On some simple geometric structure of affine Deligne–Lusztig varieties for ${{\,\textrm{GL}\,}}_n$

Abstract

In this paper we study the geometric structure of affine Deligne–Lusztig varieties $X_{\lambda }(b)$ for $\textrm{GL}_n$ and b basic. Motivated by Görtz and He (Camb J Math 3:323–353, 2015) and Chan and Ivanov (Math Ann 380:1801–1890, 2021), we consider a new condition on $\lambda$. If this is satisfied, then $X_{\lambda }(b)$ is the disjoint union of classical Deligne–Lusztig varieties times finite-dimensional affine spaces.

1 Introduction

Let F be a non-archimedean local field with finite residue field ${\mathbb {F}}_q$ of prime characteristic p, and let L be the completion of the maximal unramified extension of F. Let $\sigma$ denote the Frobenius automorphism of L/F. Further, we write ${\mathcal {O}},\ {\mathfrak {p}}$ (resp. ${\mathcal {O}}_F,\ {\mathfrak {p}}_F$) for the valuation ring and the maximal ideal of L. Finally, we denote by $\varpi$ a uniformizer of F (and L) and by $v_L$ the valuation of L such that $v_L(\varpi )=1$.

Let G be a split connected reductive group over ${\mathcal {O}}_F$ and let T be a split maximal torus of it. Fix a Borel subgroup B of G containing T. For a cocharacter $\lambda \in X_*(T)$, let $\varpi ^{\lambda }$ be the image of $\varpi \in {\mathbb {G}}_m(F)$ under the homomorphism $\lambda :{\mathbb {G}}_m\rightarrow T$.

We fix a dominant cocharacter $\lambda \in X_*(T)$ (with respect to B) and $b\in G(L)$. Then the affine Deligne–Lusztig variety $X_{\lambda }(b)$ is the locally closed reduced $\overline{{\mathbb {F}}}_q$-subscheme of the affine Grassmannian defined as

$$\begin{aligned} X_{\lambda }(b)(\overline{{\mathbb {F}}}_q)=\left\{ xG({\mathcal {O}})\in G(L)/G({\mathcal {O}})\mid x^{-1}b\sigma (x)\in G({\mathcal {O}})\varpi ^{\lambda }G({\mathcal {O}})\right\} . \end{aligned}$$

We can also define the Iwahori-level affine Deligne–Lusztig variety associated to $w\in {{\widetilde{W}}}$ and $b\in G(L)$ (see §3.2), where ${{\widetilde{W}}}$ denotes the extended affine Weyl group.

The affine Deligne–Lusztig variety $X_{\lambda }(b)$ carries a natural action (by left multiplication) by the group

$$\begin{aligned} J=\{g\in G(L)\mid g^{-1}b\sigma (g)=b\}. \end{aligned}$$

Since b is usually fixed in the discussion, we mostly omit it from the notation. We denote $J\cap G({\mathcal {O}})$ by $J_{{\mathcal {O}}}$.

The geometric properties of affine Deligne–Lusztig varieties have been studied by many people. For example, the non-emptiness criterion and the dimension formula are already known for the affine Deligne–Lusztig varieties in the affine Grassmannian (see [4, 6] and [25]). Besides them, it is known that in certain cases, the (closed) affine Deligne–Lusztig variety admits a simple description. Let P be a standard parahoric subgroup of G(L). We define

$$\begin{aligned} X(\lambda , b)_P=\left\{ g\in G(L)/P\mid g^{-1}b\sigma (g)\in \bigcup _{w\in {{\,\textrm{Adm}\,}}(\lambda )}PwP\right\} , \end{aligned}$$

where ${{\,\textrm{Adm}\,}}(\lambda )\subset {{\widetilde{W}}}$ is the $\lambda$-admissible set (see § 4.1). In [9] (see also [10]), Görtz and He studied $X(\lambda , b)_P$ in the case that $\lambda$ is minuscule and P is maximal parahoric. They introduced a notion of “Coxeter type” (which we will explain below) and proved that if $(G, \lambda , P)$ is of Coxeter type and if b is a basic element such that $X(\lambda , b)_P\ne \emptyset$, then $X(\lambda , b)_P$ is naturally a union of classical Deligne–Lusztig varieties. Furthermore, the main result in [12] proves that $X(\lambda , b)_P$ is naturally a union of classical Deligne–Lusztig varieties if and only if $\lambda$ is minute (see [11, Definition 3.2], which means that $\lambda$ and hence the elements in ${{\,\textrm{Adm}\,}}(\lambda )$ are “very small”). In particular, the existence of such a simple description is independent of P. Note also that if $(G, \lambda , P)$ is of Coxeter type, then $\lambda$ is minute. Finally the work [13] classified the cases where $(G, \lambda , P)$ is of Coxeter type in general. These simple descriptions of $X(\lambda , b)_P$ have many applications in number theory (cf. [17, Section 8]).

Chan and Ivanov [2] gave an explicit description of Iwahori level affine Deligne–Lusztig varieties associated to $w_{0,r}$ (see § 5.1) and basic b for ${{\,\textrm{GL}\,}}_n$. Each component of the disjoint decomposition described there is a classical Deligne–Lusztig variety times finite-dimensional affine space. They point out the similarity between their description and the results in [9], even though $w_{0,r}$ can have arbitrarily large length in ${{\widetilde{W}}}$. Using this description, they gave a geometric realization of the local Langlands correspondence in many interesting cases.

In the hyperspecial case, the author [24] studied affine Deligne–Lusztig varieties for ${{\,\textrm{GL}\,}}_3$ and b basic. The main theorem there determines the pairs $(\lambda , b)$ such that $X_{\lambda }(b)$ admits the simple description suggested in [2]. Such $\lambda$ is never minute.

In this paper, we introduce another variant of the “Coxeter case” to handle the cases in [2] and [24]. Both the affine Weyl group $W_a$ and the finite Weyl group $W_0$ are subgroups of ${{\widetilde{W}}}$. Moreover, we have two semidirect product decomposition ${{\widetilde{W}}}=W_a\rtimes \Omega$ and ${{\widetilde{W}}}=W_0\ltimes X_*(T)$, where $\Omega \subset {{\widetilde{W}}}$ is the subgroup of length 0 elements. Denote by ${{\tilde{S}}}$ the set of simple affine reflections, and by S the set of simple reflections. To define “Coxeter type”, we use the EKOR stratification on $X(\lambda , b)_P$, which is parametrized by a certain subset of ${{\,\textrm{Adm}\,}}(\lambda )$. More precisely, we say that $(G, \lambda , P)$ is of Coxeter type if each non-empty EKOR stratum in $X(\lambda , b)_P$ is associated to $w\in {{\widetilde{W}}}$ such that its $W_a$-part is a certain Coxeter element in the Coxeter group $(W_a, {{\tilde{S}}})$, where b is a basic element such that $X(\lambda , b)_P\ne \emptyset$ (see [9, §5.1] and [13, §2.3] for details). Analogously, we introduce a notion of “finite Coxeter type” by using a similar stratification on $X_{\lambda }(b)$ (see Definition 4.1). A different point of view is that we use the finite part of $w\in {{\widetilde{W}}}=W_0\ltimes X_*(T)$ instead of the $W_a$-part. Note that the condition on w that its finite part is a Coxeter element in the Coxeter group $(W_0, S)$ does not restrict the length of w in ${{\widetilde{W}}}$. Although the definition of finite Coxeter type also works for arbitrary G, the purpose of this paper is to study this condition for $G={{\,\textrm{GL}\,}}_n$ and a hyperspecial parahoric subgroup $G({\mathcal {O}})\subset G$. In this case, “finite Coxeter type” is just a condition on $\lambda$.

The main results are summarized below. If $n=3$ and $\lambda$ is not central, then these results coincide with [24, Corollary 6.5].

Theorem A

(see Theorem 4.6) Let $G={{\,\textrm{GL}\,}}_n$. Let $\lambda \in X_*(T)_+$, and let $\lambda _{{{\,\textrm{ad}\,}}}$ be the image of $\lambda$ in $X_*(T)/{\mathbb {Z}}\cdot (1,\ldots , 1)$. Then $\lambda$ is of finite Coxeter type if and only if $\lambda _{{{\,\textrm{ad}\,}}}$ has one of the following forms modulo ${\mathbb {Z}}\cdot (1,\ldots , 1)$:

$$\begin{aligned} ((n-1)r+\kappa , -r,\ldots , -r)&,(r,\ldots , r,-(n-1)r-\kappa ), \\ ((n-1)r+1+\kappa , -r,\ldots , -r,-r-1)&, (r+1, r,\ldots , r,-(n-1)r-1-\kappa ), \end{aligned}$$

where $0\le \kappa <n$ is arbitrary and $r\ge 1$ (resp. $r\ge 0$) if $\kappa =0$ (resp. $1\le \kappa <n$).

Theorem B

(see Theorem 5.9) Let $G={{\,\textrm{GL}\,}}_n$. Let $\lambda$ be a dominant cocharacter of finite Coxeter type, and let b be a basic element in G(L) such that $X_{\lambda }(b)\ne \emptyset$. Then the variety $X_{\lambda }(b)$ is a disjoint union, indexed by $J/J_{{\mathcal {O}}}$ or $\sqcup _{j=1}^{n-1} J/J_{{\mathcal {O}}}$, of classical Deligne–Lusztig varieties times finite-dimensional affine spaces. Moreover, these classical Deligne–Lusztig varieties are all associated to a Coxeter element.

Remark 1.1

After we finished this work, He, Nie and Yu released their preprint [19]. They studied Iwahori level affine Deligne–Lusztig varieties associated to $w\in {{\widetilde{W}}}$ with “finite Coxeter parts” in full generality. See Remark 4.3 for this condition. As mentioned in the paragraph right after [19, Theorem 1.1], we will also study some special cases of “finite Coxeter parts” along the way of proving Theorem B. See Corollary 5.5 for this result. Our method is independent of theirs. Moreover, we show that each irreducible component is the product of a classical Deligne–Lusztig variety and a finite-dimensional affine space, while each irreducible component in [19, Theorem 1.1] is an iterated fibration over a classical Deligne–Lusztig variety.

Whenever we consider a scheme or an ind-scheme, for simplicity, we pass to the perfection even in the equal characteristic case. Nevertheless, it is easy to check that in the equal characteristic case, all proofs and ingredients in this paper work for non-perfect rings if we speak of “universal homeomorphism” instead of “isomorphism”.

The paper is organized as follows. In Sect. 2 we fix notation. In Sect. 3, we recollect properties of affine Deligne–Lusztig varieties. In Sect. 4, we introduce the definition of finite Coxeter type, and give a classification using the non-emptiness criterion for affine Deligne–Lusztig varieties in the affine flag variety. In Sect. 5, we describe the geometric structure of affine Deligne–Lusztig varieties in the affine Grassmannian. For this, we first study the Iwahori level affine Deligne–Lusztig varieties using the results in [2] and the Deligne–Lusztig reduction method developed in [8]. After that, we relate the Iwahori and hyperspecial cases.

2 Notation

Throughout the paper we will use the following notation. Let F be a non-archimedean local field with finite residue field ${\mathbb {F}}_q$ of prime characteristic p, and let L be the completion of the maximal unramified extension of F with residue field $\overline{{\mathbb {F}}}_q$. Let $\sigma$ denote the Frobenius automorphism of L/F. Further, we write ${\mathcal {O}},\ {\mathfrak {p}}$ (resp. ${\mathcal {O}}_F,\ {\mathfrak {p}}_F$) for the valuation rings and the maximal ideal of L (resp. F). Finally, we denote by $\varpi$ a uniformizer of F (and L) and by $v_L$ the valuation of L such that $v_L(\varpi )=1$.

If F has positive characteristic, let ${\mathbb {W}}$ be the ring scheme over ${\mathbb {F}}_q$ where for any ${\mathbb {F}}_q$-algebra R, ${\mathbb {W}}(R)=R[[\varpi ]]$. If F has mixed characteristic, let ${\mathbb {W}}$ be the F-ramified Witt ring scheme over ${\mathbb {F}}_q$ so that ${\mathbb {W}}({\mathbb {F}}_q)={\mathcal {O}}_F$ and ${\mathbb {W}}(\overline{{\mathbb {F}}}_q)={\mathcal {O}}$. Below, we restrict to the case that R is a perfect ${\mathbb {F}}_q$-algebra. In this case, the elements of ${\mathbb {W}}(R)$ can be written in the form $\Sigma _{i\ge 0}[r_i]\varpi ^i$, where $[r_i]$ is the Teichmüller lift of $r_i\in R$ if ${{\,\textrm{ch}\,}}F=0$ and $[r_i]=r_i$ if ${{\,\textrm{ch}\,}}F>0$. For any scheme X, we write $X^\textrm{pfn}$ for the perfection of X. We identify $({\mathbb {W}}/\varpi ^h{\mathbb {W}})^\textrm{pfn}$ with the affine space ${\mathbb {A}}^{h,\textrm{pfn}}$ under this choice of coordinates.

From now and until the end of this paper, we set $G={{\,\textrm{GL}\,}}_n$. Let T be the torus of diagonal matrices, and we let B be the Borel subgroup of upper triangular matrices. Further, we set $K=G({\mathcal {O}})$. Let us define the Iwahori subgroup $I\subset K$ as the inverse image of the lower triangular matrices under the projection $G({\mathcal {O}})\rightarrow G(\overline{{\mathbb {F}}}_q),\ \varpi \mapsto 0$.

Let $\Phi =\Phi (G,T)$ denote the set of roots of T in G. We denote by $\Phi _+$ (resp. $\Phi _-$) the set of positive (resp. negative) roots distinguished by B. Let $\chi _{ij}$ be the character $T\rightarrow {\mathbb {G}}_m$ defined by $\textrm{diag}(t_1,t_2,\ldots , t_n)\mapsto t_i{t_j}^{-1}$. Using this notation, we have $\Phi =\{\chi _{ij}\mid i\ne j\}$, $\Phi _+=\{\chi _{ij}\mid i< j\}$ and $\Phi _-=\{\chi _{ij}\mid i> j\}$. We let

$$\begin{aligned} X_*(T)_+=\{\lambda \in X_*(T)| \langle \alpha , \lambda \rangle \ge 0\ \text {for all}\ \alpha \in \Phi _+\} \end{aligned}$$

denote the set of dominant cocharacters. Through the isomorphism $X_*(T)\cong {\mathbb {Z}}^n$, ${X_*(T)}_+$ can be identified with the set $\{(m_1,\ldots , m_n)\in {\mathbb {Z}}^n|m_1\ge \cdots \ge m_n\}$.

We embed $X_*(T)$ into T(L) by $\lambda \mapsto \varpi ^{\lambda }$, where by $\varpi ^{\lambda }$ we denote the image of $\varpi$ under the map

$$\begin{aligned} \lambda :L^{\times }={\mathbb {G}}_m(L)\rightarrow T(L). \end{aligned}$$

The extended affine Weyl group ${{\widetilde{W}}}$ is defined as the quotient $N_{G(L)}T(L)/T({\mathcal {O}})$. This can be identified with the semi-direct product $W_0\ltimes X_{*}(T)$, where $W_0$ is the finite Weyl group of G. Let $e_1,\ldots , e_n\in L^n$ be the canonical basis. For any permutation $\tau$ of degree n, we denote by ${{\dot{\tau }}}$ the matrix of the form $(e_{\tau (1)}\ e_{\tau (2)} \ \cdots \ e_{\tau (n)})$. In our case, we can identify the symmetric group of degree n with $W_0$ by sending $\tau$ to the element in $W_0$ represented by ${{\dot{\tau }}}$. We have a length function $\ell :{{\widetilde{W}}}\rightarrow {\mathbb {Z}}_{\ge 0}$ given as

$$\begin{aligned} \ell (w_0\varpi ^{\lambda })=\sum _{\alpha \in \Phi _+, w_0\alpha \in \Phi _-}|\langle \alpha , \lambda \rangle +1|+\sum _{\alpha \in \Phi _+, w_0\alpha \in \Phi _+}|\langle \alpha , \lambda \rangle |, \end{aligned}$$

where $w_0\in W_0$ and $\lambda \in X_*(T)$.

Let $S\subset W_0$ denote the subset of simple reflections, i.e., adjacent transpositions. In this paper, let us write $s_1=(1\ 2), s_2=(2\ 3), \ldots , s_{n-1}=(n-1\ n)$. Set $s_0=\varpi ^{\chi _{1,n}^{\vee }}(1\ n)$, where $\chi _{1,n}$ is the unique highest root. The affine Weyl group $W_a$ is the subgroup of ${{\widetilde{W}}}$ generated by ${{\tilde{S}}}=S\cup \{s_0\}$. Then we can write the extended affine Weyl group as a semi-direct product ${{\widetilde{W}}}\cong W_a\rtimes \Omega$, where $\Omega \subset {{\widetilde{W}}}$ is the subgroup of length 0 elements. Moreover, $(W_a, S\cup \{s_0\})$ is a Coxeter system. The restriction of the length function $\ell |_{W_a}$ is the length function on $W_a$ given by the fixed system of generators. We denote by ${^S{{\widetilde{W}}}}$ the set of minimal length elements for the cosets in $W_0\backslash {{\widetilde{W}}}$ (cf. [21, (2.4.5)]).

Let $\Phi _a$ denote the set of affine roots of T in G. As usual we order the affine roots in such a way that the simple affine roots are the functions

$$\begin{aligned} \chi _{i,i+1}:{\mathbb {R}}^n\rightarrow {\mathbb {R}},\quad (x_1,\ldots , x_n)\mapsto x_i-x_{i+1}\ (1\le i \le n-1) \end{aligned}$$

together with the affine linear function

$$\begin{aligned} \chi _0:{\mathbb {R}}^n\rightarrow {\mathbb {R}},\quad (x_1,\ldots , x_n)\mapsto x_n-x_1+1. \end{aligned}$$

The extended affine Weyl group ${{\widetilde{W}}}$ acts on $\Phi _a$. For example, we have

$$\begin{aligned} \tau \chi _{i,i+1}=\chi _{\tau (i),\tau (i+1)},\quad \tau \in W_0 \end{aligned}$$

and

$$\begin{aligned} \varpi ^{\lambda }\chi _{i,i+1}=\chi _{i,i+1}-\langle \chi _{i,i+1}, \lambda \rangle \delta ,\quad \lambda \in X_*(T), \end{aligned}$$

where $\delta$ is the constant function with value 1.

3 Affine Deligne–Lusztig varieties

In this section we first recall the affine Grassmannian and the affine flag variety for ${{\,\textrm{GL}\,}}_n$. After that, we will recall the definition of affine Deligne–Lusztig varieties.

3.1 The Affine Grassmannian

Let X (resp. ${\mathcal {X}}$) be a scheme over F (resp. ${\mathcal {O}}_F$). For any perfect ${\mathbb {F}}_q$-algebra R, the functor LX defined by

$$\begin{aligned} LX(R)=X({\mathbb {W}}(R)[\varpi ^{-1}]) \end{aligned}$$

is called the loop space of X. The positive loop group of ${\mathcal {X}}$ is the functor defined by

$$\begin{aligned} L^+{\mathcal {X}}(R)={\mathcal {X}}({\mathbb {W}}(R)). \end{aligned}$$

Let ${\mathcal {G}}$ be a smooth affine group scheme over ${\mathcal {O}}_F$ with generic fiber G. The affine Grassmannian of ${\mathcal {G}}$ is the fpqc quotient $LG/L^+{{\mathcal {G}}}$. It has the following representability property.

Theorem 3.1

The fpqc-sheaf $LG/L^+{{\mathcal {G}}}$ is represented by a strict ind-perfect scheme.

Proof

This is true for any connected reductive group over F. In the equal characteristic case, see [23, Theorem 1.4]. In the mixed characteristic case, see [1, Corollary 9.6]. $\square$

If ${\mathcal {G}}({\mathcal {O}})=K$ (resp. ${\mathcal {G}}({\mathcal {O}})=I$), then we call $LG/L^+{{\mathcal {G}}}$ the affine Grassmannian (resp. affine flag variety) for G, and denote it by ${\mathcal {G}}rass$ (resp. ${\mathcal {F}}lag$). Let $\pi$ denote the projection ${\mathcal {F}}lag\rightarrow {\mathcal {G}}rass$.

The affine Grassmannian for G can be interpreted as parameter spaces of lattices satisfying certain conditions. Explicitly, for any perfect ${\mathbb {F}}_q$-algebra R, ${\mathcal {G}}rass(R)$ can be seen as the set of finite projective ${\mathbb {W}}(R)$-submodules ${\mathscr {L}}$ of ${\mathbb {W}}(R)[\varpi ^{-1}]^n$ such that ${\mathscr {L}}\otimes _{{\mathbb {W}}(R)}{\mathbb {W}}(R)[\varpi ^{-1}]={\mathbb {W}}(R)[\varpi ^{-1}]^n$. For any ${\mathscr {L}}\in {\mathcal {G}}rass(R)$, we define its dual lattice ${\mathscr {L}}^*$ by ${{\,\textrm{Hom}\,}}_{{\mathbb {W}}(R)}({\mathscr {L}}, {\mathbb {W}}(R))$, which is also a finite projective ${\mathbb {W}}(R)$-module. Further, we have

$$\begin{aligned} {\mathscr {L}}^*\subset {\mathscr {L}}^*\otimes _{{\mathbb {W}}(R)}{\mathbb {W}}(R)[\varpi ^{-1}]={{\,\textrm{Hom}\,}}_{{\mathbb {W}}(R)[\varpi ^{-1}]}({\mathbb {W}}(R)[\varpi ^{-1}]^n, {\mathbb {W}}(R)[\varpi ^{-1}]). \end{aligned}$$

Let us denote the ${\mathbb {W}}(R)[\varpi ^{-1}]$-module ${{\,\textrm{Hom}\,}}_{{\mathbb {W}}(R)[\varpi ^{-1}]}({\mathbb {W}}(R)[\varpi ^{-1}]^n, {\mathbb {W}}(R)[\varpi ^{-1}])$ by $({\mathbb {W}}(R)[\varpi ^{-1}]^n)^*$. Let $e_1,\ldots , e_n\in {\mathbb {W}}(R)[\varpi ^{-1}]^n$ be the canonical basis. Let $e_1^*,\ldots , e_n^*\in ({\mathbb {W}}(R)[\varpi ^{-1}]^n)^*$ be its dual basis, i.e., the ${\mathbb {W}}(R)[\varpi ^{-1}]$-linear maps ${\mathbb {W}}(R)[\varpi ^{-1}]^n\rightarrow {\mathbb {W}}(R)[\varpi ^{-1}]$ defined as

$$\begin{aligned} e_{i^{*}}(e_{j})= \left\{ \begin{array}{ll} 1 &{} (i=j) \\ 0 &{} (i\ne j). \\ \end{array}\right. \end{aligned}$$

Then we have an isomorphism ${\mathbb {W}}(R)[\varpi ^{-1}]^n\cong ({\mathbb {W}}(R)[\varpi ^{-1}]^n)^*$ defined by sending $e_i$ to $e_i^*$. Under this isomorphism, we may consider ${\mathscr {L}}^*$ as an element of ${\mathcal {G}}rass(R)$.

Proposition 3.2

The morphism $*:{\mathcal {G}}rass\rightarrow {\mathcal {G}}rass$ defined by ${\mathscr {L}}\mapsto {\mathscr {L}}^*$ as above is an automorphism. Moreover, the image of $g{\mathcal {O}}^n\in {\mathcal {G}}rass(\overline{{\mathbb {F}}}_q)$ is ${^tg}^{-1}{\mathcal {O}}^n\in {\mathcal {G}}rass(\overline{{\mathbb {F}}}_q)$.

Proof

The first assertion is clear because $*^2={{\,\textrm{id}\,}}$. For the last assertion, let $g\in G(L)$. Then g defines an automorphism $\varphi _g:L^n\rightarrow L^n,\ x\mapsto gx$. Its dual $\varphi _g^*$ is represented by $^tg$ relative to the basis $e_1^*,\ldots , e_n^*$. The assertion follows from the equation $\varphi _g^*((g{\mathcal {O}}^n)^*)=({\mathcal {O}}^n)^*$. $\square$

3.2 Affine Deligne–Lusztig varieties

The affine Deligne–Lusztig variety is defined as follows.

Definition 3.3

The affine Deligne–Lusztig variety $X_{\lambda }(b)$ in the affine Grassmannian associated to $b\in G(L)$ and $\lambda \in X_*(T)_+$ is given by

$$\begin{aligned} X_{\lambda }(b)=\{gK\in G(L)/K\mid g^{-1}b\sigma (g)\in K\varpi ^{\lambda }K\}\subset {\mathcal {G}}rass. \end{aligned}$$

The affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety associated to $b\in G(L)$ and $w\in {{\widetilde{W}}}$ is given by

$$\begin{aligned} X_w(b)=\{gI\in G(L)/I\mid g^{-1}b\sigma (g)\in IwI\}\subset {\mathcal {F}}lag. \end{aligned}$$

Affine Deligne–Lusztig varieties are perfect schemes, locally perfectly of finite type over $\overline{{\mathbb {F}}}_q$ (cf. [16, Corollary 6.5], [15, Lemma 1.1]). Let us denote by $J_b$ the $\sigma$-centralizer of b. Then the varieties $X_{\lambda }(b)$ and $X_w(b)$ are equipped with an action of $J_b(F)$.

Let B(G) be the set of $\sigma$-conjugacy classes of elements in G(L). For a dominant cocharacter $\lambda$, we define a subset $B(G, \lambda )$ of B(G) as the set of $b\in B(G)$ satisfying $\nu _b\preceq \lambda$, where $\nu _b$ is the Newton vector of b and $\preceq$ denotes the dominance order on $X_*(T)_+$. Inside $B(G,\lambda )$, there is always a unique basic element. Moreover, we have the following criterion for non-emptiness of $X_{\lambda }(b)$:

Theorem 3.4

The variety $X_{\lambda }(b)$ is non-empty if and only if the $\sigma$-conjugacy class of b is contained in $B(G, \lambda )$.

Proof

This is [4, Theorem 1.1]. $\square$

For $X_w(b)$ in the affine flag variety, a complete answer to the non-emptiness question is not known. However, if b is basic, we have a criterion using the notion of “P-alcove”. To explain this, we need some notation. Let $\textbf{a}$ be the base alcove corresponding to I (as in the sense of [7, 1.2]). Let $S'\subset S$ and $w_0\in W_0$. Let $U_{ij}$ be the root subgroup for $\chi _{ij}\in \Phi$. We denote by $P_{S'}$ the standard parabolic subgroup corresponding to $S'$. For any $x\in {{\widetilde{W}}}$, we say $x\textbf{a}$ is a $^{w_0}P_{S'}$-alcove, if

• (i) $w_0^{-1}x w_0\in {{\widetilde{W}}}_{S'}:= X_*(T)\rtimes W_{S'}$, and

• (ii) For any $\chi _{ij}\in w_0(\Phi _+\setminus \Phi _{S'})$, $U_{ij}\cap {^xI}\subseteq U_{ij}\cap I$,

where $W_{S'}\subset W_0$ is the subgroup generated by $S'$, and $\Phi _{S'}$ is the set of roots spanned by the simple roots corresponding to $S'$. Finally, let $M_{S'}$ be the Levi component of $P_{S'}$, and let $\kappa _{M_{S'}}$ be the corresponding Kottwitz map.

Theorem 3.5

Let $b\in G(L)$ be a basic element, and let $x\in {{\widetilde{W}}}$. Then $X_x(b)\ne \emptyset$ if and only if, for every $(S', w_0)$ for which $x\textbf{a}$ is a $^{w_0}P_{S'}$-alcove, b is $\sigma$-conjugate to an element $b'\in M_{S'}(L)$ and x and $b'$ have the same image under $\kappa _{M_{S'}}$.

Proof

This is conjectured in [7, Conjecture 1.1.1], and proved in [11, Theorem A]. $\square$

Let $p_1$ be the projection ${{\widetilde{W}}}=W_0\ltimes X_*(T)\rightarrow W_0$. We define $p_2:{{\widetilde{W}}}\rightarrow W_0$ as follows: The image $p_2(w)$ is the unique element in $W_0$ such that $p_2(w)^{-1}w\textbf{a}$ is contained in the dominant chamber. We also have a more explicit criterion for the emptiness of $X_w(b)$.

Proposition 3.6

Let $b\in G(L)$ be a basic element. Let $w\in {{\widetilde{W}}}$, and write $w=\varpi ^{\lambda }w_0$ with $\lambda \in X_*(T),\ w_0\in W_0$. Assume that $\lambda \ne \nu _b$ and $p_2(w)^{-1}p_1(w)p_2(w)\in \bigcup _{S'\subsetneq S}W_{S'}$, where $\nu _b$ is the Newton vector of b and $W_{S'}\subset W_0$ is the subgroup generated by $S'$. Then $X_w(b)=\emptyset$.

Proof

This is [7, Proposition 9.5.4] (see also [7, Corollary 11.3.5]). $\square$

3.3 Deligne–Lusztig reduction method

For affine Deligne–Lusztig varieties $X_w(b)\subset {\mathcal {F}}lag$, we have the following reduction method developed in [8, 2.5] (see also [20, Proposition 3.3.1]).

Proposition 3.7

Let $w\in {{\widetilde{W}}}$, $s\in {{\tilde{S}}}$ and $b\in G(L)$.

• (i) Let $\eta \in \Omega$. Then there exists a $J_b(F)$-equivariant isomorphism

  $$\begin{aligned} X_w(b)\xrightarrow {\sim } X_{\eta w \eta ^{-1}}(b). \end{aligned}$$

• (ii) If $\ell (sws)=\ell (w)$, then there exists a $J_b(F)$-equivariant isomorphism

  $$\begin{aligned} X_w(b)\xrightarrow {\sim } X_{sws}(b). \end{aligned}$$

• (iii) If $\ell (sws)=\ell (w)-2$, then $X_w(b)$ has a $J_b(F)$-stable closed subscheme $X_1$ satisfying the following conditions:

  • (a) There exists a $J_b(F)$-equivariant morphism

    $$\begin{aligned} X_1\rightarrow X_{sws}(b) \end{aligned}$$

    which is a Zariski-locally trivial ${\mathbb {A}}^{1,\textrm{pfn}}$-bundle.

  • (b) Let $X_2$ be the $J_b(F)$-stable open subscheme of $X_w(b)$ complement to $X_1$. Then there exists a $J_b(F)$-equivariant morphism

    $$\begin{aligned} X_2\rightarrow X_{sw}(b) \end{aligned}$$

    which is a Zariski-locally trivial ${\mathbb {G}}_m^\textrm{pfn}$-bundle.

Proof

All of the statements are proved in [8, 2.5] (note that in our case, the Frobenius endmorphism $\sigma$ is an isomorphism). $\square$

Here we will give the set-theoretical description of the morphisms in Proposition 3.7. First of all, the isomorphism in (i) is given as the map

$$\begin{aligned} gI\in X_w(b)\mapsto g\eta ^{-1}I \in X_{\eta w \eta ^{-1}}(b). \end{aligned}$$

Let us denote by $\textrm{inv}$ the relative position map on ${\mathcal {F}}lag$. If $\ell (sw)<\ell (w)$, let $C_g$ be the unique element in ${\mathcal {F}}lag$ such that $\textrm{inv}(gI, C_g)=s$ and $\textrm{inv}(C_g, b\sigma (g)I)=sw$. Using this notation, we describe the isomorphism in (ii) as the map

$$\begin{aligned} gI\in X_w(b)\mapsto C_g \in X_{sws}(b). \end{aligned}$$

Note that in this case we always have $\textrm{inv}(C_g, b\sigma (C_g))=sws$ because $\ell (sws)>\ell (sw)$. If $\ell (sw)>\ell (w)$, then by exchanging w and sws, we can reduce to the case $\ell (sw)<\ell (w)$. Finally, let us explain (iii). The set $X_1$ (resp. $X_2$) consists of the elements $gI\in X_w(b)$ satisfying $\textrm{inv}(C_g, b\sigma (C_g))=sws$ (resp. $\textrm{inv}(C_g, b\sigma (C_g))=sw$), and both of the maps in (iii) are given as the map sending gI to $C_g$.

4 A simple condition on $\lambda$

In this section, we consider a simple condition on $\lambda \in X_*(T)_+$. In Sect. 5, we will show that if $\lambda$ satisfies this condition, then $X_{\lambda }(b)$ has a simple geometric structure.

4.1 A decomposition of $X_{\lambda }(b)$

For any $\lambda \in X_*(T)_+$, the $\lambda$-admissible set ${{\,\textrm{Adm}\,}}(\lambda )$ is defined as

$$\begin{aligned} {{\,\textrm{Adm}\,}}(\lambda )=\{w\in {{\widetilde{W}}}\mid w\le \varpi ^{\lambda '}\ \text {for some}\ \lambda '\in W_0\lambda \}, \end{aligned}$$

where $\le$ denotes the Bruhat order on ${{\widetilde{W}}}$. Set ${^S\textrm{Adm}}(\lambda )^\circ ={^S{{\widetilde{W}}}}\cap W_0\varpi ^{\lambda }W_0$. Using [21, (2.3.3)], we can easily check that ${^S\textrm{Adm}}(\lambda )^\circ \subset {{\,\textrm{Adm}\,}}(\lambda )$. Then, by [9, Theorem 3.2.1] (see also [14, 2.4]), we have a stratification

$$\begin{aligned} X_{\lambda }(b)=\bigsqcup _{w\in {^S\textrm{Adm}}(\lambda )^\circ }\pi (X_w(b)) \end{aligned}$$

for any $b\in G(L)$. Further, let ${^S\textrm{Adm}}(\lambda )^\circ _{{{\,\textrm{cox}\,}}}$ be the subset of ${^S\textrm{Adm}}(\lambda )^\circ$ defined as

$$\begin{aligned} \{w\in {^S\textrm{Adm}}(\lambda )^\circ \mid p_1(w)\ \text {is a Coxeter element in}\ W_0\}. \end{aligned}$$

Definition 4.1

We say an element $\lambda \in X_*(T)_+$ is of finite Coxeter type if

where b is an element in G(L) whose $\sigma$-conjugacy class is the unique basic element in $B(G, \lambda )$.

Remark 4.2

Clearly, this definition can be applied to general G.

Remark 4.3

In [19], He-Nie-Yu defined the finite part of $w\in {{\widetilde{W}}}$ and studied the affine Deligne–Lusztig variety associated to w such that its finite part is a Coxeter element. Note that any element $w\in {{\widetilde{W}}}$ can be written in a unique way as $xt^\mu y$ with $\mu$ dominant, $x,y\in W_0$ such that $t^\mu y\in {^S{{\widetilde{W}}}}$. For split groups, the finite part of w is defined to be yx. In particular, this finite part coincides with $p_1(w)$ if $w\in {^S{{\widetilde{W}}}}$.

Since affine Deligne–Lusztig varieties are isomorphic if the element b is replaced by another element $b'$ in the same $\sigma$-conjugacy class, this definition does not depend on the choice of b. Further, let $m\in {\mathbb {Z}}$, and let $\lambda _m=(m,\ldots , m), c_m=\varpi ^{\lambda _m}$. Then $X_{\lambda }(b)$ (resp. $X_w(b)$) is equal to $X_{\lambda +\lambda _m}(c_m b)$ (resp. $X_{c_m w}(c_m b)$) as a subset of the affine Grassmannian (resp. the affine flag variety). Thus, to study $X_{\lambda }(b)$, we may replace $\lambda$ by another element $\lambda '$ in $\lambda _{{{\,\textrm{ad}\,}}}$, where $\lambda _{{{\,\textrm{ad}\,}}}$ denotes the image of $\lambda$ in the quotient $X_*(T)/{\mathbb {Z}}\lambda _1$. In particular, the property of $\lambda$ to be finite Coxeter type only depends on $\lambda _{{{\,\textrm{ad}\,}}}$. By abuse of notation, we will write $\lambda _{{{\,\textrm{ad}\,}}}=(m_1,\ldots , m_n)$ if $\lambda =(m_1,\ldots , m_n)$.

4.2 Classification

In this subsection, we determine all $\lambda$ of finite Coxeter type. For this purpose, the following observation is essential:

Lemma 4.4

Let $\lambda =(m_1,\ldots , m_n)\in X_*(T)_+$. Let $w_0$ be a permutation such that $w_0(k)>w_0(l)$ implies $m_{w_0(k)}<m_{w_0(l)}$ (equivalently $m_{w_0(k)}\ne m_{w_0(l)}$) for any $k<l$. Then $w_0\varpi ^{w_0^{-1}(\lambda )}\in {{\widetilde{W}}}$ belongs to ${^S\textrm{Adm}}(\lambda )^\circ$.

Proof

By the assumption, we have

$$\begin{aligned} \ell (w_0\varpi ^{w_0^{-1}(\lambda )})=\ell (\varpi ^{w_0^{-1}(\lambda )})-\ell (w_0) \end{aligned}$$

because $w_0^{-1}\lambda =(m_{w_0(1)},\ldots , m_{w_0(n)})$. This implies both $w_0\varpi ^{w_0^{-1}(\lambda )}\in {{\,\textrm{Adm}\,}}(\lambda )$ and $w_0\varpi ^{w_0^{-1}(\lambda )}\in {^S{{\widetilde{W}}}}$. $\square$

Using the lemma above, we will show the following proposition:

Proposition 4.5

Let $\lambda =(m_1,\ldots , m_n)\in X_*(T)_+$. Assume $n\ge 4$. If there exists $2\le i\le n-2$ such that $m_i>m_{i+1}$ (equivalently $m_i\ne m_{i+1}$), then $\lambda$ is not of finite Coxeter type.

Proof

Let us first consider the case $n\ge 5$. In this case, it is enough to show that there exists a cycle $c=(j_1\ j_2\ \cdots \ j_n)$ such that $\ell (c)>n-1$, $c^{-1}(1)<\cdots < c^{-1}(i)$ and $c^{-1}(i+1)<\cdots < c^{-1}(n)$ for any $2\le i\le n-2$. Indeed, by Lemma 4.4, $c\varpi ^{c^{-1}(\lambda )}$ belongs to ${^S\textrm{Adm}}(\lambda )^\circ \setminus {^S\textrm{Adm}}(\lambda )^\circ _{{{\,\textrm{cox}\,}}}$ if $m_i>m_{i+1}$. However, the variety $X_{c\varpi ^{c^{-1}(\lambda )}}(b)\ (b\in B(G, \lambda ))$ is non-empty. This follows from Theorem 3.5 and the fact that $w_0^{-1}c\varpi ^{c^{-1}(\lambda )} w_0\in {{\widetilde{W}}}_{S'}$ implies $S'=S$.

To find such c, we will use induction on n. If $n=5$, $c=(1\ 3\ 5\ 2\ 4)$ (resp. $c=(1\ 4\ 2\ 5\ 3)$) satisfies the condition when $i=2$ (resp. $i=3$). Let us suppose $n\ge 6$. By the induction hypothesis, there exists a cycle $c=(j_1\ j_2\ \cdots \ j_{n-1})$ fixing n such that $\ell (c)>n-2$, $c^{-1}(1)<\cdots < c^{-1}(i)$ and $c^{-1}(i+1)<\cdots < c^{-1}(n-1)$ for any $2\le i\le n-3$. Set $c'=c(n-1\ n)$. Then it is easy to check that $\ell (c')=\ell (c)+1>n-1$, $c'^{-1}(1)<\cdots < c'^{-1}(i)$ and $c'^{-1}(i+1)<\cdots < c'^{-1}(n)$ for any $2\le i\le n-3$. Similarly, by the induction hypothesis, there exists a cycle $c=(j_1\ j_2\ \cdots \ j_{n-1})$ fixing 1 such that $\ell (c)>n-2$, $c^{-1}(2)<\cdots < c^{-1}(n-2)$ and $c^{-1}(n-1)< c^{-1}(n)$. Set $c''=c(1\ 2)$. Then it is easy to check that $\ell (c'')=\ell (c)+1>n-1$, $c''^{-1}(1)<\cdots < c''^{-1}(n-2)$ and $c''^{-1}(n-1)<c''^{-1}(n)$. This finishes the proof for the case $n\ge 5$.

We next consider the case $n=4$. If $m_1>m_2>m_3>m_4$ or $m_1>m_2>m_3=m_4$, set $c=(1\ 3\ 2\ 4)$. Then, by Lemma 4.4 and Theorem 3.5, we have $c\varpi ^{c^{-1}(\lambda )}\in {^S\textrm{Adm}}(\lambda )^\circ {\setminus } {^S\textrm{Adm}}(\lambda )^\circ _{{{\,\textrm{cox}\,}}}$ and $X_{c\varpi ^{c^{-1}(\lambda )}}(b)\ne \emptyset$, where $b\in B(G, \lambda )$. If $m_1=m_2>m_3>m_4$, the same is true for $c=(1\ 4\ 2\ 3)$.

If $m_1=m_2>m_3=m_4$, we may assume $\lambda =(m_1,m_1,0,0)$. Set $w_0=(1\ 3)(2\ 4)$ and consider $w_0\varpi ^{w_0^{-1}(\lambda )}\in {{\widetilde{W}}}$. Then its representative $\dot{w}_0\varpi ^{w_0^{-1}(\lambda )}\in G(L)$ is a basic element with Newton vector $(\frac{m_1}{2}, \frac{m_1}{2}, \frac{m_1}{2}, \frac{m_1}{2})(\preceq \lambda )$. Moreover, by Lemma 4.4, $w_0\varpi ^{w_0^{-1}(\lambda )}$ belongs to ${^S\textrm{Adm}}(\lambda )^\circ \setminus {^S\textrm{Adm}}(\lambda )^\circ _{{{\,\textrm{cox}\,}}}$. However, the variety $X_{w_0\varpi ^{w_0^{-1}(\lambda )}}(\dot{w}_0\varpi ^{w_0^{-1}(\lambda )})$ is non-empty because $\dot{w}_0\varpi ^{w_0^{-1}(\lambda )}I\in X_{w_0\varpi ^{w_0^{-1}(\lambda )}}(\dot{w}_0\varpi ^{w_0^{-1}(\lambda )})$. This finishes the proof for the case $n=4$. $\square$

The goal of this section is to prove the following result.

Theorem 4.6

Let $\lambda \in X_*(T)_+$. Then $\lambda$ is of finite Coxeter type if and only if $\lambda _{{{\,\textrm{ad}\,}}}$ has one of the following forms:

$$\begin{aligned} ((n-1)r+\kappa , -r,\ldots , -r)&,(r,\ldots , r,-(n-1)r-\kappa ), \\ ((n-1)r+1+\kappa , -r,\ldots , -r,-r-1)&, (r+1, r,\ldots , r,-(n-1)r-1-\kappa ), \end{aligned}$$

where $0\le \kappa <n$ is arbitrary and $r\ge 1$ (resp. $r\ge 0$) if $\kappa =0$ (resp. $1\le \kappa <n$).

Proof

We first prove that the condition is necessary. Clearly, $\lambda$ is not of finite Coxeter type if $\lambda _{{{\,\textrm{ad}\,}}}=(0,\ldots , 0)$ unless $n=1$. So, by Proposition 4.5, we are reduced to treat the following three cases:

• (i) $m_1>m_2=\cdots =m_n$,

• (ii) $m_1=\cdots =m_{n-1}>m_n$,

• (iii) $m_1>m_2=\cdots =m_{n-1}>m_n$.

If $\lambda$ satisfies (i) (resp. (ii)) and $m_1+\cdots +m_n=\kappa$ (resp. $m_1+\cdots +m_n=-\kappa$), then it is easy to see that $\lambda$ has one of the forms in the theorem.

Let $\lambda$ be a dominant cocharacter satisfying (iii). If, moreover, $\lambda$ is of finite Coxeter type, then we have $m_1-m_2=1$ or $m_{n-1}-m_n=1$. To show this, let $s=(1\ n)$ and consider $s\varpi ^{s^{-1}(\lambda )}\in {{\widetilde{W}}}$. This belongs to ${^S\textrm{Adm}}(\lambda )^\circ \setminus {^S\textrm{Adm}}(\lambda )^\circ _{{{\,\textrm{cox}\,}}}$ by Lemma 4.4. So it is enough to show that if $m_1-m_2>1$ and $m_{n-1}-m_n>1$, then $X_{s\varpi ^{s^{-1}(\lambda )}}(b)\ne \emptyset$, where $b\in B(G, \lambda )$. If $(w_0, S')$ is a pair satisfying $w_0^{-1}s\varpi ^{s^{-1}(\lambda )} w_0\in {{\widetilde{W}}}_{S'}$, then

$$\begin{aligned} \{s_{\min \{w_0^{-1}(1),w_0^{-1}(n)\}}, s_{\min \{w_0^{-1}(1),w_0^{-1}(n)\}+1},\ldots , s_{\max \{w_0^{-1}(1),w_0^{-1}(n)\}-1}\}\subseteq S'. \end{aligned}$$

If, moreover, $\chi _{1, w_0^{-1}(1)}$ and $\chi _{w_0^{-1}(n), n}$ are both contained in $\Phi _{S'}$, then $S'=S$. Indeed, $\chi _{1, w_0^{-1}(1)}\in \Phi _{S'}$ and $\chi _{w_0^{-1}(n), n}\in \Phi _{S'}$ imply $\{s_1, s_2,\ldots , s_{w_0^{-1}(1)-1}\}\subseteq S'$ and $\{s_{w_0^{-1}(n)}, s_{w_0^{-1}(n)+1},\ldots , s_{n-1} \}\subseteq S'$, respectively. Note that the $(w_0(1), 1)$-th (resp. $(n, w_0(n))$-th) entry of $^{s\varpi ^{s^{-1}(\lambda )}}I$ is ${\mathfrak {p}}^{m_{w_0(1)}-m_1+1}={\mathfrak {p}}^{m_2-m_1+1}$ (resp. ${\mathfrak {p}}^{m_n-m_{w_0(n)}+1}={\mathfrak {p}}^{m_n-m_{n-1}+1}$) when $S'\ne S$. Thus, if $m_1-m_2>1$ and $m_{n-1}-m_n>1$, there is no pair $(w_0, S')$ such that $s\varpi ^{s^{-1}(\lambda )}\textbf{a}$ is a $^{w_0}P_{S'}$-alcove unless $S'=S$. By Theorem 3.5, this implies $X_{s\varpi ^{s^{-1}(\lambda )}}(b)\ne \emptyset$ for $b\in B(G, \lambda )$.

To finish the proof for the necessity, it remains to show that $\lambda =(1,0,\ldots ,0,-1)$ is not of finite Coxeter type. By Lemma 4.4, $s_0$ belongs to ${^S\textrm{Adm}}(\lambda )^\circ {\setminus } {^S\textrm{Adm}}(\lambda )^\circ _{{{\,\textrm{cox}\,}}}$. However, $\dot{s}_0=\varpi ^{\chi _{1,n}^{\vee }}\dot{(1\ n)}\in G(L)$ is a basic element with Newton vector $(0,\ldots , 0)(\preceq \lambda )$ and $\dot{s}_0I$ is contained in $X_{s_0}(\dot{s}_0)$, i.e., $X_{s_0}(\dot{s}_0)\ne \emptyset$. This means that $\lambda$ is not of finite Coxeter type.

For the sufficiency, we have to show $X_w(b)=\emptyset$ for any $w\in {^S\textrm{Adm}}(\lambda )^\circ \setminus {^S\textrm{Adm}}(\lambda )^\circ _{{{\,\textrm{cox}\,}}}$, where $\lambda _{{{\,\textrm{ad}\,}}}$ is one of the cocharacters in the theorem and $b\in B(G, \lambda )$. As explained in §4.1, it is enough to show this for a representative in $\lambda _{{{\,\textrm{ad}\,}}}$.

If $\lambda =((n-1)r+\kappa , -r,\ldots , -r)$ with $\kappa , r$ in the theorem, then it is easy to check

$$\begin{aligned} {^S\textrm{Adm}}(\lambda )^\circ =\{w_0\varpi ^{w_0^{-1}(\lambda )}\mid w_0=(1\ 2\ \cdots \ j)\ \text {for some}\ 1\le j\le n\}. \end{aligned}$$

If $w_0=(1\ 2\ \cdots \ j)$, then $p_1(w_0\varpi ^{w_0^{-1}(\lambda )})=w_0, p_2(w_0\varpi ^{w_0^{-1}(\lambda )})=1$, and hence

$$\begin{aligned} p_2(w_0\varpi ^{w_0^{-1}(\lambda )})^{-1}p_1(w_0\varpi ^{w_0^{-1}(\lambda )})p_2(w_0\varpi ^{w_0^{-1}(\lambda )})=w_0. \end{aligned}$$

By Proposition 3.6, $X_{w_0\varpi ^{w_0^{-1}(\lambda )}}(b)=\emptyset$ for $b\in B(G, \lambda )$ and $w_0=(1\ 2\ \cdots \ j)$ with $1\le j\le n-1$. Thus $\lambda$ is of finite Coxeter type in this case. The proof for the case $\lambda =(r,\ldots , r,-(n-1)r-\kappa )$ is similar.

If $\lambda =((n-1)r+1+\kappa , -r,\ldots , -r,-r-1)$ with $\kappa , r$ in the theorem, then any element in ${^S\textrm{Adm}}(\lambda )^\circ$ can be written as $w_0\varpi ^{w_0^{-1}(\lambda )}$ for some $w_0\in W_0$. If $w_0\varpi ^{w_0^{-1}(\lambda )}\in {^S\textrm{Adm}}(\lambda )^\circ$ satisfies $w_0^{-1}(1)<w_0^{-1}(n)$, then we have

$$\begin{aligned} w_0=(1\ 2\ \cdots \ w_0^{-1}(1))(n\ n-1\ \cdots \ w_0^{-1}(n)). \end{aligned}$$

Using Proposition 3.6, we can check that $X_{w_0\varpi ^{w_0^{-1}(\lambda )}}(b)$ is empty for $b\in B(G, \lambda )$. If $w_0\varpi ^{w_0^{-1}(\lambda )}\in {^S\textrm{Adm}}(\lambda )^\circ$ satisfies $w_0^{-1}(1)>w_0^{-1}(n)$, then we have

$$\begin{aligned} w_0=(1\ 2\ \cdots \ w_0^{-1}(n)\ n\ n-1\ \cdots \ w_0^{-1}(1)). \end{aligned}$$

This is a Coxeter element if and only if $w_0^{-1}(1)=w_0^{-1}(n)+1$. Set $k=w_0^{-1}(1),\ l=w_0^{-1}(n)$. Let ${{\dot{\eta }}}$ be the matrix of the form $\begin{pmatrix} 0 &{} \varpi \\ 1_{n-1} &{} 0 \end{pmatrix}$. Then its image $\eta$ in ${{\widetilde{W}}}$ is contained in $\Omega$. Further, $\eta ^{n-k+1}w_0\varpi ^{w_0^{-1}(\lambda )}\eta ^{-(n-k+1)}$ is equal to

$$\begin{aligned} (n-k\!+\!2\ n-k\!+\!3\ \cdots \ n-k+l+1 \ n-k\!+\!1\ n-k\ \!\cdots \ 1)\varpi ^{((n-1)r\!+\!\kappa ,-r,\ldots , -r)}. \end{aligned}$$

Note that the cocharacter $((n-1)r+\kappa ,-r,\ldots , -r)$ is not central by assumption on $\kappa , r$. So, by Proposition 3.7 (i) and Proposition 3.6, we have $X_{w_0\varpi ^{w_0^{-1}(\lambda )}}(b)=\emptyset$ for $b\in B(G,\lambda )$, unless $k=l+1$. Thus $\lambda$ is of finite Coxeter type in this case. The proof for the case $\lambda =(r+1, r,\ldots , r,-(n-1)r-1-\kappa )$ is similar. $\square$

5 Geometric structure

Set $\kappa =v_L\circ \det$. Fix a basic element $b\in G(L)$ with $0\le \kappa (b)<n$. Put $n'=\textrm{gcd}(\kappa (b),n)$, and let $n_0, k_0$ be the non-negative integers such that

$$\begin{aligned} n=n'n_0,\quad \kappa (b)=n'k_0. \end{aligned}$$

To study the geometric structure, we choose a representative b of the basic $\sigma$-conjugacy class [b]. In this section, we will work with the special representative $b_{sp}$ attached to $\kappa (b)$ introduced in [2, Definition 5.2]. This is the block-diagonal matrix of size $n\times n$ with $(n_0\times n_0)$-blocks of the form ${\begin{pmatrix} 0 &{} \varpi \\ 1_{n_0-1} &{} 0\\ \end{pmatrix}}^{k_0}$.

Let $J=J_b(F)$ and let $J_{{\mathcal {O}}}=J\cap K$ be a maximal compact subgroup of J. In our case, J is isomorphic to ${{\,\textrm{GL}\,}}_{n'}(D_{k_0/n_0})$, where $D_{k_0/n_0}$ denotes the central division algebra over F with invariant $k_0/n_0$. Let ${\mathcal {O}}_{D_{k_0/n_0}}$ be the ring of integers of $D_{k_0/n_0}$. Then $J_{{\mathcal {O}}}$ is isomorphic to ${{\,\textrm{GL}\,}}_{n'}({\mathcal {O}}_{D_{k_0/n_0}})$.

5.1 The Iwahori case

We keep the notation above and assume $b=b_{sp}$. From now on we will consider the dominant cocharacters

$$\begin{aligned} \lambda _{i,r}= \left\{ \begin{array}{ll} ((n-1)r+\kappa (b), -r,\ldots ,-r) &{} (i=0) \\ ((n-1)r+1+\kappa (b), -r,\ldots ,-r,-r-1) &{} (i=1), \end{array}\right. \end{aligned}$$

where $r>0$ (resp. $r\ge 0$) if $\kappa (b)=0$ (resp. $1\le \kappa (b)<n$). Let us first study the Iwahori level affine Deligne–Lusztig varieties lying over $X_{\lambda _{i,r}}(b)$. These are (possibly empty) varieties of the form $X_w(b)$ with $w\in W_0\varpi ^{\lambda _{i,r}}W_0$. Among them, we especially consider the parameters defined as follows: Let $X_*(T)_{\lambda _{i,r}}$ be the finite set of cocharacters obtained by the permutation of coordinates of $\lambda _{i,r}$ fixing the first entry. Clearly we have

$$\begin{aligned} |X_*(T)_{\lambda _{i,r}}|= \left\{ \begin{array}{ll} 1 &{} (i=0) \\ n-1 &{} (i=1). \end{array}\right. \end{aligned}$$

In particular, $X_*(T)_{\lambda _{i,r}}$ contains an element

$$\begin{aligned} \lambda _{i,r}'= \left\{ \begin{array}{ll} \lambda _{0,r} &{} (i=0) \\ ((n-1)r+1+\kappa (b), -r-1, -r, \ldots ,-r) &{} (i=1). \end{array}\right. \end{aligned}$$

Let $\tau =(1\ 2\ \cdots \ n)\in W_0$. In this paper, affine Deligne–Lusztig varieties associated to

$$\begin{aligned} \varpi ^{\nu _{i,r}}\tau \in {{\widetilde{W}}},\quad \nu _{i,r}\in X_*(T)_{\lambda _{i,r}} \end{aligned}$$

play an important role. In the sequel, we first study the case $\nu _{i,r}=\lambda _{i,r}'$ using the results in [2, Section 6]. Write ${\dot{w}}_{i,r}=\varpi ^{\lambda _{i,r}'}{{\dot{\tau }}}$ and $w_{i,r}=\varpi ^{\lambda _{i,r}'}\tau \in {{\widetilde{W}}}$ (note that if $i=0$, then $w_{0, r}$ is the same as the parameter studied there).

Set $V=L^n$ and ${\mathscr {L}}_0={\mathcal {O}}^n$. We define

$$\begin{aligned} g_b(x)=(x\ \ b\sigma (x)\ \ \cdots \ \ (b\sigma )^{n-1}(x)). \end{aligned}$$

Then the admissible subset $V_b^{\textrm{adm}}$ of the isocrystal $(V, b\sigma )$ consists of the elements $x\in V$ satisfying $\det g_b(x)\in L^{\times }$. We also define

$$\begin{aligned} D_b=\textrm{diag}(1, \varpi ^{\lfloor k_0/n_0\rfloor }, \varpi ^{\lfloor 2k_0/n_0\rfloor }, \ldots , \varpi ^{\lfloor (n-1)k_0/n_0\rfloor }), \end{aligned}$$

and set $g^{\textrm{red}}_b(x)=g_b(x)D_b^{-1}$. Analogously, let us denote by ${\mathscr {L}}_{0,b}^{\textrm{adm}}$ the subset of ${\mathscr {L}}_0$ consisting of the elements $x\in {\mathscr {L}}_0$ satisfying $\det g^{\textrm{red}}_b(x)\in {\mathcal {O}}^{\times }$. Set

$$\begin{aligned} \mu _{i,r}= \left\{ \begin{array}{ll} (0, r, 2r,\ldots , (n-1)r) &{} (i=0) \\ (0,r+1, 2r+1,\ldots , (n-1)r+1) &{} (i=1). \end{array}\right. \end{aligned}$$

We write $g_{b,i,r}(x)=g_b(x)\varpi ^{\mu _{i,r}}$ for any $x\in V_b^{\textrm{adm}}$.

Lemma 5.1

Let $x\in V_b^{\textrm{adm}}$. Then there exist unique elements $\alpha _j\in {\mathcal {O}}$ such that $(b\sigma )^n(x)=\Sigma _{j=0}^{n-1}\alpha _j(b\sigma )^j(x)$ with $v_L(\alpha _0)=\kappa (b)$. Moreover, if $\kappa (b)>0$, we also have $v_L(\alpha _j)>0$ for $1\le j\le n-1$.

Proof

The first assertion is [2, Lemma 6.1]. If $\kappa (b)>0$, then the slope of the Newton polygon of $(V, b\sigma )$ is positive. So the last assertion also follows from the proof of [2, Lemma 6.1]. $\square$

Lemma 5.2

Let $x\in V_b^{\textrm{adm}}$. We have

$$\begin{aligned} b\sigma (g_{b,i,r}(x))=g_{b,i,r}(x){\dot{w}}_{i,r}a, \end{aligned}$$

where $a\in I$ is a matrix, which can differ from the identity matrix only in the last column.

Proof

The proof follows along the same line as [2, Lemma 6.7]. $\square$

For an integer m, let $0\le [m]_{n_0}<n_0$ denote its residue modulo $n_0$. Let $v_0\in {{\,\textrm{GL}\,}}_{n_0}(L)$ be the permutation matrix whose j-th column is $e_{1+[(j-1)k_0]_{n_0}}$. Let $v\in {{\,\textrm{GL}\,}}_n(L)$ denote the block-diagonal matrix, whose $n_0\times n_0$ blocks are each equal to $v_0$. Further, let $\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}$ be the perfection of $n'-1$-dimensional Drinfeld’s upper half-space over ${\mathbb {F}}_{q^{n_0}}$. The following statements are generalizations of [2, Theorem 6.5 (ii)], [2, Proposition 6.12], [2, Proposition 6.15] and [2, Theorem 6.17].

Proposition 5.3

• (i) The map

  $$\begin{aligned} V_b^{\textrm{adm}}\rightarrow X_{w_{i,r}}(b),\quad x\mapsto g_{b,i,r}(x) \end{aligned}$$

  is surjective.

• (ii) Let $X_{w_{i,r}}(b)_{{\mathscr {L}}_0}$ be the image of ${\mathscr {L}}_{0,b}^{\textrm{adm}}\subset V_b^{\textrm{adm}}$ by the map in (i). Then we have a scheme theoretic disjoint union decomposition

  $$\begin{aligned} X_{w_{i,r}}(b)=\bigsqcup _{h\in J/J_{{\mathcal {O}}}}hX_{w_{i,r}}(b)_{{\mathscr {L}}_0}. \end{aligned}$$

• (iii) The variety $X_{w_{i,r}}(b)_{{\mathscr {L}}_0}$ in (ii) is a locally closed subvariety of the Schubert cell $IvD_b\varpi ^{\mu _{i,r}} I/I$ and is isomorphic to

  $$\begin{aligned} \Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}. \end{aligned}$$

  Here ${\mathbb {A}}$ is a finite-dimensional affine space over ${\mathbb {F}}_q$ with dimension depending on i, r. In particular, we have a decomposition of ${\mathbb {F}}_q$-schemes

  $$\begin{aligned} X_{w_{i,r}}(b)\cong \bigsqcup _{J/J_{{\mathcal {O}}}}\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}. \end{aligned}$$

Proof

The case $i=0$ and $r>0$ is proved in [2, Section 6]. If $\kappa (b)>0$, then we can check that the case $r=0$ also follows from the same proof as in [2, Section 6], using Lemma 5.1.

We have to show the case $i=1$. Let ${{\dot{\eta }}}$ be the matrix of the form ${\begin{pmatrix} 0 &{} \varpi \\ 1_{n-1} &{} 0\\ \end{pmatrix}}$, and let $\eta$ be its image in $\Omega$. Then we have

$$\begin{aligned} {{\dot{\eta }}}^{-1} \dot{w}_{1,r}{{\dot{\eta }}}=\varpi ^{(-r,\ldots ,-r, (n-1)r+\kappa (b))}{{\dot{\tau }}}, \end{aligned}$$

where $\tau =(1\ \cdots \ n)$. Let us write $A=\varpi ^{(-r,\ldots ,-r, (n-1)r+\kappa (b))}{{\dot{\tau }}}$. By Proposition 3.7 (i), the following map is an isomorphism:

$$\begin{aligned} \varphi :X_{w_{1,r}}(b)\rightarrow X_A(b),\quad gI\mapsto g{{\dot{\eta }}}I. \end{aligned}$$

Moreover, when we see A as an element of ${{\widetilde{W}}}$, it follows that

$$\begin{aligned} \ell (s_j\cdots s_{n-2} s_{n-1} A s_{n-1}s_{n-2} \cdots s_j)=\ell (s_{j+1}\cdots s_{n-2}s_{n-1} A s_{n-1}s_{n-2} \cdots s_{j+1} )-2, \end{aligned}$$

(1)

for any $1\le j\le n-1$. Indeed, we compute

$$\begin{aligned} \ell (A)=\ell (w_{1,r})=(n-1)(nr+1+\kappa (b)) \end{aligned}$$

and

$$\begin{aligned} \ell (s_1\cdots s_{n-1}As_{n-1}\cdots s_1)=\ell (w_{0,r})=(n-1)(nr-1+\kappa (b)). \end{aligned}$$

Since $\ell (A)-\ell (s_1\cdots s_{n-1}As_{n-1}\cdots s_1)=2(n-1)$, we must have (1) for all $1\le j\le n-1$.

Next, we claim

$$\begin{aligned} X_{s_{j}\cdots s_{n-2}s_{n-1}As_{n-1}s_{n-2} \cdots s_{j+1}}(b)=\emptyset \ \text {for all}\ 1\le j \le n-1. \end{aligned}$$

(2)

By Lemma 5.4 and Proposition 3.7 (ii), this is equivalent to

where $A'={{\dot{\tau }}}\varpi ^{\lambda _{0,r}}$. To check ($*$), we use Proposition 3.6. Set

$$\begin{aligned} A_j=s_{j}\cdots s_2s_1A's_1s_2 \cdots s_{j-1}. \end{aligned}$$

Then we have to show

$$\begin{aligned} p_2(A_j)^{-1}p_1(A_j)p_2(A_j)\in \bigcup _{S'\subsetneq S}W_{S'}, \end{aligned}$$

where $W_{S'}\subset W_0$ is the subgroup generated by $S'$. For $1\le j\le n-1$, we compute

$$\begin{aligned} p_1(A_j)=s_1\cdots s_{j-1} s_{j+1}\cdots s_{n-1},\quad p_2(A_j)=s_{j+1}\cdots s_{n-1}. \end{aligned}$$

So we have

$$\begin{aligned} p_2(A_j)^{-1}p_1(A_j)p_2(A_j)=s_1\cdots s_{j-1} s_{j+1}\cdots s_{n-1}\in W_{S_j}\subset \bigcup _{S'\subsetneq S}W_{S'}, \end{aligned}$$

where $S_j=S\setminus \{s_j\}$. This proves ($*$) and hence (2).

Combining (1) and (2) and using Proposition 3.7 (iii), we can deduce that there exists a Zariski-locally trivial ${\mathbb {A}}^{1,\textrm{pfn}}$-bundle

$$\begin{aligned} \pi _{j}:X_{s_{j+1}\cdots s_{n-1} A s_{n-1} \cdots s_{j+1}}(b)\rightarrow X_{s_{j}\cdots s_{n-1} A s_{n-1} \cdots s_{j}}(b), \end{aligned}$$

for each $1\le j\le n-1$. Moreover, for any fixed $g_0I\in X_{s_{j}\cdots s_{n-1} A s_{n-1} \cdots s_{j}}(b)$, we have

$$\begin{aligned} \pi _{j}^{-1}(g_0I)=\{gI\in {\mathcal {F}}lag\mid \textrm{inv}(gI, g_0I)=\textrm{inv}(g_0I, gI)=s_j\}. \end{aligned}$$

Using these morphisms, we prove the case $i=1$. For simplicitiy, we treat the case that $\kappa (b)=0$ or $\kappa (b)=1$, so that $v=D_b=1$. The general case follows in the same way. By the case $i=0$, $X_{w_{0,r}}(b)_{{\mathscr {L}}_0}$ is contained in $I\varpi ^{\mu _{0,r}}I/I$. Moreover, the proof of [2, Theorem 6.17] shows that $gI\in {\mathcal {F}}lag$ lies in $X_{w_{0,r}}(b)_{{\mathscr {L}}_0}$ if and only if gI is represented by the element of the form

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ x_2 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ x_3 &{}\quad *&{}\quad 1 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{} \quad \vdots &{}\quad \vdots \\ x_{n-1} &{}\quad *&{}\quad \cdots &{} \quad *&{}\quad 1 &{}\quad 0 \\ x_n &{}\quad *&{}\quad \cdots &{}\quad *&{}\quad *&{}\quad 1 \end{pmatrix}\varpi ^{\mu _{0,r}}I, \end{aligned}$$

where

$$\begin{aligned} x_j=[x_{j,0}]+[x_{j,1}]\varpi +\cdots +[x_{j, (j-1)r-1}]\varpi ^{(j-1)r-1}, \\ (x_{2,0}, \ldots ,x_{n,0})\in \Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}},\quad x_{j,k}\in {\mathbb {A}}^{1, \textrm{pfn}}\ (k>0), \end{aligned}$$

and the entries marked by $*$ are certain functions of $x_2, \ldots , x_n$ lying in ${\mathcal {O}}$. The coefficient of $\varpi ^l$ in each $*$ is actually a function of only $x_{j,k}$ with $k\le l$. Then we easily verify that $gI\in X_{w_{1, r}}(b)$ lies in the inverse image of $X_{w_{0,r}}(b)_{{\mathscr {L}}_0}$ under the morphism

$$\begin{aligned} \pi =\pi _1\circ \cdots \circ \pi _{n-1}\circ \varphi :X_{w_{1, r}}(b)\rightarrow X_{w_{0, r}}(b) \end{aligned}$$

if and only if gI is represented by the element of the form

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ x_2+[t_2]\varpi ^r &{}\quad 1 &{}\quad 0 &{} \quad \cdots &{}\quad \cdots &{}\quad 0 \\ x_3+[t_3]\varpi ^{2r} &{}\quad *&{}\quad 1 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ x_{n-1}+[t_{n-1}]\varpi ^{(n-2)r} &{}\quad *&{}\quad \cdots &{}\quad *&{}\quad 1 &{}\quad 0 \\ x_n+[t_n]\varpi ^{(n-1)r} &{}\quad *&{}\quad \cdots &{}\quad *&{} \quad *&{}\quad 1 \end{pmatrix}\varpi ^{\mu _{0,r}}\dot{s}_1\cdots \dot{s}_{n-1}\eta ^{-1}I, \end{aligned}$$

where $x_j$ are as above, $t_j\in {\mathbb {A}}^{1, \textrm{pfn}}$, and $*$ are certain functions of $x_2, \ldots , x_n$, $t_2, \ldots , t_n$ lying in ${\mathcal {O}}$. This is true even for the case $r=0$ and $\kappa (b)>0$, but we need a little more attention. So, in every case, we have a decomposition

$$\begin{aligned} X_{w_{1,r}}(b)=\bigsqcup _{h\in J/J_{{\mathcal {O}}}}\pi ^{-1}(hX_{w_{0,r}}(b)_{{\mathscr {L}}_0}), \end{aligned}$$

and each component is a locally closed subvariety of

$$\begin{aligned} I\varpi ^{\mu _{0,r}}Is_1Is_2I\cdots Is_{n-1}I\eta ^{-1}I/I=I\varpi ^{\mu _{0,r}}s_1s_2\cdots s_{n-1}\eta ^{-1}I/I, \end{aligned}$$

which is isomorphic to $\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}$. For any $x_j$ and $t_j$ as above, set

$$\begin{aligned} x'={^t(1, x_2+[t_2]\varpi ^r, \ldots , x_n+[t_n]\varpi ^{(n-1)r})}. \end{aligned}$$

Then, by Lemma 5.2 and an argument similar to that of [2, Theorem 6.17], it follows that $gI\in {\mathcal {F}}lag$ lies in $\pi ^{-1}(\varpi X_{w_{0,r}}(b)_{{\mathscr {L}}_0})$ if and only if $gI=g_{b,1,r}(x')I$ for some $x'$ as above. This implies (i). Since we have

$$\begin{aligned} X_{w_{1,r}}(b)_{{\mathscr {L}}_0}=\pi ^{-1}(\varpi X_{w_{0,r}}(b)_{{\mathscr {L}}_0})\cong \Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}, \end{aligned}$$

(ii) and (iii) also follow. $\square$

For $w, w'\in {{\widetilde{W}}}$, we write $w\rightarrow w'$ if there is a sequence $w=w_0, w_1,\ldots , w_m=w'$ of elements in ${{\widetilde{W}}}$ such that for any k, $w_k=s_{j_k}w_{k-1}s_{j_k}\ (s_{j_k}\in S)$ and $\ell (w_k)\le \ell (s_{j_k}w_{k-1}s_{j_k})$. We write $w\approx w'$ if $w\rightarrow w'$ and $w'\rightarrow w$.

Lemma 5.4

We have

$$\begin{aligned} w_{0,r}s_1&\approx w_{0,r}s_{n-1}, \\ s_1w_{0,r}s_1s_2&\approx s_{n-1}w_{0,r}s_{n-1}s_{n-2}, \\&\ \, \vdots \\ s_{n-2}\cdots s_1 w_{0,r}s_1\cdots s_{n-1}&\approx s_2\cdots s_{n-1}w_{0,r}s_{n-1}\cdots s_1. \end{aligned}$$

Proof

Let us first show

$$\begin{aligned} w_{0,r}s_1&\approx w_{0,r}s_1, \\ s_1w_{0,r}s_1s_2&\approx w_{0,r}s_1s_2s_1, \\&\ \, \vdots \\ s_{n-2}\cdots s_1 w_{0,r}s_1\cdots s_{n-1}&\approx w_{0,r}s_1\cdots s_{n-1}s_{n-2}\cdots s_1. \end{aligned}$$

We have $w_{0,r}s_1\cdots s_js_{j-1}\cdots s_1=\varpi ^{\lambda _{0,r}}(1\ j+2\ \cdots \ n)(2\ \cdots \ j+1)$ and

$$\begin{aligned} \ell (w_{0,r}s_1\cdots s_js_{j-1}\cdots s_1)&=(n-1)(nr-1+\kappa (b))+2j-1 \\&=\ell (s_{j-1}\cdots s_1w_{0,r}s_1\cdots s_j). \end{aligned}$$

For any $1\le k\le j$, set $v_{j,k}=s_{j-k}\cdots s_2 s_1w_{0,r}s_1\cdots s_j s_{j-1}s_{j-2}\cdots s_{j-k+1}$. Then, for any $1\le k\le j-1$, we also have

$$\begin{aligned} v_{j,k}\chi _{j-k, j-k+1}= \left\{ \begin{array}{ll} \chi _{j-k+2, j+2} &{} (1\le j\le n-2) \\ \chi _{j-k+2, j+2}+(nr+\kappa (b))\delta &{} (j=n-1), \end{array}\right. \end{aligned}$$

where $\delta$ is the constant function with value 1. In particular, for fixed $1\le j\le n-1$, $v_{j,k}\chi _{j-k, j-k+1}$ is always positive. This implies that $v_{j,1}$ can be transformed to $v_{j, j}$ with $\ell (v_{j,1})\le \ell (v_{j,2})\le \cdots \le \ell ({v_{j,j}})$. Since $\ell (v_{j,1})=\ell (v_{j,j})$, it follows that $v_{j,1}\approx v_{j,j}$ for any $1\le j\le n-1$, as we claimed. In the same way we can show

$$\begin{aligned} w_{0,r}s_{n-1}&\approx s_{n-1}w_{0,r}, \\ s_{n-1}w_{0,r}s_{n-1}s_{n-2}&\approx s_{n-1}s_{n-2}s_{n-1}w_{0,r}, \\&\ \, \vdots \\ s_{2}\cdots s_{n-1} w_{0,r}s_{n-1}\cdots s_1&\approx s_{n-1}\cdots s_{1}s_{2}\cdots s_{n-1}w_{0,r}. \end{aligned}$$

By the discussion above, our statement is reduced to the equivalence

$$\begin{aligned} w_{0,r}s_1\cdots s_js_{j-1}\cdots s_1\approx s_{n-1}\cdots s_{n-j}s_{n-j+1}\cdots s_{n-1}w_{0,r} \end{aligned}$$

for all $1\le j\le n-1$. If $j=n-1$, we have to show the equivalence between

$$\begin{aligned} w_{0,r}s_1\cdots s_{n-1}s_{n-2}\cdots s_1=\varpi ^{\lambda _{0,r}}(2\ \cdots \ n) \end{aligned}$$

and

$$\begin{aligned} s_{n-1}\cdots s_{1}s_{2}\cdots s_{n-1}w_{0,r}=(1\ n)\varpi ^{\lambda _{0,r}}(1\ \cdots \ n). \end{aligned}$$

In this case, it is easy to check that the transformation (by the conjugation by an element of S)

$$\begin{aligned} \varpi ^{\lambda _{0,r}}(2\ \cdots \ n)&\rightarrow s_1\varpi ^{\lambda _{0,r}}(2\ \cdots \ n)s_1 \\&\rightarrow \cdots \\&\rightarrow s_{n-1}\cdots s_1\varpi ^{\lambda _{0,r}}(2\ \cdots \ n)s_1\cdots s_{n-1} \\&=(1\ n)\varpi ^{\lambda _{0,r}}(1\ \cdots \ n) \end{aligned}$$

gives this equivalence.

If $1\le j\le n-2$, we compute

$$\begin{aligned} w_{0,r}s_1\cdots s_js_{j-1}\cdots s_1=\varpi ^{\lambda _{0,r}}(1\ j+2\ \cdots \ n)(2\ \cdots \ j+1) \end{aligned}$$

and

$$\begin{aligned} s_{n-1}\cdots s_{n-j}s_{n-j+1}\cdots s_{n-1}w_{0,r}=\varpi ^{\lambda _{0,r}}(1\ \cdots \ n-j-1\ n)(n-j\ \cdots \ n-1). \end{aligned}$$

Here we used $\varpi ^{\lambda _{0,r}}s_k=s_k\varpi ^{\lambda _{0,r}}\ (2\le k\le n-1)$. Both of these elements have length $(n-1)(nr-1+\kappa (b))+2j-1$. We are going to show that the transformation

$$\begin{aligned}&\varpi ^{\lambda _{0,r}}(1\ j+2\ \cdots \ n)(2\ \cdots \ j+1) \\ \rightarrow&\varpi ^{\lambda _{0,r}}(1\ j+1\ j+3\cdots \ n)(2\ \cdots \ j\ j+2) \\ \rightarrow&\varpi ^{\lambda _{0,r}}(1\ j\ j+3\cdots \ n)(2\ \cdots \ j-1\ j+1\ j+2) \\ \rightarrow&\cdots \\ \rightarrow&\varpi ^{\lambda _{0,r}}(1\ 2\ j+3\ \cdots \ n)(3\ \cdots \ j+2) \\ \rightarrow&\varpi ^{\lambda _{0,r}}(1\ 2\ j+2\ j+4 \cdots \ n)(3\ \cdots \ j+1\ j+3) \\ \rightarrow&\cdots \\ \rightarrow&\varpi ^{\lambda _{0,r}}(1\ \cdots \ n-j-1\ n)(n-j\ \cdots \ n-1) \end{aligned}$$

gives the desired equivalence. This transformation changes $(2\ \cdots \ j+1)$ to $(n-j\ \cdots \ n-1)$ using only $s_2,\ldots s_{n-2}$. The $W_0$-part of the element appearing in each process except for the last one is of the form

$$\begin{aligned} x_{j,k}=(1\ \cdots \ k-1\ k+j\ \cdots \ n)(k\ \cdots \ k+j-1) \end{aligned}$$

or

$$\begin{aligned} y_{j,k,j'}=(1\ \cdots \ k-1\ k+j'\ k+j+1\ \cdots \ n)(k\ \cdots \ k+j'-1\ k+j'+1\ \cdots \ k+j), \end{aligned}$$

where $2\le k\le n-j-1$ and $1\le j'\le j-1$. For any $w_0\in W_0$, let $\Phi (w_0)=\{\chi \in \Phi _+\mid w_0\chi \in \Phi _-\}$. Then we can check that

$$\begin{aligned} \Phi (x_{j,k})=\{\chi _{k,k+j-1},\ldots , \chi _{k+j-2,k+j-1}, \chi _{k-1,k},\ldots ,&\chi _{k-1,k+j-1}, \chi _{1,n},\ldots , \chi _{n-1,n}\} \end{aligned}$$

and

$$\begin{aligned} \Phi (y_{j,k, j'})=\{\chi _{k-1,k+j},\ldots ,&\chi _{k+j-1, k+j}, \chi _{k-1, k},\ldots \chi _{k-1, k+j'-2} \\&\chi _{k+j', k+j'+1},\ldots ,\chi _{k+j', k+j-1}, \chi _{1, n},\ldots , \chi _{n-1, n}\}. \end{aligned}$$

In particular, both of the sets $\Phi (x_{j,k})$ and $\Phi (y_{j,k,j'})$ contain $\{\chi _{1,n}, \ldots , \chi _{n-1, n}\}$ and $|\Phi (x_{j,k}){\setminus } \{\chi _{1,n}, \ldots , \chi _{n-1, n}\}|=|\Phi (y_{j,k,j'}){\setminus } \{\chi _{1,n}, \ldots , \chi _{n-1, n}\}|=2j-1$. Therefore the length of each element appearing in the transformation above is always $(n-1)(nr-1+\kappa (b))+2j-1$. This finishes the proof. $\square$

Using Proposition 5.3, we obtain the following result.

Corollary 5.5

Let $\tau =(1\ 2\ \cdots \ n)$. Then, for any $\nu _{i,r}\in X_*(T)_{\lambda _{i,r}}$, there exists an irreducible component $X_{\varpi ^{\nu _{i,r}}\tau }(b)_0$ of $X_{\varpi ^{\nu _{i,r}}\tau }(b)$, which is a locally closed subvariety of a Schubert cell and is isomorphic to $\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}$. Here ${\mathbb {A}}$ is a finite-dimensional affine space over ${\mathbb {F}}_q$. Moreover, we have a scheme theoretic disjoint union decomposition

$$\begin{aligned} X_{\varpi ^{\nu _{i,r}}\tau }(b)=\bigsqcup _{h\in J/J_{{\mathcal {O}}}}hX_{\varpi ^{\nu _{i,r}}\tau }(b)_0\cong \bigsqcup _{J/J_{{\mathcal {O}}}}\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}. \end{aligned}$$

Proof

We have to show this only in the case $i=1$. Let $\nu _j$ be the element of $X_*(T)_{\lambda _{1, r}}$ whose $(j+1)$-th entry is $-r-1$. Then we have $\varpi ^{\nu _1}\tau =w_{1, r}$ and the assertion for $X_{\varpi ^{\nu _1}\tau }(b)$ follows from Proposition 5.3 by setting $X_{\varpi ^{\nu _1}\tau }(b)_0=X_{w_{1, r}}(b)_{{\mathscr {L}}_0}$. Furthermore, for any $1\le j\le n-1$, we have an isomorphism

$$\begin{aligned} \phi _j:X_{\varpi ^{\nu _j}\tau }(b)\xrightarrow {\sim } X_{\varpi ^{\nu _{j+1}}\tau }(b). \end{aligned}$$

Indeed, let ${{\dot{\eta }}}={\begin{pmatrix} 0 &{} \varpi \\ 1_{n-1} &{} 0\\ \end{pmatrix}}$. Then it is easy to check that

$$\begin{aligned} (\dot{s}_j\cdots \dot{s}_1{{\dot{\eta }}}\dot{s}_{n-1}\cdots \dot{s}_{j+1})\varpi ^{\nu _j}{{\dot{\tau }}}(\dot{s}_j\cdots \dot{s}_1{{\dot{\eta }}}\dot{s}_{n-1}\cdots \dot{s}_{j+1})^{-1}=\varpi ^{\nu _{j+1}}{{\dot{\tau }}}\end{aligned}$$

and

$$\begin{aligned} \ell (\varpi ^{\nu _j}\tau )&=\ell (s_{j+1}\varpi ^{\nu _j}\tau s_{j+1})\\&=\cdots \\&=\ell (s_{n-1}\cdots s_{j+1}\varpi ^{\nu _j}\tau s_{j+1}\cdots s_{n-1}) \\&=\ell (\eta s_{n-1}\cdots s_{j+1}\varpi ^{\nu _j}\tau s_{j+1}\cdots s_{n-1}\eta ^{-1}) \\&=\ell (s_1 \eta s_{n-1}\cdots s_{j+1}\varpi ^{\nu _j}\tau s_{j+1}\cdots s_{n-1}\eta ^{-1}s_1) \\&=\cdots \\&=\ell (s_j\cdots s_1 \eta s_{n-1}\cdots s_{j+1}\varpi ^{\nu _j}\tau s_{j+1}\cdots s_{n-1}\eta ^{-1}s_1\cdots s_j)\\&=\ell (\varpi ^{\nu _{j+1}}\tau ). \end{aligned}$$

So, by Proposition 3.7 (i) and (ii), we can construct $\phi _j$ for any j.

Let $X_{\varpi ^{\nu _j}\tau }(b)_0$ be the image of $X_{\varpi ^{\nu _1}\tau }(b)_0$ under the isomorphism $\phi _{j-1}\circ \cdots \circ \phi _1$. Since our assertion is true for $X_{\varpi ^{\nu _1}\tau }(b)$, the same assertion for $X_{\varpi ^{\nu _j}\tau }(b)$ follows immediately except that $X_{\varpi ^{\nu _j}\tau }(b)_0$ is contained in a Schubert cell. Let

$$\begin{aligned} \mu _j=\mu _{1, r}-(0, \overbrace{1, \ldots , 1}^{j-1}, 0, \ldots , 0). \end{aligned}$$

Then, again by Proposition 5.3 (i), any element in $X_{\varpi ^{\nu _1}\tau }(b)_0$ can be written as $g_b(x)\varpi ^{\mu _1}I$ for some $x\in {\mathscr {L}}_{0,b}^{\textrm{adm}}$. Further, using the set-theoretical description right after Proposition 3.7, we can easily verify that

$$\begin{aligned} (\phi _{j-1}\circ \cdots \circ \phi _1)(g_b(x)\varpi ^{\mu _1}I)=g_b(x)\varpi ^{\mu _j}I, \end{aligned}$$

i.e., any element in $X_{\varpi ^{\nu _j}\tau }(b)_0$ can be written as $g_b(x)\varpi ^{\mu _j}I$ for some $x\in {\mathscr {L}}_{0,b}^{\textrm{adm}}$. By this and the same argument as in [2, Proposition 6.15], we can show that $X_{\varpi ^{\nu _j}\tau }(b)_0$ is contained in $IvD_b\varpi ^{\mu _j}I/I$. This finishes the proof. $\square$

5.2 The hyperspecial case

We keep the notation and assumptions of §5.1. Next we deduce the geometric structure of the hyperspecial level affine Deligne–Lusztig varieties $X_{\lambda _{i,r}}(b)$. To complete this, we relate the Iwahori and hyperspecial cases.

Lemma 5.6

Let $\tau =(1\ 2\ \cdots \ n)$.

• (i) The projection map

  $$\begin{aligned} X_{w_{0,r}}(b)\rightarrow X_{\lambda _{0,r}}(b),\quad gI\mapsto gK \end{aligned}$$

  is injective.

• (ii) The projection map

  $$\begin{aligned} X_{\varpi ^\nu \tau }(b)\rightarrow X_{\lambda _{1, r}}(b),\quad gI\mapsto gK \end{aligned}$$

  is injective for any $\nu \in X_*(T)_{\lambda _{1, r}}$.

Proof

To show (i), recall that any element in $X_{w_{0,r}}(b)$ is of the form $g_b(x)\varpi ^{\mu _{0,r}}I$ with $x\in V_b^{\textrm{adm}}$. So it suffices to show that for any $x,y\in V_b^{\textrm{adm}}$, $g_b(x)\varpi ^{\mu _{0,r}}K=g_b(y)\varpi ^{\mu _{0,r}}K$ implies $g_b(x)\varpi ^{\mu _{0,r}}I=g_b(y)\varpi ^{\mu _{0,r}}I$. If $g_b(y)\varpi ^{\mu _{0,r}}=g_b(x)\varpi ^{\mu _{0,r}}p$ with $p=(p_{ij})_{i,j}\in K$, then

$$\begin{aligned} y=p_{1,1}x+p_{2,1}\varpi ^r b\sigma (x)+\cdots +p_{n,1}\varpi ^{(n-1)r}(b\sigma )^{n-1}(x). \end{aligned}$$

By multiplying this equation by $b\sigma$, and by Lemma 5.1, we can represent each column of $g_b(y)\varpi ^{\mu _{0,r}}$ as a linear combination of $x, \varpi ^rb\sigma (x),\ldots , \varpi ^{(n-1)r}(b\sigma )^{n-1}(x)$, and their coefficients are nothing but $p_{ij}$. This calculation shows $p_{ij}\in {\mathfrak {p}}\ (i<j)$, i.e., $p\in I$. Therefore (i) follows. The proof for (ii) is similar. $\square$

For any $w\in {^S{{\widetilde{W}}}}$, set $S_w=\max \{S'\subseteq S\mid {{\,\textrm{Ad}\,}}(w)(S')=S'\}$.

Proposition 5.7

For any $w\in {{\widetilde{W}}}$, there exist $w'\in {^S{{\widetilde{W}}}}$ and $v\in W_{S_{w'}}$ such that $w\rightarrow vw'$, where $W_{S_{w'}}\subseteq W_0$ is the subgroup generated by $S_{w'}$.

Proof

This is [9, Proposition 3.1.1]. See also [18, Theorem 2.5]. $\square$

We use the notation of the proof of Corollary 5.5.

Lemma 5.8

For any j, the image of $X_{\varpi ^{\nu _j}\tau }(b)$ in $X_{\lambda _{1, r}}(b)$ is closed.

Proof

First note that the projection ${\mathcal {F}}lag\rightarrow {\mathcal {G}}rass$ is an inductive limit of projective morphisms. In particular, it is universally closed. So the projection

$$\begin{aligned} \bigcup _{w\in W_0\varpi ^{\lambda _{1,r}} W_0} X_{w}(b)\rightarrow X_{\lambda _{1,r}}(b),\quad gI\mapsto gK \end{aligned}$$

is also a closed map. By this fact, it suffices to show that each $X_{\varpi ^{\nu _j}\tau }(b)$ is closed in $\bigcup _{w\in W_0\varpi ^{\lambda _{1,r}}W_0} X_{w}(b)$, i.e.,

$$\begin{aligned} \bigcup _{\begin{array}{c} w\in W_0\varpi ^{\lambda _{1,r}} W_0\\ w\le \varpi ^{\nu _j}\tau \end{array}}X_w(b)=X_{\varpi ^{\nu _j}\tau }(b). \end{aligned}$$

Here $\le$ denotes the Bruhat order. By Proposition 3.7 and Proposition 5.7, we are reduced to show $X_{vw}(b)=\emptyset$ for $w\in {^S{{\widetilde{W}}}}\cap W_0\varpi ^{\lambda _{1,r}} W_0={^S\textrm{Adm}}(\lambda _{1,r})^\circ$ and $v\in W_{S_w}$ such that $\ell (vw)<\ell (\varpi ^{\nu _j}\tau )$. By the proof of Theorem 4.6, any element in ${^S{{\widetilde{W}}}}\cap W_0\varpi ^{\lambda _{1,r}} W_0$ can be written as $w_0\varpi ^{w_0^{-1}(\lambda _{1,r})}$, where

$$\begin{aligned} w_0=(1\ 2\ \cdots \ k)(n\ n-1\ \cdots \ l),\quad k<l, \end{aligned}$$

or

$$\begin{aligned} w_0=(1\ 2\ \cdots \ l\ n\ n-1\ \cdots \ k),\quad k>l. \end{aligned}$$

In the first (resp. second) case, $S_{w_0\varpi ^{w_0^{-1}(\lambda _{1,r})}}$ is equal to

$$\begin{aligned} \{s_{k+1},\ldots , s_{l-2}\} \text {(resp.}\ \{s_{l+1},\ldots , s_{k-2}\}). \end{aligned}$$

Similarly as in the proof of Theorem 4.6, it is easy to check $X_{vw}(b)=\emptyset$ using $\ell (vw)<\ell (\varpi ^{\nu _j}\tau )$. $\square$

Set

$$\begin{aligned} b^*={^tb}^{-1},\quad \lambda _{i,r}^*= \left\{ \begin{array}{ll} (r,\ldots ,r, -(n-1)r-\kappa (b)) &{} (i=0) \\ (r+1, r,\ldots , r,-(n-1)r-1-\kappa (b)) &{} (i=1), \end{array}\right. \end{aligned}$$

where $r>0$ (resp. $r\ge 0$) if $\kappa (b)=0$ (resp. $1\le \kappa (b)<n$). Then $b^*$ is a basic element in G(L) with $\kappa (b^*)=-\kappa (b)$. Let $J^*=J_{b^*}(F)$ and let $J_{{\mathcal {O}}}^*=J^*\cap K$ be a maximal compact subgroup of $J^*$.

Theorem 5.9

Let ${\mathbb {A}}$ be a finite-dimensional affine space over ${\mathbb {F}}_q$ with dimension depending on i, r.

• (i) We have a decomposition of ${\mathbb {F}}_q$-schemes

  $$\begin{aligned} X_{\lambda _{0,r}}(b)\cong \bigsqcup _{J/J_{{\mathcal {O}}}}\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}, \end{aligned}$$

  where the dimension of ${\mathbb {A}}^\textrm{pfn}$ is $\frac{1}{2}((n-1)(nr+\kappa (b))-n-n')+1$.

• (ii) We have a decomposition of ${\mathbb {F}}_q$-schemes

  $$\begin{aligned} X_{\lambda _{1,r}}(b)\cong \bigsqcup _{j=1}^{n-1} \bigsqcup _{J/J_{{\mathcal {O}}}}\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}, \end{aligned}$$

  where the dimension of ${\mathbb {A}}^\textrm{pfn}$ is $\frac{1}{2}((n-1)(nr+\kappa (b))+n-n')$.

• (iii) We have a decomposition of ${\mathbb {F}}_q$-schemes

  $$\begin{aligned} X_{\lambda _{0,r}^*}(b^*)\cong \bigsqcup _{J^*/J_{{\mathcal {O}}}^*}\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}, \end{aligned}$$

  where the dimension of ${\mathbb {A}}^\textrm{pfn}$ is $\frac{1}{2}((n-1)(nr+\kappa (b))-n-n')+1$.

• (iv) We have a decomposition of ${\mathbb {F}}_q$-schemes

  $$\begin{aligned} X_{\lambda _{1,r}^*}(b^*)\cong \bigsqcup _{j=1}^{n-1} \bigsqcup _{J^*/J_{{\mathcal {O}}}^*}\Omega ^{n'-1}_{{\mathbb {F}}_{q^{n_0}}}\times {\mathbb {A}}^\textrm{pfn}, \end{aligned}$$

  where the dimension of ${\mathbb {A}}^\textrm{pfn}$ is $\frac{1}{2}((n-1)(nr+\kappa (b))+n-n')$.

Proof

For $X_{\lambda }(b)\ne \emptyset$, we have an explicit dimension formula:

$$\begin{aligned} \mathop {\textrm{dim}} X_{\lambda }(b)=\langle \rho , \lambda -\nu _b \rangle -\frac{1}{2}\textrm{def}(b), \end{aligned}$$

where $\nu _b$ is the Newton vector of b, $\rho$ is half the sum of the positive roots, and $\textrm{def}(b)$ is the defect of b. For split groups, the formula was obtained in [25, Theorem 1.1]. Once we obtain the decomposition in the theorem, we can easily compute the dimension of ${\mathbb {A}}^\textrm{pfn}$ by this formula (and $\textrm{def}(b)=n-n'$). So it suffices to show the decomposition in each case.

First we prove (i). Since the map

$$\begin{aligned} IvD_b\varpi ^{\mu _{0,r}}I/I\rightarrow KD_b\varpi ^{\mu _{0,r}}K/K,\quad gI\mapsto gK \end{aligned}$$

is an immersion (see, for example, [22, Lemme 2.2]), it follows from Corollary 5.5 that the map $X_{w_{0,r}}(b)_{{\mathscr {L}}_0}\rightarrow X_{\lambda _{0,r}}(b),\ gI\mapsto gK$ is also an immersion. So we regard each $hX_{w_{0,r}}(b)_{{\mathscr {L}}_0}(h\in J/J_{{\mathcal {O}}})$ as a locally closed subvariety of $X_{\lambda _{0,r}}(b)$. Let $C_{b,r}=\frac{n(n-1)}{2}r+\sum _{j=1}^{n-1}\lfloor \frac{jk_0}{n_0}\rfloor$. By the proof of Theorem 4.6, any element in $X_{\lambda _{0,r}}(b)$ can be written as $g_b(x)\varpi ^{\mu _{0,r}}$. So, in the same way as [2, Proposition 6.12], we can show that $X_{w_{0,r}}(b)_{{\mathscr {L}}_0}$ is equal to the set

$$\begin{aligned} \{gK\in X_{\lambda _{0,r}}(b)\mid g{\mathscr {L}}_0\subset {\mathscr {L}}_0\ \text {and}\ v_L(\det (g))=C_{b,r}\}, \end{aligned}$$

and hence is closed in $X_{\lambda _{0,r}}(b)$. By Lemma 5.6, the closed subvarieties $hX_{w_{0,r}}(b)_{{\mathscr {L}}_0} (h\in J/J_{{\mathcal {O}}})$ form a disjoint cover of $X_{\lambda _{0,r}}(b)$. Since $X_{\lambda _{0,r}}(b)$ is locally perfectly of finite type, $hX_{w_{0,r}}(b)_{{\mathscr {L}}_0}$ is open. This result and Corollary 5.5 prove (i).

Next we prove (ii). We use the notation of the proof of Corollary 5.5. By the proof of Theorem 4.6, we have

$$\begin{aligned} {^S\textrm{Adm}}(\lambda _{1,r})^\circ _{{{\,\textrm{cox}\,}}}=\{c_j\varpi ^{c_j^{-1}(\lambda _{1,r})}\mid c_j=(1\ 2\ \cdots \ j\ n\ n-1\ \cdots \ j+1), 1\le j\le n-1\}. \end{aligned}$$

It is easy to check that $c_j\varpi ^{c_j^{-1}(\lambda _{1,r})}\approx \varpi ^{\nu _{n-j}}\tau$ for all $1\le j\le n-1$. Thus we have

$$\begin{aligned} X_{\lambda _{1,r}}(b)=\bigsqcup _{j=1}^{n-1}\pi (X_{\varpi ^{\nu _j}\tau }(b)). \end{aligned}$$

In a similar way as (i), it follows that the map $X_{\varpi ^{\nu _j}\tau }(b)_0\rightarrow X_{\lambda _{1,r}}(b),\ gI\mapsto gK$ is an immersion. So we regard each $hX_{\varpi ^{\nu _j}\tau }(b)_0(h\in J/J_{{\mathcal {O}}})$ as a locally closed subvariety of $X_{\lambda _{1,r}}(b)$. Let $C_{b,r}^j=\frac{n(n-1)}{2}r+j+\sum _{j=1}^{n-1}\lfloor \frac{jk_0}{n_0}\rfloor$ ($1\le j\le n-1$). Then, we can show that $X_{\varpi ^{\nu _{n-j_0}}\tau }(b)_0$ is equal to the set

$$\begin{aligned} \{gK\in X_{\lambda _{1,r}}(b)\mid g{\mathscr {L}}_0\subset {\mathscr {L}}_0\ \text {and}\ v_L(\det (g))=C_{b,r}^{j_0}\}\setminus \bigsqcup _{j=1}^{j_0-1}X_{\varpi ^{\nu _{n-j}}}(b). \end{aligned}$$

In particular, by the case $j_0=1$, $X_{\varpi ^{\nu _{n-1}}\tau }(b)_0$ is closed in $X_{\lambda _{1,r}}(b)$. By Lemma 5.6 and Lemma 5.8, the closed subvarieties $X_{\varpi ^{\nu _1}\tau }(b),\ldots , X_{\varpi ^{\nu _{n-2}}\tau }(b)$ and $hX_{\varpi ^{\nu _{n-1}}\tau }(b)_0\ (h\in J/J_{{\mathcal {O}}})$ form a disjoint cover of $X_{\lambda _{1,r}}(b)$. Since $X_{\lambda _{1,r}}(b)$ is locally perfectly of finite type, $hX_{\varpi ^{\nu _{n-1}}\tau }(b)_0$ is open. So $X_{\varpi ^{\nu _{n-1}}\tau }(b)$ is open and $X_{\varpi ^{\nu _{n-2}}\tau }(b)_0$ is closed. Repeating the same argument, we see that each $hX_{\varpi ^{\nu _{n-j}}}(b)_0$ is closed and open in $X_{\lambda _{1,r}}(b)$. This result and Corollary 5.5 prove (ii).

Finally, (iii) (resp. (iv)) follows from (i) (resp. (ii)) by Proposition 3.2. $\square$

Remark 5.10

In [3], Chen and Viehmann define the J-stratification of affine Deligne–Lusztig varieties, which coincides with the Bruhat-Tits stratification of $X(\lambda , b)_P$ if $(G, \lambda , P)$ is of Coxeter type, see [5]. If b is superbasic, it follows from [3, Proposition 3.4] that the decomposition in Theorem 5.9 is also an example of the J-stratification.