2.1 Setup

Let F be a non-archimedean local field, fix an algebraic closure $\overline {F}$ , denote by $\breve F$ the completion of its maximal unramified extension $F^{\mathrm{un}}\subset \overline {F}$ , and denote by $\sigma $ the Frobenius automorphism of $\breve F$ over F. We usually think of F being of mixed characteristic, i.e., F is a finite extension of $\mathbb Q_p$ . Everything has an equal-characteristic counterpart, however, where F is of the form $\mathbb F_q((t))$ , the Laurent series field over a finite field $\mathbb F_q$ . In either case, we denote by p the residue characteristic of F.

We fix a connected reductive group ${\mathbf G}$ over F. Write $\breve {G} = {\mathbf G}(\breve F)$ . Let $\breve {\mathcal I} \subseteq \breve {G}$ be a $\sigma $ -invariant Iwahori subgroup, our standard Iwahori subgroup, and let T be a maximal torus of G such that the alcove $\mathfrak a$ corresponding to $\breve {\mathcal I}$ , the standard alcove in the Bruhat–Tits building of ${\mathbf G}$ over $\breve F$ , lies in the apartment attached to $T_{\breve F}$ . Attached to these data, we have the extended affine Weyl group $\tilde W$ and the (relative) finite Weyl group $W_0$ . We fix a special vertex of the base alcove and obtain a splitting $\tilde W = X_* \rtimes W_0$ , where $X_* := X_*(T)_{\Gamma _0}$ denotes the coinvariants of the cocharacter lattice of T with respect to $\Gamma _0 = \mathop {\mathrm{Gal}}(\overline {F}/F^{\mathrm{un}})$ . See [Reference Tits, Borel and Casselman55] and [Reference Görtz, He and Rapoport15, Section 2].

We denote by $\tilde {\mathbb {S}} $ the set of simple affine reflections (defined by our base alcove) inside the affine Weyl group $W_a\subseteq \tilde W$ and denote by $\Omega $ the set of length-zero elements in $\tilde W$ . The Frobenius $\sigma $ acts on $\tilde {\mathbb {S}} $ (since by assumption the base alcove is fixed by $\sigma $ ). Likewise, if $\tau \in \tilde W$ has length $0$ , then it fixes the base alcove and thus acts by conjugation on $\tilde {\mathbb {S}} $ ; we denote this action by ${\mathrm{Ad}}(\tau )$ . For an element $w = w' \tau \in W_a\tau $ , the $\sigma $ -support $\operatorname {\mathrm{supp}}_{\sigma }(w)$ is the smallest subset of $\tilde {\mathbb {S}} $ , which is ${\mathrm{Ad}}(\tau )\circ \sigma $ -stable and contains all simple affine reflections that occur in a reduced expression for $w'$ . The final condition can also be rephrased as $w'\in W_{\operatorname {\mathrm{supp}}_{\sigma }(w)}$ , where for a subset $K\subseteq \tilde {\mathbb {S}} $ we write $W_K$ for the subgroup of $W_a$ generated by the elements of K. See Example 2.10 for explicit examples. We denote by ${}^K \tilde W$ the set of minimal length representatives of the cosets in $W_K\backslash \tilde W$ .

For $b\in \breve {G}$ , we denote by ${\mathbf J}_b$ the $\sigma $ -centralizer of b, i.e.,

$$\begin{align*}{\mathbf J}_b(F) = \{ g\in \breve{G};\ g^{-1}b\sigma(g) = b\}. \end{align*}$$

If b is understood, we just write ${\mathbf J}$ instead of ${\mathbf J}_b$ .

Below, we always work with the unique reduced root system $\Phi $ underlying the relative root system of ${\mathbf G}$ over $\breve F$ (the échelonnage root system).

2.2 (Enhanced) Tits data and Coxeter data

To specify the classification results below, two types of data typically arise. On the one hand, we will refer to affine Weyl groups together with an automorphism and a coweight; these kinds of data we will call Coxeter data. On the other hand, we will refer to algebraic groups over F together with a conjugacy class of cocharacters; this is what we will call Tits data below (using Tits’s translation between isomorphism classes of such data with affine Dynkin diagrams). In both cases, we often enhance these data by including a “level structure,” i.e., a subset $K\subset \tilde {\mathbb {S}} $ with $W_K$ finite, of the set of simple affine reflections. In the group case, this gives rise to a standard parahoric subgroup. Our level structure will always be assumed to be rational, i.e., K is fixed by the automorphism $\sigma $ (the automorphism induced by the Frobenius over F in the group case).

We hope that no confusion will arise between the notions of Coxeter datum and of being of Coxeter type, the latter being a property that certain Coxeter data have, and others do not—similarly as some elements of a Coxeter group are Coxeter elements.

It is worth pointing out that the Coxeter diagram associated with an affine Dynkin diagram is obtained by disregarding the arrows in the affine Dynkin diagram. We refer to [Reference He, Pappas and Rapoport26, Section 5.2] for the discussion on the relationship and difference between the (enhanced) Coxeter data and the (enhanced) Tits data.

2.3 Fully Hodge–Newton decomposable pairs $({\mathbf G}, \mu )$

We now fix a conjugacy class $\mu $ of cocharacters ${\mathbf G}_{m, \overline {F}}\rightarrow {\mathbf G}_{\overline {F}}$ over the algebraic closure $\overline {F}$ of F. We fix the representative $\mu _+\in X_*(T)$ of this conjugacy class whose image $\underline {\mu }$ in the coweight lattice $X_* = X_*(T)_{\Gamma _0}$ , i.e., the translation lattice of the Iwahori–Weyl group, is dominant (i.e., translates the base alcove into the dominant chamber). We use here the Bruhat–Tits convention that for $\lambda \in X_*(T)$ , the element $\lambda (\pi )$ for a uniformizer $\pi $ acts by translation by $-\lambda $ . Cf. [Reference Görtz, He and Rapoport15, Section 2.2].

We also fix a length $0$ element $\tau \in \tilde W$ whose $\sigma $ -conjugacy class is the unique basic element in $B({\mathbf G}, \mu )$ . By abusing the notation, we also use $\tau $ for a chosen representative in $\breve G$ .

Denote by $X_w(b)$ the affine Deligne–Lusztig variety for $w\in \tilde W$ and $b\in \breve {G}$ , a subvariety of the affine flag variety for ${\mathbf G}$ , $X_w(b) = \{g\breve {\mathcal I}\in \breve {G}/\breve {\mathcal I};\ g^{-1}b\sigma (g)\in \breve {\mathcal I} w\breve {\mathcal I}\}$ . Here, we view $\breve {G}/\breve {\mathcal I}$ as the $\mathbf k$ -valued points of the affine flag variety, where $\mathbf k$ is the residue class field of the ring of integers of $\breve F$ . Depending on whether F has characteristic $> 0$ or characteristic $0$ , the affine flag variety is an ind-scheme, or an ind-(perfect scheme), over $\mathbf k$ (see [Reference Pappas and Rapoport40] and [Reference Bhatt and Scholze3, Reference Zhu64], respectively). All affine Deligne–Lusztig varieties are equipped with the reduced scheme structure, so that we can identify them with their $\mathbf k$ -valued points.

Let $\pi = \pi _K\colon \breve {G}/\breve {\mathcal I} \rightarrow \breve {G}/\breve {\mathcal K}$ denote the projection from the affine flag variety to the partial affine flag variety of level K ( $\breve {\mathcal K}$ denotes the standard parahoric subgroup of type K). Recall that

$$\begin{align*}\operatorname{\mathrm{Adm}}(\mu) = \{ w\in\tilde W;\ w \leqslant t^{x(\underline{\mu})}\ \text{ for some } x\in W_0 \} \end{align*}$$

denotes the $\mu $ -admissible set. By definition,

$$\begin{align*}X(\mu, \tau)_K = \{ g\breve{\mathcal K} \in \breve{G}/\breve{\mathcal K};\ g^{-1}\tau\sigma(g) \in \breve{\mathcal K}\operatorname{\mathrm{Adm}}(\mu)\breve{\mathcal K}\}. \end{align*}$$

We write ${}^{K}\!{\mathrm{Adm}}(\mu ) = \operatorname {\mathrm{Adm}}(\mu )\cap {}^K\tilde W$ for the subset of $\operatorname {\mathrm{Adm}}(\mu )$ consisting of all elements w which are of minimal length in their right $W_K$ -coset $W_K w$ . Below, we sometimes write $X_{K, w}(\tau ) = \pi (X_w(\tau ))$ .

It is shown in [Reference Görtz and He13, Reference He24] that

(2.1) $$ \begin{align} X(\mu, \tau)_K = \bigsqcup_{w\in {}^{K}\!{\mathrm{Adm}}(\mu)} \pi(X_w(\tau)). \end{align} $$

The subsets $\pi (X_w(\tau ))$ are called the EKOR strata of $X(\mu , \tau )_K$ .

Note that if $W_{\operatorname {\mathrm{supp}}_{\sigma }(w)}$ is finite, then $X_w(\tau ) \ne \emptyset $ (see [Reference He23, Lemma 3.2]). Let us recall the following characterization of fully Hodge–Newton decomposable pairs $({\mathbf G}, \mu )$ (see [Reference Görtz, He and Nie14, Definition 3.1]).

Note that this property, as shown by condition (1), is independent of K, and depends only on the Coxeter datum $(W_a, \sigma , \underline {\mu })$ . Set

$$ \begin{align*}{}^{K}\!{\mathrm{Adm}}(\mu)_0=\{w \in {}^{K}\!{\mathrm{Adm}}(\mu);\ W_{\operatorname{\mathrm{supp}}_{\sigma}(w)} \text{ is finite}\}.\end{align*} $$

Then, in the fully Hodge–Newton decomposable cases, we may rewrite (2.1) as

$$\begin{align*}X(\mu, \tau)_K = \bigsqcup_{w\in {}^{K}\!{\mathrm{Adm}}(\mu)_0} \pi(X_w(\tau)). \end{align*}$$

Note that we deviate here from the usual definition of (twisted) Coxeter elements where one asks that exactly one simple reflection from each orbit occurs. It would be more precise to say that w is a $\sigma $ -Coxeter element in $W_{\operatorname {\mathrm{supp}}_{\sigma }(w)} \tau $ , but for our purposes, it is useful to shorten this as in the above definition. See Example 2.10 for examples.

We denote by ${}^K\!{\mathrm{Cox}}(\mu ) \subset {}^{K}\!{\mathrm{Adm}}(\mu )_0$ the subset of ${}^{K}\!{\mathrm{Adm}}(\mu )_0$ consisting of $\sigma $ -Coxeter elements.Footnote ¹ If $K=\emptyset $ , then we may simply omit the superscript and write $\operatorname {\mathrm{Adm}}(\mu )_0$ and $\mathop {\mathrm{Cox}}(\mu )$ . We define the following notion.

Observe that in the definition, $(W_a, \sigma , \mu )$ being fully Hodge–Newton decomposable implies in particular that $W_{\operatorname {\mathrm{supp}}_{\sigma }(w)}$ is finite for all $w\in {}^{K}\!{\mathrm{Adm}}(\mu )_0$ . Therefore, the above is the same definition as the one given in the introduction. Furthermore, this definition is equivalent to the one in [Reference Görtz and He13, Section 5.1]. In fact, the definition in loc. cit. is easily seen to be equivalent to saying that

$$\begin{align*}X(\mu, \tau)_K = \bigsqcup_{w\in \, {}^K\!{\mathrm{Cox}}(\mu)} \pi(X_w(\tau)). \end{align*}$$

The equivalence of the two definitions then follows from (2) in Theorem 2.2.

Remark 2.5 Let $K \subset K'$ be proper $\sigma $ -stable subsets of $\tilde {\mathbb {S}} $ . If $(W_a, \sigma , \mu , K)$ is of Coxeter type, then ${}^K\!{\mathrm{Cox}}(\mu )={}^{K}\!{\mathrm{Adm}}(\mu )_0$ and

$$\begin{align*}{}^{K'}\mathop{\mathrm{Cox}}(\mu)={}^K\!{\mathrm{Cox}}(\mu) \cap {}^{K'} \tilde W={}^{K}\!{\mathrm{Adm}}(\mu)_0 \cap {}^{K'} \tilde W={}^{K'}\!\operatorname{\mathrm{Adm}}(\mu)_0. \end{align*}$$

Hence, $(W_a, \sigma , \mu , K')$ is of Coxeter type.

2.4 The Bruhat–Tits stratification

Let $({\mathbf G}, \mu )$ be fully Hodge–Newton decomposable. As pointed out above, we have

$$\begin{align*}X(\mu, \tau)_K = \bigsqcup_{w\in {}^{K}\!{\mathrm{Adm}}(\mu)_0} \pi(X_w(\tau)), \end{align*}$$

where as before $\pi \colon \breve {G}/\breve {\mathcal I} \rightarrow \breve {G}/\breve {\mathcal K}$ denotes the projection from the full affine flag variety to the partial affine flag variety of type K.

For each $w\in {}^{K}\!{\mathrm{Adm}}(\mu )_0$ , we define the following standard parahoric subgroups of $\breve G$ . Let $\mathcal P^{\flat }_w$ be the standard parahoric subgroup generated by $\operatorname {\mathrm{supp}}_{\sigma }(w)$ and the standard Iwahori subgroup $\breve {\mathcal I}$ . Let $\mathcal P_w$ be the standard parahoric generated by $\operatorname {\mathrm{supp}}_{\sigma }(w)$ and $I(K, w, \sigma )$ and $\breve {\mathcal I}$ . Here, $I(K, w, \sigma )$ is the maximal ${\mathrm{Ad}}(w)\circ \sigma $ -stable subset of K. Then [Reference Görtz, He and Nie14, Proposition 5.7] shows that $W_{\operatorname {\mathrm{supp}}_{\sigma }(w) \cup I(K, w, \sigma )}$ is finite. Finally, let $\breve {\mathcal K}\subset \breve {G}$ be the standard parahoric subgroup of type K, and let $\mathcal Q_w := \mathcal P_w^{\flat } \cap \breve {\mathcal K}$ be the intersection.

We then have

$$\begin{align*}\pi(X_w(\tau)) = \bigsqcup_{j\in {\mathbf J}(F)/({\mathbf J}(F)\cap \mathcal P_w)} jY(w), \end{align*}$$

where

$$\begin{align*}Y(w) = \{ g\in \mathcal P_w^{\flat} / \mathcal Q_w;\ g^{-1} \tau\sigma(g)\in \mathcal Q_w \cdot_{\sigma} (\breve{\mathcal I} w \breve{\mathcal I}) \}. \end{align*}$$

Here, $\cdot _{\sigma }$ means that the group on the left acts by $\sigma $ -conjugation, i.e., we have $\mathcal Q_w \cdot _{\sigma } (\breve {\mathcal I} w \breve {\mathcal I})=\{g^{-1} h \sigma (g);\ g\in \mathcal Q_w, h \in \breve {\mathcal I} w \breve {\mathcal I}\}$ . We can identify $\mathcal P_w^{\flat } / \mathcal Q_w$ with a classical partial flag variety for the maximal reductive quotient of the special fiber of the parahoric group scheme attached to $\mathcal P_w^{\flat }$ . The variety $Y(w)$ is a “fine Deligne–Lusztig variety” in that partial flag variety, i.e., the image of a classical Deligne–Lusztig variety in the corresponding full flag variety. See Example 2.10 for examples.

More precisely, in the mixed characteristic case, $Y(w)$ is the perfection of a classical Deligne–Lusztig variety; even though we sometimes drop the adjective perfect for notational convenience, we do not have an actual scheme structure on $X(\mu , \tau )_K$ and therefore cannot talk about the strata as usual schemes, if F has mixed characteristic.

If w is a twisted Coxeter element, then we have the following simple description of the set $I(K, w, \sigma )$ , which appears in the definition of $\mathcal P_w$ .

In terms of the Dynkin diagram, we can express this as saying that $I(K, w, \sigma )$ is the union of those ${\mathrm{Ad}}(\tau )\circ \sigma $ -orbits in K in which no element is connected to any vertex in $\operatorname {\mathrm{supp}}_{\sigma }(w)$ .

In particular, this implies that the projection $\pi $ restricts to an isomorphism from $\{ g\breve {\mathcal I};\ g\in {\mathcal P}^{\flat }_w, g^{-1}\tau \sigma (g)\in \breve {\mathcal I} w\breve {\mathcal I}\}$ onto its image $Y(w)$ . So $Y(w)$ is isomorphic to the classical Deligne–Lusztig variety attached to $w\tau ^{-1}$ in the (finite-dimensional) flag variety $\mathcal P^{\flat }_w/\breve {\mathcal I}$ , for the Frobenius given by ${\mathrm{Ad}}(\tau )\circ \sigma $ .

See also [Reference Görtz and He13, Corollary 4.6.2, Section 7.2] (cf. also [Reference Görtz, He and Nie14, Section 5.10] for further details).

Putting together these stratifications for all the different w, we obtain the decomposition of $X(\mu , \tau )_K$ as a union of classical Deligne–Lusztig varieties “in a natural way.” We call this stratification the weak Bruhat–Tits stratification.

The closure of the stratum $Y(w)$ (and likewise of $jY(w)$ ) in $X(\mu , \tau )_K$ is isomorphic to (the perfection of) the closure of $Y(w)$ inside $\mathcal P_w^{\flat } / (\mathcal P_w^{\flat } \cap \breve {\mathcal K})$ , because $\mathcal P_w^{\flat } / (\mathcal P_w^{\flat } \cap \breve {\mathcal K})$ is proper and hence closed in $\breve {G}/\breve {\mathcal K}$ . Even if w is a twisted Coxeter element, then this closure is typically not isomorphic to the closure of $Y(w)$ inside $\mathcal P_w^{\flat } /\breve {\mathcal I}$ (see Example 2.7).

The closure relations between strata are given as follows (cf. [Reference Görtz and He13, Sections 3.3 and 7]). The closure of a stratum is a union of strata. The closure of a stratum $j Y(w)$ contains a stratum $j' Y(w')$ if and only if:

1. (1) $w' \leqslant _{K, \sigma }w$ , which means by definition that there exists $u\in W_K$ such that $u^{-1} w' \sigma (u)\leqslant w$ , and

2. (2) $j'({\mathbf J}(F)\cap \mathcal P_{w'}) \cap j({\mathbf J}(F)\cap \mathcal P_w) \ne \emptyset $ .

We can express the second condition in an equivalent way in terms of the building, as follows (see [Reference Görtz and He13, Erratum, Proposition 7.2.2]). Let $\breve {\kappa }$ be the Kottwitz homomorphism (see Section 5.1). We identify the set

$$\begin{align*}\{ h\in {\mathbf J}(F)/{\mathbf J}(F)\cap \mathcal P_w;\ \breve{\kappa}(h) = \breve{\kappa}(j) \} \end{align*}$$

with the set of simplices of type w, i.e., with stabilizer $\mathcal P_w\cap {\mathbf J}(F)$ , in the rational building of ${\mathbf J}$ , and similarly for $w'$ . Then (2) above is equivalent to requiring that $\breve {\kappa }(j') = \breve {\kappa }(j)$ and that the simplices attached to j and $j'$ via the above identification are contained in the closure of some alcove.

If moreover $({\mathbf G}, \mu , K)$ is of Coxeter type, then this stratification has further nice properties.

Note that by definition, we automatically have $(1) \Rightarrow (2) \Rightarrow (4)$ and $(1) \Rightarrow (3) \Rightarrow (4)$ . The nontrivial part is $(4) \Rightarrow (1)$ , which follows by analyzing all Coxeter-type cases in Section 4 and the explicit description of ${}^K\!{\mathrm{Cox}}(\mu )$ . See Section 6.1 for some detailed discussion for the Drinfeld case.

We have the following consequence.

By Corollary 2.9, if $({\mathbf G}, \mu , K)$ is of Coxeter type, then the index set

$$ \begin{align*}\bigsqcup_{w \in {}^K\!{\mathrm{Cox}}(\mu)} {\mathbf J}(F)/({\mathbf J}(F)\cap \mathcal P_w)\end{align*} $$

of the weak Bruhat–Tits (BT) stratification can be seen as a subset of the set of all simplices in the Bruhat–Tits building of ${\mathbf J}$ over F (up to fixing the connected component, i.e., the image under $\breve \kappa $ ). In view of these particularly favorable properties, we call the resulting stratification the Bruhat–Tits stratification of $X(\mu , \tau )_K$ . See Example 2.10 for a discussion in a specific case.

Example 2.10 Let us illustrate some of the notions we introduced in the specific case of Tits datum $({}^2\tilde {A}^{\prime }_{n-1}, \omega _1^{\vee })$ , i.e., for ${\mathbf G}$ (say over $F=\mathbb Q_p$ to be specific) a unitary group which splits over an unramified quadratic extension and $\underline {\mu } = \omega _1^{\vee }$ . This case arises from Shimura varieties for unitary groups $GU(1, n-1)$ at primes p where the group splits over an unramified quadratic extension of $\mathbb Q_p$ (but does not split over $\mathbb Q_p$ ), and is the case that was studied by Vollaard and Wedhorn in [Reference Vollaard and Wedhorn56] for hyperspecial level structure.

The corresponding Coxeter datum is $(\tilde {A}_{n-1}, \varsigma _0, \omega _1^{\vee })$ , i.e., the affine Dynkin diagram is a circle with vertices $0$ , $1$ , …, $n-1$ , where the automorphism $\sigma = \varsigma _0$ is the “reflection” fixing $0$ and exchanging $1$ and $n-1$ , $2$ and $n-2$ , and so on. The action by ${\mathrm{Ad}}(\tau )$ is the rotation $0\mapsto 1\mapsto 2\mapsto \cdots $ . Hence, ${\mathrm{Ad}}(\tau )\circ \sigma $ has order $2$ and maps

$$\begin{align*}0 \leftrightarrow 1,\quad n-1\leftrightarrow 2,\quad n-2\leftrightarrow 3,\quad \dots .\end{align*}$$

Depending on whether n is even or odd, ${\mathrm{Ad}}(\tau )\circ \sigma $ has no fix point, or the one fix point $\frac {n+1}{2}$ .

An element $w\in W_a\tau $ is a twisted Coxeter element in the sense of Definition 2.3 if and only if at most one simple reflection from each pair $\{0, 1\}$ , $\{n-1, 2\}$ , …, occurs in a reduced expression of $w\tau ^{-1}$ .

Now, let us come back to the case of general n and consider the case of hyperspecial level structure, more specifically we set $K=\tilde {\mathbb {S}} - \{0\}$ . (If n is odd, then $\tilde {\mathbb {S}} - \{ (n+1)/2 \}$ is another choice of K that corresponds to hyperspecial level structure.) We then have

$$\begin{align*}{}^{K}\!{\mathrm{Adm}}(\mu)_0 = \left\{\tau,\ s_{0} \tau,\ s_0 s_{n-1}\tau,\ \dots,\ s_0 s_{n-1}\cdots s_{\lfloor\frac{n+4}{2}\rfloor}\tau \right\}. \end{align*}$$

For $w = s_0 s_{n-1}\cdots s_j\tau \in {}^{K}\!{\mathrm{Adm}}(\mu )_0$ , we have

$$\begin{align*}\operatorname{\mathrm{supp}}_{\sigma}(w) = \{ j,\ j+1,\ \dots,\ n-1,\ 0,\ 1,\ \dots,\ n-j+1 \} \end{align*}$$

(where we write i instead of $s_i$ ). Together with the restriction of the automorphism ${\mathrm{Ad}}(\tau )\circ \sigma $ , this is the Dynkin diagram of a unitary group over the finite field $\mathbb F_p$ . Denote by $\overline {G}$ its base change to an algebraic closure of $\mathbb F_p$ . Let $\overline {B}\subset \overline {Q}\subset \overline {G}$ denote the Borel group of $\overline {G}$ induced by the Borel of ${\mathbf G}$ and the maximal parabolic subgroup corresponding to the vertex $0$ (the complement of K in $\tilde {\mathbb {S}} $ ), viewed as a vertex of the Dynkin diagram of $\overline {G}$ . The Deligne–Lusztig variety $Y(w)$ is isomorphic to a Deligne–Lusztig variety in the “flag variety” $\overline {G}/\overline {B}$ , and also isomorphic to its image (a “fine” Deligne–Lusztig variety) in the “Grassmannian” $\overline {G}/\overline {Q}$ . The dimension of $Y(w)$ equals the length $\ell (w)$ of w, in particular, in this case all strata of a fixed dimension are translates of the same Deligne–Lusztig variety.

Using the lattice interpretation of the Bruhat–Tits building of ${\mathbf J}$ , we can describe the index set of the Bruhat–Tits stratification in terms of vertex lattices as in [Reference Vollaard and Wedhorn56]. We then recover precisely the result of Theorem B of loc. cit., after perfection.

If n is odd, then we can also consider the parahoric level structure given by $K= \tilde {\mathbb {S}} - \{ 0, \frac {n+1}{2} \}$ . The corresponding parahoric subgroup is not maximal, and a fortiori not hyperspecial. Since K is smaller than in the previous case, the set ${}^{K}\!{\mathrm{Adm}}(\mu )_0$ is larger than before (see Table 1). In particular, there exist different elements in ${}^{K}\!{\mathrm{Adm}}(\mu )_0$ of the same length. See Sections 6.1 and 6.1.

3.1 Deligne–Lusztig reduction

We first recall the Deligne–Lusztig reduction method.

Let $x, x' \in \tilde W$ and $s \in \tilde {\mathbb {S}} $ . We write $x {\xrightarrow {s}}_{\sigma } x'$ if $x' = s x \sigma (s)$ and $\ell (x') \leqslant \ell (x)$ . We write $x \rightarrow _{\sigma } x'$ if there exists a sequence $x_0, x_1, \dots , x_r$ in $\tilde W$ and a sequence $s_1, s_2, \dots , s_r$ in $\tilde {\mathbb {S}} $ such that $x = x_0 {\xrightarrow {s_1}}_{\sigma } x_1 {\xrightarrow {s_2}}_{\sigma } \cdots {\xrightarrow {s_r}}_{\sigma } x_r = x'$ . We write $x \approx _{\sigma } x'$ if $x \rightarrow _{\sigma } x'$ and $x' \rightarrow _{\sigma } x$ .

The following theorem, which is referred to as the reduction à la Deligne and Lusztig, is proved in [Reference Deligne and Lusztig8, Proof of Theorem 1.6] (parts (i) and (ii)) and [Reference He23, Theorem 4.8] (see also [Reference Görtz and He12, Corollary 2.5.3]).

3.2 Proof of Proposition 3.1

We argue by induction on the length of w. If w is of minimal length in its $\sigma $ -conjugacy class, by Theorem 3.3, $\dim X_w(\tau ) = \ell (w)$ and the statement follows. Otherwise, by Theorem 3.2, there exist $u \in \tilde W$ and $s \in \tilde {\mathbb {S}} $ such that $w \approx _{\sigma } u$ and $s u \sigma (s) < u$ . Thus, $\operatorname {\mathrm{supp}}_{\sigma }(u) = \operatorname {\mathrm{supp}}_{\sigma }(w)$ and

$$ \begin{align*}\operatorname{\mathrm{supp}}_{\sigma}(u) - \{({\mathrm{Ad}}(\omega) \circ \sigma)^i(s); i \in \mathbb {Z}\} \subseteq \operatorname{\mathrm{supp}}_{\sigma}(s u \sigma(s)).\end{align*} $$

Moreover, $\dim X_w(\tau ) = \dim X_u(\tau )$ and either $X_{s u \sigma (s)}(\tau ) \neq \emptyset $ or $X_{s u}(\tau ) \neq \emptyset $ . Let us assume that the former case occurs; the proof in the other case is basically the same. By induction hypothesis,

$$ \begin{align*} \dim X_w (\tau) &= \dim X_u (\tau) \geqslant 1 + \dim X_{s u \sigma(s)}(\tau ) \\ &\geqslant 1 + \sharp \{({\mathrm{Ad}}(\omega) \circ \sigma)\text{-orbits on } \operatorname{\mathrm{supp}}_{\sigma}(s u \sigma(s))\} \\ &\geqslant 1 + \sharp \{({\mathrm{Ad}}(\omega) \circ \sigma)\text{-orbits on } \operatorname{\mathrm{supp}}_{\sigma}(u)\} - 1 \\ &= \sharp \{({\mathrm{Ad}}(\omega) \circ \sigma)\text{-orbits on } \operatorname{\mathrm{supp}}_{\sigma}(u)\} \\ &= \sharp \{({\mathrm{Ad}}(\omega) \circ \sigma)\text{-orbits on } \operatorname{\mathrm{supp}}_{\sigma}(w)\}. \Box \end{align*} $$

3.3 A general dimension bound

For any reductive group $\mathbf H$ over F, we denote by ${\mathrm{rank}_F^{\mathrm{ss}}}(\mathbf H)$ the semisimple F-rank of $\mathbf H$ . By [Reference Kottwitz34, Section 1.9], if our group ${\mathbf G}$ is quasi-simple over F, then ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau }) = \sharp \{({\mathrm{Ad}}(\tau ) \circ \sigma )\text {-orbits on } \tilde {\mathbb {S}} \} - 1$ .

Now, we prove the main result of this section.

Theorem 3.5 Suppose that $\mu $ is noncentral in every simple factor of the adjoint group ${\mathbf G}_{\text {ad}}$ over F. Then

$$ \begin{align*}\dim X(\mu, \tau)_K \geqslant {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau}).\end{align*} $$

If moreover the equality holds, then $({\mathbf G}, \mu )$ is fully Hodge–Newton decomposable.

Proof We may reduce to the case where ${\mathbf G}$ is quasi-simple over F and that it is semisimple of adjoint type. Under this assumption, we have

(3.1) $$ \begin{align} {\mathbf G}_{\breve F} = {\mathbf G}_1 \times \cdots \times {\mathbf G}_r, \end{align} $$

where each ${\mathbf G}_i$ is a simple reductive group over $\breve F$ , and $\sigma ({\mathbf G}_j) = {\mathbf G}_{j+1}$ for all j. Here, we set ${\mathbf G}_{r+1} = {\mathbf G}_1$ . Write $\tilde {\mathbb {S}} = \tilde {\mathbb {S}} _1 \sqcup \cdots \sqcup \tilde {\mathbb {S}} _r$ and $\underline {\mu } = (\underline {\mu }_1, \dots , \underline {\mu }_r)$ with respect to the decomposition above. For $J \subseteq \tilde {\mathbb {S}} $ , we set $J_j = J \cap \tilde {\mathbb {S}} _j$ . We may write $\rho $ as $\rho =\rho _1+\cdots +\rho _r$ , where $\rho _i$ is the half sum of positive roots corresponding to the root system associated with $\tilde {\mathbb {S}} _i$ . We also have that $\Omega =\Omega _1 \times \cdots \times \Omega _r$ . We write $\tau = (\tau _1, \ldots , \tau _r)$ , where $\tau _i \in \Omega _i$ .

It is easy to see that $\ell (t^{\underline {\mu }_1}) = \langle \underline {\mu }_1, 2 \rho _1\rangle \geqslant \sharp \mathbb {S} _1 \geqslant {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ . Let $\xi =(\xi _1, \ldots , \xi _r) \in W_0 \cdot \underline {\mu }$ such that $t^{\xi } \in {}^K \tilde W$ . We choose a reduced expression $t^{\xi }=\tau s_{i_1} \cdots s_{i_k}$ of $t^{\xi }$ . Let $w=\tau s_{i_1} \cdots s_{i_m}$ , where $m={\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ . Then $w \leqslant t^{\xi }$ and $w \in {}^K \tilde W$ . Hence, $w \in {}^{K}\!{\mathrm{Adm}}(\mu )$ . Since $\ell (w)=m<\sharp \{({\mathrm{Ad}}(\tau ) \circ \sigma )\text {-orbits on } \tilde {\mathbb {S}} \}$ , the Weyl group $W_{\operatorname {\mathrm{supp}}_{\sigma }(w)}$ is finite and hence $\dim X_{K, w}(\tau )=\dim X_w(\tau )=\ell (w)$ . Thus, $\dim X(\mu , \tau )_K \geqslant \ell (w)=m$ .

Now, we assume that $\dim X(\mu , \tau )_K = {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ . Let $w \in {}^{K}\!{\mathrm{Adm}}(\mu )$ such that $\breve {\mathcal K} \cdot _{\sigma } \breve {\mathcal I} w \breve {\mathcal I} \cap [\tau ] \neq \emptyset $ , that is, $\breve {\mathcal I} w \breve {\mathcal I} \cap [\tau ] \neq \emptyset $ or in other words, $X_w(\tau ) \neq \emptyset $ . Then $ \dim X_w(\tau )=\dim X_{K, w}(\tau ) \leqslant \dim X(\mu , \tau )_K = {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ . By Corollary 3.4, we have $\operatorname {\mathrm{supp}}_{\sigma }(w) \subsetneq \tilde {\mathbb {S}} $ . By [Reference He23, Lemma 3.2], $\breve {\mathcal K} \cdot _{\sigma } \breve {\mathcal I} w \breve {\mathcal I} \subseteq \breve G \cdot _{\sigma } \breve {\mathcal I} \tau =[\tau ]$ . Noticing that

$$ \begin{align*}\breve{\mathcal K} \operatorname{\mathrm{Adm}}(\mu) \breve{\mathcal K} = \sqcup_{w \in {}^{K}\!{\mathrm{Adm}}(\mu)} \breve{\mathcal K} \cdot_{\sigma} \breve{\mathcal I} w \breve{\mathcal I},\end{align*} $$

we deduce that

(3.2) $$ \begin{align} \breve{\mathcal K} \operatorname{\mathrm{Adm}}(\mu) \breve{\mathcal K} \cap [\tau] = \bigsqcup_{w \in {}^{K}\!{\mathrm{Adm}}(\mu), \sharp W_{\operatorname{\mathrm{supp}}_{\sigma}(w)} < \infty} \breve{\mathcal K} \cdot_{\sigma} \breve{\mathcal I} w \breve{\mathcal I}. \end{align} $$

By Section 2.3, $({\mathbf G}, \mu )$ is fully Hodge–Newton decomposable.

4.1 Admissible triples

Let $s \in \tilde {\mathbb {S}} $ . If $s \in W_0$ , set $\alpha _s$ to be the simple root corresponding to s. Otherwise, set $\alpha _s = -\theta $ , where $\theta \in \Phi ^+$ is the highest root of the j-component of the decomposition (4.1), where $s\in \tilde {\mathbb {S}} _j$ ; then $s = t^{\theta ^{\vee }} s_{\theta }$ . Let $J \subseteq \tilde {\mathbb {S}} $ such that $W_J$ is finite. We denote by $\Phi _J$ the root system spanned by $\alpha _s$ for $s \in J$ .

Let $p: \tilde W \rtimes \langle \sigma \rangle \rightarrow \mathrm {GL}({X_*} \otimes \mathbb {R} )$ be the natural projection.

Let $\xi \in W_0 \cdot \underline {\mu }$ , and let $J \subsetneq \tilde {\mathbb {S}} $ be a maximal proper $\sigma $ -stable subset. Let $\xi _J \in \mathbb {R} \Phi _J^{\vee }$ be such that $\langle \xi _J, \alpha \rangle = \langle \xi , \alpha \rangle $ for $\alpha \in \Phi _J$ . We denote by $\xi _J^{\diamond }$ the $p(\sigma )$ -average of $\xi _J$ .

In this case, we define

$$ \begin{align*}K_{\xi} = \cup_C \cup_{i \in \mathbb {Z}} \sigma^i(C) \subseteq K,\end{align*} $$

where C ranges over the connected components C of K on which the $p(\sigma )$ -average $\xi ^{\diamond }$ is nonzero. In other words, $K_{\xi }$ is the minimal $\sigma $ -stable subset of K such that $\xi _J^{\diamond } \in \mathbb {R} \Phi _{K_{\xi }}^{\vee }$ .

Lemma 4.3 Let $(\xi , J, K)$ be an admissible triple. Then there exists some $\sigma $ -Coxeter element $c \in W_{K_{\xi }}$ such that

$$ \begin{align*} \ell(t^{\xi} c) = \ell(t^{\xi}) - \ell(c) = \langle \underline{\mu}, 2\rho\rangle - \sharp \{\sigma\text{-orbits of } K_{\xi}\}. \end{align*} $$

In particular, $t^{\xi } c \in {}^{K}\!{\mathrm{Adm}}(\mu )$ and the Newton point of $t^{\xi } c$ is central.

Lemma 4.4 Suppose that $\dim X(\mu , \tau )_K={\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ and that $(\xi , J, K')$ is an admissible triple such that $K' \supseteq K$ . Then

(a) $$ \begin{align} \langle \underline{\mu}, 2\rho\rangle \leqslant \sharp \{\sigma\text{-orbits of } K_{\xi}'\} + {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau}). \end{align} $$

Proof Let c be as in Lemma 4.3. Then we need to show that $\ell (t^{\xi } c) \leqslant {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ . However, our assumption implies, by Theorem 3.5, that $({\mathbf G}, \mu )$ is fully Hodge–Newton decomposable, and hence we have $\ell (t^{\xi } c) = \dim X_{t^{\xi } c}(\tau )$ (note that this is an easy consequence of (3.2) and does not require the use of classification results).

Altogether we obtain

$$\begin{align*}\ell(t^{\xi} c) = \dim X_{t^{\xi} c}(\tau) \leqslant \dim X(\mu, \tau)_K={\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau}), \end{align*}$$

as desired.

Given a $\sigma $ -stable subset $K \subseteq \tilde {\mathbb {S}} $ with $W_K$ finite, it follows from Lemma 4.1 that there always exists some admissible triple $(\xi , J, K)$ .

Corollary 4.5 If $\dim X(\mu , \tau )_K={\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ , then

(b) $$ \begin{align} \langle \underline{\mu}, 2\rho\rangle \leqslant {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf G}) + {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau}). \end{align} $$

In particular,

(c) $$ \begin{align} \langle \underline{\mu}, 2\rho\rangle \leqslant 2\,{\mathrm{rank}}^{\mathrm{ss}}_{\overline{F}}({\mathbf G}). \end{align} $$

We can now state the following equivalent characterizations of being of Coxeter type. Note that there is an obvious notion of product of Coxeter data. We call a Coxeter datum irreducible, if it cannot be decomposed as a product in a nontrivial way.

4.2 Strategy

In Table 1, we list the minimal irreducible enhanced Coxeter data (up to isomorphism) satisfying condition (3) of Theorem 4.6 together with the set ${}^{K}\!{\mathrm{Adm}}(\mu )_0$ . It is easy to see that in all these cases, ${}^{K}\!{\mathrm{Adm}}(\mu )_0={}^K\!{\mathrm{Cox}}(\mu )$ . Therefore, we also have ${}^{K'}\!\operatorname {\mathrm{Adm}}(\mu )_0={}^{K'}\mathop {\mathrm{Cox}}(\mu )$ for all $K' \supset K$ . This shows $(4) \Rightarrow (1)$ . Note that $(1) \Rightarrow (2)$ is obvious and $(2) \Rightarrow (3)$ follows from Lemma 4.4. It remains to show that $(3) \Rightarrow (4)$ .

Note that condition (3), although a bit technical, is the most elementary one among the four conditions and only involves the root system. In the rest of the section, we will analyze condition (3) and give a classification of the irreducible enhanced Coxeter data that satisfy this condition, and finally show that those cases are of Coxeter type. This finishes the direction $(3) \Rightarrow (4)$ .

Our strategy is as follows.

In Step (I), we show that condition (3) implies that the Coxeter datum $(W_a, \sigma , \underline {\mu })$ is one of those listed in Table 2.

This is done by the inequalities (b) and (c) in Corollary 4.5 in most of the cases. The only exception is in Type D, where in some cases we have to use the full strength of condition (3).

Note that the cases in this table are the fully Hodge–Newton decomposable cases. We will then further analyze these cases and give a complete classification.

In Step (II), we show that condition (3) implies that for the Coxeter datum in Table 2, the parahoric subgroup K must be as specified in Theorem 4.6/Table 1.

4.3 Step (I): The Coxeter datum $(W_a, \sigma , {\mu })$

Recall the decomposition (4.1). We will argue on the type of the irreducible affine Dynkin diagram $\tilde {\mathbb {S}} _j$ , which does not depend on j. For $i \in \tilde {\mathbb {S}} _j$ , we denote by $\omega _{i, j}^{\vee }$ the corresponding fundamental coweight in ${\mathbf G}_j$ . If $r = 1$ , we write $\omega _i^{\vee } = \omega _{i, 1}^{\vee }$ for simplicity.

4.3.2. Type $\tilde A_{n-1}$

By applying a suitable automorphism, we may assume that $K_i \subset \tilde {\mathbb {S}} _i - \{0\}$ . By (b), we deduce that (up to isomorphism) one of following cases occurs:

1. (1) $r = 1$ and $\underline {\mu } \in \{\omega _k^{\vee }; 1 \leqslant k \leqslant n-1\}$ .

2. (2) $r = 1$ and $\underline {\mu } \in \{2\omega _1^{\vee }, \omega _1^{\vee } + \omega _{n-1}^{\vee }, 2\omega _{n-1}^{\vee }\}$ .

3. (3) $r = 2$ and $\underline {\mu } = \omega _{1, 1}^{\vee } + \omega _{n-1, 2}^{\vee }$ .

In the last two cases, we have by (b) that ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf G}) = {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau }) = n-1$ , which means that $\underline {\mu } = \omega _1^{\vee } + \omega _{n-1}^{\vee }$ (which equals $2\omega ^{\vee }_1$ if $n=2$ ) and $\sigma = \mathrm {id} $ or $\underline {\mu } = \omega _{1, 1}^{\vee } + \omega _{n-1, 2}^{\vee }$ and $\sigma = {}^1\varsigma _0$ .

Now, we assume $r = 1$ . Suppose $\underline {\mu } = \omega _i^{\vee }$ for some $1 \leqslant i \leqslant n-1$ . Notice that ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_1) \leqslant \frac {n}{2} $ (resp. ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau }) \leqslant \frac {n}{2}$ ) if $\sigma \neq \mathrm {id} $ (resp. ${\mathrm{Ad}}(\tau ) \circ \sigma \neq \mathrm {id} $ ). So either $\underline {\mu } \in \{\omega _1^{\vee }, \omega _{n-1}^{\vee }\}$ or $\underline {\mu }=\omega ^{\vee }_2$ with $n=4$ .

Suppose $\underline {\mu } = \omega _1^{\vee }$ and $\sigma ={\mathrm{Ad}}(\tau _k)$ for $0 \leqslant k \neq n-1$ . Then, by (b), we have

$$ \begin{align*}n-1 = \langle \underline{\mu}, 2\rho\rangle \leqslant {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf G})+{\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau}) = \gcd(n, k) -1 + \gcd(n, k+1) - 1,\end{align*} $$

which implies $k = n-1$ or $k = 0$ . Otherwise, we deduce that (up to isomorphism) $\sigma = \varsigma _0$ .

Suppose $\underline {\mu } = \omega ^{\vee }_2$ and $n = 4$ . We have (up to isomorphism) $\sigma = \mathrm {id} $ , or $\sigma = \varsigma _0$ , or $\sigma = {\mathrm{Ad}}(\tau _1)$ , or $\sigma = {\mathrm{Ad}}(\tau _1) \circ \sigma _0$ . The last case does not occur since (b) fails.

4.3.4. Type $\tilde C_n$ for $n \geqslant 2$

Here,

$$ \begin{align*}\langle \omega^{\vee}_{i, j}, 2 \rho\rangle =\begin{cases} i (2n-i+1), & \text{ if } i \leqslant n-1, \\ \frac{n(n+1)}{2}, & \text{ if } i=n.\end{cases}\end{align*} $$

Therefore, $\langle \underline {\mu }, 2 \rho \rangle \leqslant 2n$ implies that $r = 1$ and either $\underline {\mu }=\omega ^{\vee }_1$ or $\underline {\mu }=\omega ^{\vee }_n$ with $n \leqslant 3$ .

If $\underline {\mu }=\omega ^{\vee }_1$ , then $\langle \underline {\mu }, 2 \rho \rangle =2n$ and hence ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })=n$ . Therefore, $\sigma =id$ .

For $n=2$ and $\underline {\mu }=\omega ^{\vee }_2$ , we have $\sigma =\mathrm {id} $ or $\sigma ={\mathrm{Ad}}(\tau _2)$ .

For $n=3$ and $\underline {\mu }=\omega ^{\vee }_3$ , we have $\langle \underline {\mu }, 2 \rho \rangle =6$ and hence ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })=3$ . Therefore, $\sigma ={\mathrm{Ad}}(\tau _3)$ and $\langle \underline {\mu }, 2\rho \rangle = 6> 4 = {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf G}) + {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ , contradicting (b).

4.3.5. Type $\tilde D_n$ for $n \geqslant 4$

Here,

$$ \begin{align*}\langle \omega^{\vee}_{i, j}, 2 \rho\rangle =\begin{cases} i (2n-i-1), & \text{ if } i \leqslant n-2, \\ \frac{n(n-1)}{2}, & \text{ if } i=n-1 \text{ or } n.\end{cases} \end{align*} $$

Therefore, $\langle \underline {\mu }, 2 \rho \rangle \leqslant 2n$ implies that $r = 1$ and (up to isomorphism) either $\underline {\mu }=\omega ^{\vee }_1$ or $\underline {\mu }=\omega ^{\vee }_n$ with $n=5$ .

If $n=5$ and $\underline {\mu }=\omega ^{\vee }_5$ , we have $\langle \underline {\mu }, 2 \rho \rangle =10$ and hence ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })=5$ . Therefore, $\sigma ={\mathrm{Ad}}(\tau _4)$ . However, $\langle \underline {\mu }, 2 \rho \rangle = 10> 6 = {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf G}) + {\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau })$ , contradicting (b).

If $\underline {\mu }=\omega ^{\vee }_1$ , then $\langle \underline {\mu }, 2 \rho \rangle =2(n-1)$ and ${\mathrm{rank}_F^{\mathrm{ss}}}({\mathbf J}_{\tau }) \geqslant n-2$ . Thus, we have $\sigma =\mathrm {id} $ , $\sigma =\sigma _0$ , or $\sigma ={\mathrm{Ad}}(\tau _1)$ . Suppose $\sigma ={\mathrm{Ad}}(\tau _1)$ and $i \notin K$ for some $1 \leqslant i \leqslant n-1$ . Let $J = \tilde {\mathbb {S}} - \{i, \sigma (i)\}$ . Let $\xi = \underline {\mu }$ if $i = 1$ and $\xi = s_i \cdots s_2 s_1(\underline {\mu })$ if $2 \leqslant i \leqslant n-1$ . Then $J_{\xi } = \emptyset $ if $i \in \{1, n-1\}$ and $J_{\xi } = \{i+1, \dots , n-1, n\}$ otherwise. Hence, (a) fails for the admissible triple $(\xi , J, J)$ .

In Table 2, we list the remaining cases together with the $\sigma $ -orbits and the ${\mathrm{Ad}}(\tau ) \circ \sigma $ -orbits on $\tilde {\mathbb {S}} $ . This information will be used in the case-by-case analysis in the remainder of this section.

4.4 Step (II): Exclude certain K

4.5 Step (III): Final verification

To finish the proof of Theorem 4.6, it only remains to check in each case that the set ${}^{K}\!{\mathrm{Adm}}(\mu )_0$ is the set given in Table 1. From the explicit description, we see that these cases are of Coxeter type. We omit the explicit computations.

5.1 Definitions

We consider the situation where $F = \mathbb Q_p$ , and where the pair $({\mathbf G}, \mu )$ corresponds to a Rapoport–Zink space. As before, we consider the basic case, i.e., we denote by b the basic $\sigma $ -conjugacy class in $B({\mathbf G}, \mu )$ .

Since there are different constructions of Rapoport–Zink (RZ) spaces in the PEL case and the more general case of Hodge type, we will axiomatize the properties that we require, rather than fixing one of the constructions.

In all cases, the RZ space $\mathcal M({\mathbf G}, \mu , b)_K$ is a formal scheme over $\breve O$ , the ring of integers of $\breve F$ . We denote by $\mathbf k$ the residue class field of $\breve F$ , and by a subscript $-_{\mathbf k}$ indicate the base change to $\mathbf k$ .

Note that the results in the previous sections are group-theoretic in nature and hence concern parahoric level structures, but the known constructions of RZ spaces work for stabilizers of facets in the Bruhat–Tits building. See the discussion in Section 5.5. To take this possible difference into account right from the beginning, we change notation as follows: we denote by P the stabilizers of facets of the base alcove, and by $P^{\circ }$ the corresponding parahoric.

The setup of the theory entails that P and $P^{\circ }$ are always defined over F, i.e., fixed by $\sigma $ .

We denote by $\mathrm {Gr}_{P^{\circ }}$ the partial affine flag variety for $P^{\circ }$ (i.e., with $\mathbf k$ -valued points $\breve {G}/P^{\circ }$ , and similarly by $\mathrm {Gr}_{P}$ the “partial affine flag variety” for P with $\mathbf k$ -valued points $\breve {G}/P$ ; cf. [Reference Bhatt and Scholze3], [Reference Zhu64, Theorem 1.4]). Let $\pi \colon \mathrm {Gr}_{P^{\circ }}\rightarrow \mathrm {Gr}_{P}$ denote the projection.

Since $P^{\circ } \subseteq P$ is a normal subgroup, the (finite) quotient group $P/P^{\circ }$ acts on $\mathrm {Gr}_{P^{\circ }}$ on the right by $gP^{\circ } \cdot p = gpP^{\circ }$ , $p\in P/P^{\circ }$ , and $\mathrm {Gr}_P$ is the quotient by this action. The quotient here is of a very simple nature, since the action just identifies certain connected components of $\mathrm {Gr}_{P^{\circ }}$ . In fact, let $\breve {\kappa }\colon \breve {G}\rightarrow \pi _0(\mathrm {Gr}_{P^{\circ }}) = \pi _1({\mathbf G})_{\Gamma _0}$ be the Kottwitz homomorphism (cf. [Reference Zhu64, Proposition 1.21]). Now, $P^{\circ }$ is the kernel of $\breve {\kappa }_{|P}$ (see [Reference Pappas and Rapoport40, Appendix Proposition 3], and note that the Appendix covers the case of mixed characteristic, which we need here). Correspondingly, the parahoric group scheme corresponding to $P^{\circ }$ is the connected component of the “stabilizer group scheme” corresponding to P. So the above action by $P/P^{\circ }$ permutes the connected components of $\mathrm {Gr}_{P^{\circ }}$ and is free. In particular, the restriction of $\pi $ to any connected component of $\mathrm {Gr}_{P^{\circ }}$ is an isomorphism onto a connected component of $\mathrm {Gr}_{P}$ .

In this way, we obtain a perfect ind-scheme $\mathrm {Gr}_{P}$ with $\mathbf k$ -valued points $\breve {G}/P$ and such that the projection $\pi \colon \mathrm {Gr}_{P^{\circ }}\rightarrow \mathrm {Gr}_{P}$ is an isomorphism when restricted to a connected component of $\mathrm {Gr}_{P^{\circ }}$ .

We write

$$\begin{align*}X_{P} := X(\mu, b)_{P} = \{ g\in \mathrm{Gr}_{P};\ g^{-1}b\sigma(g) \in P \operatorname{\mathrm{Adm}}(\mu) P\}, \end{align*}$$

which inherits the structure of a perfect ind-scheme and is even a scheme locally of finite type over $\mathbf k$ (cf. [Reference Hamacher and Viehmann20, Reference Rapoport and Zink47]).

Similarly as in [Reference Görtz, He and Nie14], we consider the following condition:

( $\diamondsuit $ ) For facet stabilizers $P\subset P'$ , we have a projection $\mathcal M({\mathbf G}, \mu , b)_P\rightarrow \mathcal M({\mathbf G}, \mu , b)_{P'}$ , and there are isomorphisms

$$\begin{align*}\mathcal M({\mathbf G}, \mu, b)_{P, \mathbf k}^{p^{-\infty}} \cong X(\mu, b)_P \end{align*}$$

of perfect schemes, compatible with the projections for inclusions $P\subset P'$ .

The second condition that we need to impose is the following compatibility between the RZ spaces for levels $P_i$ and the RZ space attached to the intersection $P:=\bigcap _i P_i$ . It follows from property ( $\diamondsuit $ ) that the morphism $\mathcal M({\mathbf G}, \mu , b)_{P, \mathbf k}^{\mathrm{red}}\rightarrow \prod _i \mathcal M({\mathbf G}, \mu , b)_{P_i, \mathbf k}^{\mathrm{red}}$ is a homeomorphism onto a closed subscheme of its target. In fact, by ( $\diamondsuit $ ) it is enough to check the analogous property for the affine Deligne–Lusztig varieties, where it follows from the fact that the natural morphism $\mathrm {Gr}_P \rightarrow \prod _i\mathrm {Gr}_{P_i}$ is a closed immersion. We will need the corresponding property “before perfection” on the RZ-space side and hence impose the following stronger statement as our second axiom:

( $\clubsuit $ ) The natural morphism $\mathcal M({\mathbf G}, \mu , b)_{P, \mathbf k}^{\mathrm{red}}\rightarrow \prod _i \mathcal M({\mathbf G}, \mu , b)_{P_i, \mathbf k}^{\mathrm{red}}$ is a closed immersion of $\mathbf k$ -schemes.

5.2 The PEL case

For most RZ spaces of PEL type, it is known that assumptions ( $\diamondsuit $ ) and ( $\clubsuit $ ) are satisfied.

Consider an RZ space attached to a PEL datum as in [Reference Rapoport and Zink46, Chapter 3] (see also [Reference Hartwig22, Reference Rapoport and Viehmann45] for summaries and further discussions). Let ${\mathbf G}$ and $\mu $ be the group and cocharacter attached to it. In this context, the level structure is given by a polarized chain of lattices (i.e., we use a lattice model for the relevant Bruhat–Tits building). Denote by $P \subset \breve G$ the stabilizer of this fixed standard chain.

Denote by $\mathcal M^{\mathrm{naive}}_P$ the corresponding RZ space [Reference Rapoport and Zink46, Definition 3.21] over $O_{\breve E}$ , the ring of integers of the completion of the maximal unramified extension of the local reflex field E. It depends on the choice of a “framing object” $\mathbf X$ (a p-divisible group with additional structure corresponding to the group ${\mathbf G}$ ) and parameterizes pairs $((X_{\Lambda }), (\rho _{\Lambda }))$ , where $(X_{\Lambda })$ is a chain of isogenies of p-divisible groups (over a scheme S on which p is locally nilpotent) indexed by the fixed lattice chain, and $\rho _{\Lambda }$ is a quasi-isogeny between $X_{\Lambda }$ and $\mathbf X$ over the closed subscheme $V(p)\subseteq S$ . These data are required to satisfy certain compatibilities (see [Reference Rapoport and Zink46, Definition 3.12]).

It is clear that property ( $\diamondsuit $ ) cannot in general be expected to hold for the “naive” RZ spaces, because their special fiber in general comprises strata not reflected in the set $\operatorname {\mathrm{Adm}}(\mu )$ . Therefore, we pass to the corresponding “flat RZ space.”

As shown in [Reference Rapoport and Zink46], the space $\mathcal M^{\mathrm{naive}}_P$ admits a local model diagram

$$\begin{align*}\mathcal M^{\mathrm{naive}}_P \longleftarrow \widetilde{\mathcal M^{\mathrm{naive}}_P} \longrightarrow M^{{\mathrm{naive}}, \wedge}_P, \end{align*}$$

where $M^{\mathrm{naive}}_P$ denotes the local model of [Reference Rapoport and Zink46, Definition 3.27] and $-^{\wedge }$ denotes the p-adic completion. See [Reference Rapoport and Zink46, Chapter 3], in particular Sections 3.26–3.35. Denote by $M^{\mathrm{flat}}_P$ the flat closure inside $M^{\mathrm{naive}}_P$ of its generic fiber.

Then $\mathcal M_P$ is defined by pulling back and pushing forward the inclusion $M^{\mathrm{flat}}_P \subseteq M^{\mathrm{naive}}_P$ along the local model diagram (after passing to the completion along the special fiber), i.e., we have a diagram

where the vertical arrows are closed immersions and both squares are Cartesian. Then $\mathcal M_P$ is flat over $O_{\breve E}$ .

Forgetting the endomorphism and polarization structure, we obtain a closed embedding into the corresponding RZ space for the general linear group and the same lattice chain, now considered without additional structure. We will denote this space by $\mathcal M^{\prime }_{P'}$ (imitating the notation of [Reference Hamacher and Kim19]), in particular $P'$ is the stabilizer of our lattice chain inside the general linear group $G' = GL(\mathbf N)(\breve {\mathbb Q}_p)$ of automorphisms of the rational Dieudonné module $\mathbf N$ of $\mathbf X$ .

By Dieudonné theory, we obtain a commutative diagram of inclusions:

(5.1)

Here, the right vertical map maps a point $(X_{\bullet }, \rho )\in \mathcal M^{\prime }_{P'}$ , where $X_{\bullet }$ is a chain of isogenies of p-divisible groups indexed by the fixed periodic lattice chain, and $\rho $ is a quasi-isogeny with the framing object $\mathbb X$ , to $gP'$ where g maps the fixed (partial) standard lattice chain to the chain of Dieudonné modules of $X_{\bullet }$ (considered inside the rational Dieudonné module of $\mathbb X$ via $\rho $ ). Since in the $GL_n$ case the vertical map is induced by an isomorphism of perfect schemes onto its image (cf. [Reference Zhu64, Proposition 3.11], which easily generalizes to general parahoric level structure in that case), the same is true for this map.

The map $\mathcal M^{\mathrm{naive}}_P(\mathbf k)\rightarrow \mathcal M^{\prime }_{P'}(\mathbf k)$ , which forgets the additional structure is an inclusion, because the additional structure is uniquely determined (by that structure on $\mathbf X$ and the quasi-isogenies $\rho _{\Lambda }$ ), if it exists. Cf. [Reference Rapoport and Zink46, Proof of Theorem 3.25].

The Frobenius morphism on $\mathbf N$ is given by $b\sigma $ for some $b\in \breve {G}$ .

Consider a point $(\mathcal F_{\Lambda })_{\Lambda }\in M^{\mathrm{naive}}(\mathbf k)$ . By definition, each $\mathcal F_{\Lambda }$ is a subspace of $\Lambda \otimes _{\mathbb Z_p}\mathbf k$ (with further properties which we do not state here; in comparison to [Reference Rapoport and Zink46, Definition 3.27], we switch from quotients to subspaces). Equivalently, we can record these data as a lattice $\tilde {\mathcal F}$ lying between $\Lambda $ and $p\Lambda $ . This defines an inclusion $M^{\mathrm{naive}}(\mathbf k) \rightarrow \breve {G}/P$ , and the action of P on $\breve {G}/P$ on the left preserves the subset $M^{\mathrm{naive}}(\mathbf k)$ .

By the definition of the local model diagram, we obtain a commutative diagram

where the lower horizontal map maps $gP$ to the double coset of $g^{-1}b\sigma (g)$ . Likewise, P acts on $M_P^{\mathrm{flat}}(\mathbf k)\subseteq M^{\mathrm{naive}}(\mathbf k)$ since this action comes from an action of the (smooth) stabilizer group scheme associated with P on the $O_{\breve E}$ -scheme $M^{\mathrm{naive}}$ . We write $\mathcal A(\mu )_P = P\backslash M^{\mathrm{flat}}(\mathbf k) \subset P\backslash \breve {G} / P$ . Similarly, we write $\operatorname {\mathrm{Adm}}(\mu )_P = P\backslash P\operatorname {\mathrm{Adm}}(\mu )P/P$ .

Proposition 5.1 If $\mathcal A(\mu )_P = \operatorname {\mathrm{Adm}}(\mu )_P$ , then assumption ( $\diamondsuit $ ) is satisfied. More precisely, the inclusion $\mathcal M_P(\mathbf k) \subset \breve G/P$ is induced by an isomorphism

$$\begin{align*}\mathcal M_{P, \mathbf k}^{p^{-\infty}} \cong X(\mu, b)_P \end{align*}$$

of perfect schemes, and these isomorphisms are compatible with the projections for passing to sub-lattice chains.

Proof Since we know (from the embedding into a $GL$ situation) that the map is induced from a morphism of perfect schemes, it is enough to check the claim on $\mathbf k$ -valued points.

The above discussion shows, together with our assumption $\mathcal A(\mu )_P = \operatorname {\mathrm{Adm}}(\mu )_P$ and the definition of $M^{\mathrm{flat}}_P$ , that we have a diagram

in which both squares are Cartesian. Viewing $\mathcal M^{\mathrm{naive}}_P(\mathbf k)\subset \breve {G}/P$ as before, the lower horizontal map is given by $gP\mapsto P g^{-1}b\sigma (g)P$ . So we see that $\mathcal M_P(\mathbf k) \subseteq X(\mu , b)_P$ , and that it is enough to show that $X(\mu , b)_P\subseteq \mathcal M^{\mathrm{naive}}_P(\mathbf k)$ in order to complete the proof.

For this inclusion, note that the square in the diagram (5.1), while not Cartesian in general, is close to being Cartesian. More precisely, $\mathcal M^{\mathrm{naive}}_P(\mathbf k)$ is defined inside the intersection $\mathcal M^{\prime }_{P'}(\mathbf k)\cap \breve {G}/P$ by imposing the Kottwitz determinant condition. Since this condition can be checked on the local model, and since it is satisfied by definition on $M^{\mathrm{naive}}$ , and a fortiori on $M^{\mathrm{flat}}$ , it is enough to show that $X(\mu , b)_P\subseteq \mathcal M^{\prime }_{P'}$ .

Denote by $\mu '$ the composition of $\mu $ with the inclusion ${\mathbf G}\rightarrow GL(\mathbf N)$ . Since the identification of $\mathcal M^{\prime }_{P'}(\mathbf k)$ with the corresponding generalized affine Deligne–Lusztig variety $X^{GL}(\mu ', b)_{P'}$ for the general linear group is easy to check, we see that is enough to show that $X_P$ embeds into $X^{GL}(\mu ', b)_{P'}$ under the embedding $\breve G/P\subseteq G'/P'$ . This follows once we can prove that $\operatorname {\mathrm{Adm}}(\mu )_P$ embeds into the admissible set $\operatorname {\mathrm{Adm}}(\mu ')_{P'}\subset P'\backslash G'/P'$ .

This compatibility of admissible sets follows from the inclusions $M^{\mathrm{flat}}_P(\mathbf k) \subseteq M^{\mathrm{naive}}_P(\mathbf k) \subseteq M^{GL}_{P'}(\mathbf k)$ , where $M^{GL}_{P'}$ denotes the local model for the general linear group, using once more the assumption $\mathcal A(\mu )_P = \operatorname {\mathrm{Adm}}(\mu )_P$ and the fact that for the general linear group the corresponding equality is known, as well.

It is clear that everything above is compatible with the projections arising from forgetting some of the lattices in our chain.

5.3 RZ spaces of Hodge type

In the work of Kim [Reference Kim33] and Hamacher and Kim [Reference Hamacher and Kim19] where RZ spaces for data of Hodge type are constructed, the bijection $\mathcal M({\mathbf G}, \mu , b)_{P}(\mathbf k) \cong X(\mu , b)_P(\mathbf k)$ on $\mathbf k$ -valued points is an essential feature of the construction (see [Reference Hamacher and Kim19, Proposition 4.3.5]). Zhu [Reference Zhu64, Proposition 3.11] proved that this set-theoretical equality implies the above isomorphism of perfect schemes, using results of Gabber and Lau on Dieudonné theory over perfect rings to handle the case ${\mathbf G}=GL_n$ , and then embedding the general situation into a suitable $GL_n$ -situation. While in [Reference Zhu64] it was assumed that K is hyperspecial, the only reason for this assumption is that at the time of writing RZ spaces of Hodge type had been constructed only in this special situation; the paper [Reference Hamacher and Kim19] appeared only later.

Note, however, that the previous paragraph concerns only the situation for a fixed level P. Because of the way RZ spaces are defined in the Hodge-type situation, it is not clear that these bijections are compatible with the projection maps attached to a pair $P\subset P'$ of parahoric subgroups. In fact, the definition relies on embedding the situation into an RZ space of Siegel type (i.e., associated with a group of symplectic similitudes and hyperspecial level structure). However, it is not evident whether the result is independent of the choice of embedding, and it does not seem clear whether such embeddings can be chosen in a compatible way given $P\subset P'$ .

5.4 The weak Bruhat–Tits stratification

Next, we discuss the question of defining a (weak) Bruhat–Tits stratification on Rapoport–Zink spaces. Hence, we now restrict to the fully Hodge–Newton decomposable case. Then we have the weak Bruhat–Tits stratification on $X(\mu , b)_{P^{\circ }}$ (Section 2.4).

Since property ( $\diamondsuit $ ) involves P instead of $P^{\circ }$ , we first need to discuss the question of defining a weak BT stratification on $X(\mu , b)_P$ . To this end, we impose the following additional assumption:

( $\heartsuit $ )  The projection $X_{P^{\circ }} \rightarrow X_P$ is surjective.

See Section 5.5 for two sufficient criteria for this assumption. Of course, it is trivially satisfied if $P=P^{\circ }$ . It is also satisfied in those cases that have been studied in detail on the Shimura variety side (attached to ramified unitary groups). We do not know of an example where this property fails.

Let us denote by $X_{P^{\circ }, c}$ the “component” indexed by $c\in \pi _0(\mathrm {Gr}_{P^{\circ }})$ , i.e., $X_{P^{\circ }, c} = X_{P^{\circ }}\cap {\breve \kappa }^{-1}(c)$ . As explained in Section 5.5, $X_{P^{\circ }, c} = \emptyset $ unless $c\in \pi _0(\mathrm {Gr}_{P^{\circ }})^{\sigma }$ .

For each $c\in \pi _0(\mathrm {Gr}_{P^{\circ }})^{\sigma }$ , the projection $X_{P^{\circ }}\rightarrow X_{P}$ restricts to an isomorphism

$$\begin{align*}\gamma_c\colon X_{P^{\circ}, c} {\overset{\sim}{\longrightarrow}} X_{P, \pi(c)}. \end{align*}$$

Here, we denote by $\pi (c)$ the connected component of $\mathrm {Gr}_{P}$ , which is the image of c, and by $X_{P, \pi (c)}$ its intersection with $X_{P}$ . Then $\pi ^{-1}(X_{P, \pi (c)})$ is the union of the $X_{P^{\circ }, c'}$ where $c'$ ranges over the $P/P^{\circ }$ -orbit of c in $\pi _0(\mathrm {Gr}_{P^{\circ }})$ .

To establish that $\gamma _c$ above is an isomorphism, recall that each connected component of $\mathrm {Gr}_{P^{\circ }}$ maps isomorphically onto a connected component of $\mathrm {Gr}_{P}$ . Therefore, the only question is the surjectivity, which follows from ( $\heartsuit $ ).

Since all BT strata in $X_{P^{\circ }}$ are connected, each stratum lies in one of the components $X_{P^{\circ }, c}$ . Using the isomorphisms $\gamma _c$ , we obtain a Bruhat–Tits stratification on $X_{P}$ which is independent of the choice of c, as the following lemma shows.

Lemma 5.4 Let $c, c'\in \pi _0(\mathrm {Gr}_{P^{\circ }})^{\sigma }$ such that $c'c^{-1}\in P/P^{\circ }$ . Let $j Y(w) \subseteq X_{P^{\circ }, c'}$ be a BT stratum. Then

$$\begin{align*}\gamma_{c}^{-1}(\gamma_{c'}(jY(w))) \subseteq X_{P^{\circ}, c} \end{align*}$$

is a BT stratum.

Proof The map $\gamma _{c}^{-1}\circ \gamma _ {c'}$ is given by $gP^{\circ }\mapsto g(c')^{-1}cP^{\circ }$ . We may represent $(c')^{-1}c$ by an element of $P\cap N_G(T)(\breve F)$ that induces a length $0$ element in $\tilde W$ . To simplify the notation, we change notation and denote this element by c. Then we have $\sigma (c) = c$ inside $\tilde W$ (passing to a different representative, if necessary we could achieve the same property on the level of elements of $\breve G$ ).

If $g\in jY(w)$ , then $g^{-1}\tau \sigma (g)\in P^{\circ } \cdot _{\sigma } (\breve {\mathcal I} w\tau \breve {\mathcal I})$ . But then,

$$\begin{align*}c^{-1} g^{-1} \tau\sigma(g)\sigma(c) \in c^{-1}P^{\circ} \cdot_{\sigma}(\breve{\mathcal I} w\tau \breve{\mathcal I}) \sigma(c) = P^{\circ} \cdot_{\sigma} (\breve{\mathcal I} c^{-1}w\tau \sigma(c) \breve{\mathcal I}). \end{align*}$$

Now, $c^{-1}w\tau \sigma (c) = c^{-1} w\tau c$ is again an EKOR element for $P^{\circ }$ (i.e., it lies in ${}^K\tilde W\cap \operatorname {\mathrm{Adm}}(\mu )$ , where $K\subset \tilde {\mathbb {S}} $ denotes the set of simple affine reflections generating $P^{\circ }$ ).

Since the element $c^{-1} w\tau c$ is independent of g, the lemma follows.

We hence get a well-defined “Bruhat–Tits stratification” on $X_{P}$ and the projection $X_{P^{\circ }}\rightarrow X_{P}$ is compatible with the stratifications.

Consider an inclusion $P\subset P'$ of facet stabilizers and the corresponding inclusion $P^{\circ } \subset {P'}^{\circ }$ . We obtain a commutative diagram

The sets of connected components of $\mathrm {Gr}_{P^{\circ }}$ and $\mathrm {Gr}_{{P'}^{\circ }}$ coincide. Fixing a connected component c of $\mathrm {Gr}_{P^{\circ }}$ and denoting by $c'$ its image in $\mathrm {Gr}_{{P'}^{\circ }}$ , we can restrict the above diagram to c and $c'$ , and obtain a diagram

(with $\pi '\colon \mathrm {Gr}_{{P'}^{\circ }}\rightarrow \mathrm {Gr}_{P'}$ the projection) where the vertical morphisms are isomorphisms. Since the upper horizontal map maps each BT stratum in $X_c$ isomorphically onto a BT stratum in $X_{c'}$ , the same holds true for the lower horizontal map.

We also record the following easy lemma.

Now, we can invoke property ( $\diamondsuit $ ) and obtain that in the fully Hodge–Newton decomposable case and under the above assumptions, the spaces $\mathcal M({\mathbf G}, \mu , b)_{P, \mathbf k}^{p^{-\infty }}$ carry a weak BT stratification which is compatible with the weak BT stratification on $X_P$ via ( $\diamondsuit $ ). For an inclusion $P\subset P'$ , the corresponding map of RZ spaces maps each BT stratum in the source isomorphically (in the sense of perfect schemes) onto a BT stratum in the target.

Since the morphism $\mathcal M({\mathbf G}, \mu , b)_{P, \mathbf k}^{p^{-\infty }} \rightarrow \mathcal M({\mathbf G}, \mu , b)_{P, \mathbf k}^{\mathrm{red}}$ is a homeomorphism, we likewise get the following corollary.

The goal of this section is to investigate the following property (under the assumption of full Hodge–Newton decomposability).

Property $BT_P$ : In the (weak) Bruhat–Tits stratification of $\mathcal M({\mathbf G}, \mu , b)_{P, \mathbf k}^{\mathrm{red}}$ , each stratum is isomorphic to the corresponding classical Deligne–Lusztig variety, without passing to the perfection. The closure of each stratum is isomorphic to the closure of this classical Deligne–Lusztig variety in the corresponding finite-dimensional partial flag variety.

Our result will be that if ( $BT_{P_i}$ ) holds for all i, then ( $BT_P$ ) also holds for $P=\cap _i P_i$ . Note that we need to use in the proof that we already know the result after perfection for level P.

For most maximal parahoric level structures P, property ( $BT_P$ ) has been established by now, and this result will allow us to deduce the same property for many nonmaximal level structures, and in particular for most level structures of Coxeter type. See Section 6 for a discussion of the individual cases.

5.5 Stabilizers versus parahoric subgroups

It remains to discuss in more detail in which cases assumption ( $\heartsuit $ ) is satisfied, i.e., when the map $X_{P^{\circ }}\rightarrow X_{P}$ is surjective. Let us briefly recall the setup.

We consider $({\mathbf G}, \mu )$ as in Section 5.1 (not necessarily fully Hodge–Newton decomposable) and $b\in B({\mathbf G}, \mu )$ . Let $P\subset \breve G$ be the stabilizer of a (poly-)simplex in the Bruhat–Tits building. We denote by $L\breve {G}$ the loop group attached to $\breve {G}$ , an ind-perfect scheme over $\mathbf k$ . The intersection $P^{\circ }$ of P with the kernel of the Kottwitz homomorphism $\breve {\kappa }\colon \breve G\rightarrow \pi _0(L\breve {G}) = \pi _1({\mathbf G})_{\Gamma _0}$ is a parahoric subgroup (and all parahoric subgroups arise in this way), but in general, $P^{\circ } \subset P$ is a proper normal subgroup.

In the sequel, we assume that $P^{\circ }$ is a standard parahoric subgroup, or in other words, that P is the stabilizer of a face of the base alcove. As mentioned above, the group-theoretic results obtained in this paper concern a priori the “parahoric setting,” but RZ spaces have been defined, so far, in the “stabilizer setting.” See also Remark 5.4 in the paper [Reference Hamacher and Kim19] by Hamacher and Kim.

Throughout the following discussion, we fix $\mu $ and $b\in B({\mathbf G}, \mu )$ , and we write $X_{P^{\circ }} = X(\mu , b)_{P^{\circ }}$ , and as above write $X_{P} := X(\mu , b)_{P} = \{ g\in \mathrm {Gr}_{P};\ g^{-1}b\sigma (g) \in P \operatorname {\mathrm{Adm}}(\mu ) P\}$ . We have

$$\begin{align*}X_{P^{\circ}}\subseteq \pi^{-1}(X_{P}). \end{align*}$$

Now, for $gP^{\circ }\in \mathrm {Gr}_{P^{\circ }}$ , we have

$$\begin{align*}gP^{\circ} \in X_{P^{\circ}} \quad \Longleftrightarrow \quad g^{-1}b\sigma(g) \in P^{\circ} \operatorname{\mathrm{Adm}}(\mu) P^{\circ}, \end{align*}$$

and

$$\begin{align*}gP^{\circ} \in \pi^{-1}(X_{P}) \quad \Longleftrightarrow \quad g^{-1}b\sigma(g) \in P \operatorname{\mathrm{Adm}}(\mu) P. \end{align*}$$

The lemma shows that $P \operatorname {\mathrm{Adm}}(\mu ) P = P^{\circ } \operatorname {\mathrm{Adm}}(\mu ) P$ , and we obtain that

$$\begin{align*}P^{\circ} \operatorname{\mathrm{Adm}}(\mu) P^{\circ} = P \operatorname{\mathrm{Adm}}(\mu) P \cap \breve{\kappa}^{-1}(\mu). \end{align*}$$

Recall that our assumption $b\in B({\mathbf G}, \mu )$ implies in particular that b and $\mu $ have the same image in $\pi _1({\mathbf G})_{\Gamma }$ , where $\Gamma = \operatorname {\mathrm{Gal}}(\overline {F}/F)$ is the absolute Galois group of F. Since $\breve \kappa $ takes its images in the coinvariants for the (smaller) inertia group $\Gamma _0$ , this does, however, not necessarily entail $\breve \kappa (b)=\breve \kappa (\mu )$ . We choose an element $c_{b,\mu }\in \pi _1({\mathbf G})_{\Gamma _0}$ such that $\sigma (c_{b,\mu })-c_{b,\mu } = \breve {\kappa }(\mu )-\breve {\kappa }(b)$ . Such an element exists because the difference $\breve {\kappa }(\mu )-\breve {\kappa }(b)$ vanishes in $\pi _1({\mathbf G})_{\Gamma }$ , as we just recalled. See also [Reference He and Zhou28, Section 6].