Ekedahl–Oort and Newton strata for Shimura varieties of PEL type

Abstract

We study the Ekedahl–Oort stratification for good reductions of Shimura varieties of PEL type. These generalize the Ekedahl–Oort strata defined and studied by Oort for the moduli space of principally polarized abelian varieties (the “Siegel case”). They are parameterized by certain elements $w$ in the Weyl group of the reductive group of the Shimura datum. We show that for every such $w$ the corresponding Ekedahl–Oort stratum is smooth, quasi-affine, and of dimension $\ell (w)$ (and in particular non-empty). Some of these results have previously been obtained by Moonen, Vasiu, and the second author using different methods. We determine the closure relations of the strata. We give a group-theoretical definition of minimal Ekedahl–Oort strata generalizing Oort’s definition in the Siegel case and study the question whether each Newton stratum contains a minimal Ekedahl–Oort stratum. As an interesting application we determine which Newton strata are non-empty. This criterion proves conjectures by Fargues and by Rapoport generalizing a conjecture by Manin for the Siegel case. We give a necessary criterion when a given Ekedahl–Oort stratum and a given Newton stratum meet.

A. Appendix on Coxeter groups, reductive group schemes and quotient stacks

In this appendix we fix some conventions and recall results on Coxeter groups, on reductive group schemes and on quotient stacks that are used in the main text.

1.1 A.1 Coset representatives of Coxeter groups

Let $W$ be a Coxeter group and $I$ its generating set of simple reflections. Let $\ell$ denote the length function on $W$.

Let $J$ be a subset of $I$. We denote by $W_J$ the subgroup of $W$ generated by $J$ and by $W^J$ (respectively ${}^{J}{W}$) the set of elements $w$ of $W$ which have minimal length in their coset $wW_J$ (respectively $W_Jw$). Then every $w \in W$ can be written uniquely as $w=w^{J}w_{J}= w^{\prime }_J {}^{J}{w}$ with $w_J, w^{\prime }_J \in W_J$, $w^J\in W^J$ and ${}^{J}{w} \in {}^{J}{W}$, and $\ell (w) = \ell (w_J) + \ell (w^J) = \ell (w^{\prime }_J) + \ell ({}^{J}{w})$ (see [7, Proposition 4.16]). In particular, $W^J$ and ${}^{J}{W}$ are systems of representatives for $W/W_J$ and $W_J\backslash W$ respectively.

Furthermore, if $K$ is a second subset of $I$, let ${}^{J}{W}^{K}$ be the set of $w \in W$ which have minimal length in the double coset $W_JwW_K$. Then ${}^{J}{W}^{K}={}^{J}{W}\cap W^K$ and ${}^{J}{W}^{K}$ is a system of representatives for $W_J\backslash W/W_K$ (see [7, (4.3.2)]).

1.2 A.2 Bruhat order

We let $\le$ denote the Bruhat order on $W$. This natural partial order is characterized by the following property: For $x, w\in W$ we have $x\le w$ if and only if for some (or, equivalently, any) reduced expression $w=s_{i_{1}}\cdots s_{i_{n}}$ as a product of simple reflections $s_i\in I$, one gets a reduced expression for $x$ by removing certain $s_{i_{j}}$ from this product. More information about the Bruhat order can be found in [1, Chapter 2]. The set ${}^{J}{W}$ can be described as

$$\begin{aligned} {}^{J}{W}=\{\,w \in W\ ;\ w < sw \text{ for} \text{ all} s \in J\,\} \end{aligned}$$

(12.1)

(see [1, Definition 2.4.2 and Corollary 2.4.5]).

1.3 A.3 Reductive group schemes, maximal tori, and Borel subgroups

Let $S$ be a scheme. A reductive group scheme over $S$ is a smooth affine group scheme $G$ over $S$ such that for every geometric point $s \in S$ the geometric fiber $G_{\bar{s}}$ is a connected reductive algebraic group over $\kappa (\bar{s})$.

Let $G$ be a reductive group scheme over $S$. A maximal torus of $G$ is a closed subtorus $T$ of $G$ such that $T_{\bar{s}}$ is a maximal element in the set of subtori of $G_{\bar{s}}$ for all $s \in S$. By a theorem of Grothendieck [48, Exp. XIV, 3.20] maximal tori of $G$ exist Zariski locally on $S$. A Borel subgroup of $G$ is a closed smooth subgroup scheme $B$ of $G$ such that for all $s \in S$ the geometric fiber $B_{\bar{s}}$ is a Borel subgroup of $G_{\bar{s}}$ in the usual sense (i.e., a maximal smooth connected solvable subgroup). A reductive group scheme over $S$ is called split if there exists a maximal torus $T$ of $G$ such that $T \cong \mathbb G _{m,S}^r$ for some integer $r \ge 0$.

1.4 A.4 Parabolic subgroups and Levi subgroups

A smooth closed subgroup scheme $P$ of $G$ is called parabolic subgroup of $G$ if the fppf quotient $G/P$ is representable by a smooth projective scheme or, equivalently by [48, Exp. XXII (5.8.5)], if $G_{\bar{s}}/P_{\bar{s}}$ is proper for all $s \in S$. Every Borel subgroup of $G$ is a parabolic subgroup. The unipotent radical of $P$, denoted by $U_P$, is the largest smooth normal closed subgroup scheme with unipotent and connected fibers. It exists by [48, Exp. XXII, (5.11.4)]. If $P$ contains a maximal torus $T$ of $G$, there exists a unique reductive closed subgroup scheme $L$ of $P$ containing $T$ such that the canonical homomorphism $L \rightarrow P/U_P$ is an isomorphism (loc. cit.). Any such subgroup $L$ is called a Levi subgroup of $P$.

The functor that sends an $S$-scheme $T$ to the set of Borel (resp. parabolic) subgroups of $G \times _S T$ is representable by a smooth projective $S$-scheme by [48, Exp. XXVI, 3]. We call the representing scheme $\mathrm{Bor}_G$ (resp. $\mathrm{Par}_G$). The functor that attaches to an $S$-scheme $T$ the set of pairs $(P,L)$, where $P$ is a parabolic subgroup of $G \times _S T$ and $L$ is a Levi subgroup of $P$ is representable by a smooth quasi-projective $S$-scheme (loc. cit.).

If $S$ is local, $G$ is called quasi-split if there exists a Borel subgroup of $G$. Every split reductive group scheme is quasi-split. If $S = {\mathrm{Spec}} R$, where $R$ is a henselian local ring whose residue field $k$ has cohomological dimension $\le 1$ (e.g., if $k$ is finite), then $G$ is quasi-split. Indeed, the special fiber $G_k$ has a Borel subgroup $B_k$ by [49, III, Sect. 2.3]. As the scheme of Borel subgroups $\mathrm{Bor}_G$ is smooth, there exists a lifting of $B_k$ to a Borel subgroup of $G$.

1.5 A.5 Weyl groups and types of parabolic subgroups over connected base schemes

Let $G$ be a reductive group over an algebraically closed field, let $B$ be a Borel subgroup of $G$, and let $T$ be a maximal torus of $B$. Let $W(T) := \mathrm{Norm}_G(T)/T$ denote the associated Weyl group, and let $I(B,T) \subset W(T)$ denote the set of simple reflections defined by $B$. Then $W(T)$ is a Coxeter group with respect to the subset $I(B,T)$.

A priori this data depends on the pair $(B,T)$. However, any other such pair $(B^{\prime },T^{\prime })$ is obtained by conjugating $(B,T)$ by some element $g\in G$ which is unique up to right multiplication by $T$. Thus conjugation by $g$ induces isomorphisms $W(T) \stackrel{\sim }{\rightarrow } W(T^{\prime })$ and $I(B,T) \stackrel{\sim }{\rightarrow } I(B^{\prime },T^{\prime })$ that are independent of $g$. Moreover, the isomorphisms associated to any three such pairs are compatible with each other. Thus $W := W_G := W(T)$ and $I := I(B,T)$ for any choice of $(B,T)$ can be viewed as instances of “the” Weyl group and “the” set of simple reflections of $G$, in the sense that up to unique isomorphisms they depend only on $G$.

Now let $G$ be a quasi-split reductive group scheme over a connected scheme $S$. Then we obtain for any geometric point $\bar{s}\rightarrow S$ the Weyl group and the set of simple reflections $(W_{\bar{s}},I_{\bar{s}})$ of $G_{\bar{s}}$. The algebraic fundamental group $\pi _1(S,\bar{s})$ acts naturally on $W_{\bar{s}}$ preserving $I_{\bar{s}}$ (because $G$ is quasi-split), and every étale path $\gamma$ from $\bar{s}$ to another geometric point $\bar{s}^{\prime }$ of $S$ yields an isomorphism of $(W_{\bar{s}},I_{\bar{s}}) \overset{\sim }{\rightarrow }(W_{\bar{s}^{\prime }},I_{\bar{s}^{\prime }})$ that is equivariant with respect to the isomorphism $\pi _1(S,\bar{s}) \overset{\sim }{\rightarrow }\pi _1(S,\bar{s}^{\prime })$ induced by $\gamma$ (cf. [48] Exp. XII, 2.1). In particular $(W_{\bar{s}},I_{\bar{s}})$ together with its action by $\pi _1(S,\bar{s})$ is independent of the choice of $\bar{s}$ up to isomorphism. We denote it by $(W,I)$ and call it the Weyl system of $G$.

If $P$ is a parabolic subgroup of $G$ and $s \in S$, the type $J_{\bar{s}} \subset I$ of the parabolic subgroup $P_{\bar{s}}$ of $G_{\bar{s}}$ is independent of $s \in S$ [48, Exp. XXVI, 3] and we call $J := J_{\bar{s}}$ the type of $P$. For a subset $J$ of $I$ we denote by $\mathrm{Par}_J$ the open and closed subscheme of $\mathrm{Par}$ parameterizing parabolic subgroups of type $J$. If $S$ is a semi-local scheme, $J$ and $\mathrm{Par}_J$ are defined over a finite étale covering of $S$ [48, Exp. XXIV, 4.4.1].

For simplicity assume that $S$ is local. Let $J, K \subseteq I$ be subsets and let $S_1 \rightarrow S$ be the finite étale extension over which $J$ and $K$ are defined. Let $w \in {}^{J}{W}^{K}$. For every $S_1$-scheme $S^{\prime }$ and for every parabolic subgroup $P$ of $G_{S^{\prime }}$ of type $J$ and every parabolic subgroup $Q$ of $G_{S^{\prime }}$ of type $K$ we write

$$\begin{aligned} \mathrm{relpos}(P,Q) = w \end{aligned}$$

if there exists an fppf-covering on $S^{\prime \prime } \rightarrow S^{\prime }$, a Borel subgroup $B$ of $G_{S^{\prime \prime }}$ and a split maximal torus $T$ of $B$ such that $P_{S^{\prime \prime }}$ contains $B$ and $Q_{S^{\prime \prime }}$ contains ${}^{\dot{w}}{B}$, where $\dot{w} \in \mathrm{Norm}_{G_{S^{\prime \prime }}}(T)(S^{\prime \prime })$ is a representative of $w \in W = \mathrm{Norm}_{G_{S^{\prime \prime }}}(T)(S^{\prime \prime })/T(S^{\prime \prime })$.

If $S^{\prime } = {\mathrm{Spec}} k$ for an algebraically closed field, then $(P,Q) \mapsto \mathrm{relpos}(P,Q)$ yields a bijection between $G(k)$-orbits on $\mathrm{Par}_J(k) \times \mathrm{Par}_K(k)$ and the set ${}^{J}{W}^{K}$.

1.6 A.6 One-parameter subgroups and their type

Let $G$ be a reductive group scheme over a connected scheme $S$. A one-parameter subgroup of $G$ is by definition a homomorphism of group schemes $\lambda :\mathbb G _{m,S} \rightarrow G$, where $\mathbb G _{m,S}$ denotes the multiplicative group over $S$. We identify the character group of $\mathbb G _{m,S}$ with $\mathbb Z$. Composing $\lambda$ with the adjoint representation $G \rightarrow \mathrm{GL}(\mathrm{Lie}(G))$ yields a decomposition $\mathrm{Lie}(G) = \bigoplus _{n \in \mathbb Z } \mathfrak{g }_n$. There is a unique parabolic subgroup $P(\lambda )$ of $G$ and a unique Levi subgroup $L(\lambda )$ of $P(\lambda )$ such that

$$\begin{aligned} \mathrm{Lie}P(\lambda ) = \bigoplus _{n \ge 0}\mathfrak{g }_n, \quad \mathrm{Lie}L(\lambda ) = \mathfrak{g }_0. \end{aligned}$$

Indeed, because of the uniqueness assertion we may show this locally for the étale topology and hence can assume that the image of $\lambda$ lies in a split maximal torus. Then the claim follows from [48] Exp. XXVI, 1.4 and 4.3.2. By loc. cit. we may define $L(\lambda )$ also as the centralizer of $\lambda$.

The type of $P(\lambda )$ is also called the type of $\lambda$. It depends only on the conjugacy class of $\lambda$.

1.7 A.7 The example of the symplectic group

As an example consider a symplectic space $(V,{\langle \ ,\ \rangle })$ of dimension $2g$ over a field $k$ and denote by $G = \mathrm{GSp}(V,{\langle \ ,\ \rangle })$ the group of symplectic similitudes of $(V,{\langle \ ,\ \rangle })$. Let $S_{2g}$ denote the symmetric group of the set $\{1,\cdots ,2g\}$. Then the Weyl system $(W,I)$ of $G$ is given by

$$\begin{aligned} \begin{aligned} W&= \{\,w \in S_{2g}\ ;\ w(i) + w(2g+1-i) = 2g+1 \,\text{ for} \text{ all} i = 1,\dots ,g\,\},\\ I&= \{s_1,\dots ,s_g\},\quad \text{ where}\quad s_i = \left\{ \begin{array}{l} \tau _i\tau _{2g-i},\\ \tau _g, \end{array} \begin{array}{l} \text{ for} i = 1,\dots ,g-1;\\ \text{ for} i = g. \end{array} \right. \end{aligned} \end{aligned}$$

(12.2)

Here $\tau _j$ denotes the transposition of $j$ and $j+1$. The length of an element $w \in W$ can be computed as follows

$$\begin{aligned} \ell (w)&= \#\{\,(i,j)\!\ ;\ 1 \le i < j \le g, w(i) > w(j)\,\} \\&+ \#\{\,(i,j)\!\ ;\ 1 \le i \le j \le g, w(i) + w(j) > 2g+1\,\}. \end{aligned}$$

For every $d$-tuple $(x_1,\dots ,x_d)$ of real numbers we denote by $(x_1,\dots ,x_d){\uparrow }$ the $d$-tuple $(x_{\sigma (1)},\dots ,x_{\sigma (d)})$ with $\sigma \in S_d$ such that $x_{\sigma (1)} \le \dots \le x_{\sigma (d)}$. Then the Bruhat order is given by

$$\begin{aligned} w^{\prime } \le w \iff \forall \ 1\le i \le g: (w^{\prime }(1),\dots ,w^{\prime }(i)){\uparrow } \le (w(1),\dots ,w(i)){\uparrow }, \end{aligned}$$

where we compare tuples componentwisely.

To study the Siegel case we consider the subset $J := \{s_1,\dots ,s_{g-1}\}$ of $I$. Then $W_J$ consists of those permutations $w \in W$ such that $w(\{1,\dots ,g\}) = \{1,\dots ,g\}$. The map

$$\begin{aligned} W_J \rightarrow S_g, \quad w \mapsto w{}_{\vert }{}_{\{1,\dots ,g\}} \end{aligned}$$

is a group isomorphism. The set ${}^{J}{W}$ consists in this case of those elements $w \in W$ such that $w^{-1}(1) < w^{-1}(2) < \dots < w^{-1}(g)$. Of course, this implies $w^{-1}(g+1) < \dots < w^{-1}(2g)$.

It is convenient to give an alternative description of ${}^{J}{W}$. To $w \in {}^{J}{W}$ we attach $\epsilon = (\epsilon _i)_{1 \le i \le g} \in \{0,1\}^g$ by

$$\begin{aligned}\epsilon _i := \left\{ \! \begin{array}{ll} 0,&\quad \text{ if}\, i \in \{w^{-1}(1),\dots ,w^{-1}(g)\};\\ 1,&\quad \text{ otherwise}, \end{array} \right. \quad i = 1,\dots ,g. \end{aligned}$$

This yields a bijection ${}^{J}{W} \leftrightarrow \{0,1\}^g$. The length of such an element $\epsilon = (\epsilon _1,\dots ,\epsilon _g)$ is equal to

$$\begin{aligned} \ell (\epsilon ) = \sum _{i=1}^g i\epsilon _{g+1-i}. \end{aligned}$$

(12.3)

The set ${}^{J}{W}^{J} \cong W_J \backslash W/W_J$ corresponds to the set of $W_J$-orbits of $\{0,1\}^g$, where $W_J = S_g$ acts by permuting the entries of $(\epsilon _i)_i \in \{0,1\}^g$. Thus $(\epsilon _i)_i \mapsto \#\{\,i\ ;\ \epsilon _i = 1\,\}$ yields a bijection

$$\begin{aligned} {}^{J}{W}^{J} \leftrightarrow \{0,\dots ,g\}. \end{aligned}$$

(12.4)

The set ${}^{J}{W}$ can also be described by elementary sequences in the sense of Oort (see [10, Sect. 2]; note that in loc. cit. the set $W^J$ is considered which we identify with ${}^{J}{W}$ via $w \mapsto w^{-1}$): For $w \in {}^{J}{W}$ we define

$$\begin{aligned}\varphi _w:\{0,\dots ,g\} \rightarrow \mathbb Z _{\ge 0}, \quad i \mapsto i - \#\{\,1 \le a \le g\ ;\ w^{-1}(a) \le i\,\}. \end{aligned}$$

Then $\varphi _w$ is an elementary sequence, i.e., a map $\varphi :\{0,\dots ,g\} \rightarrow \mathbb Z _{\ge 0}$ such that $\varphi (0) = 0$ and such that $\varphi (i-1) \le \varphi (i) \le \varphi (i-1) + 1$ for all $i = 1,\dots ,g$. The map $w \mapsto \varphi _w$ yields a bijection between ${}^{J}{W}$ and the set $\mathcal{S }_g$ of elementary sequences. The tuple $(\epsilon _i)_i \in \{0,1\}^g$ corresponding to an element $w \in {}^{J}{W}$ can be described via the elementary sequence $\varphi _w$ by $\epsilon _i = \varphi _w(i) - \varphi _w(i-1)$. Using (12.3) an easy calculation shows

$$\begin{aligned} \ell (w) = \sum _{i=1}^g \varphi _w(i). \end{aligned}$$

(12.5)

Finally, for $w,w^{\prime } \in {}^{J}{W}$ we have $w^{\prime } \le w$ if and only if $\varphi _{w^{\prime }}(i) \le \varphi _w(i)$ for all $i = 1,\dots ,g$.

Let us now describe parabolic subgroups of $G$ of type $J$. Let $S$ be any $k$-scheme. Then $V_S = V \otimes _k {\fancyscript{O}}_S$ is a free ${\fancyscript{O}}_S$-module of rank $2g$, and the base change of ${\langle \ ,\ \rangle }$ is a perfect alternating form on $V_S$. An ${\fancyscript{O}}_S$-submodule ${\fancyscript{L}}$ of $V_S$ is called Lagrangian if ${\fancyscript{L}}$ is locally on $S$ a direct summand of $V_S$ of rank $g$ and if ${\fancyscript{L}}$ is totally isotropic. Attaching to a Lagrangian ${\fancyscript{L}}$ its stabilizer in $G_S$ yields a bijection between the set of Lagrangians in $V_S$ and the set of parabolic subgroups $P$ of $G \times _k S$ of type $J$ (with $J = \{s_1,\dots ,s_{g-1}\}$).

Let ${\fancyscript{L}}$ and ${\fancyscript{L}}^{\prime }$ be Lagrangians in $V_S$ and let $P$ and $P^{\prime }$ be the corresponding parabolic subgroups of $G_S$ of type $J$. Then for $d \in \{0,\dots ,g\} = {}^{J}{W}^{J}$ (12.4) we have

$$\begin{aligned} \mathrm{relpos}(P,P^{\prime }) = d \iff V_S/({\fancyscript{L}}+ {\fancyscript{L}}^{\prime }) \text{ is} \text{ locally} \text{ free} \text{ of} \text{ rank} g-d. \end{aligned}$$

(12.6)

Here the condition that $V_S/({\fancyscript{L}}+ {\fancyscript{L}}^{\prime })$ is locally free (of some rank) is equivalent to the condition that fppf-locally $P$ and $Q$ contain a common maximal torus [35, Sect. 3.5].

1.8 A.8 The underlying topological space of a quotient stack

We recall some—probably well known—facts on quotient stacks. For lack of a better reference we refer to [58, (4.2)–(4.4)]. Let $k$ be a field, $\bar{k}$ an algebraic closure, $\Gamma := \mathrm{Aut}(\bar{k}/k)$ the profinite group of $k$-automorphisms of $\bar{k}$. Let $X$ be a $k$-scheme of finite type and let $H$ be a smooth affine group scheme over $k$ that acts on $X$. We assume that there are only finitely many $H(\bar{k})$-orbits in $X(\bar{k})$.

Let $[H \backslash X]$ be the algebraic quotient stack. To describe its underlying topological space we first recall that if $\le$ is a partial order on a set $Z$, we can define a topology on $Z$ by defining a subset $U$ of $Z$ to be open if for all $u \in U$ and $z \in Z$ with $u \le z$ one has $z \in U$.

Let $\Xi ^\mathrm{alg}$ be the set of $H(\bar{k})$-orbits in $X(\bar{k})$. For $\mathcal{O }, \mathcal{O }^{\prime } \in \Xi ^\mathrm{alg}$ we set $\mathcal{O }^{\prime } \le \mathcal{O }$ if the closure of $\mathcal{O }$ contains $\mathcal{O }^{\prime }$. This defines a partial order and hence a topology on $\Xi ^\mathrm{alg}$. The group $\Gamma$ acts on $\Xi ^\mathrm{alg}$ and we denote the set of $\Gamma$-orbits on $\Xi ^\mathrm{alg}$ by $\Xi$. We endow $\Xi$ with the quotient topology. Then $\Xi$ is homeomorphic to the underlying topological space of $[H \backslash X]$.