2.1.1

Fix a prime $p>2$ . Let K be a finite extension of ${\mathbb Q}_p$ or a finite extension of $\breve {\mathbb {Q}}_p$ and let G be a (connected) reductive group over K. We let ${\mathcal B}(G, K)$ denote the extended building and $\bar {\mathcal B}(G, K)={\mathcal B}(G^{\mathrm{ad}} , K)$ the reduced (“classical”) building [Reference Bruhat and TitsBTI], [Reference Bruhat and TitsBTII]. Recall that a quasi-parahoric group scheme for G is a smooth affine scheme ${\mathcal G}$ over the integers ${\mathcal O}={\mathcal O}_K$ with $G={\mathcal G}\otimes _{{\mathcal O}}K$ , whose neutral connected component is a parahoric group scheme and with $\breve {\mathcal O}$ -valued points satisfying

$$\begin{align*}{\mathcal G}_{\mathbf{x}}^\circ(\breve{\mathcal O})\subset {\mathcal G}(\breve{\mathcal O})\subset {\mathcal G}_{\mathbf{x}}(\breve{\mathcal O}), \end{align*}$$

for some point $\mathbf {x}$ in the extended building $ {\mathcal B}(G, K)$ of G over K, [Reference Bruhat and TitsBTII], [Reference Kaletha and PrasadKaP23]. Here ${\mathcal G}_{\mathbf {x}}$ is the Bruhat–Tits stabilizer group scheme associated to $\mathbf {x}$ by Bruhat-Tits in [Reference Bruhat and TitsBTII]. Then the neutral component ${\mathcal G}^\circ ={\mathcal G}^\circ _{\mathbf {x}}$ is the associated parahoric and the inclusions above give quotients which are finite abelian groups, see [Reference Haines and RapoportHR08]. Most of the time we will consider the case ${\mathcal G}={\mathcal G}_{\mathbf {x}}$ , for some $\mathbf {x}\in {\mathcal B}(G, K)$ .

2.1.2

If $\tilde K/K$ is a finite extension, then we have the building ${\mathcal B}(G,\tilde K)$ over $\tilde K$ , and for $\mathbf {x}'\in {\mathcal B}(G,\tilde K)$ , we let $\tilde {\mathcal G}_{\mathbf {x}'}$ denote the stabilizer scheme over $\tilde {\mathcal O}={\mathcal O}_{\tilde K}$ associated to $\mathbf {x}'$ . Let $H=\mathrm{Res}_{\tilde K/K}G_{\tilde K}$ . Then by [Reference Haines and RicharzHR20, Prop. 4.6], we have an identification ${\mathcal B}(G,\tilde K)\cong {\mathcal B}(H,K)$ and there is an isomorphism $\mathrm{Res}_{\tilde {\mathcal O}/\cal O}\tilde {\mathcal G}_{\mathbf {x}'}\cong {\mathcal H}_{\mathbf {x}'}$ , where ${\mathcal H}_{\mathbf {x}'}$ is the stabilizer scheme of H for $\mathbf {x}'$ considered as a point in ${\mathcal B}(H,K)$ .

Now assume $\tilde K/K$ is a finite tame Galois extension with Galois group ${\Gamma }={\mathrm{Gal}}(\tilde K/K)$ contained in an algebraic closure $\bar {K}$ . By [Reference Prasad and YuPrY02], the natural map ${\mathcal B}(G,K)\to {\mathcal B}(G,\tilde K)$ gives identifications

(2.1.3) $$ \begin{align} {\mathcal B}(G, K)={\mathcal B}(G, \tilde K)^\Gamma,\quad \bar{\mathcal B}(G, K)=\bar{\mathcal B}(G, \tilde K)^\Gamma \end{align} $$

with the fixed points by the natural action of ${\Gamma }$ .

2.1.4

We now recall the notion of R-smoothness from [Reference Kisin and ZhouKZ25] which will play an important role in what follows.

Let T be a torus over K and let $\tilde K/K$ be a finite extension. We let ${\mathcal T}$ (resp. $\tilde {\mathcal T}$ ) denote the lft Néron model for T (resp. the base change $T_{\tilde K}$ ); see [Reference Bosch, Lütkebohmert and RaynaudBLR90, §10]. Then $\mathrm{Res}_{\tilde {\mathcal O}/{\mathcal O}}{\mathcal T}_{\tilde K}$ is the lft Néron model for $\mathrm{Res}_{\tilde K/K}T_{\tilde K}$ .

Now fix a $\tilde K/K$ such that T splits over $\tilde K$ . Recall [Reference Kisin and ZhouKZ25, Def. 2.4.3] that the torus T is said to be R-smooth if the Zariski closure of T inside $\mathrm{Res}_{\tilde {\mathcal O}/{\mathcal O}}\tilde {\mathcal T}$ is smooth.Footnote ² If G is a reductive group over K, we say that G is R-smooth if the centralizer of a (equivalently any) maximal $\breve K$ -split torus in G is R-smooth. The following summarizes the main results on R-smoothness from [Reference Kisin and ZhouKZ25] that we will need.

2.2.1

We now assume that G is a classical reductive group over K (i.e., there are no exceptional factors in $G^{{\mathrm{ad}} }$ ; by convention, this also excludes triality forms.) We will show that the identification (2.1.3) allows us to realize stabilizer schemes as the Galois fixed points of hyperspecials over a tame extension.

Let $H_0$ be the Chevalley (split) form of G over ${\mathbb Z}_p$ . We assume that G is tamely ramified, that is, there is a tame finite (Galois) extension $\tilde K/K$ of ramification degree e with ${\Gamma }={\mathrm{Gal}}(\tilde K/K)$ such that $G\otimes _K\tilde K\simeq H_0\otimes _{{\mathbb Z}_p}\tilde K$ . By adjoining an unramified extension, we can always assume that $\tilde K$ contains a uniformizer $\tilde \pi $ with $\tilde \pi ^e\in K^{\mathrm{un}}$ , where $K^{\mathrm{un}}$ is the maximal unramified extension of K contained in $\tilde K$ .

Suppose $\mathbf {x}\in {\mathcal B}(G, K)$ is such that ${\mathcal G}_{\mathbf {x}}$ is connected. Then, if $\mathbf {y}\in {\mathcal B}(G, K)$ is generic in the smallest facet containing $\mathbf {x}$ , we have ${\mathcal G}_{\mathbf {y}}={\mathcal G}_{\mathbf {x}}$ . Hence, the result applies to all stabilizer group schemes that are parahoric, that is, connected.

Proof. The statement is a variation of [Reference Pappas and RapoportPR24b, Prop. 2.8]. We will explain how the proof in loc. cit. can be extended to give this result. First we note that it is enough to show:

( $*$ ) There is a point $\mathbf {x}'\in {\mathcal B}(G, K)$ such that ${\mathcal G}_{\mathbf {x}}={\mathcal G}_{\mathbf {x}'}$ and a finite tame extension $\tilde K/K$ such that $G\otimes _K \tilde K$ is split and $\mathbf {x}'$ is hyperspecial in ${\mathcal B}(G, \tilde K)$ .

A hyperspecial point remains hyperspecial after every finite field extension. Hence, assuming ( $*$ ) we can pass to the normal closure and make sure that $\tilde K/K$ is in addition Galois with group $\Gamma ={\mathrm{Gal}}(\tilde K/K)$ . Then the rest follows by the standard argument which uses the smoothness of fixed points of a smooth scheme for a tame finite group action ([Reference EdixhovenEd92, 3.4: Prop.]).

Statement ( $*$ ) is shown in the course of the proof of [Reference Pappas and RapoportPR24b, Prop. 2.7, Prop. 2.8] when G is absolutely simple and simply connected. We will show how this argument extends under our assumptions.

First let us assume that G is semi-simple. Write

$$\begin{align*}G^{\mathrm{sc}}=\prod_i \mathrm{Res}_{L_i/K} G_i \end{align*}$$

with $G_i$ over $L_i$ , simply connected and absolutely simple. This gives

$$\begin{align*}{\mathcal B}(G, K)={\mathcal B}(G^{\mathrm{sc}}, K)=\prod_i {\mathcal B}(G_i, L_i); \quad \mathbf{x}\mapsto (\mathbf{x}_i). \end{align*}$$

Since $\mathbf {x}$ is generic in its facet, each $\mathbf {x}_i\in {\mathcal B}(G_i, L_i)$ is generic in its facet. By applying the argument in the proof of [Reference Pappas and RapoportPR24b, Prop. 2.8] which considers a “tame subdivision” of the apartment with its simplicial structure, we see that there exists a “nearby” $\mathbf {x}^{\prime }_i\in {\mathcal B}(G_i, L_i)$ which is hyperspecial in ${\mathcal B}(G_i, \tilde L_i)$ , where $\tilde L_i$ is a finite tame extension of K. In fact, by enlarging $\tilde L_i$ , we can find $\mathbf {x}^{\prime }_i$ with these properties which, using the standard metric of the apartment, is as close to $\mathbf {x}_i$ as we like. The assumption for groups of type A enters in the existence of the suitable tame subdivision, see the proof of [Reference Pappas and RapoportPR24b, Prop. 2.8], also [Reference Pappas and RapoportPR24b, Rem. 2.9]. Consider $\mathbf {x}'=(\mathbf {x}^{\prime }_i)\in {\mathcal B}(G, K)$ which is then close to $\mathbf {x}$ and defines the same stabilizer group scheme as $\mathbf {x}$ . By passing to the normal closure $\tilde K$ of the join of the $\tilde L_i$ ’s in an algebraic closure of K, we can assume that $\mathbf {x}^{\prime }_i\in {\mathcal B}(G_i, \tilde K)$ is hyperspecial for all i. We now have

$$\begin{align*}{\mathcal B}(G, \tilde K)=\prod_{(i, \alpha)} {\mathcal B}(G_i, \tilde K) \end{align*}$$

the indexing set also including all $\alpha : L_i\hookrightarrow \tilde K$ over K. For $\alpha : L_i\hookrightarrow \tilde K$ , find $\tau \in \Gamma ={\mathrm{Gal}}(\tilde K/K)$ such that $\alpha =\tau _{|L_i}: L_i\hookrightarrow \tilde K$ . Then the projection of the image of $\mathbf {x}'$ to the factor indexed by $(i, \alpha )$ , is $\tau _*(\mathbf {x}^{\prime }_i)$ . In this,

$$ \begin{align*}\tau_*: {\mathcal B}(G_i\otimes_{L_i}\tilde K, \tilde K)\to {\mathcal B}(G_i\otimes_{L_i, \tau}\tilde K, \tilde K)\end{align*} $$

is induced by functoriality of buildings by the Galois automorphism $ \tau : \tilde K\to \tilde K$ . Hence, $ \tau _*(\mathbf {x}^{\prime }_i)$ is hyperspecial, and so $\mathbf {x}'=( \tau _*(\mathbf {x}^{\prime }_i))_{i, \alpha }$ is hyperspecial in ${\mathcal B}(G,\tilde K)$ . This shows ( $*$ ) when G is semi-simple.

Now we discuss the general reductive case. Note that for a split group H, a point in $\mathbf {x}\in {\mathcal B}(H, K)$ is hyperspecial if and only if its image $\bar {\mathbf {x}}\in \bar {\mathcal B}(H,K)={\mathcal B}(H^\mathrm{der}, \tilde K)$ is hyperspecial.

We have $G(\breve K)_{\mathbf {x}}=G(\breve K)_{\bar {\mathbf {x}}}\cap G(\breve K)^1$ . Here, $G(\breve K)_{\bar {\mathbf {x}}}$ is the stabilizer of $\bar {\mathbf {x}}\in \bar {\mathcal B}(G, K)\subset \bar {\mathcal B}(G, \breve K)$ under the natural action of $G(\breve K)$ on $\bar {\mathcal B}(G, \breve K)$ ; the group $G(\breve K)^1$ is the kernel of

$$\begin{align*}G(\breve K)\xrightarrow{\kappa_G} \pi_1(G)_I\to \pi_1(G)_I/\{\mathrm{torsion}\} \end{align*}$$

obtained from the Kottwitz homomorphism, see [Reference Haines and RapoportHR08, Rem. 11], [Reference Bruhat and TitsBTII, 4.2.16]. If $\bar {\mathbf {x}}$ is generic in a facet and $\bar {\mathbf {x}}'$ is nearby, $G(\breve K)_{\bar {\mathbf {x}}}=G(\breve K)_{\bar {\mathbf {x}}'}$ ; hence we also have $G(\breve K)_{\mathbf {x}}=G(\breve K)_{\mathbf {x}'}$ and so ${\mathcal G}_{\mathbf {x}}={\mathcal G}_{\mathbf {x}'}$ .

Recall that we know ( $*$ ) for $G^\mathrm{der}$ . Consider $\mathbf {x}\in {\mathcal B}(G, K)$ generic in its facet with corresponding point $\bar {\mathbf {x}}\in {\mathcal B}(G^\mathrm{der}, K)$ , also generic in its facet. By ( $*$ ) for $G^\mathrm{der}$ , there is nearby $\mathbf {x}^{\prime }_\mathrm{der}\in {\mathcal B}(G^\mathrm{der}, K)$ and a tame Galois extension $\tilde K/K$ which splits $G^\mathrm{der}$ such that $\mathbf {x}^{\prime }_\mathrm{der}\in {\mathcal B}(G^\mathrm{der}, \tilde K)$ is hyperspecial. By enlarging $\tilde K/K$ we can assure that G is also split over $\tilde K$ . Now lift $\mathbf {x}^{\prime }_\mathrm{der}$ to $\mathbf {x}'\in {\mathcal B}(G, K)$ ; that is, with $\bar {\mathbf {x}}'=\mathbf {x}^{\prime }_\mathrm{der}$ . The point $\mathbf {x}'$ is hyperspecial in ${\mathcal B}(G, \tilde K)$ and by the argument above ${\mathcal G}_{\mathbf {x}}={\mathcal G}_{\mathbf {x}'}$ . This shows ( $*$ ) for G and $\mathbf {x}$ .

2.3.2

Let $\tilde K/K$ be a finite tame Galois extension with Galois group ${\Gamma }$ , inertia subgroup $I\subset {\Gamma }$ , and ramification index $e=|I|$ . Let $\tilde {\Lambda }\subset V\otimes _K\tilde K$ be an $\tilde {\mathcal O}$ -lattice. We assume that $\tilde {\Lambda }$ is ${\Gamma }$ -stable. Let $\tilde {\mathcal L}$ be the periodic lattice chain given by all scalar multiples $\tilde \pi ^i\tilde {\Lambda }$ of $\tilde {\Lambda }$ and consider the grading function $\tilde c$ given by $\tilde c(\tilde {\pi }^i\tilde {\Lambda })=i$ . Then $(\tilde {\mathcal L}, \tilde {c})$ is a periodic graded $\tilde {{\mathcal O}}$ -lattice chain in $V\otimes _K \tilde {K}$ corresponding to a point $\tilde {\mathbf {x}}\in {\mathcal B}(\mathrm{GL}(V), \tilde {K})$ which is fixed by ${\Gamma }$ . The corresponding parahoric group scheme for $\mathrm{GL}(V\otimes _K\tilde {K})$ over $\tilde {{\mathcal O}}$ is the group scheme of $\tilde {{\mathcal O}}$ -linear automorphisms of $\tilde {\Lambda }$ ; we denote this group scheme simply by $\mathrm{GL}(\tilde {\Lambda })$ .

By tame descent on buildings (2.1.3), $\tilde {\mathbf {x}}$ is identified with a point $\mathbf {x}\in {\mathcal B}(\mathrm{GL}(V), K)$ which corresponds to a periodic graded lattice chain $({\mathcal L}, c)$ in V. We have

$$\begin{align*}\mathrm{GL}({\mathcal L})=\mathrm{Res}_{\tilde {{\mathcal O}}/{\mathcal O}}\mathrm{GL}(\tilde{\Lambda})^{\Gamma} \end{align*}$$

for the parahoric $\mathrm{GL}({\mathcal L})$ of $\mathrm{GL}(V)$ given as the stabilizer of $\mathbf {x}$ .

Lemma 2.3.3. The parahoric $\mathrm{Res}_{\tilde {{\mathcal O}}/{\mathcal O}}\mathrm{GL}(\tilde {\Lambda })^{\Gamma }$ of $\mathrm{GL}(V)$ is equal to the stabilizer $\mathrm{GL}({\mathcal L})$ of the periodic lattice chain ${\mathcal L}$ given by $\{\Lambda _i\}_{i\in {\mathbb Z}}$ where

$$\begin{align*}\Lambda_i=(\tilde{\pi}^i\tilde{\Lambda})^{\Gamma}\subset (V\otimes_K \tilde {K})^{\Gamma}=V \end{align*}$$

and $\Lambda _{i+1}\to \Lambda _i$ are the natural injective maps given by $\tilde \pi ^{i+1}\tilde {\Lambda }\subset \tilde \pi ^i\tilde {\Lambda }$ .

Note that in the above, we could have $\Lambda _{i+1}=\Lambda _i$ for some i. The periodic lattice chain ${\mathcal L}$ given by $\{\Lambda _i\}_{i\in {\mathbb Z}}$ is, by definition, the set of the lattices ${\Lambda }_i$ . Since $\tilde \pi ^e\tilde {\mathcal O}=\pi \tilde {\mathcal O}$ we have $\Lambda _{e}=(\pi \tilde {\Lambda } )^{\Gamma }=\pi \Lambda _0$ .

Proof. Both the group schemes $\mathrm{Res}_{\tilde {\mathcal O}/{\mathcal O}}\mathrm{GL}(\tilde {\Lambda })^{\Gamma }$ and $\mathrm{GL}({\mathcal L})$ are smooth affine with generic fiber $\mathrm{GL}(V)$ and, by [Reference Bruhat and TitsBTII, Prop. 1.7.6], it is enough to show they have the same $\breve {\mathcal O}$ -points. For this, we base change to $\breve {\mathcal O}$ and assume that $K=\breve K$ . So, it is enough to show

(2.3.4) $$ \begin{align} \mathrm{GL}(V)\cap \mathrm{GL}(\tilde{\Lambda})=\bigcap_{i=0}^{e-1} \mathrm{GL}({\Lambda}_i) \end{align} $$

(the intersection taking place in $\mathrm{GL}(V\otimes _K\tilde K)$ .) Let $\tilde \pi \in \tilde {\mathcal O}$ be a uniformizer with $\tilde \pi ^e\in {\mathcal O}$ . Let $\chi : I\to k^*={\mathrm{Aut}}_k((\tilde \pi )/(\tilde \pi )^2)$ be the standard inertia character. Write

$$\begin{align*}\tilde {\Lambda}= \bigoplus_{i\in {\mathbb Z}/e{\mathbb Z}} \tilde {\Lambda}_{i} \end{align*}$$

for the decomposition into eigenspaces for the action of the inertia. Here

$$\begin{align*}\tilde{\Lambda}_i=\tilde {\Lambda}_{i\,\mathrm{mod}\, e}=\{x\in \tilde{\Lambda}\ |\ \gamma(x)=[\chi(\gamma)]^i\cdot x\}, \end{align*}$$

with $[\ ]: k^*\to {\mathcal O}^*$ the Teichmüller map. The eigenspaces $\tilde {\Lambda }_i$ are ${\mathcal O}$ -modules and

$$\begin{align*}{\Lambda}_i=(\tilde\pi^i\tilde{\Lambda})^I=\tilde\pi^i \tilde{\Lambda}_{-i\,\mathrm{mod}\, e}\xrightarrow{\sim} \tilde{\Lambda}_{-i\,\mathrm{mod}\, e}, \end{align*}$$

the last map given by multiplying by $\tilde \pi ^{-i}$ . So, we have

(2.3.5) $$ \begin{align} \tilde{\Lambda}=\bigoplus_{i=0}^{e-1} \tilde\pi^{-i}{\Lambda}_i\subset V\otimes_K \tilde K=\bigoplus_{i=0}^{e-1} \tilde\pi^{-i}V. \end{align} $$

Multiplication by $g\in \mathrm{GL}(V)\cap \mathrm{GL}(\tilde {\Lambda })$ respects the eigenspace decomposition of $\tilde {\Lambda }$ and commutes with scaling by $\tilde \pi $ , so the LHS of (2.3.4) is contained in the RHS. Suppose $g\in \mathrm{GL}({\Lambda }_i)$ , for all i. Then, by the above, g considered in $\mathrm{GL}(V\otimes _K \tilde K)$ gives an automorphism of $\tilde \pi ^i \tilde {\Lambda }_{-i\,\mathrm{mod}\, e}$ and hence of $\tilde {\Lambda }$ . This shows that the RHS is contained in the LHS.

2.3.6

In the lemma above, $\{{\Lambda }_i\}_{i\in {\mathbb Z}}$ is given by the $\pi ^{\mathbb Z}$ multiples of its segment

$$\begin{align*}\pi{\Lambda}_0\subset {\Lambda}_{e-1}\subset \cdots\subset {\Lambda}_1\subset {\Lambda}_0. \end{align*}$$

Assuming $\tilde \pi ^e\in {\mathcal O}$ and that $K=K^{\mathrm{un}}$ , that is, $\tilde K/K$ is totally ramified, the proof of the lemma gives

(2.3.7) $$ \begin{align} {\Lambda}_0\oplus{\Lambda}_1\oplus\cdots \oplus {\Lambda}_{e-1}\xrightarrow{\sim}\tilde{\Lambda}_{0}\oplus\tilde{\Lambda}_{-1}\oplus\cdots\oplus \tilde{\Lambda}_{-(e-1)}=\tilde{\Lambda} \end{align} $$

as ${\mathcal O}$ -modules, with the map given by multiplication by $(1,\tilde \pi ^{-1},\ldots , \tilde \pi ^{-(e-1)})$ . Hence,

$$\begin{align*}{\mathrm{tot}}({\mathcal L})\subset {\Lambda}_0\oplus{\Lambda}_1\oplus\cdots \oplus {\Lambda}_{e-1}\simeq \tilde{\Lambda} \end{align*}$$

and it is a direct summand. (The inclusion is proper when we have ${\Lambda }_i={\Lambda }_{i+1}$ , for some i.) It follows that multiplication by corresponding powers of $\tilde \pi $ on the graded pieces gives an isomorphism

(2.3.8) $$ \begin{align} L\oplus {\mathrm{tot}}({\mathcal L})\xrightarrow{\ \sim\ } \tilde{\Lambda}. \end{align} $$

where L is a certain direct sum of ${\Lambda }_i$ .

2.4.1

Let $p>2$ and $\rho :G\rightarrow \mathrm{GL}(V)$ a faithful representation of a reductive group over K. We have the following proposition which generalizes [Reference KisinKi10, Lem. 2.3.1] with a similar proof.

Proposition 2.4.2. Let $\tilde K/K$ be a finite Galois extension with ${\Gamma }={\mathrm{Gal}}(\tilde K/K)$ and with the following property:

There is a split reductive group scheme $\tilde {\mathcal G}$ over $\tilde {\mathcal O}$ such that

1. 1) $\tilde {\mathcal G}\otimes _{\tilde {\mathcal O}}\tilde K=G\otimes _{K}\tilde K$ (in particular, G splits over $\tilde K$ ),

2. 2) $\tilde {\mathcal G}$ supports an $\tilde {\mathcal O}$ -semilinear ${\Gamma }$ -action which extends the standard $\tilde K$ -semilinear $\Gamma $ -action on $G\otimes _K\tilde K$ .

Then there is a $\tilde {\mathcal O}$ -lattice $\tilde {\Lambda }$ in $V\otimes _K\tilde K$ which is ${\Gamma }$ -invariant and such that the base change $\rho \otimes _K\tilde K: G\otimes _K\tilde K\to \mathrm{GL}(V\otimes _K\tilde K)$ extends to a closed group scheme immersion

$$\begin{align*}\tilde{\mathcal G}\hookrightarrow \mathrm{GL}(\tilde {\Lambda}). \end{align*}$$

Proof. Let M be the maximal unramified extension of $\tilde K$ in an algebraic closure of $\tilde K$ . Then $M/K$ is an (infinite) Galois extension. The natural homomorphism ${\mathrm{Gal}}(M/K)\to {\Gamma }={\mathrm{Gal}}(\tilde K/K)$ is a surjection with kernel ${\mathrm{Gal}}(M/\tilde K)$ which identifies inertia subgroups. We first show that there is a ${\mathrm{Gal}}(M/K)$ -invariant ${\mathcal O}_M$ -lattice ${\Lambda }_M$ in $V\otimes _K M$ which is also preserved by the action of $\rho (\tilde {\mathcal G}({\mathcal O}_M))$ . Observe that $\tilde {\mathcal G}({\mathcal O}_M)$ is bounded and the Galois group ${\mathrm{Gal}}(M/K)$ is compact. We can consider the semi-direct product $\tilde {\mathcal G}({\mathcal O}_M)\rtimes {\mathrm{Gal}}(M/K)$ (obtained by the ${\mathrm{Gal}}(M/K)$ -action on $\tilde {\mathcal G}({\mathcal O}_M)$ given by the semi-linear ${\Gamma }$ -action on $\tilde {\mathcal G}$ ) and apply the same argument as in the proof of [Reference KisinKi10, Lem. 2.3.1] to obtain the existence of ${\Lambda }_M$ . Then, by [Reference Bruhat and TitsBTII, 1.7.6], $\rho $ extends to a group scheme homomorphism

$$\begin{align*}\tilde{\mathcal G}\otimes_{\tilde {\mathcal O}}{\mathcal O}_M\to \mathrm{GL}({\Lambda}_M). \end{align*}$$

Since $\tilde {\mathcal G}$ is reductive and p is odd, this is a closed immersion by [Reference Prasad and YuPrY06, 1.3]. We can then take

$$\begin{align*}\tilde {\Lambda}:=( {\Lambda}_M)^{{\mathrm{Gal}}(M/\tilde K)}. \end{align*}$$

This is an $\tilde {\mathcal O}$ -lattice in $V\otimes _K\tilde K$ by étale descent along ${\mathcal O}_M/\tilde {\mathcal O}$ and the rest follows.

Remark 2.4.3. a) After applying restriction of scalars and then ${\Gamma }$ -fixed points to $\tilde {\mathcal G}\hookrightarrow \mathrm{GL}(\tilde {\Lambda })$ , we obtain a closed immersion of group schemes

$$\begin{align*}(\mathrm{Res}_{\tilde{\mathcal O}/{\mathcal O}}\tilde{\mathcal G})^{\Gamma}\hookrightarrow \mathrm{Res}_{\tilde{\mathcal O}/\tilde O}\mathrm{GL}(\tilde {\Lambda})^{\Gamma} \end{align*}$$

which gives $\rho : G\to \mathrm{GL}(V)$ on generic fibers.

b) Note that we do not need that $\tilde K/K$ is tame in Proposition 2.4.2. However, under this additional assumption, we see, using Edixhoven’s lemma [Reference EdixhovenEd92, 3.4:Prop.], that both the target and the source of the closed immersion in (a) above are smooth affine schemes over ${\mathcal O}$ . By Lemma 2.3.3 and étale descent, the target is a parahoric group scheme for $\mathrm{GL}(V)$ . In fact, it is the parahoric group scheme given as the stabilizer of the chain of ${\mathcal O}$ -lattices $\{(\tilde \pi ^i\tilde {\Lambda })^{\Gamma }\}_{i\in {\mathbb Z}}$ .

2.4.4

We now assume that G and $\tilde K/K$ are as in §2.2 and let $\rho : G\to \mathrm{GL}(V)$ be a faithful representation over K. Suppose $\mathbf {x}\in {\mathcal B}(G, K)$ is generic in its facet and that after replacing $\mathbf {x}$ by a nearby point with the same stabilizer group scheme, $\mathbf {x}$ is hyperspecial in ${\mathcal B}(G, \tilde K)$ and hence the corresponding parahoric group scheme $\tilde {\mathcal G}=\tilde {\mathcal G}_{\mathbf {x}}$ of $G\otimes _K\tilde K$ is reductive. This is possible by Prop. 2.2.2, under the assumptions stated there.

By Proposition 2.4.2, there is a ${\Gamma }$ -stable $\tilde {\mathcal O}$ -lattice $\tilde {\Lambda }$ in $V\otimes _K\tilde K$ such that $\rho $ extends to a closed immersion of group schemes

$$\begin{align*}\rho: \tilde{\mathcal G}_{\mathbf{x}}\hookrightarrow \mathrm{GL}(\tilde{\Lambda}). \end{align*}$$

Taking restriction of scalars and then ${\Gamma }$ -fixed points gives a closed immersion

(2.4.5) $$ \begin{align} \rho: {\mathcal G}_{\mathbf{x}}=(\mathrm{Res}_{\tilde{\mathcal O}/{\mathcal O}}\tilde{\mathcal G}_{\mathbf{x}})^{\Gamma} \hookrightarrow (\mathrm{Res}_{\tilde{\mathcal O}/{\mathcal O}}\mathrm{GL}(\tilde{\Lambda})^{\Gamma}\hookrightarrow \mathrm{GL}(\tilde{\Lambda}) \end{align} $$

where in the target we consider $\tilde {\Lambda }$ as an ${\mathcal O}$ -module by restriction of scalars.

3.1.1

In this section, we let F be a finite extension of ${\mathbb Q}_p$ . Let $(G, \{\mu \}, {\mathcal G} )$ be a local model triple over F. By definition, in these triples

• • G is a (connected) reductive group over F,

• • $\{\mu \}$ is the $G(\overline F)$ -conjugacy class of a minuscule cocharacter $\mu : {{\mathbb G}_{\mathrm{m}}}_{\overline F}\to G_{\overline F}$ , where $\overline F$ is an algebraic closure of F,

• • ${\mathcal G}$ is a quasi-parahoric stabilizer group scheme over ${\mathcal O}_F$ for G.

A morphism of local model triples $(G,\{\mu \},{\mathcal G}) \rightarrow (G',\{\mu '\},{\mathcal G}')$ is a group scheme homomorphism ${\mathcal G} \rightarrow {\mathcal G}'$ taking $\{\mu \}$ to $\{\mu '\}.$

As usual, we denote by E the reflex field of the pair $(G, \{\mu \})$ . It is a subfield of $\bar {F}$ containing F. To simplify notation, we often write $(G, \mu )$ for $(G, \{\mu \})$ and $({\mathcal G},\mu )$ instead of $(G, \{\mu \}, {\mathcal G} )$ .

By definition, an integral Hodge embedding for $({\mathcal G},\mu )$ is a closed immersion of group schemes ${\mathcal G}\hookrightarrow \mathrm{GL}({\Lambda })$ over ${\mathcal O}_F$ , where ${\Lambda }$ is an ${\mathcal O}_F$ -lattice in V, such that the homomorphism of generic fibers $G\hookrightarrow \mathrm{GL}(V)$ is a Hodge embedding in the sense above.

In this situation, G is classical when $G^{\mathrm{ad}} \simeq \prod _{i=1}^s\mathrm{Res}_{K_i/F}H_i$ , with each $H_i$ splitting over a tamely ramified extension of $K_i$ , and of classical type. (By definition, “classical type” excludes triality groups.)

Remark 3.1.6. a) Suppose $p>2$ and $(G, \mu )$ is of abelian type. Write

$$ \begin{align*}(G^{\mathrm{ad}} ,\mu^{\mathrm{ad}} )\simeq (\prod_{i=1}^s\mathrm{Res}_{K_i/F}H_i, \{\mu_i\}), \end{align*} $$

where, for all i, $K_i/F$ is a finite extension and $H_i$ is absolutely simple over $K_i$ . As we will explain below, if $\mu _i$ is nontrivial, then $H_i$ is of classical type, and also splits over a tamely ramified extension of $K_i$ . Hence, the additional condition “essentially tame and classical” in the standard assumptions above, is only relevant when $\mu _i=1$ , for some i.

Indeed, when $\mu _i\neq 1$ , $H_i$ is of type A, B, C, or D: This follows from Deligne’s argument classifying Hodge embeddings which also applies in this local case. The triality forms of type $D_4$ are excluded: Indeed, the existence of a rational minuscule embedding implies that the Galois group cannot act transitively on the set of end vertices in the Dynkin diagram of a simple factor of type $D_4$ . Now, if $p>3$ any reductive group G over F is essentially tame. If $p=3$ , there are G which are not essentially tame: However, they are all triality forms and these are excluded. For details, see [Reference Pappas and RapoportPR26, Prop. 7.2.1 and its proof], cf. [Reference DeligneDe79, §2.3.8] and Prop. 3.2.11 below.

b) If $p>2$ and $(G, \mu _h)$ is obtained, by completion at p, from a (global) Shimura datum $(\mathbf {G}, X)$ of abelian type in the sense of [Reference DeligneDe79], then the pair $(G, \mu _h)$ satisfies the standard assumptions.

3.1.7

In what follows, we write ${\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu }$ for the local model associated to the local model triple $(G,\{\mu \},{\mathcal G})$ . By definition, ${\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu }={\mathbb M}^{\mathrm{loc}}_{{\mathcal G}^\circ ,\mu }$ and is the unique, up to unique isomorphism, proper flat ${\mathcal O}_E$ -scheme with ${\mathcal G}$ -action, with generic fiber $G/P_\mu $ and reduced special fiber, which represents the v-sheaf ${\mathrm{M}}^v_{{\mathcal G},\mu }$ over ${\mathrm{Spd}}({\mathcal O}_E)$ defined in [Reference Scholze and WeinsteinSW20]. (This is denoted by ${\mathrm{Gr}}_{{\mathcal G},{\mathrm{Spd}}({\mathcal O}_E),\mu }$ in [Reference Scholze and WeinsteinSW20, Lect. 21].)

The existence of ${\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu }$ was conjectured by Scholze-Weinstein [Reference Scholze and WeinsteinSW20, Conj. 21.4.1] and is shown in [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22] under mild assumptions (which are weaker than the standard assumptions above), and in general in [Reference Gleason and LourençoGL22].

In fact, under the above standard assumptions, we will construct ${\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu }$ following the work in [Reference Pappas and ZhuPZ13], [Reference LevinLe16], [Reference He, Pappas and RapoportHPR20], independently of the arguments of [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22], [Reference Gleason and LourençoGL22], see Theorem 3.2.15. Our specific construction of ${\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu }$ is important for the rest of the argument, and is intertwined with the construction of a suitable embedding of the local model in a Grassmannian, see Theorem 3.3.25.

3.1.8

The perfection of the special fiber of the local model ${\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu }$ is a closed subscheme of the (perfect) Witt vector affine Grassmannian ${\mathrm{Gr}}_{\mathcal G}^W=L^W{\mathcal G}/L^{W+}{\mathcal G}$ ([Reference ZhuZhu17], [Reference Bhatt and ScholzeBS17]), see [Reference Anschütz, Gleason, Lourenço and RicharzAGLR22, Thm 2.1, Thm. 7.23]. If ${\mathcal K}$ is an algebraically closed field of characteristic p,

$$\begin{align*}{\mathrm{Gr}}_{\mathcal G}^W({\mathcal K})=\frac{G(W({\mathcal K})[1/p])}{{\mathcal G}(W({\mathcal K}))}, \end{align*}$$

hence there is a natural equivariant embedding

$$\begin{align*}{\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu}({\mathcal K})\subset {\mathrm{Gr}}_{\mathcal G}^W({\mathcal K})=\frac{G(W({\mathcal K})[1/p])}{{\mathcal G}(W({\mathcal K}))}. \end{align*}$$

3.1.9

Now consider local model data $({\mathcal G}, \mu )$ of Hodge type and integral Hodge embeddings ${\mathcal G}\hookrightarrow \mathrm{GL}({\Lambda })$ extending $\rho : (G,\mu )\hookrightarrow (\mathrm{GL}(V),\mu _d)$ . By functoriality and by using the full-faithfulness result of [Reference Scholze and WeinsteinSW20, Prop. 18.4.1], we see that there is a canonical equivariant morphism

$$\begin{align*}\rho_*: {\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu}\to {\mathrm{Gr}}(d,{\Lambda})_{{\mathcal O}_E}={\mathbb M}^{\mathrm{loc}}_{\mathrm{GL}({\Lambda}),\mu_d}\otimes_{{\mathcal O}}{\mathcal O}_E \end{align*}$$

attached to $({\mathcal G},\mu )\hookrightarrow (\mathrm{GL}({\Lambda }),\mu _d)$ , where ${\mathrm{Gr}}(d,\Lambda )$ is the smooth Grassmannian classifying d-dimensional subspaces of $\Lambda $ . This morphism identifies ${\mathbb M}^{\mathrm{loc}}_{{\mathcal G},\mu }$ with the normalization of its scheme theoretic image. Note that by [Reference Scholze and WeinsteinSW20, Cor. 21.6.10 and its proof], we have ${\mathrm{Gr}}(d,{\Lambda })^\Diamond ={\mathrm{M}}^v_{\mathrm{GL}({\Lambda }),\mu _d}$ , and so ${\mathbb M}^{\mathrm{loc}}_{\mathrm{GL}({\Lambda }),\mu _d}={\mathrm{Gr}}(d,{\Lambda })$ (this proves the Scholze-Weinstein conjecture for $\mathrm{GL}_n$ ).

Suppose that ${\mathcal K}$ is an algebraically closed field extension of $k_E$ . Then, by combining with §3.1.8, we obtain a commutative diagram of inclusions

(3.1.10)

with the vertical arrows induced by ${\mathcal G}\hookrightarrow \mathrm{GL}({\Lambda })$ .

3.2.1

Let G be a (connected) reductive group over a field $\kappa $ . We let ${\mathrm{Gr}}_G:=LG/L^+G$ denote the affine Grassmannian for G; thus ${\mathrm{Gr}}_{G}$ is the ind-scheme over ${\mathrm{Spec} \, } (\kappa )$ which represents the fpqc sheaf associated to the functor given by $R\mapsto G(R(\!(t)\!))/G(R[\![t]\!])$ on $\kappa $ -algebras R. The affine Grassmannian ${\mathrm{Gr}}_G$ also represents the functor on $\kappa $ -algebras which sends R to the isomorphism classes of pairs $({\mathcal E},\varphi )$ where

• • ${\mathcal E}$ is a G-torsor over ${\mathrm{Spec} \, } R[\![ t]\!] $ ,

• • $\varphi :{\mathcal E}[1/t]\xrightarrow {\sim } {\mathcal E}^0[1/t]$ is a trivialization of the restriction ${\mathcal E}[1/t]$ of the G-torsor ${\mathcal E}$ to ${\mathrm{Spec} \, } (R(\!( t)\!))$ .

Here, ${\mathcal E}^0$ denotes the trivial G-torsor.

3.2.2

Let $K_0/F$ be a finite unramified extension. Let $P(u)\in {\mathcal O}_{K_0}[u]$ be a monic polynomial and $\underline {{\mathcal G}}$ a smooth affine group scheme over ${\mathcal O}_{K_0}[u]$ with geometrically connected fibers. We consider the functor $\mathrm{Fl}^{P(u)}_{\underline {{\mathcal G}},0}$ on ${\mathcal O}_{K_0}$ -algebras R given by

$$ \begin{align*}\mathrm{Fl}^{P(u)}_{\underline{{\mathcal G}},0}(R)=\{\text{iso. classes of pairs }({\mathcal E},\beta)\},\end{align*} $$

where ${\mathcal E}$ is a $\underline {{\mathcal G}}$ -torsor over $R[u]$ and $\beta :{\mathcal E}|_{R[u][1/P(u)]}\xrightarrow {\sim } {\mathcal E}^0$ is an isomorphism of $\underline {{\mathcal G}}$ -torsors, where ${\mathcal E}^0$ denotes the trivial $\underline {{\mathcal G}}$ -torsor. We then define the mixed characteristic affine Grassmannian

$$ \begin{align*}\mathrm{Fl}^{P(u)}_{\underline{{\mathcal G}}}:=\mathrm{Res}_{{\mathcal O}_{K_0}/{\mathcal O}_{F}}\mathrm{Fl}^{P(u)}_{\underline{{\mathcal G}},0}. \end{align*} $$

By embedding $\underline {\mathcal G}$ into a general linear group, one deduces as in [Reference LevinLe16, Prop. 4.1.4], that $\mathrm{Fl}^{P(u)}_{\underline {{\mathcal G}}}$ is representable by an ind-scheme over ${\mathcal O}_F$ .

3.2.3

Let $(G,\{\mu \},{\mathcal G})$ be a local model triple with $G\cong \mathrm{Res}_{K/F}H$ . Assume that ${\mathcal G}$ is the stabilizer of a point $\mathbf {x}\in {\mathcal B}(G, F)$ . Then by [Reference Haines and RicharzHR20, Prop. 4.7], we have ${\mathcal G}\cong \mathrm{Res}_{{\mathcal O}_K/{\mathcal O}_F}{\mathcal H}$ .

Assume now that H splits over a tamely ramified extension of K. Let $K_0$ denote the maximal unramified extension of F contained in K and write ${\mathcal O}_{K_0}$ (resp. $k_0$ ) for its ring of integers (resp. residue field). We let ${\mathcal O}_{K_0}[u^\pm ]$ denote the ring ${\mathcal O}_{K_0}[u,u^{-1}]$ . We fix a uniformizer $\pi $ of K and we write $E(u)\in {\mathcal O}_{K_0}[u]$ for the Eisenstein polynomial which is the minimal polynomial for $\pi $ over $K_0$ . Fix also a rigidification $(H, A, S, P)$ of H in the sense of [Reference Pappas and ZhuPZ13, Def. 2.7], cf. [Reference LevinLe16, §3.1], in which A is a maximal split torus of H over K such that $\mathbf {x}\in {\mathcal B}(H, K)$ lies in the apartment corresponding to A. Denote by $\underline H$ the reductive group scheme over ${\mathcal O}_{K_0}[u^{\pm }]$ constructed by [Reference LevinLe16, Prop. 3.1.2]. This extends the group H in the sense that the base change of $\underline H$ by ${\mathcal O}_{K_0}[u^{\pm }]\to {\mathcal O}_K$ , $u\mapsto \pi $ , is H. Then, [Reference LevinLe16, Thm 3.3.3], cf. [Reference Pappas and ZhuPZ13, Thm. 4.1], gives a smooth affine group scheme $\underline {\mathcal H}^\circ $ over ${\mathcal O}_{K_0}[u]$ , with geometrically connected fibers, extending $\underline H$ which also specializes to ${\mathcal H}^\circ $ under the map ${\mathcal O}_{K_0}[u]\rightarrow {\mathcal O}_{K}, u\mapsto \pi $ .

Applying the construction of §3.2 we obtain the ind-scheme $\mathrm{Fl}_{\underline {{\mathcal H}}^\circ }^{E(u)}$ over ${\mathcal O}_F$ which is ind-projective by [Reference LevinLe16, Thm. 4.2.11].

3.2.5

For a $K_0$ -algebra R, the completion $\widehat {R[u]}$ of $R[u]$ along the ideal $(E(u))$ , contains the completion of $K_0[u]$ along $(E(u))$ . The latter ring may be identified with $K[\![ t]\!]$ , by a map sending t to $E(u)$ and inducing the identity on residue fields. Then $\widehat {R[u]}$ may be identified with $(K\otimes _{K_0} R)[\![ t]\!]$ by sending t to $E(u)$ . This induces an isomorphism from the generic fiber of $\mathrm{Fl}_{\underline {{\mathcal H}}^\circ ,0}^{E(u)}$ to the affine Grassmannian $\mathrm{Gr}_{\mathrm{Res}_{K/K_0}H}$ (cf. [Reference Haines and RicharzHR20, Cor. 3.5]), and hence an isomorphism from the generic fiber of $\mathrm{Fl}^{E(u)}_{\underline {{\mathcal H}}^\circ }$ to $\mathrm{Gr}_{\mathrm{Res}_{K/F}H}\cong \mathrm{Gr}_{G}$ .

A representative $\mu $ of $\{\mu \}$ over $\bar {F}$ determines an element of $G(\bar {F}(\!( t)\!))$ and hence a point $e_\mu := \mu (t)\in \mathrm{Gr}_G(\bar {F})$ . The (affine) Schubert variety $S_\mu $ is the closure of the $G(\bar {F}[\![ t]\!])$ -orbit of $e_\mu $ in $\mathrm{Gr}_G$ . The conjugacy class $\{\mu \}$ has the reflex field E as a minimal field of definition and the Schubert variety $S_\mu \subset \mathrm{Gr}_G$ is defined over E.

3.2.8

In general, if G is quasi-tame, choose an isomorphism $G\cong \prod _{i=1}^r\mathrm{Res}_{K_i/F}H_i$ , with $H_i$ splitting over a tame extension, and set

$$\begin{align*}{\mathrm{M}}_{{\mathcal G}, \mu }:=\prod_{i=1}^r{\mathrm{M}}_{{\mathcal G}_i, \mu_i}\otimes_{{\mathcal O}_{E_i}}{\mathcal O}_E. \end{align*}$$

Here ${\mathcal G}_i$ with generic fiber $\mathrm{Res}_{K_i/F}H_i$ is determined by ${\mathcal G}\cong \prod _{i=1}^r{\mathcal G}_i$ , $\{\mu _i\}$ is the $\mathrm{Res}_{K_i/F}H_i$ -factor of the G-conjugacy class $\{\mu \}$ , and $E_i$ (resp. E) is the field of definition of $\{\mu _i\}$ (resp. $\{\mu \}$ ). The following theorem follows immediately from [Reference LevinLe16, Thm. 4.2.7].

3.2.10

We now extend this construction of local models to a more general situation.

3.2.12

Assume now $(G,\{\mu \}, {\mathcal G})$ satisfies the standard assumptions 3.1.5. We construct a local model ${\mathrm{M}}^{\mathrm{loc}}_{{\mathcal G},\mu }$ for $(G,\{\mu \}, {\mathcal G})$ as follows: We write $G^{\mathrm{ad}} _1\times G^{\mathrm{ad}} _2$ , where $G^{\mathrm{ad}} _1$ (resp. $G^{\mathrm{ad}} _2$ ) is the product of the F-simple factors of $G^{\mathrm{ad}} $ where $\mu ^{\mathrm{ad}} $ is nontrivial (resp. trivial). Let $G_1$ be the kernel of $G\to G^{\mathrm{ad}} _2$ . Then $\{\mu \}$ factors through $G_1$ and we denote by $\{\mu _1\}$ for the induced conjugacy class of cocharacters. The morphism $G_1\to G^{\mathrm{ad}} _1$ is a central extension and $(G_1,\mu _1)$ is of abelian type and satisfies the assumptions of Proposition 3.2.11 above. Let $(G',\mu ')$ be as in the conclusion of Proposition 3.2.11 applied to $(G_1,\mu _1)$ . Now define

(3.2.13) $$ \begin{align} {\mathrm{M}}^{\mathrm{loc}}_{{\mathcal G}, \mu}:={\mathrm{M}}_{{\mathcal G}',\mu'}\otimes_{{\mathcal O}_{E'}}{\mathcal O}_E. \end{align} $$

This is a flat projective ${\mathcal O}_E$ -scheme with reduced special fiber, by Theorem 3.2.9.

We will show:

This will follow as a consequence of Theorem 3.3.25 below. This implication is shown in §3.4.2.

3.3.1

Let $(G,\{\mu \},{\mathcal G})$ be a local model triple over F with $G\simeq \mathrm{Res}_{K/F}H$ , where H splits over a tamely ramified extension of K. We fix the isomorphism above and just write $G=\mathrm{Res}_{K/F}H$ . Assume $\mathbf {x}\in {\mathcal B}(H, K)={\mathcal B}(G, F)$ is generic in its facet and let ${\mathcal H}={\mathcal H}_{\mathbf {x}}$ , resp. ${\mathcal G}={\mathcal G}_{\mathbf {x}}$ , be the Bruhat-Tits stabilizer group schemes for H, resp. G, over ${\mathcal O}_F$ , resp. ${\mathcal O}_K$ . We have

$$\begin{align*}{\mathcal G}\cong\mathrm{Res}_{{\mathcal O}_K/{\mathcal O}_F}{\mathcal H}. \end{align*}$$

Now suppose that the reductive group H over K and $\mathbf {x}\in {\mathcal B}(H, K)$ satisfies all the assumptions of Proposition 2.2.2. Let $\mathbf {x}'$ , $\tilde K/K$ , $\Gamma ={\mathrm{Gal}}(\tilde K/K)$ be as in the conclusion of Proposition 2.2.2: Then $\tilde H:=H\otimes _K \tilde K\simeq H_0\otimes _{{\mathbb Z}_p}\tilde K$ is split and the point $\mathbf {x}'$ is hyperspecial over $\tilde K$ . In this, $H_0$ is the Chevalley form of the split group $\tilde H$ . Again, $\tilde {\mathcal H}=\tilde {\mathcal H}_{\mathbf {x}'}\simeq H_0\otimes _{{\mathbb Z}_p}{\mathcal O}_{\tilde K}$ is the corresponding hyperspecial group scheme for $\tilde H$ over ${\mathcal O}_{\tilde K}$ and we have

$$\begin{align*}{\mathcal H}\simeq (\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_K}\tilde{\mathcal H})^\Gamma. \end{align*}$$

Consider the map

(3.3.2) $$ \begin{align} G=\mathrm{Res}_{K/F}H\to \mathrm{Res}_{{\tilde K}/F}(H_0\otimes_{{\mathbb Z}_p}\tilde K)=\mathrm{Res}_{K/F}(\mathrm{Res}_{{\tilde K}/K}(H_0\otimes_{{\mathbb Z}_p}\tilde K)). \end{align} $$

given by applying restriction of scalars to

$$ \begin{align*}H\to \mathrm{Res}_{\tilde K/K}(H\otimes_K\tilde K)\simeq \mathrm{Res}_{{\tilde K}/K}(H_0\otimes_{{\mathbb Z}_p}\tilde K).\end{align*} $$

This extends to the closed immersion of group schemes

(3.3.3) $$ \begin{align} {\mathcal G}=\mathrm{Res}_{{\mathcal O}_{K}/{\mathcal O}_F}{\mathcal H}\to \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}). \end{align} $$

by Proposition 2.1.5. We let $\tilde \mu $ be the geometric cocharacter of $\mathrm{Res}_{{\tilde K}/F}(H_0\otimes _{{\mathbb Z}_p}\tilde K)$ which is given by composing $\mu $ with the map (3.3.2). Then

$$ \begin{align*}(\mathrm{Res}_{{\tilde K}/F}(H_0\otimes_{{\mathbb Z}_p}\tilde K), \{\tilde\mu\}, \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}))\end{align*} $$

is a local model triple with reflex field $\tilde {E}$ and

(3.3.4) $$ \begin{align} (G, \{\mu\}, {\mathcal G})\to (\mathrm{Res}_{{\tilde K}/F}(H\otimes_K\tilde K), \{\tilde\mu\}, \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K})) \end{align} $$

a morphism of local model triples.

3.3.5

We will show (3.3.4) induces a closed immersion of local models

$$ \begin{align*}{{\mathrm{M}}}_{{\mathcal G},\mu}\to ({{\mathrm{M}}}_{\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}),\tilde \mu})\otimes_{{\mathcal O}_{\tilde E}}{\mathcal O}_E.\end{align*} $$

To do this, we recall some aspects of the construction of the group schemes $\underline {\mathcal H}^\circ $ from §3.2.2. We let $K_0$ (resp. $\tilde K_0$ ) be the maximal unramified extension of F in K (resp. $\tilde K$ ), and we set

$$\begin{align*}\underline{\tilde{\mathcal H}}=H_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K_0}[\tilde u]. \end{align*}$$

If e is the ramification degree of the tame extension $\tilde K/K$ , then, after possibly enlarging $\tilde K$ , we can find a uniformizer $\pi $ of ${\mathcal O}_{K}$ and a uniformizer $\tilde \pi $ of ${\mathcal O}_{\tilde K}$ such that $\tilde \pi ^e=\pi $ . We can then identify $\Gamma ={\mathrm{Gal}}(\tilde K/K)$ with the Galois group of the cover ${\mathcal O}_{\tilde K_0}[\tilde u^{\pm }]/{\mathcal O}_{K_0}[u^{\pm }]$ given by $u\mapsto \tilde u^e$ ; this identification is compatible with the specializations $u\mapsto \pi $ , $\tilde u\mapsto \tilde \pi $ . For typesetting simplicity, in what follows we will write

$$\begin{align*}{\mathcal O}_0={\mathcal O}_{K_0},\quad \tilde {\mathcal O}_0:={\mathcal O}_{\tilde K_0}. \end{align*}$$

According to the construction in [Reference LevinLe16], [Reference Pappas and ZhuPZ13], there is a semi-linear action of $\Gamma $ on the group scheme $H_0\otimes _{{\mathbb Z}_p}\tilde {\mathcal O}_{0}[v]$ and one considers

$$\begin{align*}\underline{\mathcal H}:= (\mathrm{Res}_{\tilde {\mathcal O}_{0}[\tilde u]/{\mathcal O}_{0}[u]}(H_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u]) )^\Gamma. \end{align*}$$

This is an affine group scheme over ${\mathcal O}_{0}[u]$ which is smooth by Edixhoven’s lemma. It now follows from the construction in the proof or by the uniqueness statement in [Reference LevinLe16, Thm. 3.3], cf. [Reference Pappas and ZhuPZ13, §4.2.1], that, as the notation suggests, the group scheme $\underline {\mathcal H}^\circ $ given by [Reference LevinLe16, Thm. 3.3] is isomorphic to the neutral connected component of $\underline {\mathcal H}$ . Then

$$\begin{align*}\underline{\mathcal H}\to \mathrm{Res}_{\tilde{\mathcal O}_{0}[\tilde u]/{\mathcal O}_{0}[u]}(H_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u]) \end{align*}$$

is a closed immersion of group schemes over ${\mathcal O}_{0}[u]$ lifting (3.3.3), and

$$\begin{align*}\underline{\mathcal H}^\circ\to \mathrm{Res}_{\tilde{\mathcal O}_{ 0}[\tilde u]/{\mathcal O}_{0}[u]}(H_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u]) \end{align*}$$

is a locally closed immersion. This gives a natural morphism

(3.3.6) $$ \begin{align} {\mathrm{Fl}}^{E(u)}_{\underline{\mathcal H}^\circ}\to {\mathrm{Fl}}^{E(u)}_{\mathrm{Res}_{\tilde{\mathcal O}_{ 0}[\tilde u]/{\mathcal O}_{ 0}[u]}(H_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{ 0}[\tilde u])} \end{align} $$

between the Beilinson-Drinfeld style affine Grassmannians of [Reference LevinLe16] over ${\mathcal O}_{F}$ .

Proposition 3.3.7. The natural morphism

(3.3.8) $$ \begin{align} {{\mathrm{M}}}_{{\mathcal G},\mu}={{\mathrm{M}}}_{\mathrm{Res}_{{\mathcal O}_K/{\mathcal O}_F}{\mathcal H},\mu}\to ({{\mathrm{M}}}_{\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}),\tilde \mu})\otimes_{{\mathcal O}_{\tilde E}}{\mathcal O}_E, \end{align} $$

induced by (3.3.6), is a closed immersion.

3.3.9

We now slightly digress to give a result about minuscule representations which will be useful later.

Let $H_0$ be a split reductive group scheme over ${\mathbb Z}_p$ . Let L be a field extension of ${\mathbb Q}_p$ and let ${\rho : H_0\otimes _{{\mathbb Z}_p}L\to \mathrm{GL}(V)}$ be a representation over L. Choose a maximal torus $T_0\simeq {\mathbb G}_m^r$ and a Borel $B_0$ of $H_0$ containing $T_0$ . Let $\{\lambda _1,\ldots , \lambda _n\}$ be the (distinct) highest weights of $T_0$ that appear in the highest weight decomposition of V and denote by $V_{{\mathbb Z}_p}(\lambda _i)$ the Weyl module with highest weight $\lambda _i$ over ${\mathbb Z}_p$ . Then there is an $H_0\otimes _{{\mathbb Z}_p}L$ -equivariant isomorphism

$$\begin{align*}V\simeq \bigoplus_{i=1}^n V_{{\mathbb Z}_p}(\lambda_i)^{\oplus m_i}\otimes_{{\mathbb Z}_p} L \end{align*}$$

where $m_i\geq 1$ are corresponding multiplicities. Set

$$\begin{align*}{\Lambda}_0=\bigoplus_{i=1}^n V_{{\mathbb Z}_p}(\lambda_i)^{\oplus m_i} \end{align*}$$

which supports an $H_0$ -representation, that is, a group scheme homomorphism

$$\begin{align*}\rho_0: H_0\to \mathrm{GL}({\Lambda}_0). \end{align*}$$

If $\rho _0\otimes _{{\mathbb Z}_p}L\simeq \rho $ is faithful, by [Reference Prasad and YuPrY06, Cor. 1.3], $\rho _0$ is a closed immersion.

Proof. As above, we fix a maximal torus $T_0$ and a Borel subgroup $B_0$ of $H_0$ . Let $\{\lambda _1,\ldots , \lambda _n\}$ be the (distinct) highest weights that appear in the highest weight decomposition of V. Then, since $\rho $ is minuscule, all the weights appearing in V are of the form $w\cdot \lambda _i$ , $w\in W=N_{H_0}(T_0)/T_0$ . Write

$$\begin{align*}{\Lambda}=\bigoplus_{\lambda\in X^*(T_0)} {\Lambda}_{\lambda},\quad {\Lambda}'=\bigoplus_{\lambda\in X^*(T_0)} {\Lambda}^{\prime}_{\lambda}, \end{align*}$$

for the direct sum decompositions induced by the action of the torus $T_0$ via $\rho ({\Lambda })$ , $\rho ({\Lambda }')$ ; in these, ${\Lambda }_{\lambda }$ , ${\Lambda }^{\prime }_{\lambda }\subset V_{\lambda }$ are both lattices in the corresponding L-vector space $V_{\lambda }$ . For each w pick a representative $n_w\in N_{H_0}(T_0)$ . Then we have ${\Lambda }_{w\cdot {\lambda }}=\rho (n_w){\Lambda }_{{\lambda }}$ , ${\Lambda }^{\prime }_{w\cdot {\lambda }}=\rho (n_w){\Lambda }^{\prime }_{{\lambda }}$ .

If $g\in \mathrm{GL}(V)$ centralizes $\rho (H_0\otimes _{{\mathbb Z}_p} L)$ , then we can consider $g_{|V_{\lambda }}\in \mathrm{GL}(V_{\lambda })$ and set $g_i=g_{|V_{\lambda _i}}$ . By Schur’s lemma, $g\mapsto (g_i)_i$ gives an isomorphism of the centralizer $Z(H):=Z_{\mathrm{GL}(V)}(\rho (H_0\otimes _{{\mathbb Z}_p}F))$ with the group $\prod _{i=1}^n \mathrm{GL}(V_{\lambda _i})$ . Choose $g\in Z(H)\subset \mathrm{GL}(V)$ that corresponds to $(g_i)_i$ with $g_{i}:V_{{\lambda }_i}\xrightarrow {\sim } V_{{\lambda }_i}$ such that $g_i\cdot {\Lambda }_{{\lambda }_i}={\Lambda }^{\prime }_{{\lambda }_i}$ . Then, since ${\Lambda }_{w\cdot {\lambda }_i}=\rho (n_w){\Lambda }_{{\lambda }_i}$ , ${\Lambda }^{\prime }_{w\cdot {\lambda }_i}=\rho (n_w){\Lambda }^{\prime }_{{\lambda }_i}$ , we also have $g\cdot {\Lambda }={\Lambda }'$ .

3.3.11

Let us now combine this with the set-up of §3.3.1. We consider a faithful minuscule representation $\rho : H\to \mathrm{GL}(V)$ over K with base change

$$\begin{align*}\rho\otimes_K \tilde K: H\otimes_K\tilde K\to \mathrm{GL}(V\otimes_K\tilde K). \end{align*}$$

Recall that $H\otimes _K\tilde K\simeq H_0\otimes _{{\mathbb Z}_p} \tilde K$ is split. We assume that the composition of $\tilde \mu $ with $\rho \otimes _K \tilde K$ is minuscule. We have a group scheme homomorphism

$$\begin{align*}\underline \rho_0:= \rho_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u]: H_0\otimes_{{\mathbb Z}_p} \tilde{\mathcal O}_{0}[\tilde u]\to \mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u]) \end{align*}$$

over $\tilde {\mathcal O}_{ 0}[\tilde u]$ . By restriction of scalars, this induces

(3.3.12) $$ \begin{align} \mathrm{Res}_{\tilde{\mathcal O}_{ 0}[\tilde u]/{\mathcal O}_{ 0}[u]}(H_0\otimes_{{\mathbb Z}_p} \tilde{\mathcal O}_{0}[\tilde u])\to \mathrm{Res}_{\tilde{\mathcal O}_{0}[\tilde u]/{\mathcal O}_{0}[u]}(\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u])) \end{align} $$

over ${\mathcal O}_{0}[u]$ . Since $\rho _0$ is a closed immersion [Reference Prasad and YuPrY06, Cor. 1.3], $ \underline \rho _0$ and $ \mathrm{Res}_{\tilde {\mathcal O}_{ 0}[\tilde u]/{\mathcal O}_{ 0}[u]}( \underline \rho _0)$ are also closed immersions of group schemes.

Base changing the morphism (3.3.12) along ${\mathcal O}_{0}[u]\to {\mathcal O}_K$ , $u\mapsto \pi $ , gives

$$\begin{align*}\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_K}(\rho_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}): \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_K}(H_0\otimes_{{\mathbb Z}_p} {\mathcal O}_{\tilde K})\to \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_K}\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}) \end{align*}$$

over ${\mathcal O}_K$ .

Since (3.3.12) is a closed immersion, it follows that the corresponding morphism

(3.3.13) $$ \begin{align} {\mathrm{Fl}}^{E(u)}_{\mathrm{Res}_{\tilde{\mathcal O}_{0}[\tilde u]/{\mathcal O}_{0}[u]}(H_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u])}\to {\mathrm{Fl}}^{E(u)}_{\mathrm{Res}_{\tilde{\mathcal O}_{0}[\tilde u]/{\mathcal O}_{0}[u]}(\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u]))} \end{align} $$

of affine Grassmannians is a monomorphism and hence a closed immersion of ind-projective schemes over ${\mathcal O}_{F}$ . As above, this implies

Proposition 3.3.14. The morphism

(3.3.15) $$ \begin{align} {\mathrm{M}}_{ \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes_{{\mathbb Z}_p} {\mathcal O}_{\tilde K}),\tilde\mu}\to ({\mathrm{M}}_{\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}),\tilde\mu'})\otimes_{{\mathcal O}_{\tilde E'}}{\mathcal O}_{\tilde E} \end{align} $$

of local models obtained from (3.3.13) is a closed immersion.

In the above, $\tilde \mu '$ is the geometric cocharacter of $\mathrm{Res}_{{\tilde K}/F}\mathrm{GL}(V\otimes _K \tilde K)$ obtained by composing $\mathrm{Res}_{\tilde K/F}(\rho \otimes _K\tilde K)$ with $\tilde \mu $ .

Remark 3.3.16. Note that [Reference Haines and RicharzHR20, Cor. 3.6] applied to the finite flat morphism ${\mathrm{Spec} \, } ( \tilde {\mathcal O}_0[\tilde u])\to {\mathrm{Spec} \, } ({\mathcal O}_0[u])$ given by $u\mapsto \tilde u^e$ , gives a natural isomorphism

(3.3.17) $$ \begin{align} {\mathrm{Fl}}^{E(u)}_{\mathrm{Res}_{\tilde{\mathcal O}_{0}[\tilde u]/{\mathcal O}_{0}[u]}(\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u]))}\xrightarrow{\sim} {\mathrm{Fl}}^{\tilde E(\tilde u)}_{\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}\tilde{\mathcal O}_{0}[\tilde u])} \end{align} $$

of ind-schemes over ${\mathcal O}_F$ . Here $\tilde E(\tilde u)=E(\tilde u^e)$ is the Eisenstein polynomial of $\tilde \pi $ in $\tilde {\mathcal O}_{0}[\tilde u]$ . This reflects the identifications

$$\begin{align*}\mathrm{Res}_{K/F}(\mathrm{Res}_{\tilde K/K}\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}\tilde K))= \mathrm{Res}_{\tilde K/F}\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}\tilde K), \end{align*}$$

$$\begin{align*}\mathrm{Res}_{{\mathcal O}_K/{\mathcal O}_F}(\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_K}\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}))= \mathrm{Res}_{\tilde K/F}\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}). \end{align*}$$

Indeed, since $\tilde K/K$ is tame, $\mathrm{Res}_{\tilde K/K}\mathrm{GL}({\Lambda }_0\otimes _{{\mathbb Z}_p}\tilde K)$ splits over the tame extension $\tilde K/K$ and the two sides in this identification lead to two -a priori different- constructions as in [Reference LevinLe16]. However, the isomorphism (3.3.17) above gives an identification between the two possible definitions for the local model ${\mathrm{M}}_{\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}\mathrm{GL}({\Lambda }_0\otimes _{{\mathbb Z}_p}{\mathcal O}_{\tilde K}),\tilde \mu '}$ . A similar comment applies to the local model ${\mathrm{M}}_{ \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes _{{\mathbb Z}_p} {\mathcal O}_{\tilde K}),\tilde \mu }$ .

3.3.18

For S an R-algebra and V a module over S, we write $V^{(R)}$ for the R-module obtained by restriction of structure. Let ${\Lambda }$ be any ${\mathcal O}_{K}$ -lattice in a finite dimensional K-vector space V and $K/F$ a finite extension. (We will eventually apply this to K replaced by $\tilde K$ , to connect with the previous set-up.) Consider the natural homomorphism

(3.3.19) $$ \begin{align} \mathrm{Res}_{{\mathcal O}_{K}/{\mathcal O}_F} \mathrm{GL}({\Lambda})\to \mathrm{GL}({\Lambda}^{({\mathcal O}_F)}) \end{align} $$

of group schemes over ${\mathcal O}_F$ . We can easily see that this is a closed immersion by writing down the equations giving this morphism. Consider a geometric minuscule cocharacter $\mu $ of $ \mathrm{Res}_{K/F} \mathrm{GL}(V)$ with reflex field E.

Proposition 3.3.20. There is a closed immersion

(3.3.21) $$ \begin{align} {\mathrm{M}}_{\mathrm{Res}_{{\mathcal O}_{K}/{\mathcal O}_{F}} \mathrm{GL}({\Lambda} ), \mu}\hookrightarrow {\mathrm{M}}_{\mathrm{GL}({\Lambda}^{({\mathcal O}_F)} ), \mu} \otimes_{{\mathcal O}_F}{\mathcal O}_{E} \end{align} $$

equivariant for the homomorphism (3.3.19) above which extends the natural morphism between Grassmannians on the generic fibers.

Proof. Lift ${\Lambda }$ to a finite free ${\mathcal O}_0[u]$ -module $\underline {{\Lambda }}$ and consider the smooth affine group scheme $\underline {\mathcal {GL}}=\mathrm{GL}(\underline {{\Lambda }})$ over ${\mathcal O}_0[u]$ . This is the ${\mathcal O}_0[u]$ -group scheme associated to $\mathrm{GL}({\Lambda })$ and the extension $K/F$ as in §3.2.3. Write $\underline {\mathcal {GL}}_F$ for the group scheme of linear automorphisms of $\underline {{\Lambda }}$ considered as a ${\mathcal O}_F[v]$ -module by restriction of scalars by ${\mathcal O}_F[v]\to {\mathcal O}_0[u]$ , $v\mapsto E(u)+\pi _F$ . The group scheme $\underline {\mathcal {GL}}_F$ is split reductive over ${\mathcal O}_F[u]$ and so the local model ${\mathrm{M}}_{\mathrm{GL}({\Lambda } ), \mu }$ above is naturally a closed subscheme of ${\mathrm{Fl}}_{\underline {\mathcal {GL}}_F}^{v-\pi _F}$ . Here, ${\mathrm{Fl}}_{\underline {\mathcal {GL}}_F}^{v-\pi _F}$ is defined by applying the definition in §3.2.2 with $K=F$ . We will show that there is a map

$$\begin{align*}{\mathrm{Fl}}^{E(u)}_{\underline{\mathcal {GL}}}\to {\mathrm{Fl}}_{\underline {\mathcal {GL}}_F}^{v-\pi_F}. \end{align*}$$

Consider the ${\mathcal O}_0$ -algebra homomorphism

$$\begin{align*}r: {\mathcal O}_0[v]\to {\mathcal O}_0[u],\quad v\mapsto E(u)+\pi_F \end{align*}$$

which lifts the inclusion ${\mathcal O}_0\hookrightarrow {\mathcal O}_K$ , via $v\mapsto \pi _F$ , $u\mapsto \pi $ . Then r is finite and flat. Let $\underline {\mathcal {GL}}_{K/K_0}$ the group scheme obtained by Weil restriction of $\underline {\mathcal {GL}}$ along r; then the base change of $\underline {\mathcal {GL}}_{K/K_0}$ along ${\mathcal O}_0[v]\to {\mathcal O}_0=$ , $v\mapsto \pi _F$ , is identified with $\mathrm{Res}_{{\mathcal O}_{K}/{\mathcal O}_0}\mathrm{GL}({\Lambda })$ . Denote by $\underline {\mathcal {GL}}_{K_0}$ the group scheme of linear automorphisms of $\underline {\Lambda }$ regarded as an ${\mathcal O}_0[v]$ -module via $r: {\mathcal O}_0[v]\to {\mathcal O}_0[u]$ . We will first give a map

$$\begin{align*}{\mathrm{Fl}}^{E(u)}_{\underline{\mathcal {GL}},0}\to {\mathrm{Fl}}_{\underline {\mathcal {GL}}_{K_0}, 0}^{u-\pi_F} \end{align*}$$

over ${\mathcal O}_0$ . See §3.2.2 for the definition of these ind-schemes. (This amounts to constructing the map in the special case $F=K_0$ .)

We start by giving a morphism

$$\begin{align*}i: \underline{\mathcal {GL}}_{K/K_0}\to \underline{\mathcal {GL}}_{K_0} \end{align*}$$

over ${\mathcal O}_0[v]$ extending the morphism $\mathrm{Res}_{{\mathcal O}_{K}/{\mathcal O}_0}\mathrm{GL}({\Lambda })\to \mathrm{GL}({\Lambda }^{({\mathcal O}_0)})$ of ${\mathcal O}_0$ -group schemes under the specialization $v\mapsto \pi _F$ . This morphism is obtained by viewing an ${\mathcal O}_0[u]$ -automorphism of $\underline {{\Lambda }}$ as an ${\mathcal O}_0[v]$ -automorphism of $\underline {{\Lambda }}$ viewed as an ${\mathcal O}_0[v]$ -module via r. The base change of i to $k[\![ v]\!]$

$$\begin{align*}i_{k[\![ v]\!]}: \underline{\mathcal {GL}}_{K/K_0, k[\![ v]\!]}\to \underline{\mathcal {GL}}_{K_0, k[\![ v]\!]} \end{align*}$$

is a closed immersion since it is induced by restriction of scalars from $k[\![ u]\!]$ -lattices to $k[\![ v]\!]$ -lattices under the map $v\mapsto u^{[K:K_0]}$ .

By [Reference Haines and RicharzHR20, Cor. 3.6], the Weil restriction of torsors along r induces an isomorphism

$$\begin{align*}{\mathrm{Fl}}^{E(u)}_{\underline{\mathcal {GL}},0}\xrightarrow{\sim} {\mathrm{Fl}}^{u-\pi_F}_{ \underline{\mathcal {GL}}_{K/K_0},0}. \end{align*}$$

Combining this isomorphism with the map given by taking push-outs of torsors along i, we obtained the required map

$$\begin{align*}\iota_0: {\mathrm{Fl}}^{E(u)}_{\underline{\mathcal {GL}},0}\simeq {\mathrm{Fl}}^{v-\pi_F}_{ \underline{\mathcal {GL}}_{K/K_0},0}\to {\mathrm{Fl}}^{v-\pi_F}_{\underline{\mathcal {GL}}_{K_0},0}. \end{align*}$$

Applying $\mathrm{Res}_{{\mathcal O}_0/{\mathcal O}_F}$ we obtain a map

$$\begin{align*}\iota: {\mathrm{Fl}}^{E(u)}_{\underline{\mathcal {GL}}}\to \mathrm{Res}_{{\mathcal O}_0/{\mathcal O}_F}{\mathrm{Fl}}^{v-\pi_F}_{\underline{\mathcal {GL}}_{K_0}}. \end{align*}$$

A standard argument ([Reference Pappas and RapoportPR08, Thm. 1.4]) shows that $\iota \otimes _{{\mathcal O}_F}k$ is a locally closed immersion. Since the domain of this map is ind-projective, it follows that $\iota \otimes _{{\mathcal O}_F}k$ is a closed immersion. We now compose this with the map

$$\begin{align*}\iota': \mathrm{Res}_{{\mathcal O}_0/{\mathcal O}_F}{\mathrm{Fl}}^{v-\pi_F}_{\underline{\mathcal {GL}}_{K_0}}\to {\mathrm{Fl}}^{v-\pi_F}_{\underline{\mathcal {GL}}_F} \end{align*}$$

obtained by the construction of [Reference LevinLe16] applied to $\mathrm{Res}_{{\mathcal O}_0/{\mathcal O}_F}\mathrm{GL}({\Lambda }^{({\mathcal O}_0)})\to \mathrm{GL}({\Lambda }^{({\mathcal O}_F)})$ . We can easily see that $\iota '\otimes _{{\mathcal O}_F}k$ is a closed immersion, cf. [Reference Pappas and ZhuPZ13, Prop. 8.1]. It follows that the composite map $\iota '\cdot \iota $ is a closed immersion on special fibers.

Restricting to the local models we obtain a map

(3.3.22) $$ \begin{align} {\mathrm{M}}_{\mathrm{Res}_{{\mathcal O}_{K}/{\mathcal O}_{F}} \mathrm{GL}({\Lambda} ), \mu}\hookrightarrow {\mathrm{M}}_{\mathrm{GL}({\Lambda}^{({\mathcal O}_F)}), \mu} \otimes_{{\mathcal O}_F}{\mathcal O}_{E} \end{align} $$

which is a closed immersion on special fibers. An argument involving Nakayama’s lemma as in [Reference Pappas and ZhuPZ13, Prop. 8.1], shows that (3.3.22) is itself a closed immersion. Finally, it remains to check that (3.3.22) extends to the natural morphism on generic fibers. This follows from the definitions of local models in §3.2.5 and the fact that r takes $v-\pi _F$ to $E(u)$ .

3.3.23

We now combine the previous results to show that a suitable Hodge embedding induces a closed immersion of local models.

We will consider local model triples $(G, \{\mu \}, {\mathcal G})$ over F with G quasi-tame, $G\simeq \prod _{i=1}^r\mathrm{Res}_{K_i/F}H_i$ , with $H_i$ split over a tame extension of $K_i$ . Then the local Shimura pair $(G,\{\mu \})$ over F arises as a product of local Shimura pairs $(\mathrm{Res}_{K_i/F}H_i, \{\mu _i\})$ , $1\leq i \leq r$ . Suppose we are given faithful minuscule representations $\rho _i: H_i\to \mathrm{GL}(V_i)$ over $K_i$ , such that the compositions

$$\begin{align*}\mathrm{Res}_{K_i/F}H_i \xrightarrow{\ \mathrm{Res}_{K_i/F}(\rho_i)\ } \mathrm{Res}_{K_i/F}\mathrm{GL}(V_i)\hookrightarrow \mathrm{GL}(V_{i}^{(F)}) \end{align*}$$

give Hodge embeddings for $(\mathrm{Res}_{K_i/F}H_i, \{\mu _i\})$ over F, for each i.

We consider

(3.3.24) $$ \begin{align} \rho: G\simeq \prod_{i=1}^r\mathrm{Res}_{K_i/F}H_i\xrightarrow{\ \prod_i \mathrm{Res}_{K_i/F}(\rho_i)\ } \prod_{i=1}^r\mathrm{Res}_{K_i/F}\mathrm{GL}(V_i)\hookrightarrow \mathrm{GL}(V) \end{align} $$

where $V=\oplus _{i=1}^rV_i^{(F)}$ is considered as an F-vector space with F-structure given by restriction from the $K_i$ -structure on each summand. Then $\rho $ also gives a Hodge embedding $\rho : (G,\{\mu \})\to (\mathrm{GL}(V),\{\mu _d\})$ . In particular, $ (G,\{\mu \})$ is of Hodge type. Note that then, for any $m\geq 1$ , the direct sum representation

$$\begin{align*}\rho^{\oplus m}: G\to \mathrm{GL}(V)\times\cdots \times \mathrm{GL}(V)\hookrightarrow \mathrm{GL}(V^{\oplus m}) \end{align*}$$

also gives a Hodge embedding that factors as in (3.3.24).

Theorem 3.3.25. Let $(G, \{\mu \}, {\mathcal G})$ be a local model triple over F. Assume G quasi-tame, $G\simeq \prod _{i=1}^r\mathrm{Res}_{K_i/F}H_i$ , with $H_i$ split over a tame extension of $K_i$ . Assume that p is odd and that all the $H_i$ are of classical type. Suppose $ (G,\{\mu \})$ admits a Hodge embedding $\rho $ of the form (3.3.24) as above. After replacing the Hodge embedding $\rho $ by a direct sum $\rho ^{\oplus m}$ as above, there exists a lattice ${\Lambda }\subset V$ and a quasi-parahoric group scheme ${\mathcal G}'$ of G with $({\mathcal G}')^\circ ={\mathcal G}^\circ $ such that $\rho $ extends to a closed immersion ${\mathcal G}'\hookrightarrow \mathrm{GL}({\Lambda })$ and there is a closed immersion

$$\begin{align*}\rho_*: {\mathrm{M}}_{{\mathcal G}', \mu}={\mathrm{M}}_{{\mathcal G},\mu}\to {\mathrm{Gr}}(d,{\Lambda})_{{\mathcal O}_E} \end{align*}$$

extending the natural map on the generic fiber.

Proof. We can reduce to the case $G=\mathrm{Res}_{K/F}H$ , with H split over a tamely ramified extension of K; the general case is obtained by taking products. We may assume ${\mathcal G}={\mathcal G}_{\mathbf {x}}=\mathrm{Res}_{{\mathcal O}_K/{\mathcal O}_F}{\mathcal H}_{\mathbf {x}}$ and $\mathbf {x}\in {\mathcal B}(G, F)={\mathcal B}(H, K)$ which we can assume is generic in its facet. We have the Hodge embedding $\rho : G\to \mathrm{GL}(V^{(F)})$ given as a composition

$$\begin{align*}G=\mathrm{Res}_{K/F}H\xrightarrow{\mathrm{Res}_{K/F}\rho_1} \mathrm{Res}_{K/F}\mathrm{GL}(V)\to \mathrm{GL}(V^{(F)}), \end{align*}$$

starting from $\rho _1: H\to \mathrm{GL}(V)$ , cf. (3.3.24). For notational simplicity, in what follows, we will often drop the superscript ${(F)}, {({\mathcal O}_F)}$ , from the notation for the restriction of structure, as it should be clear from context over which ring the modules are being taken.

We first assume that $H^{\mathrm{ad}} $ does not involve division algebras of degree divisible by p. Then the assumption of Proposition 2.2.2 for H is satisfied. Hence, we can find a finite tame Galois extension $\tilde K/K$ that splits H and a point $\mathbf {x}'\in {\mathcal B}(H, K)$ with ${\mathcal H}={\mathcal H}_{\mathbf {x} }={\mathcal H}_{\mathbf {x}'}$ which is hyperspecial in ${\mathcal B}(H, \tilde K)$ . Now we can apply the construction of §3.3.1. In this, we consider the composition of the natural map

$$\begin{align*}G=\mathrm{Res}_{K/F}H\to \mathrm{Res}_{K/F}(\mathrm{Res}_{\tilde K/K}(H\otimes_K\tilde K))=\mathrm{Res}_{\tilde K/F}(H\otimes_K\tilde K) \end{align*}$$

with

$$\begin{align*}\mathrm{Res}_{\tilde K/F}(H\otimes_K\tilde K)\xrightarrow{\mathrm{Res}_{\tilde K/F}(\rho_1\otimes_K\tilde K)} \mathrm{Res}_{\tilde K/F}\mathrm{GL}(V\otimes_K{\tilde K})\xrightarrow{\ } \mathrm{GL}(V\otimes_K\tilde K), \end{align*}$$

as a representation over F which is isomorphic to a direct sum of $[\tilde K:K]$ -copies of $\rho $ . This extends to a morphism of ${\mathcal O}_F$ -group schemes

$$\begin{align*}{\mathcal G}\hookrightarrow \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F} (H_0\otimes_{{\mathbb Z}_p} {\mathcal O}_{\tilde K})\hookrightarrow \mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F} \mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K})\hookrightarrow \mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}) .\end{align*}$$

Here we fix an isomorphism $\tilde {\mathcal H}\cong H_0\otimes _{{\mathbb Z}_p} {\mathcal O}_{\tilde K}$ , and identify $\rho _1\otimes _K\tilde K$ with the base to $\tilde K$ of a represention $H_0\hookrightarrow \mathrm{GL}(\Lambda _0)$ over ${\mathbb Z}_p$ . This morphism is a closed immersion by Proposition 2.1.5 and [Reference Prasad and YuPrY06, Cor. 1.3].

Correspondingly, by composing (3.3.8), (3.3.15) and the morphism (3.3.21) of Proposition 3.3.20 applied to $\tilde K/F$ and the lattice ${\Lambda }_0\otimes _{{\mathbb Z}_p}{\mathcal O}_{\tilde K}$ , we obtain equivariant maps

(3.3.27) $$ \begin{align} {{\mathrm{M}}}_{{\mathcal G},\mu}\to {\mathrm{M}}_{\mathrm{Res}_{{\mathcal O}_{\tilde K}/{\mathcal O}_F}(H_0\otimes_{{\mathbb Z}_p} {\mathcal O}_{\tilde K}), \tilde\mu}\to ({\mathrm{M}}_{\mathrm{GL}({\Lambda}_0\otimes_{{\mathbb Z}_p}{\mathcal O}_{\tilde K}), \tilde\mu'})={\mathrm{Gr}}(d,\Lambda). \end{align} $$

with $\Lambda =\Lambda _0\otimes _{{\mathbb Z}_p}{\mathcal O}_{\tilde K}$ as ${\mathcal O}_F$ -modules. These extend the natural morphisms on the generic fibers and are all closed immersions. The result follows in this case.

We now deal with the general case (i.e., when $H^{{\mathrm{ad}} }$ could involve division algebras of index divisible by p). We may assume $K/F$ is totally ramified; the general case is easily reduced to this. Let $F^\natural /F$ be a finite unramified extension with ring of integers ${\mathcal O}_{F^\natural }$ such that H is quasi-split after base changing to $K^\natural =KF^\natural $ . We let ${\mathcal G}^\natural _{\mathbf {x}}$ denote the stabilizer scheme of $G^\natural :=G\otimes _{F}F^\natural $ for the image of $\mathbf {x}$ in ${\mathcal B}(G,F^\natural )$ . Then we have an identification ${\mathcal G}^{\natural }_{\mathbf {x}}\cong {\mathcal G}_{\mathbf {x}}\otimes _{{\mathcal O}_F}{\mathcal O}_{F^\natural }$ . By construction, we also have an isomorphism

$$ \begin{align*}{\mathrm{M}}_{{\mathcal G},\mu}\otimes_{{\mathcal O}_F}{\mathcal O}_{F^\natural}\cong{\mathrm{M}}_{{\mathcal G}^\natural,\mu^\natural} \end{align*} $$

where $ {\mathrm{M}}_{{\mathcal G}^\natural ,\mu ^\natural }$ is the local model associated to the local model triple

$$\begin{align*}(G^\natural, \{\mu^\natural\}, {\mathcal G}^\natural):=(G\otimes_FF^\natural,\{\mu\otimes_{F}{F^\natural}\}, {\mathcal G}\otimes_{{\mathcal O}_F}{\mathcal O}_{F^\natural}), \end{align*}$$

over $F^\natural $ .

Let $\Omega \subset {\mathcal B}(G^\natural ,F^\natural )$ be the facet containing $\mathbf x$ and $\mathbf {y}\in \Omega $ a point which is generic in $\Omega $ . Then $\Omega $ is stable under $\Gamma ^\natural ={\mathrm{Gal}}(F^\natural /F)$ and ${\mathcal G}^\natural _{\mathbf {y}}$ has the same neutral component as ${\mathcal G}^\natural $ . Since $G^\natural $ is quasi-split, its adjoint group does not involve division algebras of degree divisible by p, and so the above argument applied to the base changed embedding $\rho ^\natural $ gives (upon replacing $\rho ^\natural $ by a direct sum) closed immersions

$$ \begin{align*}{\mathcal G}^{\natural}_{\mathbf{y}}\hookrightarrow \mathrm{GL}(\Lambda^\natural),\ \ \ \ {\mathrm{M}}_{{\mathcal G}^{\natural}_{\mathbf{y}},\mu^\natural}={\mathrm{M}}_{{\mathcal G} , \mu }\otimes_{{\mathcal O}_F}{\mathcal O}_{F^\natural}\hookrightarrow {\mathrm{Gr}}( d, {\Lambda}^\natural)_{{\mathcal O}^\natural_E} \end{align*} $$

for $\Lambda ^{\natural }\subset V\otimes _{{\mathcal O}_F}{\mathcal O}^\natural $ an ${\mathcal O}^\natural $ -lattice. By étale descent, the natural morphisms

$$\begin{align*}{\mathrm{M}}_{{\mathcal G} , \mu } \to \mathrm{Res}_{{\mathcal O}_{F^\natural}/{\mathcal O}_F}({\mathrm{M}}_{{\mathcal G} , \mu }\otimes_{{\mathcal O}_F}{\mathcal O}_{F^\natural}), \quad \mathrm{Res}_{{\mathcal O}_{F^\natural}/{\mathcal O}_F}{\mathrm{Gr}}( d, {\Lambda}^\natural)_{{\mathcal O}^\natural_E}\to {\mathrm{Gr}}(d f, \Lambda)_{{\mathcal O}^\natural_E}, \end{align*}$$

are closed immersions where $f=[F^\natural :F]$ and $\Lambda $ is $\Lambda ^\natural $ considered as an ${\mathcal O}_F$ -module. We thus obtain a closed immersion ${\mathrm{M}}_{{\mathcal G} , \mu }\hookrightarrow {\mathrm{Gr}}(d f, \Lambda ^\natural )_{{\mathcal O}^\natural _E}$ .

Now note that ${\mathcal G}^\natural _{\mathbf {y}}$ is equal to the stabilizer $\widehat {{\mathcal G}}_\Omega $ of $\Omega $ , hence ${\mathcal G}^\natural _{\mathbf {y}}$ is $\Gamma ^\natural $ -invariant, hence arises as the base change to ${\mathcal O}_{F^\natural }$ of a quasi parahoric ${\mathcal G}'$ of G with ${\mathcal G}^{\prime \circ }={\mathcal G}^\circ $ . Thus the composition

$$ \begin{align*}{\mathcal G}'\rightarrow \mathrm{Res}_{{\mathcal O}_{F^\natural}/{\mathcal O}_F}{\mathcal G}_{{\mathcal O}_{F^\natural}}'=\mathrm{Res}_{{\mathcal O}_{F^\natural}/{\mathcal O}_F}{\mathcal G}^\natural_{\mathbf{y}}\rightarrow \mathrm{Res}_{{\mathcal O}_{F^\natural}/{\mathcal O}_F}\mathrm{GL}(\Lambda^\natural)\rightarrow \mathrm{GL}(\Lambda) \end{align*} $$

is a closed immersion as desired.