $$

Then, we have that

$$ \begin{align*} \overline{B}^{\curlyvee}=gw_{\Lambda}\overline{B}_0^{\curlyvee},\text{ } \overline{A}^{\curlyvee}=gw_{\Lambda}\overline{A}_0^{\curlyvee}, \text{ } \overline{A}=g\overline{A}_0, \text{ } \overline{B}=g\overline{B}_0. \end{align*} $$

Then, we can prove (1) by following the previous case.

Similarly, for any $(A \subset B) \in R_{\Lambda }(k)$ , it satisfies

$$ \begin{align*} \begin{array}{cccccccc} &p B^{\vee} &\overset{1}{\subset} &A &\subset& \Lambda &\subset &p \Lambda^{\vee} \\ &\cup & &\cup & &&&\\ p^2 \Lambda^{\vee} \subset&p^2 A^{\vee}&\overset{1}{\subset}&pB. \end{array} \end{align*} $$

Therefore, the map $R_{\Lambda }(k)\rightarrow X_{I_{\Lambda }}(id)(k) \sqcup X_{I_{\Lambda }}(w_{\Lambda })(k)$ is given by $(A,B) \rightarrow (pB/p^2 \Lambda ^{\vee } \subset A/ p^2 \Lambda ^{\vee })$ . The remaining proofs of (2), (3) and (4) are similar.

Definition 3.11. For $i=0,1$ , let $\Lambda \in {\mathcal L}_i^+$ . If $1 \leq h \leq n-1$ , then we define an ${\mathbb F}_{q^2}$ -scheme

$$ \begin{align*} Y_{\Lambda}:=\overline{X_{I_{\Lambda}}(w_{\Lambda})}=X_{I_{\Lambda}}(id) \cup X_{I_{\Lambda}}(w_{\Lambda}). \end{align*} $$

The second equality is from the property of the Bruhat order (see [Reference Howard and PappasHP14, Lemma 3.7]). If $h=0$ and $\Lambda \in {\mathcal L}_0^+$ , then we define $Y_{\Lambda }:=X_{I_{\Lambda }}(id)$ . Similarly, if $h=n$ and $\Lambda \in {\mathcal L}_1^+$ , then we define $Y_{\Lambda }:=X_{I_{\Lambda }}(id)$ .

By Proposition 3.4 and Proposition 3.5, we have the following proposition.

Proposition 3.12. For $\Lambda \in {\mathcal L}_i^+$ ( $i=0,1$ ), $Y_{\Lambda }$ is irreducible, and

(1) if $\Lambda \in {\mathcal L}_0^+$ , the dimension of $Y_{\Lambda }$ is

$$ \begin{align*} \dfrac{t(\Lambda)-1-h}{2}+h, \end{align*} $$

(2) if $\Lambda \in {\mathcal L}_1^+$ , the dimension of $Y_{\Lambda }$ is

$$ \begin{align*} \dfrac{t(\Lambda)-1-(n-h)}{2}+n-h. \end{align*} $$