2.1.1.

Let p be an odd prime and F be a local field over $\mathbb{Q}_p$ . We use $O_F$ to denote the ring of integers of F and $k_F$ its residue field. Let $\breve F$ denote the completion of the maximal unramified extension $F^{{\rm ur}}$ of F, $W=O_{\breve{F}}$ its ring of integers and k the residue field of W. We always fix a uniformizer $\pi$ of F. We let V be a quadratic space over F of dimension n with non-degenerate symmetric form $({\kern2pt}, )$ and quadratic form $q(\,)$ .

We define the determinant of V to be

\begin{align*}\operatorname{det}(V):=\operatorname{det}((x_i, x_j)_{1\leqslant i, j\leqslant n}) \in F^{\times} /(F^{\times})^2,\end{align*}

where $\{x_1, \ldots, x_n\}$ is a basis of V, and the discriminant of V to be

\begin{align*}\operatorname{disc}(V):=(-1)^{\binom{n}{2}} \cdot \operatorname{det}(V) \in F^{\times} /(F^{\times})^2.\end{align*}

Let $\operatorname{val}(V)=\operatorname{val}(\operatorname{det}(V))\in \mathbb{Z}/2\mathbb{Z}$ and $\chi=(\cdot/\pi)_{F}: F^{\times} /(F^{\times})^2 \rightarrow\{ \pm 1,0\}$ be the quadratic residue symbol. Thus, $\chi(\alpha)=0$ if ${{\rm val}}(\alpha)$ is odd and $\chi(\alpha)$ is identified with the image of $\alpha/\pi^{{{\rm val}}(\alpha)}$ under the isomorphism $O_F^\times/ (O_F^\times)^2 \cong \{\pm\}$ if ${{\rm val}}(\alpha)$ is even. Define

\begin{align*}\chi(V):=\chi(\operatorname{disc}(V)) \in\{ \pm 1,0\},\quad\chi'(V):=\chi(\pi \operatorname{disc}(V)) \in\{ \pm 1,0\}.\end{align*}

Then, ${{\rm disc}}(V)$ is determined by $\chi(V)$ and $\chi'(V)$ . Note that only one of $\chi(V)$ and $\chi'(V)$ is nontrivial.

Let $\{v_1,\ldots,v_n\}$ be an orthogonal basis of V. We define the Hasse invariant of V to be

\begin{align*} \epsilon(V):= \prod_{1\leqslant i\lt j\leqslant n} (q(v_i),q(v_j))_{F}\in \{\pm 1\}.\end{align*}

Here, $({\kern2pt}, )_{F}$ is the Hilbert symbol. It is well known that quadratic spaces over F are classified by its dimension, discriminant and Hasse invariant.

For a quadratic lattice $L\subset V$ of rank n, we define $\chi(L)=\chi(V),\chi'(L)=\chi'(V)$ and $\epsilon(L)=\epsilon(V)$ .

2.2.1.

Let V be a quadratic space over F. Consider the following decomposition of V:

where $(e_i,e_j)=(f_i,f_j)=0$ , $(e_i,f_j)=\delta_{ij}$ , $V_{{{{\rm an}}}}$ is anisotropic and denotes the orthogonal sum. Note that $\dim_{F}(V_{{{{\rm an}}}})\leqslant 4$ , since any quadratic space over F of dimension at least 5 contains an isotropic vector.

Let $L\subset V$ be a lattice. We let $L^\vee=\{x\in L\otimes_{O_F} F\mid (x,y) \in O_F, \, \forall y \in L\}$ denote the dual lattice. We say that L is a vertex lattice if we have $\pi L \subset L \subset L^{\vee}$ . If L is a vertex lattice, its type $t=t(L)$ is defined to be the dimension of the $k_F$ -vector space $L^\vee/L$ . Note that our convention of vertex lattice is different with the one used in [Reference Howard and PappasHP17]. The dual lattice of a vertex lattice in [Reference Howard and PappasHP17] is our vertex lattice.

We will describe all the possible $V_{{{{\rm an}}}}$ and the vertex lattices in $V_{{{{\rm an}}}}$ following [Reference Howard and PappasHP17, § 5.1].

2.2.2.

If $\operatorname{dim}(V_{{{{\rm an}}}})=1$ , then it is isomorphic to F with the quadratic form $q(x)=d x^2$ for some $d \in F^{\times} /(F^{\times})^2$ . So there are four cases. Then, V contains a self-dual lattice if and only if ${{\rm val}}(d)$ is even. If we write $d=u\pi^{{{\rm val}}(d)}$ , then $\chi(V)=\chi(u)$ . The vertex lattice in $V_{{{{\rm an}}}}$ is ${\rm Span}_{O_F}\{x\}$ for ${{\rm val}}(q(x))=0$ (respectively 1) if ${{\rm val}}(d)$ is even (respectively odd).

If $\operatorname{dim}(V_{{{{\rm an}}}})=2$ , then we can identify $V_{{{{\rm an}}}}$ with a quadratic extension $E/F$ whose quadratic form is given by $q(x)=d\cdot {\rm Nm}(x)$ for some $d \in E^{\times} / {\rm Nm}(E^{\times})$ . Without loss of generality, we can choose d such that ${{\rm val}}(d)=0$ or 1 when E is unramified and $d\in (O_F^\times)^2$ or $d\in O_F^\times \setminus (O_F^\times)^2$ when E is ramified. If E is an unramified extension, then the vertex lattice in $V_{{{{\rm an}}}}$ has moment matrix ${\rm Diag}(u_1 d, u_2 d)$ such that $\chi(-u_1u_2)=-1$ . If E is a ramified extension, then the vertex lattice in $V_{{{{\rm an}}}}$ has moment matrix ${\rm Diag}(d , -d \pi )$ .

If $\operatorname{dim}(V_{{{{\rm an}}}})=3$ , there are exactly four anisotropic quadratic spaces over F of dimension 3. Let B denote the quaternion division algebra over F and $\bar{x}$ denote the main involution of B. Then the subspace of elements $B^0=\{x \in B: x+\bar{x}=0\}$ is a three-dimensional quadratic space with quadratic form given by the reduced norm on $B^0$ . More explicitly,

(2.1) \begin{eqnarray} (B^0, {\rm Nrd}) \stackrel{\sim}{\rightarrow}(F^3,-u x_1^2-\pi x_2^2+u \pi x_3^2)\end{eqnarray}

for any nonsquare $u \in O_F^{\times}$ . Now the four possible $V_{{{{\rm an}}}}$ of dimension 3 are the spaces $(B_0, d\cdot {\rm Nrd})$ with $d \in F^{\times} /(F^{\times})^2$ . Again, we assume ${{\rm val}}(d)=0$ or 1 without loss of generality. Then, according to (2.1), when ${{\rm val}}(d)$ is even, a vertex lattice has moment matrix given by ${\rm Diag}(-ud,-d\pi,ud\pi )$ . When ${{\rm val}}(d)$ is odd, a vertex lattice has moment matrix given by ${\rm Diag}(-ud,-\pi^{-1}d,u\pi^{-1}d)$ . In particular, $V_{{{{\rm an}}}}$ can never contain self-dual lattices when $\dim(V_{{{{\rm an}}}})=3$ .

If $\operatorname{dim}(V_{{{{\rm an}}}})=4$ , then $V\stackrel{\sim}{\to} B $ with quadratic form given by the reduced norm. In this case, a vertex lattice has ${\rm Diag}(u_1,u_2,u_3\pi,u_4\pi)$ as moment matrix, where $u_i \in O_F^\times$ and $\chi(-u_1u_2)=\chi(-u_3u_4)=-1$ .

2.2.3.

Combining the above discussion, we summarize all the possible vertex lattices of maximal type and minimal type up to isomorphism in the following proposition. We set

\begin{align*} H_{n}^{\pm}:= \text{a self-dual lattice of rank} n \text{with} \chi(H_{n}^\pm)=\pm 1.\end{align*}

We also use the same notation to denote the corresponding moment matrix. We set

\begin{align*} \Lambda_n^{\pm}:= \text{a} \pi\text{-modular lattice with moment matrix} \pi\cdot H_{n}^\pm.\end{align*}

1. (i) If $\operatorname{dim}(V_{{{{\rm an}}}})=0$ , then

   \begin{align*} \Lambda_{\rm max}=\Lambda_{n}^+ \quad \text{and} \quad \Lambda_{\rm min}=H_{n}^+. \end{align*}

2. (ii) If $\operatorname{dim}(V_{{{{\rm an}}}})=1$ , then

   

3. (iii) If $\operatorname{dim}(V_{{{{\rm an}}}})=2$ , then

   

4. (iv) If $\operatorname{dim}(V_{{{{\rm an}}}})=3$ , then

   

5. (v) If $\operatorname{dim}(V_{{{{\rm an}}}})=4$ , then

   

Here, if , then

(2.2) \begin{eqnarray}\notag \chi(V)&=&\begin{cases} \chi_0\chi_1 & \text{if }\textit{t}\text{ is even},\\ 0 & \text{if }\textit{t}\text{ is odd}, \end{cases}\\ \notag \chi'(V)&=& \chi(-1)^{n-1}\cdot\begin{cases} 0 & \text{if }\textit{t}\text{ is even},\\ \chi_0\chi_1 & \text{if }\textit{t}\text{ is odd}, \end{cases}\\ \epsilon(V)&=&\begin{cases}\chi_1& \text{if }\textit{t}\text{ is even},\\ \chi((-1)^{{(n-1)(n-2)}/{2}}) \chi_0 & \text{if }\textit{t}\text{ is odd}. \end{cases}\end{eqnarray}

2.3.1.

Note that $O_{{{\breve{F}}}}^\times=(O_{{{\breve{F}}}}^\times)^2$ . In particular, if $V=E$ with $q(x)={\rm Nm}(x)$ , where $E/F$ is an unramified quadratic extension, then ${{\breve{V}}}=V\otimes_{F}{{\breve{F}}}$ becomes isotropic. Combining this with the discussion in the previous section, we have the following classification of quadratic spaces over ${{\breve{F}}}$ .

Let be a quadratic space over ${{\breve{F}}}$ , where

\begin{align*}{{\breve{V}}}_0=\langle e_1,\dotsc,e_n,f_1,\dotsc,f_n\rangle\end{align*}

with $(e_i,e_j)=(f_i,f_j)=0$ , $(e_i,f_j)=\delta_{ij}$ and ${{\breve{V}}}_{{{{\rm an}}}}$ is anisotropic. Then ${{\breve{V}}}_{{{{\rm an}}}}$ is of the following form:

1. (1) ${{\breve{V}}}_{{{{\rm an}}}}=0$ ;

2. (2a) ${{\breve{V}}}_{{{{\rm an}}}}={{\breve{F}}} u$ , with $(u,u)=1$ ;

3. (2b) ${{\breve{V}}}_{{{{\rm an}}}}={{\breve{F}}} u$ , with $(u,u)=\pi$ ;

4. (3) ${{\breve{V}}}_{{{{\rm an}}}}={{\breve{F}}} u\oplus {{\breve{F}}} v$ , with $(u,u)=\pi$ , $(v,v)=1$ .

3.1.1.

Let G be a reductive group over F. We set $\Gamma=\operatorname{Gal}(\bar{F} /F)$ and let $\sigma\in \Gamma$ denote a lift of the Frobenius of k over $k_F$ . Let S be a maximal ${{\breve{F}}}$ -split torus of G and T its centralizer (cf. [Reference TitsTit79, 1.10] for the existence of S). By Steinberg’s theorem, G is quasi-split over ${{\breve{F}}}$ and T is a maximal torus of G. The relative Weyl group over ${{\breve{F}}}$ and the Iwahori Weyl group of G are defined as

\begin{align*}{{\widetilde{W}}}_0=N({{\breve{F}}})/T({{\breve{F}}}),\quad {{\widetilde{W}}}=N({{\breve{F}}})/{{\mathcal{T}}}_0({{\mathcal{O}}}_{{{\breve{F}}}}),\end{align*}

where N is the normalizer of T and ${{\mathcal{T}}}_0$ is the connected Néron model for T. We have the following exact sequence.

(3.1)

Here, $I = {\rm Gal}(\bar{F}/F^{{\rm ur}})$ denotes the inertia subgroup of the absolute Galois group of F, and $X_*(T)_I$ represents the group of coinvariants of the cocharacter lattice $X_*(T)$ under the action of I. For an element $\lambda\in X_*(T)_I$ , we write $t^\lambda$ for the corresponding element in ${{\widetilde{W}}}$ ; such elements are called translation elements.

Let ${{\mathcal{B}}}(G,F)$ (respectively ${{\mathcal{B}}}(G,{{\breve{F}}})$ ) be the (extended) Bruhat–Tits building for G over F (respectively ${{\breve{F}}}$ ), and fix ${{\mathfrak{a}}}\subset {{\mathcal{B}}}(G,F)$ a $\sigma$ -stable alcove; this determines an Iwahori group scheme ${{\mathcal{I}}}$ defined over F. Fix also a special vertex ${{\mathfrak{s}}}\in {{\mathcal{B}}}(G,{{\breve{F}}})$ lying in the closure of ${{\mathfrak{a}}}$ , which determines a splitting of the exact sequence (3.1) and gives an identification

\begin{align*} A:=X_*(T)_I\otimes_{{{\mathbb{Z}}}}{{\mathbb{R}}}\cong {{\mathcal{A}}}(G,S,{{\breve{F}}}),\end{align*}

where ${{\mathcal{A}}}(G,S,{{\breve{F}}})\subset {{\mathcal{B}}}(G,{{\breve{F}}})$ is the apartment corresponding to S. This identification determines a chamber $C^+\subset X_*(T)_I\otimes_{{{\mathbb{Z}}}}{{\mathbb{R}}}$ , namely the one containing ${{\mathfrak{a}}}$ . We write B for the corresponding Borel subgroup defined over ${{\breve{F}}}$ .

Let ${{\mathbb{S}}}$ denote the set of simple reflections about the walls of ${{\mathfrak{a}}}$ . We let ${{\widetilde{W}}}_a$ denote the affine Weyl group; it is the subgroup of ${{\widetilde{W}}}$ generated by the reflections in ${{\mathbb{S}}}$ . Then we have an exact sequence

(3.2)

where $T_{{\rm sc}}$ is the preimage of T in the simply connected cover of the derived group of G. The group $({{\widetilde{W}}}_a,{{\mathbb{S}}})$ has the structure of a Coxeter group, and hence we have a notion of length and Bruhat order. The Iwahori Weyl group and affine Weyl group are related by the following exact sequence.

The choice of ${{\mathfrak{a}}}$ induces a splitting of this exact sequence and $\pi_1(G)_I$ can be identified with the subgroup $\Omega\subset {{\widetilde{W}}}$ consisting of elements which map ${{\mathfrak{a}}}$ to itself. The length function $\ell$ and Bruhat order extend to ${{\widetilde{W}}}$ via the splitting.

By [Reference He and RapoportHR17], there is a reduced root system $\Sigma$ such that

\begin{align*}{{\widetilde{W}}}_a\cong Q^\vee(\Sigma)\rtimes {{\widetilde{W}}}(\Sigma),\end{align*}

where $Q^\vee(\Sigma) $ and ${{\widetilde{W}}}(\Sigma)$ denote the coroot lattice and Weyl group of $\Sigma$ , respectively. The root system $\Sigma$ is known as the échelonnage root system.

3.1.2.

Let $X_*(T)_I^+$ denote the set of dominant elements with respect to B. For $\mu\in X_*(T)^+_I$ , we have the $\mu$ -admissible set, which is defined as follows:

\begin{align*}{{\operatorname{Adm}}}(\mu):=\{w \in \widetilde{W} \mid w \leqslant t^{\mu^{\prime}} \text { for some } \mu^{\prime} \in {{\widetilde{W}}}_0 \mu\}.\end{align*}

This set has a minimal element $\tau_\mu$ , which is the unique length-zero element of ${{\operatorname{Adm}}}(\mu).$

Let $J\subset {{\mathbb{S}}}$ be a subset. We write ${{\widetilde{W}}}_J\subset {{\widetilde{W}}}$ for the subgroup generated by the reflections in J. If ${{\widetilde{W}}}_J$ is finite, then J determines a facet ${{\mathfrak{f}}}_J$ lying in the closure of ${{\mathfrak{a}}}$ and hence corresponds to a parahoric group scheme ${{\mathcal{G}}}^\circ_J$ . The corresponding parahoric $\mu$ -admissible set

\begin{align*}{{\operatorname{Adm}}}(\mu)_J\subset {{\widetilde{W}}}_J\backslash {{\widetilde{W}}}/{{\widetilde{W}}}_J\end{align*}

is defined to be the image of ${{\operatorname{Adm}}}(\mu)$ under the map ${{\widetilde{W}}}\rightarrow {{\widetilde{W}}}_J\backslash{{\widetilde{W}}}/{{\widetilde{W}}}_J$ . We also set

\begin{align*}{{\operatorname{Adm}}}(\mu)^J:={{\widetilde{W}}}_J\operatorname{Adm}(\mu){{\widetilde{W}}}_J\subset {{\widetilde{W}}}.\end{align*}

3.2.1.

Let V be a quadratic space over F. We now specialize to the case $G={\rm SO}(V).$ We let ${{\breve{V}}}=V\otimes_{F}{{\breve{F}}}$ . Then we can write , where

\begin{align*}{{\breve{V}}}_0=\langle e_1,\dotsc,e_n,f_1,\dotsc,f_n\rangle\end{align*}

with $(e_i,e_j)=(f_i,f_j)=0$ , $(e_i,f_j)=\delta_{ij}$ and ${{\breve{V}}}_{{{{\rm an}}}}$ is of the following form:

1. (1) ${{\breve{V}}}_{{{{\rm an}}}}=0$ ;

2. (2a) ${{\breve{V}}}_{{{{\rm an}}}}={{\breve{F}}} u$ , with $(u,u)=1$ ;

3. (2b) ${{\breve{V}}}_{{{{\rm an}}}}={{\breve{F}}} u$ , with $(u,u)=\pi$ ;

4. (3) ${{\breve{V}}}_{{{{\rm an}}}}={{\breve{F}}} u\oplus {{\breve{F}}} v$ , with $(u,u)=\pi$ , $(v,v)=1$ .

The local Dynkin diagram is of type $D_n$ in case (1), type $B_n$ in case (2) and type C- $B_n$ in case (3); here we use the notation of Tits’ table [Reference TitsTit79, § 4].

Let $L_{{{{\rm an}}}}\subset {{\breve{V}}}_{{{{\rm an}}}}$ denote the ${{\mathcal{O}}}_{{{\breve{F}}}}$ -submodule

\begin{align*}L_{{{{\rm an}}}}=\begin{cases}0 &\text{in case (1)},\\{{\mathcal{O}}}_{{{\breve{F}}}}u& \text{in case (2)},\\{{\mathcal{O}}}_{{{\breve{F}}}}u+{{\mathcal{O}}}_{{{\breve{F}}}}v& \text{in case (3)}.\end{cases}\end{align*}

For $s=0,\dotsc,n$ , we let $L_0^{(s)}\subset{{\breve{V}}}_0$ be the ${{\mathcal{O}}}_{{{\breve{F}}}}$ -module given by

\begin{align*}L_0^{(s)}={{{\rm Span}}}_{{{\mathcal{O}}}_{{{\breve{F}}}}}(\pi e_1,\dotsc,\pi e_s,e_{s+1},\dotsc,e_n,f_1,\dotsc,f_n)\end{align*}

and set

Then, $L^{(s)}$ is a vertex lattice of type

\begin{align*}t(s):=\begin{cases} 2s & \text{in case (1) and (2a)},\\2s+1 &\text{in case (2b) and (3)}.\end{cases}\end{align*}

We also set

which are vertex lattices of type t(0) and t(n), respectively.

3.2.2.

We take S to be the maximal ${{\breve{F}}}$ -split torus of ${\rm SO}({{\breve{V}}})$ given by ${{\mathbb{G}}}_m^n\subset{\rm SO}({{\breve{V}}})$ such that, for $t=(t_1,\dotsc,t_n)\in {{\mathbb{G}}}_m^n$ , we have

\begin{align*}t:{{\breve{V}}} \mapsto {{\breve{V}}} \text{ is given by } e_i\mapsto t_ie_i,\ f_i\mapsto t_i^{-1}f_i.\end{align*}

Let $\nu:{{\mathbb{G}}}_m\rightarrow {{\mathbb{G}}}_m^n$ be the cocharacter $t\mapsto (t,1,\dotsc,1)$ and let $\mu:{{\mathbb{G}}}_m\rightarrow {\rm SO}({{\breve{V}}})$ denote the composition of $\nu$ and the above inclusion. Then $\mu$ is a minuscule cocharacter of ${\rm SO}({{\breve{V}}})$ . We then have the associated $\mu$ -admissible set ${{\operatorname{Adm}}}(\mu)$ and its parahoric analogues. Our goal is to give a more explicit description of the set ${{\operatorname{Adm}}}(\mu)_J$ , where J corresponds to a vertex lattice in ${{\breve{V}}}$ . For this, we use the description of the building of ${\rm SO}({{\breve{V}}})$ in [Reference GarrettGar97, § 20.3]; we translate the necessary results to our setting. We first describe the base alcove, and facets corresponding to vertex lattices.

In each case, we have $X_*(T)_I\otimes_{{{\mathbb{Z}}}}{{\mathbb{R}}}\cong X_*(S)\otimes_{{{\mathbb{Z}}}}{{\mathbb{R}}}$ , which we identify with ${{\mathcal{A}}}(G,S,{{\breve{F}}})$ by sending 0 to the vertex ${{\mathbf{x}}}^{(0)}$ corresponding to the stabilizer of $L^{(0)}$ . We let $\epsilon_1,\dotsc,\epsilon_n$ denote the standard basis for ${{\mathbb{Z}}}^n$ .

Case (1). We have $S=T$ and $X_*(T)_I=X_*(S)$ , which we identify with ${{\mathbb{Z}}}^n$ so that the roots are given by $\pm\epsilon_i\pm\epsilon_j$ , $1\leqslant i\lt j\leqslant n$ . We take the n roots

\begin{align*} \epsilon_i-\epsilon_{i+1}, 1\leqslant i\leqslant n-1\quad \text{and}\quad \epsilon_{n-1}+\epsilon_n\end{align*}

as simple roots.

The base alcove in ${{\mathbb{R}}}^n$ is given by

\begin{align*}{{\mathfrak{a}}}=\{(a_1,\dotsc,a_n)\in {{\mathbb{R}}}^n\mid a_1\gt a_2\gt \cdots \gt\pm a_n, a_1+a_2\lt 1\}.\end{align*}

Let

\begin{align*}{{\mathbf{x}}}^{(0)} &=(0,\dotsc,0){,}\quad{{\mathbf{x}}}^{\prime (0)}=(1,0\dotsc,0),\\{{\mathbf{x}}}^{(s)}&=\sum_{i=1}^s\frac{\epsilon_i}{2}\quad \text{for } 1 \leqslant s \leqslant n-1,\\{{\mathbf{x}}}^{(n)}&=(\tfrac{1}{2},\dotsc,\tfrac{1}{2}){,}\quad{{\mathbf{x}}}^{\prime (n)}=(\tfrac{1}{2},\dotsc,\tfrac{1}{2},-\tfrac{1}{2}),\end{align*}

which lie in the closure of ${{\mathfrak{a}}}$ , with ${{\mathbf{x}}}^{(0)}$ a special vertex. The points ${{\mathbf{x}}}^{(0)},{{\mathbf{x}}}^{\prime (0)},{{\mathbf{x}}}^{(n)},{{\mathbf{x}}}^{\prime (n)}$ and ${{\mathbf{x}}}^{(s)}$ for $2\leqslant s\leqslant n-2$ are zero-dimensional facets, while ${{\mathbf{x}}}^{(1)}$ and ${{\mathbf{x}}}^{(n-1)}$ lie on one-dimensional facets given by the line segments between ${{\mathbf{x}}}^{(0)},{{\mathbf{x}}}^{\prime (0)}$ and ${{\mathbf{x}}}^{(n)},{{\mathbf{x}}}^{\prime (n)}$ , respectively. The parahoric corresponding to ${{\mathbf{x}}}^{(s)}, 0\leqslant s\leqslant n$ , is the connected stabilizer of $L^{(s)}$ ; similarly, ${{\mathbf{x}}}^{\prime (0)}$ corresponds to $L^{\prime (0)}$ and ${{\mathbf{x}}}^{\prime (n)}$ corresponds to $L^{\prime (n)}$ .

Case (2). We have $S=T$ and $X_*(T)_I=X_*(S)$ , which we identify with ${{\mathbb{Z}}}^n$ so that the roots are given by $\pm\epsilon_i\pm\epsilon_j, 1\leqslant i\lt j\leqslant n$ , $\pm\epsilon_i, 1\leqslant i \leqslant n$ . We take the n roots

\begin{align*} \epsilon_i-\epsilon_{i+1}, 1\leqslant i\leqslant n-1\quad \text{and}\quad \epsilon_n\end{align*}

as simple roots.

We first assume we are in case (2a). The base alcove in ${{\mathbb{R}}}^n$ is given by

\begin{align*}{{\mathfrak{a}}}=\{(a_1,\dotsc,a_n)\in {{\mathbb{R}}}^n \mid a_1\gt a_2\gt\cdots\gt a_n\gt 0, a_1+a_2\lt 1\}.\end{align*}

Let

\begin{align*}{{\mathbf{x}}}^{(0)}&=(0,\dotsc,0),\quad{{\mathbf{x}}}^{\prime (0)}=(1,0\dotsc,0),\\{{\mathbf{x}}}^{(s)}&=\sum_{i=1}^s\frac{\epsilon_i}{2}\quad \text{for } 1 \leqslant s \leqslant n,\end{align*}

which lie in the closure of ${{\mathfrak{a}}}$ , with ${{\mathbf{x}}}^{(0)}$ a special vertex. The points ${{\mathbf{x}}}^{(0)},{{\mathbf{x}}}^{\prime (0)}$ , ${{\mathbf{x}}}^{(s)},2\leqslant s\leqslant n$ , are zero-dimensional facets, while ${{\mathbf{x}}}^{(1)}$ lies on the one-dimensional facet given by the line segment between ${{\mathbf{x}}}^{(0)}$ and ${{\mathbf{x}}}^{\prime (0)}$ . The parahoric corresponding to ${{\mathbf{x}}}^{(s)}, 0\leqslant s\leqslant n$ , is the connected stabilizer of $L^{(s)}$ , and similarly ${{\mathbf{x}}}^{\prime (0)}$ corresponds to $L^{\prime (0)}$ .

Now assume we are in case (2b); in this case, the point ${{\mathbf{x}}}^{(0)}={(0,\dotsc,0)}$ is not a special vertex. The base alcove in ${{\mathbb{R}}}^n$ is given by

\begin{align*}{{\mathfrak{a}}}=\{(a_1,\dotsc,a_n)\in {{\mathbb{R}}}^n \mid \tfrac{1}{2}\gt a_1\gt a_2\gt\cdots\gt a_n, a_{n-1}+a_n\gt 0\}.\end{align*}

Let

\begin{align*}{{\mathbf{x}}}^{(s)}&=\sum_{i=1}^s\frac{\epsilon_i}{2}\quad \text{for }1 \leqslant s \leqslant n-1,\\{{\mathbf{x}}}^{(n)}&=(\tfrac{1}{2},\dotsc,\tfrac{1}{2},\tfrac{1}{2}),\quad {{\mathbf{x}}}^{\prime (n)}=(\tfrac{1}{2},\dotsc,\tfrac{1}{2},-\tfrac{1}{2}),\end{align*}

which lie in the closure of ${{\mathfrak{a}}}$ , with ${{\mathbf{x}}}^{(n)}$ a special vertex. The points ${{\mathbf{x}}}^{(n)}, {{\mathbf{x}}}^{\prime (n)}$ , ${{\mathbf{x}}}^{(s)}, 1\leqslant s \leqslant n-2$ , are zero-dimensional facets, while ${{\mathbf{x}}}^{(n-1)}$ lies on the one-dimensional facet given by the line segment between ${{\mathbf{x}}}^{(n)}$ and ${{\mathbf{x}}}^{\prime (n)}.$

Case (3). In this case, we have $T=S\times({\rm Res}_{{{\breve{F}}}'/{{\breve{F}}}}{{\mathbb{G}}}_m)^{{\rm Nm}=1}$ , where ${{\breve{F}}}'/{{\breve{F}}}$ is a degree-2 ramified extension, and hence $X_*(T)_I=X_*(S)\oplus{{\mathbb{Z}}}/2{{\mathbb{Z}}}$ . The échelonnage root system is of type $C_n$ (cf. [Reference TitsTit79, Example 1.16]), so we fix an identification $X_*(S)\cong {{\mathbb{Z}}}^n$ so that the roots are given by $\pm\epsilon_i\pm \epsilon_j, 1 \leqslant i\lt j\leqslant n$ , $\pm2\epsilon_i, i=1,\dotsc,n$ . We take the n roots

\begin{align*} \epsilon_i-\epsilon_{i+1}, 1\leqslant i\leqslant n-1\quad \text{and}\quad 2\epsilon_n\end{align*}

as simple roots.

The base alcove in ${{\mathbb{R}}}^n$ is given by

\begin{align*}\{{{\mathfrak{a}}}=\{(a_1,\dotsc,a_n)\mid\tfrac{1}{2}\gt a_1\gt\cdots\gt a_n\gt0\}.\end{align*}

Let

\begin{align*}{{\mathbf{x}}}^{(s)}=\sum_{i=1}^s\frac{\epsilon_i}{2}\quad \text{for } 0 \leqslant s \leqslant n,\end{align*}

which lie in the closure of ${{\mathfrak{a}}}$ ; these are all zero-dimensional facets. The parahoric corresponding to ${{\mathbf{x}}}^{(s)}$ is the connected stabilizer of $L^{(s)}$ .

3.2.3.

The relative Weyl group $W_0$ has the following description:

\begin{align*}W_0\cong\begin{cases}(({{\mathbb{Z}}}/2{{\mathbb{Z}}})^n\rtimes S_n)^\circ &\text{in case (1)},\\({{\mathbb{Z}}}/2{{\mathbb{Z}}})^n\rtimes S_n &\text{in case (2) and (3)}.\end{cases}\end{align*}

Here, $(({{\mathbb{Z}}}/2{{\mathbb{Z}}})^n\rtimes S_n)^\circ $ denotes the kernel of the homomorphism

\begin{align*}({{\mathbb{Z}}}/2{{\mathbb{Z}}})^n\rtimes S_n&\rightarrow {{\mathbb{Z}}}/2{{\mathbb{Z}}}\\((z_1,\dotsc,z_n),\sigma)&\mapsto \sum_{i=1}^n z_i.\end{align*}

In case (1), (2a) and (3), the groups $({{\mathbb{Z}}}/2{{\mathbb{Z}}})^n\cong \{\pm1\}^n$ and $S_n$ act on ${{\mathbb{Z}}}^n$ via factorwise multiplication and permutation of coordinates. In case (2b), they act via the same linear transformations, but centered at ${{\mathbf{x}}}^{(n)}$ . Thus, we see in each case that the ${{\widetilde{W}}}$ -orbit of ${{\mathbf{x}}}^{(s)}$ in ${{\mathcal{A}}}(G,S,{{\breve{F}}})\cong {{\mathbb{R}}}^n$ is given by

(3.3) \begin{eqnarray}{{\widetilde{W}}} {{\mathbf{x}}}^{(s)}=\{(a_1,\dotsc, a_n)\in {{\mathbb{Z}}}[1/2]^n \mid \text{ exactly} {s} \text{ of the } a_i \text{ are in} \tfrac{1}{2}+{{\mathbb{Z}}}\}.\end{eqnarray}

For $y=(a_1,\dotsc,a_n)\in {{\widetilde{W}}} {{\mathbf{x}}}^{(s)}$ , define

\begin{align*}L_0(y)={{{\rm Span}}}_{{{\mathcal{O}}}_F}(\pi^{\lceil a_1\rceil} e_1,\dotsc,\pi^{\lceil a_n\rceil} e_n, \pi^{\lceil -a_1\rceil}f_1,\dotsc, \pi^{\lceil -a_n\rceil}f_n)\subset {{\breve{V}}}_0\end{align*}

and set . Then, in each case, the parahoric associated to y is the connected stabilizer of L(y), and the association $y\mapsto L(y)$ induces a ${{\widetilde{W}}}$ -equivariant bijection between the ${{\widetilde{W}}}$ -orbit of ${{\mathbf{x}}}^{(s)}$ and the ${{\widetilde{W}}}$ -orbit of $L^{(s)}$ . Here, ${{\widetilde{W}}}$ acts on vertex lattices of the form L(y) via $w:L(y)\mapsto \dot{w}L(y)$ , where $\dot{w}\in N({{\breve{F}}})$ is a lift of w, noting that $\dot{w}L(y)$ does not depend on the choice of lift $\dot{w}$ .

3.2.4.

Under the identification

\begin{align*}X_*(T)_I\cong\begin{cases} {{\mathbb{Z}}}^n & \text{ in case (1) and (2)},\\ {{\mathbb{Z}}}^n\times {{\mathbb{Z}}}/2{{\mathbb{Z}}} &\text{ in case (3),}\end{cases}\end{align*}

the element $\mu$ corresponds to the element $(1,0,\dotsc,0)$ . In particular, its image in $X_*(T)_I\otimes_{{{\mathbb{Z}}}}{{\mathbb{R}}}\cong{{\mathbb{R}}}^n$ is $\epsilon_1.$

1. (I) We have

   \begin{align*}y={{\mathbf{x}}}^{(s)}\pm \epsilon_i,\quad i=1,\dotsc,n\end{align*}
   or
   \begin{align*}y={{\mathbf{x}}}^{(s)}\pm\tfrac{1}{2}\epsilon_j\pm \tfrac{1}{2}\epsilon _i,\quad 1\leqslant j\leqslant s,\ s+1\leqslant i\leqslant n.\end{align*}

2. (ii) Let $J^{(s)}\subset {{\mathbb{S}}}$ be the subset of simple reflections corresponding to the facet containing ${{\mathbf{x}}}^{(s)}$ . Then there exists $w\in {{\operatorname{Adm}}}(\mu)^{J^{(s)}}$ such that $wx^{(s)}=y$ .

(i) Let $y=(a_1,\dotsc,a_n)$ . Then, using the description of L(y) above, we see that exactly one of the integers

\begin{align*}\lceil a_1\rceil-1,\dotsc, \lceil a_s\rceil-1, \lceil a_{s+1}\rceil , \dotsc, \lceil a_n\rceil, \lceil -a_1\rceil,\dotsc, \lceil -a_n\rceil \end{align*}

is equal to 1, and the others are all $\leqslant 0$ .

Now, let use write ${{\mathbf{x}}}_k^{(s)}$ for the kth coordinate in ${{\mathbf{x}}}^{(s)}$ . Suppose first that $\lceil a_i\rceil-1 =1$ for some $i\in \{1,\dotsc,s\}$ , then $a_i=\frac{3}{2}$ or $a_i=2$ . If $a_i=2$ , then, since $y\in {{\widetilde{W}}} {{\mathbf{x}}}^{(s)}$ , there exists $j\in \{s+1,\dotsc,n\}$ with $a_j\in \frac{1}{2}+{{\mathbb{Z}}}$ . But then either $\lceil a_j\rceil\gt0$ or $\lceil -a_j\rceil\gt0$ , which is a contradiction, and hence $a_i=\frac{3}{2}$ . Suppose there exists $k\neq i$ with $a_k\neq {{\mathbf{x}}}^{(s)}_k$ . If $k\leqslant s$ , then we must have $a_k=0$ or 1, otherwise $\lceil a_k\rceil\gt0$ or $\lceil -a_k\rceil\gt0$ . Then, as before, there exists $j\in \{s+1,\dotsc,n\}$ with $a_j\in \frac{1}{2}+{{\mathbb{Z}}}$ . Thus, upon replacing k by j, we may assume $k\gt s+1$ . In this case, we have $\lceil a_k\rceil\gt0$ or $\lceil -a_k\rceil \lt0$ , which is a contradiction. Therefore, $y={{\mathbf{x}}}^{(s)}+\epsilon_i$ .

Now, suppose $\lceil a_i\rceil=1$ for some $i\in \{s+1,\dotsc,n\}$ . Then, $a_i=\frac{1}{2}$ or $a_i=1$ . If $a_i=\frac{1}{2}$ then, since $y\in {{\widetilde{W}}} {{\mathbf{x}}}^{(s)}$ , there exists $j\in \{1,\dotsc,s\}$ such that $a_j\in{{\mathbb{Z}}}$ and, since $\lceil a_j\rceil-1,\lceil -a_j\rceil \leqslant 0$ , we must have $a_j=0$ or 1. The same argument as in the previous paragraph shows that $a_k={{\mathbf{x}}}_k^{(s)}$ for $k\neq i,j$ , and hence $y={{\mathbf{x}}}^{(s)}\pm\frac{1}{2}\epsilon_j+\frac{1}{2}\epsilon_i$ . If $a_i=1$ , then similarly we have $a_k={{\mathbf{x}}}^{(s)}_k$ for $k\neq i$ , and hence $y={{\mathbf{x}}}^{(s)}+\epsilon_i$ .

The cases when $\lceil -a_i\rceil=1$ for $i\in \{1,\dotsc,s\}$ and $i\in \{s+1,\dotsc,n\}$ can be handled analogously to the previous two cases. When $\lceil -a_i\rceil=1$ for $i\in \{1,\dotsc,s\}$ , we have $y={{\mathbf{x}}}^{(s)}-\epsilon_i$ , and when $\lceil -a_i\rceil=1$ for $i\in \{s+1,\dotsc,n\}$ , we have $y={{\mathbf{x}}}^{(s)}-\epsilon_i$ or $y={{\mathbf{x}}}^{(s)}\pm\frac{1}{2}\epsilon_j-\frac{1}{2}\epsilon_i$ .

(ii) It suffices to consider the two cases from (i). Let $y={{\mathbf{x}}}^{(s)}\pm\epsilon_i$ . Then there is $u\in {{\widetilde{W}}}_0$ with $u(\epsilon_1)=\pm \epsilon_i$ . Set $\lambda=u(\mu)$ . Then $t^\lambda {{\mathbf{x}}}^{(s)}=y$ and we have $t^\lambda\in {{\operatorname{Adm}}}(\mu)$ , so the result follows in this case.

Now let $y={{\mathbf{x}}}^{(s)}\pm\frac{1}{2}\epsilon_j\pm \frac{1}{2}\epsilon _i, 1\leqslant j\leqslant s, s+1\leqslant i\leqslant n$ ; note that this case occurs only if $1\leqslant s\leqslant n-1$ . Let $u\in {{\widetilde{W}}}_{J^{(s)}}$ be the element which switches the $j{{\rm th}}$ and $s{{\rm th}}$ coordinates, and switches the $i{{\rm th}}$ and $(s+1){{\rm th}}$ coordinates. Then, upon replacing y by u(y), we may assume $y={{\mathbf{x}}}^{(s)}\pm\frac{1}{2}\epsilon_s\pm \frac{1}{2}\epsilon _{s+1}.$

Let $w^+\in S_n\subset {{\widetilde{W}}}_0$ denote the reflection switching $\epsilon_i$ and $\epsilon_j$ , and let $w^-\in {{\widetilde{W}}}_0$ be the composition of $w^+$ with multiplication by $-1$ in the $s{{\rm th}}$ and $(s+1){{\rm th}}$ factors; thus, $w^-:\epsilon_s\mapsto -\epsilon_{s+1}, \epsilon_{s+1}\mapsto -\epsilon_s$ . Let $\lambda_s,\lambda_{s+1}\in X_*(T)_I$ be the ${{\widetilde{W}}}_0$ conjugates of $\mu$ whose images in ${{\mathbb{R}}}^n$ are $\epsilon_s,\epsilon_{s+1}$ , respectively.

We define a pair $(u',\lambda)\in {{\widetilde{W}}}_0\times X_*(T)_I$ as follows:

\begin{align*}(u',\lambda)=\begin{cases} (w^+_{s,s+1},{\lambda_{s+1}})& \text{if }y={{\mathbf{x}}}^{(s)}+\tfrac{1}{2}\epsilon_s+\tfrac{1}{2}\epsilon_{s+1},\\[4pt](w^-_{s,s+1},{-\lambda_{s+1}}) &\text{if }y={{\mathbf{x}}}^{(s)}+\tfrac{1}{2}\epsilon_s-\tfrac{1}{2}\epsilon _{s+1},\\[4pt](w^-_{s,s+1},{-\lambda_{s}})&\text{if }y={{\mathbf{x}}}^{(s)}-\tfrac{1}{2}\epsilon_s+\tfrac{1}{2}\epsilon _{s+1},\\[4pt](w^+_{s,s+1},{-\lambda_{s}}) &\text{if } y={{\mathbf{x}}}^{(s)}-\tfrac{1}{2}\epsilon_s-\tfrac{1}{2}\epsilon _{s+1}, \end{cases}\end{align*}

and we set $w=u't^\lambda\in {{\widetilde{W}}}$ . Then, we have $w({{\mathbf{x}}}^{(s)})=y$ . Moreover, we have $w\leqslant t^\lambda$ in the Bruhat order, since ${{\mathfrak{a}}}$ and $t^\lambda{{\mathfrak{a}}}$ lie on different sides of the hyperplane corresponding to the reflection u’. Since $\lambda\in{{\widetilde{W}}}_0(\mu)$ it follows that $t^\lambda$ , and hence w, lie in ${{\operatorname{Adm}}}(\mu).$

Proof. Again, using Remark 3.1, we may assume we are in case (1), (2a) or (3). We compute that $\tau_\mu:{{\mathbb{Z}}}^{n}\rightarrow {{\mathbb{Z}}}^n$ has the following form:

\begin{align*}w:(a_1,a_2,\dotsc,a_{n-1},a_n)\mapsto\begin{cases} (1-a_1,a_2,\dotsc,a_{n-1},-a_n) &\text{in case (1)},\\ (1-a_1,a_2,\dotsc,a_{n-1},a_n) & \text{in case (2a)},\\(a_1,a_2,\dotsc,a_{n-1},a_n) &\text{in case (3)}.\end{cases}\end{align*}

It is then clear that $\tau_\mu$ fixes ${{\mathbf{x}}}^{(s)}$ if $t(s)\neq 0,\dim V$ .

3.2.5.

We now prove the main result of this subsection, which gives an explicit description of ${{\operatorname{Adm}}}(\mu)_{J^{(s)}}$ in terms of relative positions of lattices in ${{\breve{V}}}$ .

1. (i) $\dim_{k}(L+L')/L\leqslant1$ ;

2. (ii) there exists $g\in G({{\breve{F}}}), w\in {{\operatorname{Adm}}}(\mu)_{J^{(s)}}$ , such that

   \begin{align*}gL^{(s)}=L,\quad g\dot{w} L^{(s)}=L',\end{align*}
   where $\dot{w}\in N({{\breve{F}}})$ is a lift of w.

If $\dim (w' L^{(s)}+L^{(s)})/ L^{(s)}\leqslant 1$ , then by Lemmas 3.2 and 3.3, there exists $w\in {{\operatorname{Adm}}}(\mu)^{J^{(s)}}$ such that $wL^{(s)}=w'L^{(s)}$ and the result follows since ${{\operatorname{Adm}}}(\mu)^{J^{(s)}}$ is the preimage of ${{\operatorname{Adm}}}(\mu)_{J^{(s)}}$ in ${{\widetilde{W}}}$ .

(ii) $\Rightarrow$ (i): It suffices to show, for $w\in {{\operatorname{Adm}}}(\mu)_{J^{(s)}}$ , $\dim(L^{(s)}+wL^{(s)})/L^{(s)}\leqslant 1$ . We first consider $w=t^{w_0(\mu)}$ for $w_0\in {{\widetilde{W}}}_0$ . Then $w_0(\mu)$ is of the form $\pm \epsilon_i$ , and hence L’ is of the form

which satisfies $\dim_{k}(L+L')/L\leqslant 1$ .

Let ${\rm Fl}_{{{\mathcal{G}}}^\circ_s}$ denote the Witt-vector affine partial flag variety over k associated to the parahoric ${{\mathcal{G}}}^\circ_s$ (see [Reference Bhatt and ScholzeBS17], [Reference ZhuZhu17]). For $g\in {\rm Fl}_{{{\mathcal{G}}}^\circ_s}(k)=G({{\breve{F}}})/{{\mathcal{G}}}^\circ_s({{\mathcal{O}}}_{{{\breve{F}}}})$ , we associate to it the vertex lattice $gL^{(s)}$ of type t(s) (this association is not a bijection in case (3) as the parahoric is not equal to the stabilizer group in this case). The orbits of the positive loop group $L^+{{\mathcal{G}}}^\circ_{s}$ on ${\rm Fl}_{{{\mathcal{G}}}^\circ_s}\times {\rm Fl}_{{{\mathcal{G}}}^\circ_s}$ are indexed by ${{\widetilde{W}}}_{J^{(s)}}\backslash {{\widetilde{W}}}/{{\widetilde{W}}}_{J^{(s)}}$ , and the closure relations are given by the Bruhat order.

Now, for $(g,g')\in {\rm Fl}_{{{\mathcal{G}}}_s^\circ}\times {\rm Fl}_{{{\mathcal{G}}}_s^\circ}(k)$ , the condition that

\begin{align*}\dim (gL^{(s)}+g'L^{(s)})/gL^{(s)}\leqslant 1\end{align*}

defines a closed subscheme of ${\rm Fl}_{{{\mathcal{G}}}_s^\circ}\times {\rm Fl}_{{{\mathcal{G}}}_s^\circ}$ . Indeed, it is the preimage of the quasi-minuscule Schubert variety in the Witt-vector affine Grassmannian ${\rm Gr}(L^{(s)})$ for ${\rm GL}(L^{(s)})$ under the natural map ${\rm Fl}_{{{\mathcal{G}}}^\circ_s}\rightarrow {\rm Gr}(L^{(s)})$ induced by the locally closed immersion ${{\mathcal{G}}}_s^\circ\rightarrow {\rm GL}(L^{(s)})$ . Since the elements $t_{w_0(\mu)}, w_0\in {{\widetilde{W}}}_0$ are the maximal elements of ${{\operatorname{Adm}}}(\mu)_{J^{(s)}}$ , it follows that for any $w\in {{\operatorname{Adm}}}(\mu)_{J^{(s)}}$ , we have $\dim(L^{(s)}+wL^{(s)})/L^{(s)}\leqslant 1$ .

4.1.1.

We now specialize to the case $F={{\mathbb{Q}}}_p$ , so let V be a quadratic space over ${{\mathbb{Q}}}_p$ of dimension n. For a $\mathbb{Q}_p$ -algebra R, the tensor product $V_R=V \otimes_{\mathbb{Q}_p} R$ is a quadratic space over R; let $C(V_R )=C(V) \otimes_{\mathbb{Q}_p} R$ denote the Clifford algebra of $V_R$ . Then $C(V_R)$ is a semisimple algebra over R of rank $2^{\dim_{{{\mathbb{Q}}}_p} V}$ , and is equipped with a ${{\mathbb{Z}}}/2{{\mathbb{Z}}}$ -grading $C(V_R)=C^+(V_R)\oplus C^-(V_R)$ . We define $G=\operatorname{GSpin}(V)$ to be the group scheme over $\mathbb{Q}_p$ with R-points

\begin{align*}G(R)=\{g \in C^{+}(V_R)^{\times}: g V_R g^{-1}=V_R,\, g^* g \in R^{\times}\}.\end{align*}

We use $g \bullet v=g v g^{-1}$ to denote the conjugation action of G on V. Then there is a short exact sequence of group schemes

\begin{align*}1 \rightarrow \mathbb{G}_m \rightarrow G \stackrel{g \mapsto g \bullet}{\longrightarrow} {\rm SO}(V) \rightarrow 1\end{align*}

over ${{\mathbb{Q}}}_p$ . When V is clear in the context, we simply denote $C(V_R)$ as $C_R$ .

4.1.2.

Let $\{x_1,\ldots, x_n \}$ be a basis of $V_{\breve{\mathbb{Q}}_p}$ with moment matrix

\begin{align*} \left(\begin{array}{cccccc} 0\;\;\; & 1\;\;\; & & & & \\ 1\;\;\; & 0\;\;\; & & & & \\[-2pt] & & *\;\;\; & & & \\[-2pt] & & & \ddots\;\;\; & & \\[-2pt] & & & & * \end{array}\!\right)\!. \end{align*}

Given this choice of basis, we define a cocharacter $\mu: \mathbb{G}_m \rightarrow G$ by

(4.1) \begin{eqnarray} \mu(t)=t^{-1} x_1 x_2+x_2 x_1, \end{eqnarray}

where we regard the right-hand side of the above as an element in $C(V_{\breve{\mathbb{Q}}_p})$ .

We remark that the reflex field of $\mu$ is always $\mathbb{Q}_p$ . Indeed, when V has a split subspace of rank 2, then we can choose $\{x_1,x_2\}$ to be the basis of this subspace and $\mu$ is even defined over $\mathbb{Q}_p$ . Now we assume V does not have a split subspace of rank 2. Then V contains a subspace generated by $\{v_1,v_2\}$ with moment matrix $(\begin{smallmatrix} 1\;&0\\[1pt] 0\;&\nu \end{smallmatrix}{\kern-1pt})$ or $(\begin{smallmatrix} p\;&0\\[1pt] 0\;&p\nu \end{smallmatrix}{\kern-1pt})$ , where $\chi(-\nu)=-1$ . Assume $\{v_1,v_2\}$ has moment matrix $(\begin{smallmatrix} 1\;&0\\[1pt] 0\;&\nu \end{smallmatrix}{\kern-1pt})$ . Let $\alpha\in \mathbb{Q}_{p^2}^\times$ such that $\alpha^2=-\nu$ . Then we may choose $x_1=\alpha v_1+v_2$ and $x_2={-1}/{2\nu} (\alpha v_1-v_2)$ so that the moment matrix of $\{x_1,x_2\}$ is $(\begin{smallmatrix} 0\;&1\\[1pt] 1\;&0 \end{smallmatrix}{\kern-1pt})$ . Therefore, in terms of $v_1,v_2$ , we have

\begin{align*} \mu(t)=t^{-1}(\alpha v_1+v_2)\bigg(\frac{-1}{2\nu} (\alpha v_1-v_2)\!\bigg)+\frac{-1}{2\nu} (\alpha v_1-v_2)(\alpha v_1+v_2). \end{align*}

Let $\tau$ denote the nontrivial element in $\operatorname{Gal}(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ such that $\tau(\alpha)=-\alpha$ .

Although $\mu$ is only defined over $\mathbb{Q}_{p^2}$ , we claim that

\begin{align*} \tau\cdot \mu(t)= t^{-1}(-\alpha v_1+v_2)\bigg(\frac{-1}{2\nu} (-\alpha v_1-v_2)\!\bigg)+\frac{-1}{2\nu} (-\alpha v_1-v_2)(-\alpha v_1+v_2)\end{align*}

lies in the $G(\bar{\mathbb{Q}}_p)$ -conjugacy class of $\mu$ . Indeed, note that the moment matrix of $\{-\alpha v_1+v_2, -\alpha v_1-v_2\}$ is still $(\begin{smallmatrix} 0\;&1\\[1pt] 1\;&0 \end{smallmatrix}{\kern-1pt})$ . Therefore, we can find $h\in{\rm O}(V)(\bar{\mathbb{Q}}_p)$ so that $\rho(\mu)=h\circ \rho(\tau\cdot \mu) \circ h^{-1}$ by Witt’s theorem, where $\rho$ denotes the morphism from G to ${\rm SO}(V)$ . If $\det(h)=-1$ , we can consider $h'\in {\rm O}(V)(\bar{\mathbb{Q}}_p)$ such that $h'(x_3)=-x_3$ and fixes $x_i$ for $i\neq 3$ . Then $h'h\in {\rm SO}(V)(\bar{\mathbb{Q}}_p)$ and $\rho(\mu)=(h'h)\circ \rho(\tau \cdot \mu) \circ (h'h)^{-1}$ . Therefore, we can assume h lies in ${\rm SO}(V)(\bar{\mathbb{Q}}_p)$ without loss of generality. Now let $g\in G(\bar{\mathbb{Q}}_p)$ be a lifting of h. Since $\ker(\rho)=\mathbb{G}_m$ is central and $\rho(g\circ (\tau\cdot \mu)\circ g^{-1})=\rho(\mu)$ , we have $g\circ (\tau\cdot \mu(t))\circ g^{-1}=\mu(t) \circ t^m$ for some $m\in \mathbb{Z}$ . Since

\begin{align*}(g\circ (\tau\cdot \mu(t))\circ g^{-1})(g\circ (\tau\cdot \mu(t))\circ g^{-1})^*=t^{-1}\end{align*}

is equal to

\begin{align*}(\mu(t) \circ t^m)(\mu(t) \circ t^m)^*=t^{2m-1},\end{align*}

we have $m=0$ and our claim is proved.

The case $\{v_1,v_2\}$ has moment matrix $(\begin{smallmatrix} p\;&0\\[1pt] 0\;&p\nu \end{smallmatrix}{\kern-1pt})$ and is the same, so we omit the details.

4.1.3.

Recall that $\breve{\mathbb{Q}}_p$ denotes the completion of the maximal unramified extension of ${{\mathbb{Q}}}_p$ , and we let $\sigma\in {\rm Aut}(\breve{\mathbb{Q}}_p/{{\mathbb{Q}}}_p)$ be the Frobenius element. For an element $b\in G(\breve{\mathbb{Q}}_p)$ , we consider the following isocrystals:

\begin{align*} (V_{\breve{\mathbb{Q}}_p}, \Phi=b\circ \sigma) \quad \text{and} \quad (C_{\breve{\mathbb{Q}}_p}, F=b\circ \sigma).\end{align*}

Here, $\Phi$ and F are related by $\Phi x=F\circ x \circ F^{-1}$ . According to [Reference KottwitzKot85], we can associate to every $b\in G(\breve{\mathbb{Q}}_p)$ its Newton cocharacter

\begin{align*}\nu_b: \mathbb{{{\mathbb{D}}}} \rightarrow G,\end{align*}

which is characterized by the following condition. For any representation $\rho: G \rightarrow {\rm GL}(N)$ on a $\mathbb{Q}_p$ -vector space N, we require that the decomposition

\begin{align*}N_{\breve{\mathbb{Q}}_p}=\bigoplus_{s / t \in \mathbb{Q}} N^{s / t}\end{align*}

induced by $\rho\circ \nu_b$ is identified with the slope decomposition of the isocrystal $(N_{\breve{\mathbb{Q}}_p},\rho(b)\circ \sigma)$ . We call an element of $b\in G(\breve{\mathbb{Q}}_p)$ basic if $\nu_b$ factors through the center of G.

Let B(G) denote the set of $\sigma$ -conjugacy classes of $G(\breve{\mathbb{Q}}_p)$ . Given $\mu$ , Kottwitz introduced a subset $B(G,\mu)\subset B(G)$ in [Reference KottwitzKot97]:

\begin{align*} B(G, \mu)=\{[b] \in B(G) ; \kappa([b])=\kappa(\mu), \bar{\nu}_b \leqslant \mu^{\diamond}\},\end{align*}

where $\kappa([b])\in \pi_1(G)_{\Gamma}$ is the Kottwitz invariant of b, $\kappa(\mu)$ is the image of $\mu$ in $\pi_1(G)_{\Gamma}$ , $\bar{\nu}(b)$ is the dominant Newton cocharacter of [b] and $\mu^{\diamond}$ is the Galois average of $\mu$ . We refer the reader to [Reference KottwitzKot97] and [Reference HeHe16] for details. From now on, b will always denote the unique basic element of $B(G,\mu).$

Later on, we will need to consider $\mathbb{V}:= V_{\breve{\mathbb{Q}}_p}^{\Phi}$ , which will be regarded as the space of special endomorphisms of the framing object of the GSpin RZ space in later sections (see § 7.2). The following description of $\mathbb{V}$ is a generalization of [Reference Howard and PappasHP17, Proposition 4.2.5]. We also determine the vertex lattices of maximal and minimal types in $\mathbb{V}$ . The description of a general vertex lattice follows from this easily.

To simplify the notation, we use $U_n^{+}$ (respectively $U_n^-$ ) to denote the split (respectively non-split) non-degenerate quadratic space over $\mathbb{F}_p$ of dimension n.

Proposition 4.1. Assume V is a quadratic space over $\mathbb{Q}_p$ with Frobenius $\Phi$ as in § 2. Then the $\mathbb{Q}_p$ -quadratic space

\begin{align*} \mathbb{V}:= V_{\breve{\mathbb{Q}}_p}^{\Phi}=\{x \in V_{\breve{\mathbb{Q}}_p}: \Phi x=x\} \end{align*}

has the same dimension and discriminant as V, and has Hasse invariant $\epsilon(\mathbb{V})=-\epsilon(V)$ .

More precisely, if we let $\Lambda_{\rm max}\subset \mathbb{V}$ (respectively $\Lambda_{\rm min}$ ) denote the vertex lattice in $\mathbb{V}$ of maximal (respectively minimal) type, then we have the following.

1. (i) If $\operatorname{dim}(V_{{{{\rm an}}}})=0$ , then

   

2. (ii) If $\operatorname{dim}(V_{{{{\rm an}}}})=1$ , then

   

3. (iii) If $\operatorname{dim}(V_{{{{\rm an}}}})=2$ , then

   

4. (iv) If $\operatorname{dim}(V_{{{{\rm an}}}})=3$ , then

   

5. (v) If $\operatorname{dim}(V_{{{{\rm an}}}})=4$ , then

   \begin{align*} \Lambda_{\rm max}=\Lambda_{n}^+ \quad \text{and} \quad \Lambda_{\rm min}=H_{n}^+. \end{align*}

Now we assume V does not contain a self-dual lattice. In particular, since $n\geqslant 3$ , we may assume V has a basis $\{x_1,\ldots,x_n\}$ with diagonal moment matrix such that $q(x_1)=u_1 p$ and $q(x_2)=u_2$ , where $u_1, u_2\in \mathbb{Z}_p^\times$ . Then we can choose $b=x_2x_1$ so that $b^2=-u_2u_1p\in p\mathbb{Z}_p^\times$ . We can check that, for $\Phi=b\circ \sigma$ , we have

\begin{align*} \Phi \bullet x_1=-x_1, \quad \Phi\bullet x_2=-x_2, \quad \Phi\bullet x_i=x_i \quad \text{for $3\leqslant i\leqslant n$.} \end{align*}

Choose $\eta \in \mathbb{Z}_{p^2}^\times\setminus \mathbb{Z}_p^\times$ such that $\sigma(\eta)=-\eta$ and $\eta^2=\delta$ , where $\delta\in \mathbb{Z}_p^\times\setminus (\mathbb{Z}_p^{\times})^2$ . Set

\begin{align*} v_1=\eta x_1, \quad v_2=\eta x_2, \quad v_i =x_i \quad \text{for $3\leqslant i\leqslant n$.}\end{align*}

Then we can easily check that $\Phi\bullet v_i=v_i$ for any $i\in\{1,\ldots,n\}$ . Hence, $\{v_1,\ldots,v_n\}$ is a set of basis vectors of $\mathbb{V}$ . Moreover, we have

\begin{align*} q(v_1)=\delta q(x_1), \quad q(v_2)=\delta q(x_2).\end{align*}

Therefore, we have $\chi_0(V)=-\chi_0(\mathbb{V})$ and $\chi_1(V)=-\chi_1(\mathbb{V})$ . In particular, V and ${{\mathbb{V}}}$ have the same discriminant. Moreover, one can check that $\epsilon(V)=-\epsilon(\mathbb{V})$ by (2.2).

Then, applying Proposition 2.1, the proposition follows.

4.2.1.

Let $G={\rm GSpin}(V)$ and $\mu$ be as above. We now fix a vertex lattice $L\subset V$ of type h. This determines a parahoric group scheme ${{\mathcal{G}}}^\circ$ of G. We fix S a maximal $\breve{\mathbb{Q}}_p$ -split torus of G which is in good relative position with respect to ${{\mathcal{G}}}^\circ$ , so that L corresponds to a facet ${{\mathfrak{f}}}$ in the apartment $A(G,S,\breve{\mathbb{Q}}_p)$ corresponding to S, and we fix an alcove ${{\mathfrak{a}}}\subset A(G,S,\breve{\mathbb{Q}}_p)$ whose closure contains ${{\mathfrak{f}}}$ . Then L corresponds to a subset $J\subset{{\mathbb{S}}}$ of the set of simple reflections in ${{\mathfrak{a}}}$ . We then have the associated admissible set ${{\operatorname{Adm}}}(\mu)_J.$

Recall that we have fixed an element $b\in G(\breve{\mathbb{Q}}_p)$ whose image $[b]\in B(G)$ is the unique basic element of $B(G,\mu)$ . We let $K={{\mathcal{G}}}^\circ(\mathbb{Z}_p)$ , and we set

\begin{align*}X(\mu^\sigma,b)_{K}(k):=\bigg\{g \cdot K \in G(\breve{\mathbb{Q}}_p) / K \mid g^{-1} b \sigma(g) \in \bigcup_{w \in \operatorname{Adm}(\mu)_J} K \sigma(\dot{w}) K\bigg\},\end{align*}

where k is the residue field of $\breve{\mathbb{Z}}_p$ . Then $X(\mu^\sigma,b)_K(k)$ arises as the k-points of a closed subscheme $X(\mu^\sigma,b)_K$ of the Witt-vector affine Grassmannian constructed in [Reference ZhuZhu17, Reference Bhatt and ScholzeBS17], and $X(\mu^\sigma,b)_K$ is locally of perfectly finite type. By [Reference HeHe16, Theorem A], we have $X(\mu^\sigma,b)_K(k)\neq \emptyset$ .

Definition 4.2. A special lattice $M\subset V_{\breve{\mathbb{Q}}_p}$ of type h is a $\breve{\mathbb{Z}}_p$ -lattice such that

\begin{align*} p M^\vee \subset M \stackrel{h}{\subset}M^\vee \end{align*}

and

\begin{align*}\begin{cases} [M+\Phi(M):M]=1 & \text{if $h= 0,n$,}\\ [M+\Phi(M):M]\leqslant 1 & \text{if $h\neq 0,n$.} \end{cases} \end{align*}

Here, the notation $M \stackrel{h}\subset M^\vee$ means that $M^\vee/M$ is an h-dimensional vector space over the residue field ${{\mathbb{F}}}_p$ .

4.2.2.

For any $g\in X(\mu^\sigma,b)_K(k)$ , we define the following vertex lattices:

\begin{align*}L_g:= \sigma^{-1}(b^{-1}g)\bullet L \quad \text {and} \quad L_g':= g\bullet L.\end{align*}

We have

\begin{align*} \Phi(L_g)=(bb^{-1}g)\bullet L =L_g'.\end{align*}

In addition, note that $L_g$ and $L_g'$ determine each other. Indeed, if $L_g = L_{g'} $ for a different g’, then one can directly check $L_g'=L_{g'}'$ and vice versa.

Proposition 4.3. The map $g \mapsto L_g$ induces a bijection between $p ^{\mathbb{Z}} \backslash X(\mu^\sigma,b)_K(k) $ and

\begin{align*}\begin{cases} \text {\{special lattices of type }\textit{h}\text { in } V_{\breve{\mathbb{Q}}_p}\} & \text{if} h= 0,n,\\[2pt] \text {two copies of \{special lattices of type }\textit{h}\text { in } V_{\breve{\mathbb{Q}}_p}\} & \text{if $h\neq 0,n$.} \end{cases} \end{align*}

First, we check $L_g$ is indeed a special lattice. We have

\begin{align*} [L_g+\Phi(L_g):L_g]=[L_g+L_g':L_g]&=[g\bullet L+\sigma^{-1}(b^{-1}g)\bullet L:\sigma^{-1}(b^{-1}g)\bullet L]\\{} &=[(\sigma^{-1}(g^{-1}b)g)\bullet L+L:L]. \end{align*}

Note that the map $G\rightarrow {\rm SO}(V)$ induces a bijection

\begin{align*}{{\operatorname{Adm}}}(\mu)_J\cong {{\operatorname{Adm}}}(\mu')_{J'},\end{align*}

where $\mu'$ is the cocharacter of ${\rm SO}(V)$ induced by $\mu$ , and ${{\operatorname{Adm}}}(\mu')_{J'}$ is the corresponding admissible set of ${\rm SO}(V)$ . Thus, since $g^{-1}b\sigma(g) \in K\sigma(\dot{w})K$ , $w\in {{\operatorname{Adm}}}(\mu)_J$ , we have $[L_g+\Phi(L_g):L_g]\leqslant 1$ by Proposition 3.6.

Now we show that the map $g\mapsto L_g$ is surjective. Let L’ be any special lattice. Then, by Proposition 3.6, there exists $g\in G(\breve{\mathbb{Q}}_p)$ and $w\in {{\operatorname{Adm}}}(\mu)_{J}$ such that

\begin{align*}(\Phi(L'),L')=(g\bullet L,g\dot{w}^{-1}\bullet L).\end{align*}

It follows that $g^{-1}b\sigma(g)\sigma(\dot{w}^{-1})$ fixes L, and hence lies in $p^{{\mathbb{Z}}} K$ . Since $\kappa([b])=\kappa(\dot{w})=\kappa(\mu)\in \pi_1(G)_\Gamma$ , it follows that $g^{-1}b\sigma(g)\sigma(\dot{w}^{-1})\in K$ , and hence $g\in X(\mu^\sigma,b)_K(k).$

The stabilizer of L in $G(\breve{\mathbb{Q}}_p)$ is $p^\mathbb{Z} K \sqcup \delta p^\mathbb{Z} K$ , where the induced actions of $\delta\in G({{\breve{F}}})$ on $L^\vee/L$ and $L/p L^\vee$ have determinant $-1$ (see [Reference Pappas and ZachosPZ22, § 2.5]). Hence, the map $g\to L_g$ is a two-to-one map.

4.3.1.

Let $M\subset V_{\breve{\mathbb{Q}}_p}$ denote a special lattice of type h. Let

\begin{align*} S_i(M):= M+\Phi(M)+\cdots + \Phi^i(M).\end{align*}

From now on, we always assume c (respectively d) is the smallest non-negative integer such that $S_c(M)$ (respectively $S_d(M^\vee)$ ) is $\Phi$ -invariant (c and d are finite by a standard argument). It is not hard to show that

(4.2) \begin{eqnarray} M\stackrel{1}{\subset}S_1(M) \stackrel{1}{\subset} S_2(M) \cdots \stackrel{1}{\subset} S_c(M)=S_{c+1}(M).\end{eqnarray}

In addition, M is $\Phi$ -invariant if and only if $M^\vee$ is $\Phi$ -invariant by the following lemma.

We also remark that the equality $[S_c(M):M]=[S_c(M):\Phi(M)]$ combined with (4.2) implies that $[M+\Phi(M):M]=[M+\Phi(M):\Phi(M)]$ , which further implies that

\begin{align*} \Phi(M)\stackrel{1}{\subset} S_1(M) \stackrel{1}{\subset} S_2(M)\cdots \stackrel{1}{\subset} S_c(M).\end{align*}

1. (i) $S_c(M)\subset \bigcap_{i\in \mathbb{Z}_{\geqslant 0}}\Phi^i(p^{-1} M)$ and $ S_d(M^\vee)\subset \bigcap_{i\in \mathbb{Z}_{\geqslant 0}}\Phi^i(p^{-1} M^\vee)$ ;

2. (ii) if $\Phi(M)\subset M^\vee$ and $c\leqslant d$ , then

   \begin{align*} S_c(M)\subset \bigcap_{i\in \mathbb{Z}_{\geqslant 0}}\Phi^i(M^\vee ); \end{align*}

3. (iii) if $\Phi(p M^\vee)\subset M$ and $d\leqslant c$ , then

   \begin{align*} S_d(p M^\vee)\subset \bigcap_{i\in \mathbb{Z}_{\geqslant 0}}\Phi^i(M ). \end{align*}

First, note that

\begin{align*} \Phi(M)\subset p^{-1} M. \end{align*}

Hence,

\begin{align*} S_c(M)\subset p^{-1} S_{c-1}(M). \end{align*}

Since $S_c(M)$ is $\Phi$ -invariant, we have

\begin{align*} S_c(M)\subset \bigcap_{i\in \mathbb{Z}_{\geqslant 0}}p^{-1}\Phi^i(S_{c-1}(M)). \end{align*}

Note that

\begin{align*} S_c(M)\subset p^{-1} \Phi(S_{c-1}(M))\cap p^{-1} S_{c-1}(M)=p^{-1}\Phi(S_{c-2}(M)). \end{align*}

Here, for the last equality, we use the fact that

\begin{align*}\Phi(S_{c-1}(M))&\neq S_{c-1}(M),\\\Phi(S_{c-2}(M))&\subset \Phi(S_{c-1}(M))\cap S_{c-1}(M)\end{align*}

and

\begin{align*}[S_{c-1}(M):S_{c-2}(M)]=[\Phi(S_{c-1}(M)):\Phi(S_{c-2}(M))]=1.\end{align*}

Since $S_c(M)$ is $\Phi$ -invariant,

\begin{align*} S_c(M)\subset p^{-1}S_{c-2}(M). \end{align*}

Then, by an induction argument, we have $S_c(M)\subset p^{-1} M$ . Since $S_c(M)$ is $\Phi$ -invariant, we have

\begin{align*} S_c(M)\subset p^{-1} \bigcap_{i\in \mathbb{Z}_{\geqslant 0}}\Phi^i(M)\subset p^{-1} \bigcap_{0\leqslant i\leqslant f}\Phi^i(M) \end{align*}

for any $f\in \mathbb{Z}_{\geqslant 0}$ .

Parts (ii) and (iii) are essentially the same as (i) once we observe that the assumption $c\leqslant d$ (respectively $d\leqslant c$ ) implies that $\Phi(S_{c-1}(M^\vee))\cap S_{c-1}(M^\vee)=\Phi(S_{c-2}(M^\vee))$ (respectively $\Phi(S_{d-1}(M))\cap S_{c-1}(M)=\Phi(S_{c-2}(M))$ ).

The following is a first classification of special lattices.

The main result of this subsection is the following crucial proposition.

1. (i) $p (S_c(M))^\vee \subset p M^\vee \subset M\subset S_c(M) \subset (S_c(M))^\vee \subset M^\vee$ . In particular, $S_c(M)$ is a vertex lattice of type $t\leqslant h$ ;

2. (ii) $p M^\vee \subset p S_d(M^\vee)\subset S_d(M^\vee)^\vee \subset M\subset M^\vee\subset S_d(M^\vee)$ . In particular, $S_d(M^\vee)^\vee$ is a vertex lattice of type $t\geqslant h$ .

More precisely, we have the following.

1. (a) If $\Phi(p M^\vee)\not\subset M$ , then M satisfies (i).

2. (b) If $\Phi(M)\not\subset M^\vee$ , then M satisfies (ii).

3. (c) If $\Phi(p M^\vee) \subset M$ and $\Phi(M) \subset M^\vee$ , then M satisfies (i) if $c\leqslant d$ .

4. (d) If $\Phi(p M^\vee) \subset M$ and $\Phi(M) \subset M^\vee$ , then M satisfies (ii) if $d\leqslant c$ .

Since $ M \stackrel{\leqslant 1}{\subset} M + \Phi(M)$ , $\Phi(p M^\vee)\not\subset M $ and $\Phi(p M^\vee)\subset \Phi(M)$ , we have $M+\Phi(M)= M+\Phi(p M^\vee)$ . In fact, an inductive argument shows that

(4.3) \begin{eqnarray} S_c(M)=M+\Phi(p M^\vee)+\cdots +\Phi^c(p M^\vee). \end{eqnarray}

Equivalently,

(4.4) \begin{eqnarray} S_c(M)^\vee=M^\vee\cap \bigg(\bigcap_{1\leqslant i\leqslant c} \Phi^i(p^{-1} M)\!\bigg) . \end{eqnarray}

According to Lemma 4.5, we have

(4.5) \begin{eqnarray}\begin{split} S_c(p M^\vee)&\subset \bigcap_{0\leqslant i\leqslant c}\Phi^i(M^\vee) \subset M^\vee\cap \bigg(\bigcap_{1\leqslant i\leqslant c}\Phi^i(p^{-1}M)\!\bigg),\\ M&\subset M^\vee\cap \bigg( \bigcap_{1\leqslant i\leqslant c}\Phi^i(p^{-1}M)\!\bigg). \end{split} \end{eqnarray}

Therefore,

${\rm (b)}$ Now we assume $\Phi(M)\not\subset M^\vee$ . By construction, we have $S_d(M^\vee)^\vee\subset M\subset M^\vee\subset S_d(M^\vee)$ . Therefore, we only need to show $p S_d(M^\vee)\subset S_d(M^\vee)^\vee$ .

Since $M^\vee \stackrel{\leqslant 1}{\subset} M^\vee +\Phi(M^\vee)$ and $\Phi(M)\not\subset M^\vee$ , we have $M^\vee +\Phi(M^\vee)=M^\vee +\Phi(M)$ . In fact, an inductive argument shows that

(4.6) \begin{eqnarray} S_d(M^\vee)=M^\vee+\Phi(M)+\cdots +\Phi^d(M). \end{eqnarray}

Equivalently,

(4.7) \begin{eqnarray} S_d(M^\vee)^\vee=M\cap \bigg(\bigcap_{1\leqslant i\leqslant d}\Phi^i(M^\vee)\!\bigg). \end{eqnarray}

According to Lemma 4.5, we have

(4.8) \begin{eqnarray}\begin{split} S_d(p M)&\subset \bigcap_{0\leqslant i\leqslant d}\Phi^i(M) \subset M \cap \bigg(\bigcap_{1\leqslant i\leqslant d}\Phi^i( M^\vee)\!\bigg),\\ p M^\vee &\subset M \cap \bigg( \bigcap_{1\leqslant i\leqslant d}\Phi^i( M^\vee)\!\bigg). \end{split} \end{eqnarray}

Therefore,

${\rm (c)}$ Assuming $\Phi(M) \subset M^\vee$ and $c\leqslant d$ , we will show that M satisfies (i) (a stronger statement than (c)). Note that $M+\Phi(M)\subset M^\vee$ . Inductively, we see that $S_c(M)\subset S_{c-1}(M^\vee)$ . Since $S_c(M)$ is $\Phi$ -invariant, we have $S_c(M) \subset \bigcap_{0\leqslant i\leqslant c} \Phi^i(M^\vee) \subset (S_c(M))^\vee $ by Lemma 4.5. Hence, we have

\begin{align*} M\subset S_c(M) \subset (S_c(M))^\vee \subset M^\vee. \end{align*}

Since $p M^\vee \subset M \subset M^\vee$ , we have

\begin{align*} p (S_c(M))^\vee \subset p M^\vee \subset M\subset S_c(M) \subset (S_c(M))^\vee \subset M^\vee. \end{align*}

${\rm (d)}$ Assuming $\Phi(p M^\vee) \subset M$ and $d\leqslant c$ , we will show that M satisfies (ii). Note that $p M^\vee+\Phi(p M^\vee)\subset M$ . Inductively, we see that $S_d(p M^\vee)\subset S_{d-1}(M)$ . Since $S_d(p M^\vee)$ is $\Phi$ -invariant, we have $S_d(p M^\vee) \subset \bigcap_{0\leqslant i\leqslant d} \Phi^i(M) \subset p^{-1}(S_d(p M^\vee))^\vee $ by Lemma 4.5. Hence, we have

\begin{align*} p M^\vee \subset p S_d(M^\vee) \subset S_d(M^\vee)^\vee \subset M. \end{align*}

Since $p M^\vee \subset M \subset M^\vee$ , we have

\begin{align*} p M^\vee \subset p S_d(M^\vee)\subset S_d(M^\vee)^\vee \subset M\subset M^\vee\subset S_d(M^\vee). \end{align*}

5.1.1.

First, we can associate to a vertex lattice certain quadratic spaces defined over $\mathbb{F}_p$ as follows. Let $\Lambda\subset {{\mathbb{V}}}$ be a vertex lattice of type $t_\Lambda$ and rank n over $O_F$ with quadratic form $({\kern2pt}, )$ . We set

\begin{align*} V_{\Lambda}:= \Lambda^\vee/ \Lambda \quad \text{and} \quad V_{\Lambda^\vee} := \Lambda/p \Lambda^\vee\end{align*}

with quadratic form induced from $p(\, , \, )$ and $ (\, , \, )$ , respectively. Note that $\dim_{\mathbb{F}_p}V_{\Lambda}=t_\Lambda$ and $\dim_{\mathbb{F}_p}V_{\Lambda^\vee}=n-t_\Lambda$ .

Set

\begin{align*} \Omega_{\Lambda}&:= V_{\Lambda} \otimes_{\mathbb{F}_p} k \stackrel{\sim}{\rightarrow} \Lambda_{\breve{\mathbb{Z}}_p}^\vee / \Lambda_{\breve{\mathbb{Z}}_p},\\ \Omega_{\Lambda^\vee}&:= V_{\Lambda^\vee} \otimes_{\mathbb{F}_p} k \stackrel{\sim}{\rightarrow} \Lambda_{\breve{\mathbb{Z}}_p} /p \Lambda_{\breve{\mathbb{Z}}_p}^{\vee}, \end{align*}

with its Frobenius operator ${\rm Id}\otimes \sigma=\Phi$ .

5.1.2.

For a quadratic space $\Omega$ over k, we let ${\rm OGr}(i,\Omega)$ be the orthogonal Grassmannian that parametrizes i-dimensional isotropic subspaces of $\Omega$ . For $\Lambda\in \mathcal{V}^{\geqslant h}$ , let $S_{\Lambda}$ be the reduced closed subscheme of ${\rm OGr}(({t_\Lambda-h})/{2},\Omega_\Lambda)$ whose k-points are described as follows:

\begin{align*} S_{\Lambda}(k) & = \bigg\{ \mathscr{V} \in {\rm OGr}\bigg(\frac{t_\Lambda-h}{2},\Omega_\Lambda\!\bigg)(k)\mid \operatorname{dim}_k (\mathscr{V}+\Phi(\mathscr{V}))\leqslant \operatorname{dim}_k (\mathscr{V})+1 \bigg\} \\ & \stackrel{\sim}{\rightarrow} \{\text{special lattices } M \subset V_{\breve{\mathbb{Q}}_p}: \Lambda_{\breve{\mathbb{Z}}_p} \subset M \subset M^\vee \subset \Lambda_{\breve{\mathbb{Z}}_p}^\vee\}. \end{align*}

Similarly, for $\Lambda \in\mathcal{V}^{\leqslant h}$ , let $R_{\Lambda} $ be the reduced closed subscheme of ${\rm OGr}({h-t_\Lambda}/{2},\Omega_{\Lambda^\vee})$ with k-points

\begin{align*} R_{\Lambda}(k) & = \bigg\{ \mathscr{V} \in {\rm OGr}\bigg(\frac{h-t_\Lambda}{2},\Omega_{\Lambda^\vee}\!\bigg)(k) \mid \operatorname{dim}_k (\mathscr{V}+\Phi(\mathscr{V}))\leqslant \operatorname{dim}_k (\mathscr{V})+1 \bigg\} \\ & \stackrel{\sim}{\rightarrow} \{\text{special lattices } M \subset V_{\breve{\mathbb{Q}}_p}: p\Lambda_{\breve{\mathbb{Z}}_p}^\vee \subset p M^\vee \subset M \subset \Lambda_{\breve{\mathbb{Z}}_p}\}. \end{align*}

Note that when $h=0$ , $S_\Lambda$ has two connected components (see [Reference Howard and PappasHP17, Proposition 5.3.2]). In this case, we write $S_\Lambda$ as $S_\Lambda=S_\Lambda^+\cup S_\Lambda^-$ . Note that the definition of $S_\Lambda$ and $R_\Lambda$ also depends on the fixed h. We will denote it as $S_\Lambda^{[h]}$ and $R_\Lambda^{[h]}$ if we want to emphasize the h here.

Moreover, for certain $\mathbb{V}$ and h, $\mathcal{V}^{\geqslant h}$ (respectively $\mathcal{V}^{\leqslant h}$ ) might only contain vertex lattices of type h so that $S_\Lambda$ (respectively $R_\Lambda$ ) is simply a point. For example, when $h=2$ and $\epsilon(\mathbb{V})=-1$ , $\mathcal{V}^{\leqslant h}$ is the set of vertex lattices of type 2 since there is no self-dual lattice in ${{\mathbb{V}}}$ by $\epsilon(\mathbb{V})=-1$ .

For a quadratic lattice $\Lambda$ , we use $\Lambda^{(p)}$ to denote the same lattice with quadratic form $q(\,)=p\cdot q(\,)_\Lambda$ . We use $\Lambda^\sharp$ to denote $(\Lambda^\vee)^{(p)}$ .

Lemma 5.1. Assume $\Lambda $ is a vertex lattice of type $ t_\Lambda\leqslant h$ in $\mathbb{V}^\sharp$ so that $\Lambda^\sharp \subset \mathbb{V}$ is a vertex lattice of type $t_{\Lambda^\sharp}=n-t_\Lambda\geqslant n- h$ . Then we have

\begin{align*} R_{\Lambda}^{[h]} \cong S_{\Lambda^\sharp}^{[n-h]}. \end{align*}

Proof. Note that we have the following natural identification of quadratic spaces that preserves the quadratic forms:

\begin{align*}\Lambda/p\Lambda^\vee \cong (p^{-1}\Lambda )/\Lambda^\vee\cong (p^{-1}\Lambda )^{(p)}/(\Lambda^\vee)^{(p)}= (\Lambda^\sharp)^\vee/\Lambda^\sharp.\end{align*}

This identification induces an identification of

\begin{align*} {\rm OGr}\bigg(\frac{h-t_\Lambda}{2},\Omega_{\Lambda^\vee}\!\bigg) \quad \text{and} \quad {\rm OGr}\bigg(\frac{t_{\Lambda^\sharp}-(n-h)}{2},\Omega_{\Lambda^\sharp}\!\bigg) , \end{align*}

which restricts to an identification between $R_\Lambda^{[h]}$ and $S_{\Lambda^\sharp}^{n-h}$ .

Because of the above lemma, we will focus on $S_\Lambda$ from now on. First, we consider a stratification of $S_\Lambda$ . Assume $\Lambda\subset \Lambda'$ and $t(\Lambda')\geqslant h$ , then we may regard $S_{\Lambda'}$ as a closed subscheme of $S_{\Lambda}$ as follows. Let $\pi: \Lambda_{\breve{\mathbb{Z}}_p}^\vee/\Lambda_{\breve{\mathbb{Z}}_p}\to \Lambda_{\breve{\mathbb{Z}}_p}^\vee/\Lambda_{\breve{\mathbb{Z}}_p}'$ be the natural quotient map. Then, for ${{\mathscr{V}}}'\in S_{\Lambda'}$ , one can check that $\pi^{-1}({{\mathscr{V}}}')\in S_{\Lambda}$ . Now let

\begin{align*} S_\Lambda^\circ := S_\Lambda \setminus \bigcup_{\Lambda \subsetneq\Lambda'}S_{\Lambda'}.\end{align*}

By construction, we have

\begin{align*} S_\Lambda = \bigcup_{\Lambda\subset\Lambda'}S_{\Lambda'}^\circ.\end{align*}

5.1.3.

We need to consider the following natural decomposition of $S_{\Lambda}$ later:

(5.1)

where is the open subvariety of $S_\Lambda$ whose k-points are

is the subvariety of $S_\Lambda$ such that

and $S_\Lambda^\Phi$ is the subvariety of $S_\Lambda$ such that

\begin{align*} S_\Lambda^\Phi(k)=\{{{\mathscr{V}}} \in S_{\Lambda}(k) \mid \Phi({{\mathscr{V}}})={{\mathscr{V}}} \}.\end{align*}

Let $:= \bigcap_{\ell=0}^i\Phi^\ell({{\mathscr{V}}})$ and $:= \sum_{\ell=0}^i\Phi^\ell({{\mathscr{V}}})$ for $i\geqslant 0$ . Let $S_\Lambda^0$ be the subvariety of such that

\begin{align*} S_\Lambda^0(k)=\{{{\mathscr{V}}} \in S_{\Lambda}(k) \mid F_i^{{\dagger}} \text{ is totally isotropic for all} i \}.\end{align*}

Assume $\Lambda'$ is another vertex lattice that contains $\Lambda$ and set $\Omega_{\Lambda'/\Lambda}:=\Lambda'/\Lambda\otimes_{{{\mathbb{F}}}_p}k$ . Let $S_{\Lambda',\Lambda}$ denote the subvariety of $S_{\Lambda}$ such that

\begin{align*} S_{\Lambda',\Lambda}(k)=\{{{\mathscr{V}}} \in S_{\Lambda}\mid {{\mathscr{V}}} \subset \Omega_{\Lambda'/\Lambda}\}.\end{align*}

The subvariety of the form $S_{\Lambda',\Lambda}$ will naturally show up later. Note that if ${{\mathscr{V}}} \in S_{\Lambda',\Lambda}$ , then $\Phi^i({{\mathscr{V}}})\subset \Omega_{\Lambda'/\Lambda}$ for any $i\geqslant 0$ . In fact, since $\Omega_{\Lambda'/\Lambda}$ is already totally isotropic, any subspace of it is automatically totally isotropic. So we can simply define $S_{\Lambda',\Lambda}$ as a subvariety of the usual Grassmannian ${\rm Gr}(({t_{\Lambda}-h})/{2},\Omega_{\Lambda'/\Lambda})$ such that

\begin{align*} S_{\Lambda',\Lambda}(k)=\bigg\{{{\mathscr{V}}} \in {\rm Gr}\bigg(\frac{t_{\Lambda}-h}{2},\Omega_{\Lambda'/\Lambda}\!\bigg)(k) \mid \operatorname{dim}_k (\mathscr{V}+\Phi(\mathscr{V}))\leqslant \operatorname{dim}_k (\mathscr{V})+1 \bigg\}.\end{align*}

Since, for each ${{\mathscr{V}}}\in S_\Lambda^0(k)$ , $({{\mathscr{V}}})$ will eventually stabilize to the base change of some rational subspace of $\Omega_{\Lambda}$ , we have the following decomposition:

\begin{align*} S_\Lambda^0=\bigcup_{\substack{\Lambda\subset \Lambda',\\ t(\Lambda')\leqslant h}} S_{\Lambda',\Lambda}.\end{align*}

5.1.4.

We will need the following description of tangent spaces of in § 7.

Proposition 5.2. Assume , and let $T_z{S_{\Lambda}}$ denote the tangent space of $S_\Lambda$ at z. Then we have

\begin{align*} T_{z}S_{\Lambda}\cong \operatorname{Hom}( {{\mathscr{V}}}+\Phi({{\mathscr{V}}})/\Phi({{\mathscr{V}}}),\Omega_{\Lambda}/({{\mathscr{V}}}+\Phi({{\mathscr{V}}}))). \end{align*}

Proof. Let $\widetilde{{{\mathscr{V}}}}\in T_zS_\Lambda$ , which we consider as an element of with $\widetilde{{{\mathscr{V}}}}\otimes_{k[\epsilon]} k = {{\mathscr{V}}}$ ; here $k[\epsilon]$ is the ring of dual numbers. Then $\widetilde{{{\mathscr{V}}}}+\Phi(\widetilde{{{\mathscr{V}}}})$ is finite free of rank $({t_\Lambda-h})/{2}+1$ over $k[\epsilon]$ and lifts ${{\mathscr{V}}}+\Phi({{\mathscr{V}}})$ , hence gives us an element of the tangent space of the usual Grassmannian ${\rm Gr}(({t_\Lambda-h})/{2}+1,\Omega_\Lambda)$ at the point ${{\mathscr{V}}}+\Phi({{\mathscr{V}}})$ . It follows that we obtain a map of vector spaces

(5.2) \begin{eqnarray} T_z S_\Lambda\rightarrow\operatorname{Hom}_k({{\mathscr{V}}}+\Phi({{\mathscr{V}}}), \Omega_\Lambda/({{\mathscr{V}}}+\Phi({{\mathscr{V}}}))).\end{eqnarray}

Note that since $\epsilon^p=0$ , we have $\Phi(\widetilde{{{\mathscr{V}}}})=\Phi({{\mathscr{V}}})\otimes_kk[\epsilon]$ . Thus, (5.2) factors through a map

(5.3) \begin{eqnarray}T_zS_\Lambda\rightarrow \operatorname{Hom}_k( {{\mathscr{V}}}+\Phi({{\mathscr{V}}})/\Phi({{\mathscr{V}}}),\Omega_{\Lambda}/({{\mathscr{V}}}+\Phi({{\mathscr{V}}}))).\end{eqnarray}

We claim that this map is injective. Let $\widetilde{{{\mathscr{V}}}}$ be an element of $T_zS_\Lambda$ , which we identify with a homomorphism $\lambda:{{\mathscr{V}}}\rightarrow \Omega_\Lambda/{{\mathscr{V}}}$ by considering $\widetilde{{{\mathscr{V}}}}$ as a $k[\epsilon]$ -point of ${\rm Gr}(({t_\Lambda-h})/{2},\Omega_\Lambda)$ . Then, $\lambda$ factors through ${{\mathscr{V}}}/{{\mathscr{V}}}\cap\Phi({{\mathscr{V}}})$ , and its image $\gamma$ under (5.3) is given by composing with the natural isomorphism ${{\mathscr{V}}}/{{\mathscr{V}}}\cap\Phi({{\mathscr{V}}})\cong {{\mathscr{V}}}+\Phi({{\mathscr{V}}})/\Phi({{\mathscr{V}}})$ and the projection $\Omega_\Lambda/{{\mathscr{V}}}\rightarrow\Omega_\Lambda/({{\mathscr{V}}}+\Phi({{\mathscr{V}}}))$ . We will show that $\lambda$ is determined by $\gamma.$

Let $v\in {{\mathscr{V}}}$ be a generator of the one-dimensional vector space ${{\mathscr{V}}}/{{\mathscr{V}}}\cap\Phi({{\mathscr{V}}})\cong {{\mathscr{V}}}+\Phi({{\mathscr{V}}})/\Phi({{\mathscr{V}}})$ and fix a lifting $\tilde{\gamma}:{{\mathscr{V}}}+\Phi({{\mathscr{V}}})/\Phi({{\mathscr{V}}})\rightarrow \Omega_\Lambda/{{\mathscr{V}}}$ of $\gamma$ . Then ${{\mathscr{V}}}+\Phi({{\mathscr{V}}})$ is generated by ${{\mathscr{V}}}$ and $\Phi(v)$ , and for any $u\in {{\mathscr{V}}}$ , we have $\lambda(u)=c_u\Phi(v)+\tilde{\gamma}(u)$ for some $c_u\in k$ . By assumption, ${{\mathscr{V}}}+\Phi({{\mathscr{V}}})$ is not isotropic, and hence $(\Phi(v),v)\neq 0$ . Since $\widetilde{{{\mathscr{V}}}}$ is isotropic, we have $(u+\epsilon\lambda(u),v+\epsilon\lambda(v))=0$ for all $u\in {{\mathscr{V}}}$ , or equivalently

\begin{align*}(u,c_v\Phi(v)+\tilde{\gamma}(v))+(c_u\Phi(v)+\tilde{\gamma}(u),v)=0.\end{align*}

Taking $u=v$ , we find that $c_v=-{(v,\tilde{\gamma}(v))}/{(\Phi(v),v)}$ , and for general $u\in {{\mathscr{V}}}$ , we have

\begin{align*}c_u=-\frac{(u,c_v\Phi(v)+\tilde{\gamma}(v))+(\tilde{\gamma}(u),v)}{(\Phi(v),v)}.\end{align*}

This shows that (5.3) is injective. By Theorem 5.6, we have $\dim_kS_\Lambda=({t_\Lambda+h})/{2}-1$ and hence

\begin{align*}\dim_kT_zS_\Lambda\geqslant \dim_k\operatorname{Hom}_k({{\mathscr{V}}}+\Phi({{\mathscr{V}}})/\Phi({{\mathscr{V}}}),\Omega_\Lambda/({{\mathscr{V}}}+\Phi({{\mathscr{V}}})))=\frac{t_\Lambda+h}{2}-1.\end{align*}

It follows that (5.3) is an isomorphism.

5.2.1 The even-dimensional orthogonal case.

Let $\Lambda\subset \mathbb{V}$ be a vertex lattice of even type $t=2d\geqslant h$ . In particular, according to Proposition 4.1, h is also even. Recall that $\Omega_\Lambda=V_{\Lambda}\otimes_{\mathbb{F}_p} k$ is a quadratic space over k, and we fix a basis $\{e_1, \ldots, e_d, f_1, \ldots, f_d\}$ of $\Omega_\Lambda$ such that $\operatorname{Span}_k\{e_1, \ldots, e_d\}$ and $\operatorname{Span}_k\{f_1, \ldots, f_d\}$ are totally isotropic, $(e_i, f_j)=\delta_{i, j}$ . If $V_{\Lambda}$ is split, then we may choose this basis so that Frobenius $\Phi$ fixes $e_1, \ldots, e_{d}, f_1, \ldots, f_{d}$ . If $V_{\Lambda}$ is non-split, then $\Phi$ fixes $e_1, \ldots, e_{d-1}, f_1, \ldots, f_{d-1}$ but interchanges $e_d \leftrightarrow f_d$ .

Consider the isotropic flags $\mathscr{F}_{\bullet}^{+}$ and $\mathscr{F}_{\bullet}^{-}$ in $\Omega$ defined by

\begin{align*}\begin{aligned}\mathscr{F}_i^{ \pm} & =\operatorname{Span}_k\{e_1, \ldots, e_i\} \text { for } 1 \leqslant i \leqslant d-1, \\\mathscr{F}_d^{+} & =\operatorname{Span}_k\{e_1, \ldots, e_{d-1}, e_d\}, \\\mathscr{F}_d^{-} & =\operatorname{Span}_k\{e_1, \ldots, e_{d-1}, f_d\} .\end{aligned}\end{align*}

Hence, when $V_{\Lambda}$ is split, we have $\Phi(\mathscr{F})_{\bullet}^{ \pm}= \mathscr{F}_{\bullet}^{\pm }$ . When $V_{\Lambda}$ is non-split, we have $\Phi(\mathscr{F})_{\bullet}^{ \pm}= \mathscr{F}_{\bullet}^{\mp }$ . The stabilizers of $\Phi(\mathscr{F})_{\bullet}^{\pm}$ are the same, which we denote as $B \subset H={\rm SO}(\Omega_\Lambda)$ . Then B is a $\Phi$ -stable Borel subgroup containing T, where T is a maximal $\Phi$ -stable torus $T \subset {\rm SO}(\Omega_\Lambda)$ .

We use $\Delta^*=\{s_1, \ldots, s_{d-2}, t^{+}, t^{-}\}$ to denote the set of corresponding simple reflections in the Weyl group $W=N(T) / T$ , where:

1. (i) $s_i$ interchanges $e_i \leftrightarrow e_{i+1}$ and $f_i \leftrightarrow f_{i+1}$ , and fixes the other basis elements;

2. (ii) $t^{+}$ interchanges $e_{d-1} \leftrightarrow e_d$ and $f_{d-1} \leftrightarrow f_d$ , and fixes the other basis elements;

3. (iii) $t^{-}$ interchanges $e_{d-1} \leftrightarrow f_d$ and $f_{d-1} \leftrightarrow e_d$ , and fixes the other basis elements.

For $0\leqslant r\leqslant ({t-h})/{2}-1$ , let

\begin{align*} I_{r}:=\begin{cases} \{s_1,\ldots,s_{d-2-r}\} & \text{if $h\leqslant 2$},\\ \{s_1,\ldots,s_{(t-h)/2 -1-r}, s_{(t-h)/2 +1},\ldots,s_{d-2},t^+,t^-\} & \text{if $h\gt2$}, \end{cases}\end{align*}

and $P_{r}$ be the corresponding parabolic subgroup. Here, when $r= ({t-h})/{2}-1$ , we mean $I_r$ is empty and $P_{r}=B$ .

For $1\leqslant r\leqslant ({t-h})/{2}$ ( $r\leqslant d-1$ when $h=0$ ), let

(5.4) \begin{align} &\begin{cases} w_r^{\pm}= t^{\mp} s_{d-2} \cdots s_{d-r} & \text{if $h=0$},\\ w_r= t^{-}t^{+} s_{d-2} \cdots s_{(t-h)/2 -r+1} & \text{if $h=2$},\\ w_r= s_{(t-h)/2 }\cdots s_{d-2}t^{-}t^{+}s_{d-2} \cdots s_{(t-h)/2 -r+1} & \text{if $h\gt2$,} \end{cases} \end{align}

(5.5) \begin{align} &\begin{cases} w_r^{\prime\pm}=t^\pm s_{(t-h)/2 -1}\cdots s_{(t-h)/2 -r+1} & \text{if $h=2$},\\ w_r'=s_{(t-h)/2 }\cdots s_{(t-h)/2 -r+1} & \text{if $h\gt2$}. \end{cases}\end{align}

Here, we interpret $w_{1}^{\pm}$ as $t^\mp$ when $h=0$ and $w_{1}$ as $t^{-}t^+=t^+t^-$ when $h=2$ . Moreover, we define $w_0$ and $w_0^\pm$ as the identity. In addition, note that $w_1^{\prime \pm}=t^{\pm}$ if $h=2$ and $w_1'=s_{({t-h})/{2}}$ when $h\gt2$ .

5.2.2 The odd-dimensional orthogonal case

In this subsection, we let $\Lambda\subset \mathbb{V}$ be a vertex lattice of odd type $t=2d+1\geqslant h$ . In particular, according to Proposition 4.1, h is also odd. Let $\Omega_\Lambda=V_{\Lambda}\otimes_{\mathbb{F}_p} k$ be the associated quadratic space over k. Since $V_{\Lambda}$ is of odd dimension, we can choose a basis $\{e_1, \ldots, e_d, f_1, \ldots, f_d, e_{2d+1}\}$ of $\Omega_\Lambda$ such that $\operatorname{Span}_k\{e_1, \ldots, e_d\}$ and $\operatorname{Span}_k\{f_1, \ldots, f_d\}$ are totally isotropic, $(e_i, f_j)=\delta_{i, j}$ , $(e_{2d+1},e_{2d+1})=1$ , $e_{2d+1}$ is orthogonal to all the other basis vectors and $\Phi$ fixes all the basis vectors.

Consider the isotropic flags $\mathscr{F}_{\bullet}$ in $\Omega_\Lambda$ defined by

\[\mathscr{F}_i =\operatorname{Span}_k\{e_1, \ldots, e_i\} \quad \text{for } 1 \leqslant i \leqslant d.\]

The stabilizer of $\mathscr{F}_{\bullet}$ is denoted as $B \subset H={\rm SO}(\Omega_\Lambda)$ . Hence, B is a $\Phi$ -stable Borel subgroup containing T, where T is a maximal $\Phi$ -stable torus $T \subset {\rm SO}(\Omega_\Lambda)$ .

We use $\Delta^*=\{s_1, \ldots, s_{d}\}$ to denote the set of corresponding simple reflections in the Weyl group $W=N(T) / T$ , where:

1. (i) for $i=1,\dotsc,d-1$ , $s_i$ interchanges $e_i \leftrightarrow e_{i+1}$ and $f_i \leftrightarrow f_{i+1}$ , and fixes the other basis elements;

2. (ii) $s_d$ interchanges $e_{d} \leftrightarrow f_d$ , sends $e_{2d+1}$ to $-e_{2d+1}$ and fixes the other basis elements.

For $0\leqslant r \leqslant ({t-h})/{2}-1$ , let

\begin{align*} I_{r}:= \{s_1,\ldots,s_{(t-h)/2 -1-r}, s_{(t-h)/2 +1},\ldots,s_{d-2},s_{d-1},s_{d}\}\end{align*}

and $P_{r}$ be the corresponding parabolic subgroup. Here, when $r\geqslant d-1$ , we mean $I_r$ is empty and $P_{r}=B$ .

For $1\leqslant r\leqslant ({t-h})/{2}$ , let

(5.6) \begin{align} w_r&=\begin{cases} s_d \cdots s_{d-r+1} & \text{if $h=1$},\\ s_{(t-h)/2 }\cdots s_{d-1}s_ds_{d-1} \cdots s_{(t-h)/2 -r+1} & \text{if $h\gt1$,} \end{cases} \end{align}

(5.7) \begin{align} w_r'&=s_{(t-h)/2 }\cdots s_{(t-h)/2 -r+1} \quad \text{if $h\gt1$.}\end{align}

Here, we interpret $w_{1}$ as $s_{d}$ when $h=1$ and $w_{1} $ as $s_{d-1}s_{d}s_{d-1}$ when $h=3$ . Moreover, we define $w_0$ as the identity. In addition, note that $w_1'=s({t-h})/{2}$ when $h\gt1$ .