Normalization of closed Ekedahl-Oort strata

TL;DR

This paper uses partial flag space theory to explicitly determine the normalization of Zariski closures of Ekedahl-Oort strata in Shimura varieties of Hodge-type, linking group theory with algebraic geometry.

Contribution

It introduces a group-theoretical approach to normalize closures of Ekedahl-Oort strata, extending previous theories of partial flag spaces.

Findings

Explicit description of the normalization of strata closures

Connection between partial flag spaces and Shimura varieties

Generalization of canonical filtration for truncated Barsotti-Tate groups

Abstract

We apply our theory of partial flag spaces developed in previous articles to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

Equations

$$\zeta :X⟶{\text{G-Zip}}^{\mathcal{Z}}$$

$$X_{P_0}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E:=\{(p,q)\in P\times Q,\phi (\overline{p})=\overline{q}\}$$

$${\text{G-Zip}}^{\mathcal{Z}}\simeq \left[E\backslash G\right].$$

$$w=x{w}_J,\text{with}{w}_J\in {}^{I_x}{W}_J.$$

$$w\in {}^{I}W⟺{}^{z}B\cap L={}^{zw}B\cap L.$$

$${}^{I}W\to \{E\text{-orbits in }G\},w\mapsto G_w:=\mathcal{O}(z\dot{w}).$$

$$\dim (G_w)=\ell (w)+\dim (P).$$

$$L_w$$

$$P_w$$

$$Q_w$$

$$A:=\{(x,\phi (x)),x\in L_w,{}^{z\dot{w}}\phi (x)=x\}$$

$$dp\in \mathrm{Lie}(P)\cap {\mathrm{Ad}}_{z\dot{w}}(\mathrm{Lie}(V))=\mathrm{Lie}(P\cap {}^{zw}V)$$

$${\text{G-ZipFlag}}^{(\mathcal{Z},P_0)}\simeq \left[E\backslash \left(G\times P/P_0\right)\right]$$

$$M_0$$

$$Q_0$$

$${\Psi }_{P_0}:{\text{G-ZipFlag}}^{(\mathcal{Z},P_0)}\to {\text{G-Zip}}^{{\mathcal{Z}}_0}$$

$$Z_{P_0,w}:={\Psi }_{P_0}^{-1}({\mathfrak{o}}_{{\mathcal{Z}}_0}(z\dot{w}))$$

$$G_{P_0,w}:=\{(g,a{P}_0)\in G\times P/P_0,a^{-1}g\phi (\overline{a})\in {\mathcal{O}}_{{\mathcal{Z}}_0}(z\dot{w})\}.$$

$${\psi }_{P_0}:{\text{G-ZipFlag}}^{(\mathcal{Z},P_0)}\to {\mathrm{Brh}}^{{\mathcal{Z}}_0}.$$

$${\mathbf{Z}}_{P_0,x}:={\psi }_{P_0}^{-1}(\mathfrak{b}(x))$$

$${\mathbf{G}}_{P_0,x}:=\{(g,a{P}_0)\in G\times P/P_0,ag\phi (\overline{a}{)}^{-1}\in P_{0}z\dot{x}{Q}_0\}.$$

$$Z_{P_0,w}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\pi :{\overline{Z}}_{P_0,w}\to {\overline{Z}}_w,\tilde{\pi }:{\overline{G}}_{P_0,w}\to {\overline{G}}_w$$

$${\overline{\mathbf{G}}}_{P_0,x}={\overline{G}}_{P_0,w}=\{(g,a{P}_0)\in G\times P/P_0,a^{-1}g\phi (\overline{a})\in \overline{P_{0}z\dot{w}{Q}_0}\}$$

$$R/R_{P_0}\to S/S_{P_0}.$$

$$A_1:=\{x\in L_w,{}^{z\dot{w}}\phi (x)=x\}.$$

$$({}^a{L}_w)({\mathbf{F}}_p)\subset {}^a{P}_0.$$

$$H:=L\cap \bigcap _{i\in \mathbf{Z}}{\sigma }^i(P)=\bigcap _{i\in \mathbf{Z}}L\cap {\sigma }^i(P).$$

$$\bigcap _{i=1}^m{P}_i=L_0⋉U_0$$

$$\dim ({\mathcal{O}}_{E_{{\mathcal{Z}}_0}}(z\dot{w}))=\dim (P_0)+\ell (w).$$

$$\dim (P_{0}z\dot{w}{Q}_0)=2\dim (P_0)-\dim ({\mathrm{Stab}}_{P_0\times Q_0}(z\dot{w})).$$

Full text

Introduction

Let $H$ be a truncated Barsotti-Tate group of level 1 (in short BT1), over an algebraically closed field $k$ of characteristic $p$, and let $\sigma:k\to k$ denote the map $x\mapsto x^{p}$. Denote by $D:=\mathbf{D}(H)$ its Dieudonné module, which is a finite-dimensional $k$-vector space $D$ endowed with a $\sigma$-linear endomorphism $F$ and a $\sigma^{-1}$-linear endomorphism $V$ satisfying $FV=VF=0$ and ${\rm Im}(F)=\operatorname{Ker}(V)$. Oort showed in [Oor01] that there exists a flag of $D$ which is stable by $V$ and $F^{-1}$ and which is coarsest among all such flags, called the canonical filtration of $D$. After choosing a basis of $D$, we obtain a filtration of $k^{n}$ (where $n=\dim_{k}(D)$ is the height of $H$). The stabilizer of this flag is a parabolic subgroup $P(H)\subset GL_{n}$, well-defined up to conjugation. We want to emphasize that this construction attaches a group-theoretical object $P(H)$ to a truncated Barsotti-Tate group of level 1.

The theory of $F$-zips developed in [MW04], [PWZ11] and [PWZ15] establishes the precise link between BT1’s and group theory. Specifically, isomorphism classes of BT1’s of height $n$ and dimension $d$ correspond bijectively to $E$-orbits in $GL_{n}$, where $E$ is the zip group (see section 4.1). The stack of $F$-zips of type $(n,d)$ can be defined as the quotient stack \mathop{\text{F-{\tt Zip}}}\nolimits^{n,d}=\left[E\backslash GL_{n}\right]. Moreover, there is a natural morphism of stacks BT^{n,d}_{1}\to\mathop{\text{F-{\tt Zip}}}\nolimits^{n,d}, where $BT^{n,d}_{1}$ is the stack of BT1’s of height $n$ and dimension $d$ over $k$. More generally, let $G$ be a connected reductive group over $\mathbf{F}_{p}$, and $P,Q\subset G$ parabolic subgroups (defined over some finite extension of $\mathbf{F}_{p}$). Let $L\subset P$ and $M\subset Q$ be Levi subgroups and assume that $\varphi(L)=M$, where $\varphi:G\to G$ is the Frobenius homomorphism. One can define the stack of $G$-zips of type ${\mathcal{Z}}=(G,P,L,Q,M,\varphi)$ as the quotient stack \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}=\left[E\backslash G\right], where $E\subset P\times Q$ is the zip group (see section 1.1). For example, if $G$ is the automorphism group of a PEL-datum, then \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}(k) classifies BT1’s over $k$ of type ${\mathcal{Z}}$ endowed with this additional structure. If $W$ denotes the Weyl group of $G$, the $E$-orbits in $G$ are parametrized by a subset ${}^{I}W\subset W$ (see section 1.3). Denote by $G_{w}\subset G$ the $E$-orbit corresponding to $w\in{}^{I}W$ and put $Z_{w}:=[E\backslash G_{w}]$ the corresponding zip stratum.

In [GKb], we defined for each parabolic $P_{0}\subset P$ the stack of partial zip flags \mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P_{0})} endowed with a natural projection \pi:\mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P_{0})}\to\mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}} which makes it a $P/P_{0}$-bundle over \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}. This defines a tower of stacks above \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}. Moreover, the stack \mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P_{0})} admits two natural stratifications. In general, one is finer than the other, but they coincide when $P_{0}$ is a Borel subgroup. The fine strata $Z_{P_{0},w}$ are parametrized by $w\in{}^{I_{0}}W$, where ${}^{I_{0}}W\subset W$ is a subset containing ${}^{I}W$ (see section 2.1). The strata $Z_{P_{0},w}$ attached to elements $w\in{}^{I}W$ are called minimal and satisfy $\pi(Z_{P_{0},w})=Z_{w}$ and the restriction $\pi:Z_{P_{0},w}\to Z_{w}$ is finite étale. When $P_{0}=P$, the stack \mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P)} coincides with \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}} and the fine stratification is the stratification by $E$-orbits, whereas the coarse stratification is given by $P\times Q$-orbits. In general, we say that a stratum $Z_{P_{0},w}$ has coarse closure if it is open in the coarse stratum containing it. If $Z_{P_{0},w}$ has coarse closure, its Zariski closure $\overline{Z}_{P_{0},w}$ is normal.

In the formalism of $G$-zips, one can attach to each $w\in{}^{I}W$ a parabolic subgroup $P_{w}\subset P$. In the case $G=GL_{n}$, if $H$ is a BT1 corresponding to $w\in{}^{I}W$ under the correspondence between BT1’s and $E$-orbits, then $P_{w}$ is the parabolic $P(H)$ defined above. This is proved in Prop. 4.3.1. Since $P_{w}$ is canonically attached to $w$, it is natural to ask what special property is satisfied by the stratum $Z_{P_{w},w}$ of \mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P_{w})}. Our main theorem answers this question:

Theorem 1**.**

[Th. 3.1.3] Let $w\in{}^{I}W$. The following properties hold:

1. (i)

$\pi:Z_{P_{w},w}\to Z_{w}$* is an isomorphism.* 2. (ii)

$Z_{P_{w},w}$* has coarse closure.*

Furthermore, among all parabolic subgroups $P_{0}$ such that ${}^{z}B\subset P_{0}\subset P$, the parabolic $P_{w}$ is the smallest parabolic satisfying (i) and the largest one satisfying (ii).

The Borel subgroup ${}^{z}B$ of the above theorem is defined in section 1.2. Note that property (i) is obviously satisfied for $P_{0}=P$ and property (ii) is satisfied for $P_{0}={}^{z}B$ because fine and coarse strata coincide in this case. Hence the canonical parabolic $P_{w}$ is the unique intermediate parabolic such that both properties are satisfied. As a consequence, we deduce that the normalization of the Zariski closure $\overline{Z}_{w}$ is $\tilde{Z}_{w}:=\operatorname{Spec}({\mathcal{O}}(\overline{Z}_{P_{w},w}))$ (see Corollary 3.3.2).

Let $X$ be the special fiber of a good reduction Hodge-type Shimura variety, and let $G$ be the attached reductive $\mathbf{F}_{p}$-group (see section 3.4). In [Zha], Zhang has constructed a smooth map of stacks

[TABLE]

where ${\mathcal{Z}}$ is the zip datum attached to $X$ as in [GKb] §6.2. The Ekedahl-Oort stratification of $X$ is defined as the fibers of $\zeta$. For $w\in{}^{I}W$, set $X_{w}:=\zeta^{-1}(Z_{w})$. For any parabolic ${}^{z}B\subset P_{0}\subset P$, define the partial flag space $X_{P_{0}}$ as the fiber product

[TABLE]

For $w\in{}^{I_{0}}W$, define the fine stratum $X_{P_{0},w}:=\zeta_{P_{0}}^{-1}(Z_{P_{0},w})$ of $X_{P_{0}}$. The space $X_{P_{0}}$ is a generalization of the flag space considered by Ekedahl and Van der Geer in [EvdG09], where they consider flags refining the Hodge filtration of an abelian variety.

Corollary 1**.**

[Cor. 3.4.1] Let $w\in{}^{I}W$. The normalization of $\overline{X}_{w}$ is the Stein factorization of the map $\pi:\overline{X}_{P_{w},w}\to\overline{X}_{w}$. It is isomorphic to \overline{X}_{w}\times_{\mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}}\tilde{Z}_{w}.

For Siegel-type Shimura varieties, an analogous result to Cor. 1 was proved by Boxer in [Box15, Th.5.3.1] using different methods.

We now give an overview of the paper. In section 1, we review the theory of $G$-zips and prove a result on point stabilizers for later use. Section 2 is devoted to the stack of partial $G$-zips and its stratifications. We define minimal strata and give an explicit form for the restriction of the map $\pi$ to a minimal stratum (Prop. 2.2.1). In section 3, we define the notion of canonical parabolic and explain its relevance with respect to the normalization of a closed stratum of \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}. We prove Theorem 3.1.3 after giving criteria for properties (i) and (ii) above. Finally, we explain in section 4 the correspondence between the classical theory of BT1’s and the theory of G-zips, following [PWZ11]. We establish the link between the parabolic $P_{w}$ and the canonical parabolic of a BT1.

Acknowledgements

The author would like to thank Wushi Goldring for many fruitful discussions about this work and Torsten Wedhorn for helpful comments on this paper. I am grateful to the reviewer for suggesting improvements.

1. Review of $G$-zips

We will need to review some facts about the stack of $G$-zips found in [PWZ11] and prove a result on the stabilizer of an element by the group $E$.

1.1. The stack \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}

We fix an algebraic closure $k$ of $\mathbf{F}_{p}$. A zip datum is a tuple ${\mathcal{Z}}=(G,P,L,Q,M,\varphi)$ where $G$ is a connected reductive group over $\mathbf{F}_{p}$, $\varphi:G\to G$ is the Frobenius homomorphism, $P,Q\subset G$ are parabolic subgroups of $G_{k}$, $L\subset P$ and $M\subset Q$ are Levi subgroups of $P$ and $Q$ respectively. One imposes the condition $\varphi(L)=M$. One can attach to ${\mathcal{Z}}$ a zip group $E$ defined by

[TABLE]

where $\overline{p}\in L$ and $\overline{q}\in M$ denote the projections of $p$ and $q$ via the isomorphisms $P/R_{u}(P)\simeq L$ and $Q/R_{u}(Q)\simeq M$. We let $G\times G$ act on $G$ via $(a,b)\cdot g:=agb^{-1}$ and we obtain by restriction an action of $E$ on $G$. The stack of $G$-zips is then isomorphic to the following quotient stack:

[TABLE]

When we want to specify the zip datum ${\mathcal{Z}}$, we write sometimes $E_{\mathcal{Z}}$ for the zip group $E$.

1.2. Frame

A frame for ${\mathcal{Z}}$ is a triple $(B,T,z)$ where $(B,T)$ is a Borel pair and $z\in G(k)$ satisfying the following conditions:

1. (i)

$B\subset Q$ 2. (ii)

${}^{z}T\subset L$ 3. (iii)

${}^{z}B\subset P$ 4. (iv)

$\varphi({}^{z}B\cap L)=B\cap M$ 5. (v)

$\varphi({}^{z}T)=T$.

We fix throughout a frame $(B,T,z)$ and we define:

1. (1)

$\Phi\subset X^{*}(T)$ : the set of $T$-roots of $G$. 2. (2)

$\Phi_{+}$ : the set of positive roots with respect to $B$. 3. (3)

$\Delta\subset\Phi_{+}$ : the set of positive simple roots. 4. (4)

For $\alpha\in\Phi$, let $s_{\alpha}\in W$ be the corresponding reflection. Then $(W,\{s_{\alpha}\}_{\alpha\in\Delta})$ is a Coxeter group and we denote by $\ell:W\to\mathbf{N}$ the length function. 5. (5)

For $K\subset\Delta$, Let $W_{K}\subset W$ be the subgroup generated by the $s_{\alpha}$ for $\alpha\in K$. Let $w_{0}\in W$ be the longest element and $w_{0,K}$ the longest element in $W_{K}$. 6. (6)

If $R\subset G$ is a parabolic subgroup containing $B$ and $D$ is the unique Levi subgroup of $R$ containing $T$, then the type of $R$ (or of $D$) is the unique subset $K\subset\Delta$ such that $W(D,T)=W_{K}$. The type of an arbitrary parabolic $R$ is the type of its unique conjugate containing $B$. Let $I\subset\Delta$ (resp. $J\subset\Delta$) be the type of $P$ (resp. $Q$). 7. (7)

For $K\subset\Delta$, ${}^{K}W$ (resp. $W^{K}$) : the subset of elements $w\in W$ which are minimal in the coset $W_{K}w$ (resp. $wW_{K}$). 8. (8)

For $K,R\subset\Delta$, ${}^{K}W^{R}:={}^{K}W\cap W^{R}$. 9. (9)

For an element $x\in{}^{I}W^{J}$, define $I_{x}:=J\cap{}^{x^{-1}}I$. By Proposition 2.7 in [PWZ11], any element $w\in W_{I}xW_{J}$ can be uniquely written as

[TABLE]

For $w\in W$, one has an equivalence:

[TABLE]

1.3. Stratification

For $w\in W$, choose a representative $\dot{w}\in N_{G}(T)$, such that $(w_{1}w_{2})^{\cdot}=\dot{w}_{1}\dot{w}_{2}$ whenever $\ell(w_{1}w_{2})=\ell(w_{1})+\ell(w_{2})$ (this is possible by choosing a Chevalley system, see [ABD*+*66], Exp. XXIII, §6). For $h\in G(k)$, denote by ${\mathcal{O}}_{\mathcal{Z}}(h)$ the $E$-orbit of $h$ in $G$ and define $\mathfrak{o}_{\mathcal{Z}}(h):=[E\backslash{\mathcal{O}}_{\mathcal{Z}}(h)]$. By Theorem 7.5 in [PWZ11], there is a bijection:

[TABLE]

Furthermore, for all $w\in{}^{I}W$, one has:

[TABLE]

For $w\in{}^{I}W$, we endow the locally closed subset $G_{w}$ with the reduced structure, and we define the corresponding zip stratum of \mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}} by $Z_{w}:=\left[E\backslash G_{w}\right]$.

1.4. Point stabilizers

Definition 1.4.1** ([PWZ11, Def. 5.1]).**

Let $w\in{}^{I}W$. There is a largest subgroup $M_{w}$ of ${}^{w^{-1}z^{-1}}L$ satisfying $\varphi({}^{zw}M_{w})=M_{w}$.

In *loc. cit. *§5.1, this subgroup is denoted by $H_{w}$. It is a Levi subgroup of $G$ contained in $M$. We define also:

[TABLE]

Since $\varphi(L_{w})=M_{w}$, we obtain a zip datum ${\mathcal{Z}}_{w}:=(G,P_{w},L_{w},Q_{w},M_{w},\varphi)$. Note that $(B,T,z)$ is again a frame for ${\mathcal{Z}}_{w}$. If an algebraic group $G$ acts on a $k$-scheme $X$ and $x\in X(k)$, we denote by $\operatorname{Stab}_{G}(x)$ the scheme-theoretical stabilizer of $x$. For an algebraic group $H$, we denote by $H_{\rm red}$ the underlying reduced algebraic group and by $H^{\circ}$ the identity component of $H$.

Lemma 1.4.2**.**

1. (1)

One has $\operatorname{Stab}_{E}(z\dot{w})_{\rm red}=A\ltimes R$ where $A\subset L_{w}\times M_{w}$ is the finite group

[TABLE]

and $R$ is a unipotent smooth connected normal subgroup. 2. (2)

One has $\operatorname{Stab}_{E}(z\dot{w})^{\circ}\subset{}^{z}B\times B$.

Proof.

The first part is Th. 8.1 in *loc. cit. *To prove (2), it suffices to show that $\operatorname{Stab}_{E}(z\dot{w})^{\circ}\subset{}^{z}B\times G$, or equivalently $\operatorname{Lie}(\operatorname{Stab}_{E}(z\dot{w}))\subset\operatorname{Lie}({}^{z}B)\times\operatorname{Lie}(G)$. We follow the proof of Th. 8.5 of *loc. cit. *An arbitrary tangent vector of $E$ at $1$ has the form $(1+dp,1+dv)$ for $dp\in\operatorname{Lie}(P)$ and $dv\in\operatorname{Lie}(V)$. This element stabilizes $z\dot{w}$ if and only if $dp=\operatorname{Ad}_{z\dot{w}}(dv)$. Hence

[TABLE]

Hence it suffices to show $P\cap{}^{zw}V\subset{}^{z}B$. This amounts to $L\cap{}^{zw}V\subset L\cap{}^{z}B$ and equivalently $M\cap\varphi({}^{zw}V)\subset M\cap\varphi({}^{z}B)=M\cap B$. This is proved in Prop.4.12 of *loc. cit. *More precisely, the authors define in construction 4.3 a group $V_{x}$ (note that the element $z$ is denoted by $g$ there), where $w=xw_{J}$ is a decomposition as in (1.2.1). One has $V_{x}=M\cap\varphi({}^{z\dot{x}}V)=M\cap\varphi({}^{zw}V)$ because $w_{J}\in W_{J}$, so ${}^{w_{J}}V=V$. Prop. 4.12 of *loc. cit. *shows that $(M\cap B,T,1)$ is a frame for ${\mathcal{Z}}_{\dot{x}}$, so in particular one has $V_{x}\subset M\cap B$. This terminates the proof of the lemma.

∎

2. The stack of partial zip flags

We recall in this section some of the results of [GKb].

2.1. Fine and coarse flag strata

For each parabolic subgroup $P_{0}$ satisfying ${}^{z}B\subset P_{0}\subset P$, we defined in *loc. cit. *§2 a stack \mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P_{0})} which parametrizes $G$-zips of type ${\mathcal{Z}}$ endowed with a compatible $P_{0}$-torsor. There is an isomorphism

[TABLE]

where $E$ acts on $G\times P/P_{0}$ by $(a,b)\cdot(g,xP_{0}):=(agb^{-1},axP_{0})$. Furthermore, there is a natural projection map \pi:\mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P_{0})}\to\mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}} which is a $P/P_{0}$-bundle.

Denote by $L_{0}\subset P_{0}$ the Levi subgroup containing ${}^{z}T$ (note that $L_{0}\subset L$). We define a second zip datum ${\mathcal{Z}}_{0}=(G,P_{0},L_{0},Q_{0},M_{0},\varphi)$ by setting:

[TABLE]

Note that $(B,T,z)$ is again a frame of ${\mathcal{Z}}_{0}$. By *loc. cit. *§3.1, there is a natural morphism of stacks

[TABLE]

which is an $\mathbf{A}^{r}$-bundle for $r=\dim(P/P_{0})$ (Proposition 3.1.1 in loc. cit.). It is induced by the map $G\times P\to G$, $(g,a)\mapsto a^{-1}g\varphi(\overline{a})$. Let $I_{0}$ and $J_{0}$ denote respectively the types of $P_{0}$ and $Q_{0}$. For $w\in{}^{I_{0}}W$, we define the fine flag stratum $Z_{P_{0},w}$ of \mathop{\text{G-{\tt ZipFlag}}}\nolimits^{({\mathcal{Z}},P_{0})} as the locally closed substack

[TABLE]

endowed with the reduced structure. Explicitly, one has $Z_{P_{0},w}=\left[E\backslash G_{P_{0},w}\right]$ where $G_{P_{0},w}$ is the algebraic subvariety of $G\times P/P_{0}$ defined by

[TABLE]

Denote by $\mathop{\text{{\tt Brh}}}\nolimits^{{\mathcal{Z}}_{0}}$ the quotient stack $\left[P_{0}\backslash G/Q_{0}\right]$, called the Bruhat stack. Since $E_{{\mathcal{Z}}_{0}}\subset P_{0}\times Q_{0}$, there is a natural projection morphism \beta:\mathop{\text{G-{\tt Zip}}}\nolimits^{{\mathcal{Z}}_{0}}\to\mathop{\text{{\tt Brh}}}\nolimits^{{\mathcal{Z}}_{0}}. The composition $\Psi_{P_{0}}\circ\beta$ gives a smooth map of stacks

[TABLE]

By [Wed14, Lem. 1.4], the set $\{z\dot{x},x\in{}^{I_{0}}W^{J_{0}}\}$ is a set of representatives of the $P_{0}\times Q_{0}$-orbits in $G$ (pay attention to the fact that ${}^{z}B\subset P$). For $x\in{}^{I_{0}}W^{J_{0}}$, write ${\mathfrak{b}}(x):=[P_{0}\backslash(P_{0}z\dot{x}Q_{0})/Q_{0}]$ (locally closed substack of $\mathop{\text{{\tt Brh}}}\nolimits^{{\mathcal{Z}}_{0}}$) and define the coarse flag stratum $\mathbf{Z}_{P_{0},x}$ as

[TABLE]

endowed with the reduced structure. Explicitly, one has $\mathbf{Z}_{P_{0},x}=\left[E\backslash\mathbf{G}_{P_{0},x}\right]$ where $\mathbf{G}_{P_{0},x}$ is the subvariety of $G\times P/P_{0}$ defined by

[TABLE]

All fine and coarse flag strata are smooth. A coarse stratum is a union of fine strata and the Zariski closure of a coarse flag stratum is normal. In each coarse stratum there is a unique open fine stratum.

Definition 2.1.1**.**

We say that a fine flag stratum $Z$ has coarse closure if it is open in the coarse stratum that contains it, equivalently, if its Zariski closure coincides with the Zariski closure of a coarse flag stratum.

In particular, the Zariski closure $\overline{Z}$ of such a stratum is normal ([GKb, Prop. 2.2.1(1)]).

2.2. Minimal strata

Recall that we defined in [GKb] a minimal flag stratum as a flag stratum $Z_{P_{0},w}$ parametrized by an element $w\in{}^{I}W$. For a minimal stratum one has $\pi(Z_{P_{0},w})=Z_{w}$ and the induced morphism $\pi:Z_{P_{0},w}\to Z_{w}$ is finite (Prop. 3.2.2 of loc. cit.). The following proposition shows that it is also étale. For $w\in{}^{I}W$, denote by $\tilde{\pi}:G_{P_{0},w}\to G_{w}$ the first projection, it is an $E$-equivariant map.

Proposition 2.2.1**.**

Let $w\in{}^{I}W$ and denote by $S:=\operatorname{Stab}_{E}(z\dot{w})$ the stabilizer of $z\dot{w}$ in $E$ and define $S_{P_{0}}:=S\cap(P_{0}\times G)$.

1. (1)

There is a commutative diagram

[TABLE]

where the horizontal maps are isomorphisms and the right-hand side vertical map is the natural projection. 2. (2)

The map $\pi:Z_{P_{0},w}\to Z_{w}$ is finite étale. 3. (3)

The map $\pi:Z_{P_{0},w}\to Z_{w}$ is an isomorphism if and only if the inclusion $S\subset P_{0}\times G$ holds.

Proof.

We first prove (1). There is a natural identification $G_{w}\simeq\left[E/S\right]$ because $G_{w}$ is the $E$-orbit of $z\dot{w}$. It follows that $Z_{w}\simeq[E\backslash E/S]\simeq[1/S]$. Similarly, we claim that the variety $G_{P_{0},w}$ consists of a single $E$-orbit. This was proved in [GKa] Prop. 5.4.5 in the case when $P_{0}$ is a Borel subgroup. For a general $P_{0}$, we may reduce to the Borel case as follows: By Proposition 3.2.2 of [GKb], we have a natural $E$-equivariant surjective projection map $G_{{}^{z}B,w}\to G_{P_{0},w}$, hence $G_{P_{0},w}$ consists of a single $E$-orbit. We thus can identify $Z_{P_{0},w}\simeq\left[E\backslash E/S^{\prime}\right]$ where $S^{\prime}=\operatorname{Stab}_{E}(z\dot{w},1)$. It is clear that $S^{\prime}=S_{P_{0}}$, so the result follows.

We now show (2). By Prop. 3.2.2 of loc. cit., we know that $\pi:Z_{P_{0},w}\to Z_{w}$ is finite. By (1), it is equivalent to show that $S/S_{P_{0}}$ is an étale scheme. By Lemma 1.4.2 (2), we have $S^{\circ}\subset S\cap({}^{z}B\times G)\subset S_{P_{0}}$. Hence the quotient map $S\to S/S_{P_{0}}$ factors through a surjective map $\pi_{0}(S)\to S/S_{P_{0}}$, which shows that $S/S_{P_{0}}$ is étale.

Finally, the last assertion follows immediately from (1). ∎

3. The canonical parabolic

We fix an element $w\in{}^{I}W$. Recall that we defined in (1.4.3) a parabolic subgroup $P_{w}\subset P$.

3.1. Definition

Let $P_{0}$ be a parabolic subgroup of $G$ such that ${}^{z}B\subset P_{0}\subset P$.

Definition 3.1.1**.**

We say that $P_{0}$ is a canonical parabolic subgroup for $w$ if the following properties are satisfied:

1. (i)

The map $\pi:Z_{P_{0},w}\to Z_{w}$ is an isomorphism. 2. (ii)

The stratum $Z_{P_{0},w}$ has coarse closure.

Using the notations of Prop. 2.2.1, property (i) is equivalent to $S\subset P_{0}\times G$. The justification of this definition is the following: For $P_{0}=P$, Condition (i) is obviously satisfied. On the other hand, if $P_{0}={}^{z}B$, then (ii) is satisfied because coarse and fine strata coincide. For a given $w$, a canonical parabolic for $w$ is an intermediate parabolic subgroup $P_{0}$ satisfying both conditions. A priori neither the existence nor the uniqueness of such a parabolic is clear.

We give justification for this definition. Let $P_{0}$ be a canonical parabolic subgroup for $w$. We have morphisms:

[TABLE]

which yield isomorphisms $Z_{P_{0},w}\simeq Z_{w}$ and $G_{P_{0},w}\simeq G_{w}$. Since $Z_{P_{0},w}$ has coarse closure, the stack (resp. variety) $\overline{Z}_{P_{0},w}$ (resp. $\overline{G}_{P_{0},w}$) is normal. We deduce the following proposition:

Proposition 3.1.2**.**

Let $P_{0}$ be a canonical parabolic subgroup for $w\in{}^{I}W$. Write $w=xw_{J}$ as in (1.2.1). Then the normalization of $\overline{G}_{w}$ is the Stein factorization of the map $\tilde{\pi}:\overline{G}_{P_{0},w}\to\overline{G}_{w}$. It is isomorphic to $\operatorname{Spec}({\mathcal{O}}(\overline{\mathbf{G}}_{P_{0},x}))$, where

[TABLE]

and the first projection induces an isomorphism $G_{P_{0},w}\simeq G_{w}$.

The following theorem is the main result of this paper. Its proof will follow from the results of §3.2 and §3.3.

Theorem 3.1.3**.**

Let $w\in{}^{I}W$. The parabolic subgroup $P_{w}$ is the unique canonical parabolic subgroup for $w$. More precisely, among all parabolic subgroups ${}^{z}B\subset P_{0}\subset P$, the following holds:

1. (a)

$P_{w}$* is the smallest parabolic $P_{0}$ such that $\pi:Z_{P_{0},w}\to Z_{w}$ is an isomorphism.* 2. (b)

$P_{w}$* is the largest parabolic $P_{0}$ such that $Z_{P_{0},w}$ has coarse closure.*

3.2. A criterion for Condition (i)

Lemma 3.2.1**.**

Let ${}^{z}B\subset P_{0}\subset P$ be a parabolic subgroup. The following assertions are equivalent:

1. (1)

The map $\pi:Z_{P_{0},w}\to Z_{w}$ is an isomorphism. 2. (2)

One has $P_{w}\subset P_{0}$.

Proof.

Using the notation of Prop. 2.2.1, we know that $\pi:Z_{P_{0},w}\to Z_{w}$ is an isomorphism if and only if $S_{P_{0}}:=S\cap(P_{0}\times G)=S$. By the same proposition, we know that the quotient $S/S_{P_{0}}$ is a finite affine étale scheme over $k$. In particular, we have $S\subset P_{0}\times G$ if and only if $S_{\rm red}\subset P_{0}\times G$.

By Lemma 1.4.2, we can write $S_{\rm red}=A\ltimes R$ with $A$ the finite group given by equation (1.4.4) of Lemma 1.4.2 and $R$ a smooth unipotent connected normal subgroup. Write $R_{P_{0}}:=R\cap(P_{0}\times G)$. The inclusion $R\subset S$ induces a closed embedding

[TABLE]

Hence $R/R_{P_{0}}$ is a finite, smooth, connected $k$-scheme, so $R/R_{P_{0}}=\operatorname{Spec}(k)$, hence $R\subset P_{0}\times G$. It follows that $S\subset P_{0}\times G$ if and only if $A\subset P_{0}\times G$, which is equivalent to $A_{1}\subset P_{0}$, where

[TABLE]

By Steinberg’s theorem we can write $z\dot{w}=a^{-1}\varphi(a)$ with $a\in G(k)$. Then it is easy to see that the subgroup ${}^{a}L_{w}$ is defined over $\mathbf{F}_{p}$ and the inclusion $A_{1}\subset P_{0}$ is equivalent to

[TABLE]

Note that both ${}^{a}L_{w}$ and ${}^{a}P_{0}$ contain the torus ${}^{az}T$, which is defined over $\mathbf{F}_{p}$. Thus Lemma 3.2.2 below shows that (3.2.3) is equivalent to ${}^{a}L_{w}\subset{}^{a}P_{0}$, hence $L_{w}\subset P_{0}$, which is the same as $P_{w}\subset P_{0}$. This terminates the proof. ∎

Lemma 3.2.2**.**

Let $G$ be a connected reductive group over $\mathbf{F}_{p}$. Let $L$ be a Levi $\mathbf{F}_{p}$-subgroup of $G$ and $P$ be a parabolic subgroup of $G_{k}$. Assume that there exists a maximal $\mathbf{F}_{p}$-torus $T$ contained in $L\cap P$ and that $L(\mathbf{F}_{p})\subset P$. Then one has $L\subset P$.

Proof.

Define a subgroup of $G$ by

[TABLE]

It is clear that $H\subset L$, $H$ is defined over $\mathbf{F}_{p}$, and $L(\mathbf{F}_{p})=H(\mathbf{F}_{p})$. Furthermore, $H$ is an intersection of parabolic subgroups of $L$ containing $T$. Hence it suffices to prove the following claim: Let $G$ be a connected reductive group over $\mathbf{F}_{p}$, $T\subset G$ a maximal $\mathbf{F}_{p}$-torus, and $T\subset H\subset G$ an $\mathbf{F}_{p}$-subgroup which is an intersection of parabolic subgroups of $G_{k}$ containing $T$, and assume that $H(\mathbf{F}_{p})=G(\mathbf{F}_{p})$. Then one has $H=G$.

We now prove the claim. Using inductively [DM91, Prop. 2.1], one shows that an intersection of parabolic subgroups $P_{1},...,P_{m}$ containing $T$ is connected and can be written as a semidirect product

[TABLE]

where $L_{0}$ is a Levi subgroup of $G$ containing $T$ and $U_{0}$ is a unipotent connected subgroup of $G$, normalized by $L_{0}$. Applying this to $H$, we can write $H=L_{0}\ltimes U_{0}$. Since $H$ is defined over $\mathbf{F}_{p}$, so are $L_{0}$ and $U_{0}$.

By [Car93, Th. 3.4.1], the highest power of $p$ dividing $|G(\mathbf{F}_{p})|$ is $p^{N}$, where $N=|\Phi_{+}|$ is the dimension of any maximal unipotent subgroup of $G_{k}$. Since $G(\mathbf{F}_{p})=H(\mathbf{F}_{p})=L_{0}(\mathbf{F}_{p})\times U_{0}(\mathbf{F}_{p})$, we deduce that for all maximal unipotent subgroup $U^{\prime}$ in $L_{0}$, the subgroup $U^{\prime}\times U_{0}$ is unipotent maximal in $G$. In particular, $H$ contains a Borel subgroup, so $H$ is a parabolic subgroup. Then $H=G$ follows from [ABD*+*66, XXVI, 5.11]

∎

3.3. A criterion for Condition (ii)

We examine Condition (ii) of Definition 3.1.1. Let ${}^{z}B\subset P_{0}\subset P$ be a parabolic subgroup and let $L_{0}$, $M_{0}$, $Q_{0}$, ${\mathcal{Z}}_{0}$ as defined in section 2.1.

Lemma 3.3.1**.**

Let ${}^{z}B\subset P_{0}\subset P$ be a parabolic subgroup. The following assertions are equivalent:

1. (1)

$Z_{P_{0},w}$* has coarse closure.* 2. (2)

One has ${}^{z\dot{w}}M_{0}=L_{0}$.

Proof.

The stratum $Z_{P_{0},w}$ has coarse closure if and only if ${\mathcal{O}}_{E_{{\mathcal{Z}}_{0}}}(z\dot{w})\subset P_{0}z\dot{w}Q_{0}$ is an open embedding, which is equivalent to the equality of their dimensions. Note that $(B,T,z)$ is again a frame of ${\mathcal{Z}}_{0}$, so formula (1.3.2) shows that

[TABLE]

On the other hand, we have:

[TABLE]

The stabilizer $\operatorname{Stab}_{P_{0}\times Q_{0}}(z\dot{w})$ is the subgroup:

[TABLE]

Hence $Z_{P_{0},w}$ has coarse closure if and only if $\dim\left(P_{0}/(P_{0}\cap{}^{z\dot{w}}Q_{0})\right)=\ell(w)$. Since the property is satisfied when $P_{0}={}^{z}B$, we have $\dim\left({}^{z}B/({}^{z}B\cap{}^{z\dot{w}}B)\right)=\ell(w)$, so we can rewrite the property as

[TABLE]

Since $(B,T,z)$ is a frame for ${\mathcal{Z}}_{0}$ and ${}^{I}W\subset{}^{I_{0}}W$, equation (1.2.2) shows that $P_{0}\cap{}^{z}B=P_{0}\cap{}^{z\dot{w}}B$, thus the inclusion $P_{0}\cap{}^{z\dot{w}}Q_{0}\subset P_{0}$ induces an embedding

[TABLE]

Hence (3.3.3) is satisfied if and only if the image of $P_{0}\cap{}^{z\dot{w}}Q_{0}$ is open in $P_{0}/{}^{z}B$. Since $P_{0}/{}^{z}B\simeq L_{0}/({}^{z}B\cap L_{0})$ it is also equivalent to $L_{0}\cap{}^{z\dot{w}}Q_{0}$ having open image in $L_{0}/({}^{z}B\cap L_{0})$.

Denote by $B^{\prime}$ the opposite Borel in $G$ of $B$ with respect to $T$. Then ${}^{z}B^{\prime}\cap L_{0}$ is the opposite Borel of ${}^{z}B\cap L_{0}$ in $L_{0}$ with respect to ${}^{z}T$. Thus the image of $L_{0}\cap{}^{z\dot{w}}Q_{0}$ is open in $L_{0}/({}^{z}B\cap L_{0})$ if and only if ${}^{z}B^{\prime}\cap L_{0}\subset{}^{z\dot{w}}Q_{0}$. It follows immediately from equation (1.2.2) that ${}^{z}B^{\prime}\cap L_{0}={}^{z\dot{w}}B^{\prime}\cap L_{0}$. Finally, we find that $Z_{P_{0},w}$ has coarse closure if and only if

[TABLE]

The groups $B^{\prime}\cap{}^{(z\dot{w})^{-1}}L_{0}$ and $B\cap{}^{(z\dot{w})^{-1}}L_{0}$ are opposite Borel subgroups of ${}^{(z\dot{w})^{-1}}L_{0}$ containing $T$. Since $B\subset Q_{0}$, equation (3.3.5) is simply equivalent to ${}^{(z\dot{w})^{-1}}L_{0}\subset Q_{0}$, which is equivalent to ${}^{(z\dot{w})^{-1}}L_{0}=M_{0}$. This terminates the proof. ∎

Proof of Theorem 3.1.3.

The result follows immediately by combining Lemmas 3.2.1 and 3.3.1. ∎

Corollary 3.3.2**.**

Write $w=xw_{J}$ as in (1.2.1). The normalization of the Zariski closure $\overline{G}_{w}$ is the Stein factorization of the map $\tilde{\pi}:\overline{G}_{P_{w},w}\to\overline{G}_{w}$. It is isomorphic to $\tilde{Z}_{w}:=\operatorname{Spec}({\mathcal{O}}(\overline{\mathbf{G}}_{P_{w},x}))$, where

[TABLE]

and the first projection induces an isomorphism $G_{P_{w},w}\simeq G_{w}$.

3.4. Shimura varieties and Ekedahl-Oort strata

Let $X$ be the special fiber of a Hodge-type Shimura variety attached to a Shimura datum $(\mathbf{G},\mathbf{X})$ with hyperspecial level at $p$. Write $G:=G_{\mathbf{Z}_{p}}\times\mathbf{F}_{p}$, where $G_{\mathbf{Z}_{p}}$ is a reductive $\mathbf{Z}_{p}$-model of $\mathbf{G}_{\mathbf{Q}_{p}}$. By Zhang [Zha], there exists a smooth morphism of stacks

[TABLE]

where ${\mathcal{Z}}$ is the zip datum attached to $(\mathbf{G},\mathbf{X})$ as in [GKb] §6.2. The Ekedahl-Oort stratification of $X$ is defined as the fibers of $\zeta$. For $w\in{}^{I}W$, set:

[TABLE]

By the smoothness of $\zeta$, this defines a stratification of $X$. Let ${}^{z}B\subset P_{0}\subset P$ be a parabolic subgroup and define the partial flag space $X_{P_{0}}$ as the fiber product

[TABLE]

The map $\pi:X_{P_{0}}\to X$ is a $P/P_{0}$-bundle. For $w\in{}^{I_{0}}W$ and $x\in{}^{I_{0}}W^{J_{0}}$ define

[TABLE]

We call $X_{P_{0},w}$ the fine stratum attached to $w\in{}^{I_{0}}W$ and $\mathbf{X}_{P_{0},x}$ the coarse stratum attached to $x\in{}^{I_{0}}W^{J_{0}}$. All coarse and fine strata are smooth and locally closed, they define stratifications of $X_{P_{0}}$, and the Zariski closure of a coarse stratum is normal. Recall that we defined $\tilde{Z}_{w}:=\operatorname{Spec}({\mathcal{O}}(\overline{G}_{P_{w},w}))$ (Cor. 3.3.2).

Corollary 3.4.1**.**

Let $w\in{}^{I}W$. The normalization of $\overline{X}_{w}$ is the Stein factorization of the map $\pi:\overline{X}_{P_{w},w}\to\overline{X}_{w}$. It is isomorphic to \overline{X}_{w}\times_{\mathop{\text{G-{\tt Zip}}}\nolimits^{\mathcal{Z}}}\tilde{Z}_{w}.

4. The canonical filtration

Most of the content of this section can be found in [PWZ11]. We merely unwind their proofs to make the link between the canonical filtration of a Dieudonné space and the group $L_{w}$ defined previously. See also [Moo01, §4.4] and [Box15] for related results.

4.1. Dieudonné spaces and $GL_{n}$-zips

Let $H$ be a truncated Barsotti-Tate groups of level $1$ over $k$ of height $n$. Set $d:=\dim(\operatorname{Lie}(H))$ and write $D:=\mathbf{D}(H)$ for its Dieudonné space. It is a $k$-vector space of dimension $n$ endowed with a $\sigma$-linear endomorphism ${\mathcal{F}}:D\to D$, a $\sigma^{-1}$-linear endomorphism ${\mathcal{V}}:D\to D$ satisfying the conditions:

1. (1)

$\operatorname{Ker}({\mathcal{F}})={\rm Im}({\mathcal{V}})$, 2. (2)

$\operatorname{Ker}({\mathcal{V}})={\rm Im}({\mathcal{F}})$, 3. (3)

$\mbox{ rk }({\mathcal{V}})=d.$

We say that $(D,{\mathcal{F}},{\mathcal{V}})$ is a Dieudonné space of height $n$ and dimension $d$. Let $M^{(r)}_{n}(k)$ be the set of matrices in $M_{n}(k)$ of rank $r$. After choosing a $k$-basis of $D$, we may write ${\mathcal{F}}=a\otimes\sigma$ and ${\mathcal{V}}=b\otimes\sigma^{-1}$, where $(a,b)$ is in the set

[TABLE]

Note that for $(a,b)\in{\mathscr{X}}$, we have $\operatorname{Ker}(a)={\rm Im}(\sigma(b))=\sigma({\rm Im}(b))$ and ${\rm Im}(a)=\operatorname{Ker}(\sigma(b))=\sigma(\operatorname{Ker}(b))$.

It is easy to see that two such pairs $(a,b)$ and $(a^{\prime},b^{\prime})$ yield isomorphic Dieudonné spaces if and only if there exists $M\in GL_{n}(k)$ such that

[TABLE]

This defines an action of $GL_{n}(k)$ on ${\mathscr{X}}$ and we obtain a bijection between isomorphism classes of Dieudonné spaces of height $n$ and dimension $d$ and the set of $GL_{n}(k)$-orbits in ${\mathscr{X}}$.

Let $(e_{1},...,e_{n})$ the canonical basis of $k^{n}$ and define

[TABLE]

Define $P:=\operatorname{Stab}(V_{2})$, $Q:=\operatorname{Stab}(V_{1})$, $L:=P\cap Q$, $U:=R_{u}(P)$ and $V:=R_{u}(Q)$. Consider the set

[TABLE]

The action of $GL_{n}(k)$ on ${\mathscr{X}}$ restricts to an action of $P(k)$ on ${\mathscr{Y}}$ and the inclusion ${\mathscr{Y}}\subset{\mathscr{X}}$ induces a bijection between $P(k)$-orbits in ${\mathscr{Y}}$ and $GL_{n}(k)$-orbits in ${\mathscr{X}}$.

Lemma 4.1.1**.**

There is a natural bijection

[TABLE]

Proof.

Let $(a,b)\in{\mathscr{Y}}$ and choose a subspace $H\subset k^{n}$ such that ${\rm Im}(a)\oplus H=k^{n}$. Define a matrix $f_{H}\in GL_{n}(k)$ by the following diagram

[TABLE]

In other words, $f_{H}v=av$ for $v\in V_{1}$, and if $v\in V_{2}$, then $f_{H}v$ is the only element $h\in H$ such that $\sigma(b)h=v$ (note that ${\rm Im}(a)=\operatorname{Ker}(\sigma(b))$, so this element is well-defined). It is clear that $f_{H}$ is invertible.

If $H^{\prime}$ denotes another subspace such that ${\rm Im}(a)\oplus H^{\prime}=k^{n}$, then we can write $f_{H^{\prime}}=f_{H}\alpha$, for some $\alpha\in GL_{n}(k)$. It is clear that $\alpha(v)=v$ for all $v\in V_{1}$. Furthermore, for $v\in V_{2}$, one must have $\sigma(b)(f_{H^{\prime}}v-f_{H}v)=0$, thus $f_{H}(\alpha(v)-v)\in\operatorname{Ker}(\sigma(b))={\rm Im}(a)$, so $\alpha(v)-v\in V_{1}$. This shows that $\alpha\in V$. It follows that $(a,b)\mapsto f_{H}$ induces a well-defined map $\Psi:{\mathscr{Y}}\longrightarrow GL_{n}(k)/V$. We leave it to the reader to check that this map is a bijection. ∎

Define a subgroup of $P\times Q$ by

[TABLE]

Let this group acts on $GL_{n}(k)$ by the rule $(M_{1},M_{2})\cdot g:=M_{1}gM_{2}^{-1}$.

Proposition 4.1.2**.**

The map $\Psi$ induces a bijection

[TABLE]

Hence there is a bijection between isomorphism classes of Dieudonné spaces of height $n$ and dimension $d$ and the set of $E$-orbits in $GL_{n}(k)$.

Proof.

Let $M\in P(k)$, $(a,b)\in{\mathscr{Y}}$ and set $(a^{\prime},b^{\prime}):=(Ma\sigma(M)^{-1},Mb\sigma^{-1}(M)^{-1})$. Note that ${\rm Im}(a^{\prime})=M({\rm Im}(a))$. Choose a subspace $H$ such that ${\rm Im}(a)\oplus H=k^{n}$ and set $H^{\prime}:=M(H)$. Let $\overline{M}\in L(k)$ denote the Levi component of $M\in P(k)$. Finally, write $f_{H}$ and $f^{\prime}_{H^{\prime}}$ for the maps attached to $(a,b,H)$ and $(a^{\prime},b^{\prime},H^{\prime})$ respectively by the previous construction. We claim that one has the relation:

[TABLE]

First assume $v\in V_{1}$. Then $f^{\prime}_{H^{\prime}}\sigma(\overline{M})v=Ma\sigma(M)^{-1}\overline{M}v$. Since $\sigma(M)^{-1}\overline{M}\in U$, we have $\sigma(M)^{-1}\overline{M}v-v\in V_{2}$, hence $f^{\prime}_{H^{\prime}}\sigma(\overline{M})v=Mav=Mf_{H}v$.

Now if $v\in V_{2}$, the element $f_{H}v$ is the only element $h\in H$ satisfying $\sigma(b)h=v$. Similarly, $f_{H^{\prime}}\sigma(\overline{M})v$ is the only element $h^{\prime}\in H^{\prime}=M(H)$ such that $\sigma(b^{\prime})h^{\prime}=\sigma(\overline{M})v$. Hence $\sigma(\overline{M})^{-1}\sigma(M)\sigma(b)M^{-1}h^{\prime}=v$. But $\sigma(\overline{M})^{-1}\sigma(M)\in U$ and $\sigma(b)M^{-1}h^{\prime}\in V_{2}$, so we deduce $\sigma(b)M^{-1}h^{\prime}=v$, and finally $M^{-1}h^{\prime}=h$ as claimed.

This shows that $\Psi$ induces a well-defined map $P(k)\backslash{\mathscr{Y}}\to E\backslash GL_{n}(k)$. We leave it to the reader to check that it is bijective. ∎

4.2. The canonical filtration

Let $(D,{\mathcal{F}},{\mathcal{V}})$ be a Dieudonné space. The operators ${\mathcal{V}}$ and ${\mathcal{F}}^{-1}$ act naturally on the set of subspaces of $D$. It can be shown that there exists a flag of $D$ which is stable by ${\mathcal{V}}$ and ${\mathcal{F}}^{-1}$ and which is coarsest among all such flags. This flag is called the canonical filtration of $D$. It is obtained by applying all finite combinations of ${\mathcal{V}},{\mathcal{F}}^{-1}$ to the flag $0\subset D$.

Choose a basis of $D$ and write ${\mathcal{F}}=a\otimes\sigma$ and ${\mathcal{V}}=b\otimes\sigma^{-1}$ with $(a,b)\in{\mathscr{X}}$. By choosing an appropriate basis, we will asssume that $(a,b)\in{\mathscr{Y}}$.

Remark 4.2.1**.**

Actually, there exists a basis such that $(a,b)\in{\mathscr{Y}}$ and such that the coefficents of $a,b$ are either [math] or $1$ and each column and each row has at most one non-zero coefficient.

Let $H\subset k^{n}$ be a subspace such that ${\rm Im}(a)\oplus H=k^{n}$ and let $f_{H}\in GL_{n}(k)$ be the element defined in diagram (4.1.6). We have the following easy lemma:

Lemma 4.2.2**.**

For any subspace $W\subset k^{n}$, one has the following relations:

[TABLE]

In particular, the right-hand terms are independent of the choice of $H$. This observation suggests the following definition:

Definition 4.2.3**.**

For $f\in GL_{n}(k)$, there exists a unique coarsest flag $\operatorname{Fl}(f)$ of $k^{n}$ satisfying the following properties:

1. (1)

For any $W\in\operatorname{Fl}(f)$, the following inclusions hold

[TABLE] 2. (2)

For any $W\in\operatorname{Fl}(f)$, all subspaces appearing in (4.2.3) are in $\operatorname{Fl}(f)$.

The flag $\operatorname{Fl}(f)$ is simply the canonical flag attached to the Dieudonné space corresponding to the left-coset $fV$ under the bijection $\Psi$.

Lemma 4.2.4**.**

Let $f\in GL_{n}(k)$. The following assertions hold

1. (1)

For all $v\in V$, one has $\operatorname{Fl}(fv)=\operatorname{Fl}(f)$. 2. (2)

For $M\in P$, one has $\operatorname{Fl}(Mf\sigma(\overline{M})^{-1})=M\operatorname{Fl}(f)$.

We leave the verification of the lemma to the reader. In particular, the conjugation class of $\operatorname{Fl}(f)$ depends only on the $E$-orbit of $f$. Denote by $P(f):=\operatorname{Stab}(\operatorname{Fl}(f))$. Since $\operatorname{Fl}(f)$ contains ${\rm Im}(V)=V_{2}$, we have $P(f)\subset P$. Furthermore, for $v\in V$ and $M\in P$, one has

[TABLE]

4.3. The canonical flag versus $P_{w}$

Denote by $T$ the diagonal torus of $G:=GL_{n}$, and let $B$ be the Borel subgroup of upper-triangular matrices. The Weyl group $W(G,T)$ is the symmetric group $S_{n}$, which we identify with a subgroup of $G(k)$ be letting it act on $k^{n}$ by $\tau(e_{i})=e_{\tau(i)}$ for all $\tau\in S_{n}$ and $i\in\{1,...,n\}$.

Using the notations of section 1.2, define a permutation

[TABLE]

Then $(B,T,z)$ is a frame for the zip datum $(G,P,Q,L,M,\varphi)$. For $w\in{}^{I}W$, set $f_{w}:=zw$. By the parametrization (1.3.1), the set $\{zw,w\in{}^{I}W\}$ is a set of representatives of the $E$-orbits in $G$. For $w\in{}^{I}W$, it is easy to see that any $W\in\operatorname{Fl}(f_{w})$ is spanned by $(e_{i})_{i\in C_{W}}$ for some subset $C_{W}\subset\{1,...,n\}$. In particular we have $T\subset P(f_{w})$. Note that for all $w\in{}^{I}W$, we have simplified formulas:

[TABLE]

There is a unique Levi subgroup $L(f_{w})\subset P(f_{w})$ containing $T$. Finally, for $w\in{}^{I}W$, denote by $L_{w}\subset L$ and $P_{w}\subset P$ the subgroups defined in (1.4.1) and (1.4.3).

Proposition 4.3.1**.**

We have $P(f_{w})=P_{w}$ and $L(f_{w})=L_{w}$.

Proof.

We will show first that ${}^{z}B\subset P(f_{w})$. Clearly it suffices to show ${}^{z}B\cap L\subset P(f_{w})$. Note that since $w\in{}^{I}W$, we have ${}^{z}B\cap L=B\cap L={}^{zw}B\cap L$. From this it follows that if $W\subset k^{n}$ is a subspace such that ${}^{z}B\cap L\subset\operatorname{Stab}(W)$, then ${}^{z}B\cap L$ stabilizes also $\sigma^{-1}(f_{w}^{-1}(W))$. From this it follows easily by induction that ${}^{z}B\cap L$ stabilizes $\operatorname{Fl}(f_{w})$, hence ${}^{z}B\cap L\subset P(f_{w})$ as claimed.

To finish the proof, it suffices to show the second assertion. By definition we have ${}^{f_{w}}\varphi(L_{w})=L_{w}$. Hence if $W\subset k^{n}$ is a subspace such that $L_{w}\subset\operatorname{Stab}(W)$ then $L_{w}\subset\operatorname{Stab}(\sigma^{-1}(f_{w}^{-1}(W)))$. From this, it follows again by an easy induction that $L_{w}$ stabilizes $\operatorname{Fl}(f_{w})$, so $L_{w}\subset P(f_{w})$. Since $L_{w}$ contains the torus $T$, we deduce that $L_{w}\subset L(f_{w})$.

Finally, we must show that ${}^{f_{w}}\varphi(L(f_{w}))=L(f_{w})$. Since $L(f_{w})$ is clearly defined over $\mathbf{F}_{p}$, this is the same as ${}^{f_{w}}L(f_{w})=L(f_{w})$. Let $k^{n}=D_{1}\oplus...\oplus D_{m}$ denote the decomposition attached to $L(f_{w})$, numbered so that the filtration $\operatorname{Fl}(f_{w})$ is composed of the subspaces $W_{j}:=\bigoplus_{j=1}^{i}D_{j}$ for $1\leq i\leq m$. There exists an integer $1\leq r\leq m$ such that $W_{r}=V_{2}$ (and then necessarily $V_{1}=\bigoplus_{j=r+1}^{m}D_{j}$). We need to show that $f_{w}$ permutes the $(D_{i})_{1\leq i\leq m}$. For this, it suffices to show that if $f_{w}(D_{i})\cap D_{j}\neq 0$, then $D_{j}\subset f_{w}(D_{i})$ for all $1\leq i,j\leq m$.

First assume $1\leq j\leq r$ and let $1\leq i\leq m$ be the smallest integer such that $D_{j}\cap f_{w}(D_{i})\neq 0$. He have $0\neq D_{j}\cap f_{w}(D_{i})\subset V_{2}\cap f_{w}(W_{i})$, which implies $D_{j}\subset V_{2}\cap f_{w}(W_{i})$. By minimality of $i$, we deduce $D_{j}\subset f_{w}(D_{i})$.

Now assume $r<j\leq m$ and let $1\leq i\leq m$ be the smallest integer such that $D_{j}\cap f_{w}(D_{i})\neq 0$. Then $0\neq D_{j}\cap f_{w}(D_{i})\subset V_{2}+f_{w}(W_{i})$, which implies $D_{j}\subset(V_{2}+f_{w}(W_{i}))\cap V_{1}=f_{w}(W_{i})\cap V_{1}\subset f_{w}(W_{i})$. By minimality of $i$, we deduce $D_{j}\subset f_{w}(D_{i})$, which terminates the proof.

∎