On Families of Weakly Admissible Filtered

φ-Modules and the Adjoint Quotient of GL_{d}

Eugen Hellmann

Received: April 8, 2011 Revised: November 20, 2011

Communicated by Peter Schneider

Abstract. We study the relation of the notion of weak admissibility in families of filtered φ-modules, as considered in [He], with the adjoint quotient. We show that the weakly admissible subset is an open subvariety in the fibers over the adjoint quotient. Further we determine the image of the weakly admissible set in the adjoint quotient generalizing earlier work of Breuil and Schneider.

2010 Mathematics Subject Classification: 11F80, 11F85, 14G20, 14G22 Keywords and Phrases: Filtered φ-modules, p-adic Hodge theory, rigid geometry

1. Introduction

Filtered φ-modules appear in p-adic Hodge-theory as a category of linear algebra data describing crystalline representations of the absolute Galois group of a local p-adic field. More precisely, there is an equivalence of categories between crystalline representations and weakly admissible filtered φ-modules, see [CF]. Here weak admissibility is a semi-stability condition relating the slopes of the φ-linear endomorphism Φ with the filtration. In our companion paper [He] we define and study arithmetic families of filtered φ-modules and crystalline representations. Our families are parameterized by rigid analytic spaces or adic spaces in the sense of Huber, see [Hu2] for example. We show that the condition of being weakly admissible is an open condition [He, Theorem 1.1] and that there is an open subset of the weakly admissible locus over which there exists a family of crystalline representations giving rise to the family of filtered φ-modules [He, Theorem 1.3]. In this paper we study the weakly admissible locus in more detail. In the setting of period domains in the sense of Rapoport and Zink [RZ], the weakly admissible locus is an admissible open subset of a flag variety. Contrarily, the weakly admissible locus in our set up has an algebraic nature as soon as we fix the Frobenius Φ, or even the conjugacy class of its semi-simplification. Further we analyze the image of the weakly admissible locus in the adjoint quotient. The question whether there exists a weakly admissible filtration for a fixed conjugacy class of the semisimplification of the Frobenius already appears in work of Breuil and Schneider [BS] on the p-adic Langlands correspondence. Unlike the characterization in [BS], our characterization of the set of automorphisms Φ for which there exists a weakly admissible filtration is purely in terms of the adjoint quotient of GL_{d}. Our main results are as follows: Fix a finite extension K of ℚ_{p} and write K_{0} for the maximal unramified extension of ℚ_{p} inside K. Let d > 0 be an integer and denote by A ⊂ GL_{d} the diagonal torus. For a dominant cocharacter

ν : 𝔾_{m,ℚ̄_{p}} ⟶ (Res_{K/ℚ_{p}} A)ℚ̄_{p}

we write Gr_{ν} for the partial flag variety of Res_{K/ℚ_{p}} GL_{d} parametrizing flags of "type ν", see section 3.1 for the precise definition. This variety is defined over the reflex field E of ν. As in [He, 4.1] we denote by

𝔇_{ν} = ((Res_{K_{0}/ℚ_{p}} GL_{d})_{E} × Gr_{ν} )/(Res_{K_{0}/ℚ_{p}} GL_{d})_{E}

the stack of filtered φ-modules with filtration of "type ν" on the category of adic spaces locally of finite type. The action of (Res_{K_{0}/ℚ_{p}} GL_{d})_{E} is explicitly described in (3.4). Let W denote the Weyl group of GL_{d}. We will define a morphism

α : 𝔇_{ν} ⟶ (A/W)^{ad}

to the adification (see [Hu2, Remark 4.6 (i)]) of the adjoint quotient A/W and prove the following theorem.

Theorem 1.1. Let x ∈ (A/W)_{E}^{ad} and form the 2-fiber product

\xymatrix{α^{−1}(x)^{wa} \ar[r] \ar[d] & 𝔇_{ν}^{wa} \ar[d] \\ x \ar[r] & (A/W)_{E}^{ad}.}

Then there Artin stack in schemes 𝔄 over the field k(x) such that

α^{−1}(x)^{wa} = 𝔄^{ad}.

The stack 𝔄 is the stack quotient of a quasi-projective k(x)-variety.

Further we determine the image of the weakly admissible locus 𝔇_{ν}^{wa} under the morphism α. The description of this image works in the category of analytic spaces in the sense of Berkovich.

Theorem 1.2. Let ν be a dominant coweight as above. There is a dominant coweight μ(ν) of GL_{d} associated to ν such that

α^{−1}(x)^{wa} ≠ ∅ ⟺ x ∈ (A/W)_{≤μ(ν)}.

Here (A/W)_{≤μ(ν)} is a Newton-stratum in the sense of Kottwitz [Ko].

Filtered φ-modules and the adjoint quotient 971

The coweight μ(ν) which appears in the theorem is explicit and defined in Definition 5.4 Acknowledgements: I thank my advisor M. Rapoport for his interest and advice. Further I thank the referee for his careful reading and his comments. This work was supported by the SFB/TR 45 "Periods, Moduli Spaces and Arithmetic of Algebraic Varieties" of the DFG (German Research Foundation).

2. Filtered φ-modules

Throughout this section we denote by F a topological field containing ℚ_{p} with a continuous valuation v_{F} : F → Γ_{F} ∪ {0} in the sense of [Hu1, 2, Definition] that is Γ_{F} is a totally ordered abelian group (written multiplicative) and

v_{F} (0) = 0
v_{F} (1) = 1
v_{F} (ab) = v_{F} (a)v_{F} (b)
v_{F} (a + b) ≤ max{v_{F} (a), v_{F} (b)},

where the order on Γ_{F} is extended to Γ_{F} ∪{0} by 0 < γ for all γ ∈ Γ_{F}. We will introduce the notion of a filtered φ-module with coefficients in F and define weak admissibility for these objects. Recall that K_{0} is an unramified extension of ℚ_{p} with residue field k and write f = [K_{0} : ℚ_{p}]. We write φ for the lift of the absolute Frobenius to K_{0}.

2.1. φ-modules with coefficients. In this subsection we define and study what we call isocrystals over k with coefficients in F.

Definition 2.1. An isocrystal over k with coefficients in F is a free F ⊗_{ℚ_{p}} K_{0}- module D of finite rank together with an automorphism Φ : D → D that is semi-linear with respect to id ⊗φ : F ⊗_{ℚ_{p}} K_{0} → F ⊗_{ℚ_{p}} K_{0}. A morphism f : (D, Φ) → (D′, Φ′) is an F ⊗_{ℚ_{p}} K_{0}-linear map f : D → D′ such that

f ∘ Φ = Φ′ ∘ f.

The category of isocrystals over k with coefficients in F is denoted by Isoc(k)_{F}.

It is easy to see that Isoc(k)_{F} is an F -linear abelian ⊗-category with the obvious notions of direct sums and tensor products.

Remark 2.2. (i) Given an F ⊗_{ℚ_{p}} K_{0}-module D of finite type, the existence of a semi-linear automorphism Φ : D → D implies that D is free over F ⊗_{ℚ_{p}} K_{0}. This fact will be used in the sequel. (ii) In the classical setting an isocrystal over k is a finite-dimensional K_{0}-vector space with φ-linear automorphism Φ, i.e. an object in Isoc(k)_{ℚ_{p}}. If F is finite over ℚ_{p}, then an isocrystal over k with coefficients in F is the same as an object (D, Φ) ∈ Isoc(k)_{ℚ_{p}} together with a map

F ⟶ End_{Φ}(D),

where the subscript Φ on the right hand side indicates that the endomorphisms commute with Φ (compare [DOR, VIII, 5] for example). This is clearly equivalent to our definition.

Let F′ be an extension of F with valuation v_{F′} : F′ → Γ_{F′} ∪ {0} extending the valuation v_{F}. The extension of scalars from F to F′ is the functor

(2.1)  − ⊗_{F} F′ : Isoc(k)_{F} ⟶ Isoc(k)_{F′}

that maps (D, Φ) ∈ Isoc(k)_{F} to the object (D ⊗_{F} F′, Φ ⊗ id). If F′ is a finite extension of F, then we also define the restriction of scalars

(2.2)  ϵ_{F′/F} : Isoc(k)_{F′} ⟶ Isoc(k)_{F} .

This functor maps (D′, Φ′) ∈ Isoc(k)_{F′} to itself, forgetting the F′-action but keeping the F -action. We write Γ_{F} ⊗ ℚ for the localisation of the abelian group Γ_{F}. Then every element γ′ ∈ Γ_{F} ⊗ ℚ can be written as a single tensor γ ⊗ r and we extend the total order of Γ_{F} to Γ_{F} ⊗ ℚ by

a ⊗ \frac{1}{m} < b ⊗ \frac{1}{n} ⇔ a^{n} < b^{m}.

Definition 2.3. Let (D, Φ) ∈ Isoc(k)_{F} and d = rk_{F ⊗_{ℚp} K_{0}} D. The map Φ^{f} : D → D is an F -linear automorphism of the fd-dimensional F -vector space D.

(i) Define the Newton slope of (D, Φ) as

λ_{N}^{(F)}(D, Φ) = v_{F} (det_{F} Φ^{f}) ⊗ \frac{1}{f^{2}d} ∈ Γ_{F} ⊗ ℚ.

Here det_{F} means that we take the determinant of an F -linear map on an F - vector space. (ii) Let λ ∈ Γ_{F} ⊗ ℚ. An object (D, Φ) ∈ Isoc(k)_{F} is called purely of Newton-slope λ if for all Φ-stable F ⊗_{ℚ_{p}} K_{0}-submodules D′ ⊂ D we have λ_{N}^{(F)}(D′, Φ|_{D′}) = λ.

Lemma 2.4. Let F′ be an extension of F with valuation v_{F′} extending v_{F} and (D, Φ) ∈ Isoc(k)_{F}. Then

λ_{N}^{(F′)}(D ⊗_{F} F′, Φ ⊗ id) = λ_{N}^{(F)}(D, Φ).

If in addition F′ is finite over F and (D′, Φ′) ∈ Isoc(k)_{F′}, then

λ_{N}^{(F)}(ϵ_{F′/F} (D′, Φ′)) = λ_{N}^{(F′)}(D′, Φ′).

Proof. These are straightforward computations. □

As the Newton slope is preserved under extension and restriction of scalars we will just write λ_{N} in the sequel.

Remark 2.5. Let (D, Φ) ∈ Isoc(k)_{ℚ_{p}} be an object of rank d and denote for the moment by v_{p} the usual p-adic valuation on ℚ_{p}. Write | − | = p^{−v_{p}(−)}. Then the value group of the absolute value is Γ_{ℚ_{p}} = p^{ℤ} and we identify Γ_{ℚ_{p}} ⊗ ℚ with the subgroup p^{ℚ} of ℝ\{0}. Our definitions then imply

(2.3)  λ_{N}(D, Φ) = p^{− \frac{1}{d} v_{p}(det_{K_{0}} Φ)}.

Here v_{p}(det_{K_{0}} Φ) is the p-adic valuation of the determinant over K_{0} of any matrix representing the semi-linear map Φ in some chosen basis. This matrix is well defined up to φ-conjugation and hence the valuation of the determinant is independent of choices. Note that (the negative of) the exponent in (2.3) is the usual Newton slope of the isocrystal (D, Φ) over k, compare [Zi] for example.

Proposition 2.6. Let (D, Φ) ∈ Isoc(k)_{F}, then there exist unique elements λ_{1} < λ_{2} < ⋯ < λ_{r} ∈ Γ_{F} ⊗ ℚ and a unique decomposition

D = D_{1} ⊕ D_{2} ⊕ ⋯ ⊕ D_{r}

of D into Φ-stable F ⊗_{ℚ_{p}} K_{0}-submodules such that (D_{i}, Φ|_{D_{i}}) is purely of Newton slope λ_{i}.

Proof. First we show the existence of such a decomposition. The uniqueness will then follow from Lemma 2.9 below. Step 1: Assume first that there exists an embedding ψ_{0} : K_{0} ↪ F. We obtain an isomorphism

\xymatrix{F ⊗_{ℚ_{p}} K_{0} \ar[r]^{≅} & ∏_{ψ:K_{0}→F} F.}

The endomorphism id ⊗φ on the left hand side translates to the shift of the factors on the right hand side. Further we obtain the corresponding decomposition

D = ∏_{ψ} V_{ψ}

into F -vector spaces V_{ψ} and F -linear isomorphisms

\xymatrix{Φ_{ψ} = Φ|_{V_{ψ}} : V_{ψ} \ar[r]^{≅} & V_{ψ∘φ}.}

There is a bijection between the Φ-stable subspaces D′ of D and the Φ^{f}|_{V_{ψ_{0}}} - stable subspaces of V_{ψ_{0}} given by D′ ↦ D′ ∩ V_{ψ_{0}}. Given D′ ⊂ D and U = D′ ∩ V_{ψ_{0}} we have

λ_{N}(D′, Φ|_{D′}) = v_{F} (det_{F} Φ^{f}|_{U}) ⊗ \frac{1}{f dim_{F} U} ∈ Γ_{F} ⊗ ℚ.

Hence the desired decomposition of D is induced by the decomposition of V_{ψ_{0}} into the maximal Φ^{f}|_{V_{ψ_{0}}} -stable subspaces U ⊂ V_{ψ_{0}} such that

v_{F} (det_{F} Φ^{f}|_{U′}) ⊗ \frac{1}{f dim_{F} U′} = λ_{i}

for all Φ^{f}-stable subspaces U′ ⊂ U. Step 2: If there is no embedding ψ of K_{0} into F, then we find a finite extension F′ = F K_{0} of F such that K_{0} embeds into F′. We want to deduce the result from Step 1 by Galois descent. The extension F′ is Galois over F, as K_{0} is Galois over ℚ_{p}. Further we extend the valuation from F to F′ by setting v_{F} (O_{K_{0}}^{×}) = {1}, where O_{K_{0}} ⊂ K_{0} is the ring of integers.

Write (D′, Φ′) for the extension of scalars of (D, Φ) ∈ Isoc(k)_{F} to Isoc(k)_{F′}. Then there exists λ_{1} < λ_{2} < ⋯ < λ_{r} ∈ Γ_{F} ⊗ ℚ and a decomposition

(2.4)  D′ = D′_{1} ⊕ D′_{2} ⊕ … D′_{r}

such that the D′_{i} are Φ′-stable and (D′_{i}, Φ′|_{D′_{i}}) is purely of slope λ_{i}. Now the action of the Galois group Gal(F′/F) preserves the valuation on F′ and hence also the Newton slope of a Φ′-stable subobject of D′. It follows that Gal(F′/F) preserves the decomposition (2.4) and hence this decomposition descends to D. □

Remark 2.7. Proposition 2.6 replaces the slope decomposition in the classical context (c. f. [Zi, VI, 3] for example).

Definition 2.8. Let (D, Φ) ∈ Isoc(k)_{F} and denote by D = ⨁ D_{i} a decomposition of D into Φ-stable submodules purely of slope λ_{i} ∈ Γ_{F} ⊗ ℚ as in Proposition 2.6. We will refer to this as the slope decomposition. Further, for λ ∈ Γ_{F} ⊗ ℚ we define

D_{λ} = \begin{cases}D_{i} & , λ = λ_{i} \\ 0 & otherwise. \end{cases}

Lemma 2.9. Let f : (D, Φ) → (D′, Φ′) be a morphism in Isoc(k)_{F}. Consider slope decompositions D = ⨁ D_{i} and D′ = ⨁ D′_{j} as in Proposition 2.6. Then for all λ ∈ Γ_{F} ⊗ ℚ

f(D_{λ}) ⊂ D′_{λ}.

Proof. This is an immediate consequence of f ∘ Φ = Φ′ ∘ f. □

2.2. Filtered Isocrystals with coefficients. Recall that K is a totally ramified extension of K_{0}. We denote by e = [K : K_{0}] the ramification index of K. In this section we define the basic object of our study.

Definition 2.10. A K-filtered isocrystal over k with coefficients in F is a triple (D, Φ, F^{•}), where (D, Φ) ∈ Isoc(k)_{F} and F^{•} is a descending, separated and exhaustive ℤ-filtration on D_{K} = D ⊗_{K_{0}} K by (not necessarily free) F ⊗_{ℚ_{p}} Ksubmodules. A morphism

f : (D, Φ, F^{•}) ⟶ (D′, Φ′, F^{′•})

is a morphism f : (D, Φ) → (D′, Φ′) in Isoc(k)_{F} such that f ⊗ id : D_{K} → D′_{K} respects the filtrations. The category of K-filtered isocrystals over k with coefficients in F is denoted by Fil Isoc(k)_{F}^{K}.

It is easy to see that Fil Isoc(k)_{F}^{K} is an F -linear ⊗-category. Further it has obvious notions of kernels, cokernels and exact sequences. For an extension F′ of F we again have an extension of scalars like in (2.1),

− ⊗_{F} F′ : Fil Isoc(k)_{F}^{K} ⟶ Fil Isoc(k)_{F′}^{K}.

If F′ is finite over F, we also have a restriction of scalars like in (2.2),

ϵ_{F′/F} : Fil Isoc(k)_{F′}^{K} ⟶ Fil Isoc(k)_{F}^{K}.

In the following we will often shorten our notation and just write D for an object (D, Φ, F^{•}) ∈ Fil Isoc(k)_{F}^{K}. We now want to develop a slope theory for filtered isocrystals and define weakly admissible objects.

Definition 2.11. Let (D, Φ, F^{•}) ∈ Fil Isoc(k)_{F}^{K}. We define

deg F^{•} = ∑_{i∈ℤ} \frac{1}{ef} i dim_{F} gr_{i} F^{•}
deg_{F} (D) = (v_{F} (det_{F} Φ^{f}) ⊗ \frac{1}{f^{2}} )^{−1} v_{F} (p)^{deg(F^{•})} ∈ Γ_{F} ⊗ ℚ
μ_{F} (D) = deg_{F} (D)(1 ⊗ \frac{1}{d}) ∈ Γ_{F} ⊗ ℚ.

We call μ_{F} (D) the slope of D.

Remark 2.12. As in Lemma 2.4, one easily sees that the slope μ_{F} is preserved under extension and restriction of scalars. Hence we will just write μ in the sequel.

Now we have a Harder-Narasimhan formalism as in [DOR, Chapter 1]. The only difference is that our valuations are written multiplicatively, while in the usual theory they are written additively. We will only sketch the proofs and refer to [DOR] for the details.

Lemma 2.13. Let

\xymatrix{0 \ar[r] & D′ \ar[r] & D \ar[r] & D′′ \ar[r] & 0}

be a short exact sequence in Fil Isoc(k)_{F}^{K}. Then

deg_{F} (D) = deg_{F} (D′) deg_{F} (D′′).

Further

max{μ(D′), μ(D′′)} ≥ μ(D) ≥ min{μ(D′), μ(D′′)}.

The sequence μ(D′), μ(D), μ(D′′) is either strictly increasing or strictly decreasing or stationary.

Proof. The first assertion is obvious from the definitions and the second is a direct consequence. □

Lemma 2.14. Let f : D → D′ be a morphism in Fil Isoc(k)_{F}^{K}. Then

deg_{F} (coim f) ≥ deg_{F} (im f).

Proof. Replacing D by coim f and D′ by im f, we may assume that f is an isomorphism in Isoc(k)_{F}. Now the assertion follows easily from

(f ⊗ id)(F^{i}) ⊂ F^{′i}.
□

Definition 2.15. An object (D, Φ, F^{•}) ∈ Fil Isoc(k)_{F}^{K} is called semi-stable if, for all Φ-stable subobjects D′ ⊂ D, we have μ(D′) ≥ μ(D). It is called stable if the inequality is strict for all proper subobjects. Finally D is called weakly admissible if it is semi-stable of slope 1.

Note that semi-stability is defined using "≥" instead of "≤" (as in [DOR]), since our valuations are written multiplicatively.

Remark 2.16. Let (D, Φ, F^{•}) ∈ Fil Isoc(k)_{ℚ_{p}}^{K}. Using the notations of Remark 2.5, we find

μ(D) = p^{\frac{1}{d}(v_{p}(det_{K_{0}} Φ)−∑_{i} i dim_{K}(F^{i}/F^{i+1}))}.

Hence we see that D is weakly admissible if and only if it is weakly admissible in the sense of [CF, 3.4].

Proposition 2.17. Let D, D′ ∈ Fil Isoc(k)_{F}^{K} be semi-stable objects.

(i) If μ(D) < μ(D′), then Hom(D, D′) = 0. (ii) If μ(D) = μ(D′) = μ, then for all f ∈ Hom(D, D′) we have im f ≅ coim f and the objects ker f, coker f and im f are semi-stable of slope μ.

Proof. The proof is the same as in [DOR, Proposition 1.1.20] □

Corollary 2.18. Let μ ∈ Γ ⊗ ℚ, then the full subcategory of Fil Isoc(k)_{F}^{K} consisting of semi-stable objects of slope μ is an abelian, artinian and noetherian category which is stable under extensions. The simple objects are exactly the stable ones.

Proof. The proof is the same as the proof of [DOR, Corollary 1.2.21]. □

The main result of this section is the existence of a Harder-Narasimhan filtration for the objects in Fil Isoc(k)_{F}^{K}. The existence of this filtration will also imply that semi-stability (and hence weak admissibility) is preserved under extension and restriction of scalars.

Proposition 2.19. Let D ∈ Fil Isoc(k)_{F}^{K}, then there exist unique elements μ_{1} < μ_{2} < ⋯ < μ_{r} ∈ Γ_{F} ⊗ ℚ and a unique filtration

0 = D_{0} ⊂ D_{1} ⊂ D_{2} ⊂ ⋯ ⊂ D_{r} = D

of D in Fil Isoc(k)_{F}^{K} such that D_{i}/D_{i−1} is semi-stable of slope μ_{i}.

Proof. The proof is similar to the proof of [DOR, Proposition 1.3.1 (a)]. First we prove the existence of the filtration. The uniqueness will then follow from Lemma 2.21 below. By the existence of the slope decomposition in Proposition 2.6, the set

{μ(D′) | D′ ⊂ D stable under Φ}

is finite. Hence there is a unique minimal element μ_{1} and we claim that there is a maximal subobject D_{1} ⊂ D of slope μ_{1} which then must be semi-stable. This follows, as the sum of two subobjects of slope μ_{1} has again slope μ_{1}, by Lemma 2.13 and the minimality of μ_{1}. Proceeding with D/D_{1} the claim follows by induction. □

Definition 2.20. Let D ∈ Fil Isoc(k)_{F}^{K} and denote by

0 = D_{0} ⊂ D_{1} ⊂ D_{2} ⊂ ⋯ ⊂ D_{r} = D

a filtration as in Proposition 2.19. This filtration is called the Harder-Narasimhan filtration of D. For μ ∈ Γ_{F} ⊗ ℚ we define

D_{(μ)} = \begin{cases}0 & if μ < μ_{1} \\ D_{i} & if μ_{i} ≤ μ < μ_{i+1} \\ D & if μ ≥ μ_{r}. \end{cases}

Lemma 2.21. Let f : D → D′ be a morphism in Fil Isoc(k)_{F}^{K} and fix filtrations of D and D′ as in Proposition 2.19. Let μ ∈ Γ_{F} ⊗ ℚ, then (with the notation of Definition 2.20):

f(D_{(μ)}) ⊂ D′_{(μ)}.

Proof. The proof is the same as in [DOR, Proposition 1.3.1 (b)]. □

Corollary 2.22. Let F′ be an extension of F with valuation v_{F′} extending v_{F} and D ∈ Fil Isoc(k)_{F}^{K}. Then D is semi-stable of slope μ, if and only if D′ = D ⊗_{F} F′ is semi-stable of slope μ. If in addition F′ is finite over F, then D′ ∈ Fil Isoc(k)_{F′}^{K} is semi-stable of slope μ, if and only if ϵ_{F′/F} (D′) ∈ Fil Isoc(k)_{F}^{K} is semi-stable of slope μ.

Proof. First it is clear that semi-stability of D′ implies semi-stability of D, as every Φ-stable F -subspace of D defines a Φ-stable F′-subspace of D′ which has the same slope. Now assume that D is semi-stable. We may assume that F′ is finitely generated over F, as every counterexample for the semi-stability condition is defined over a finitely generated extension. Then F′ is an algebraic extension of a purely transcendental extension and we can treat both cases separately. Assume first that F′ is an algebraic extension of F. We may replace it by its Galois hull and denote by G = Gal(F′/F) the Galois group of F′ over F. Then the action of G preserves the valuation on F′. We denote by

0 = D′_{0} ⊂ D′_{1} ⊂ D′_{2} ⊂ ⋯ ⊂ D′_{r} = D′

the Harder-Narasimhan filtration of D′. The action of G commutes with Φ and preserves the filtration F^{•} ⊗_{F} F′ of D′ ⊗_{K_{0}} K. It follows that it preserves the slope of a Φ-stable subobject and hence preserves the Harder-Narasimhan filtration. It follows that the filtration descends to F and hence it can only have one step, as D is semi-stable. Assume now that F′ is purely transcendental over F. By the above discussion of algebraic extensions we may also assume that F is algebraically closed. Again we write G = Aut(F′/F) for the group of F -automorphisms of F′. As above we only need to check that G preserves the slope of a Φ-stable subobject of D′. Let U ⊂ D′ be such a Φ-stable subspace. Then U is a direct sum of indecomposable Φ-modules U_{i} such that the isomorphism class of U_{i} is defined over F. This can be seen as follows: We decompose D as a product

D = V_{1} × ⋯ × V_{f}

of F -vector spaces such that Φ induces a linear map Φ_{i} : V_{i} → V_{i+1}, where V_{f+1} := V_{1}. We can choose bases of these vector spaces such that the matrix of Φ is represented by the tuple (id, …, id, A) for some A ∈ GL_{d}(F) in Jordan canonical form. If we choose a similar canonical form for Φ|_{U}, then it is clear that every Jordan-Block for Φ|_{U} is a sub-Jordan-block of A. It follows that det_{F′}(Φ^{f}|_{U}) ∈ F and hence the action of G preserves the slope of U, which yields the claim by the same descent argument as above. Now assume that F′ is finite over F and D′ is a semi-stable object of Fil Isoc(k)_{F′}^{K}. Consider the Harder-Narasimhan filtration of ϵ_{F′/F} (D′). By Lemma 2.21 the filtration steps are stable under the operation of F′. Hence the filtration can have only one step. If conversely ϵ_{F′/F} (D′) is known to be semi-stable, then every Φ-stable F′- subspace of D′ is a Φ-stable F -subspace of ϵ_{F′/F} (D′), and hence semi-stability of D′ follows. □

3. Families of filtered φ-modules

It is shown in [He, 4] that the stack of weakly admissible filtered φ-modules is an open substack of the stack of filtered φ-modules. We briefly recall this result before we study the weakly admissible locus in the fibers over the adjoint quotient. We write Rig_{E} for the category of rigid analytic spaces over a finite extension E of ℚ_{p} (see [BGR] and also [Bo] for an introduction to rigid geometry) and Ad_{E}^{lft} for the category of adic spaces locally of finite type over E, see [Hu2].

3.1. Stacks of filtered φ-modules. Let d be a positive integer and ν an algebraic cocharacter

(3.1)  ν : ℚ̄_{p}^{×} ⟶ (Res_{K/ℚ_{p}} A_{K})(ℚ̄_{p}),

where A ⊂ GL_{d} is the diagonal torus. We assume that this cocharacter is dominant with respect to the restriction B of the Borel subgroup of upper triangular matrices in (GL_{d})_{K}. We write E ⊂ ℚ̄_{p} for the reflex field of ν, i.e. the field of definition of the cocharacter ν, see below for a precise characterization. Let Δ denote the set of simple roots (defined over ℚ̄_{p}) of Res_{K/ℚ_{p}} GL_{d} with respect to B and denote by Δ_{ν} ⊂ Δ the set of all simple roots α such that ⟨α, ν⟩ = 0. Here ⟨−, −⟩ is the canonical pairing between characters and cocharacters. We write P_{ν} for the parabolic subgroup of (Res_{K/ℚ_{p}} GL_{d}) containing B and corresponding to Δ_{ν} ⊂ Δ. This parabolic subgroup is defined over E, and the quotient by this parabolic is a projective E-variety

(3.2)  Gr_{K,ν} = (Res_{K/ℚ_{p}} GL_{d})_{E}/P_{ν}

representing the functor

S ↦ {filtrations F^{•} of O_{S} ⊗_{ℚ_{p}} K^{d} of type ν}

on the category of E-schemes. Here the filtrations are locally on S direct summands. Being of type ν means the following. Assume that the cocharacter

ν : ℚ̄_{p}^{×} ⟶ ∏_{ψ:K→ℚ̄_{p}} GL_{d}(ℚ̄_{p})

is given by cocharacters

ν_{ψ} : λ ↦ diag((λ^{i_{1}(ψ)})^{(m_{1}(ψ))}, …, (λ^{i_{r}(ψ)})^{(m_{r}(ψ))})

for some integers i_{j}(ψ) ∈ ℤ, with i_{j}(ψ) ≠ i_{j′}(ψ) for j ≠ j′, and multiplicities m_{j}(ψ) > 0. Then any point F^{•} ∈ Gr_{K,ν}(ℚ̄_{p}) is a filtration ∏_{ψ} F_{ψ}^{•} of ∏_{ψ} ℚ̄_{p}^{d} such that

dimℚ̄_{p} gr_{i}(F_{ψ}^{•}) = \begin{cases}0 & if i ∉ \{i_{1}(ψ), …, i_{r}(ψ)\} \\ m_{j}(ψ) & if i = i_{j}(ψ). \end{cases}

In terms of the integers i_{j}(ψ) and m_{j}(ψ) the fact that ν is dominant means that i_{j}(ψ) ≥ i_{j+1}(ψ) for all j and ψ. The reflex field E of the character ν is characterized by the requirement

Gal(ℚ̄_{p}/E) = {σ ∈ Gal(ℚ̄_{p}/ℚ_{p}) | i_{j}(ψ) = i_{j}(σψ), m_{j}(ψ) = m_{j}(σψ)}.

We denote by Gr_{K,ν}^{rig} resp. Gr_{K,ν}^{ad} the associated rigid space, resp. the associated adic space (cf. [BGR, 9.3.4] and [Hu2, Remark 4.6 (i)]). Given ν as in (3.1) and denoting as before by E the reflex field of ν, we consider the following fpqc-stack 𝔇_{ν} on the category Rig_{E} (resp. on the category Ad_{E}^{lft}). For X ∈ Rig_{E} (resp. Ad_{E}^{lft}) the groupoid 𝔇_{ν}(X) consists of triples (D, Φ, F^{•}), where D is a coherent O_{X} ⊗_{ℚ_{p}} K_{0}-module which is locally on X free over O_{X} ⊗_{ℚ_{p}} K_{0} and Φ : D → D is an id ⊗φ-linear automorphism. Finally F^{•} is a filtration of D_{K} = D ⊗_{K_{0}} K of type ν, i.e. after choosing fpqc-locally on X a basis of D, the filtration F^{•} induces a map to Gr_{K,ν}^{rig} (resp. Gr_{K,ν}^{ad}), compare also [PR, 5.a].

One easily sees that the stack 𝔇_{ν} is the stack quotient of the rigid space

(3.3)  X_{ν} = (Res_{K_{0}/ℚ_{p}} GL_{d})_{E}^{rig} × Gr_{K,ν}^{rig}

by the φ-conjugation action of (Res_{K_{0}/ℚ_{p}} GL_{d})_{E}^{rig} given by

(3.4)  (A, F^{•}) · g = (g^{−1}Aφ(g), g^{−1}F^{•}).

Here the canonical map X_{ν} → 𝔇_{ν} is given by

(A, F^{•}) ↦ (O_{X_{ν}} ⊗_{ℚ_{p}} K_{0}^{d}, A(id ⊗φ), F^{•}).

3.2. The weakly admissible locus. Fix a cocharacter ν with reflex field E as in the previous section. If X ∈ Ad_{E}^{lft} and x ∈ X, then our definitions imply that, given (D, Φ, F^{•}) ∈ 𝔇_{ν}(X), we have

(D ⊗ k(x), Φ ⊗ id, F^{•} ⊗ k(x)) ∈ Fil Isoc(k)_{k(x)}^{K}

One of the main results of [He] is concerned with the structure of the weakly admissible locus in the stacks 𝔇_{ν} defined above.

Theorem 3.1. Let ν be a cocharacter as in (3.1) and X be an adic space locally of finite type over the reflex field of ν. If (D, Φ, F^{•}) ∈ 𝔇_{ν}(X), then the weakly admissible locus

X^{wa} = {x ∈ X | (D ⊗ k(x), Φ ⊗ id, F^{•} ⊗ k(x)) is weakly admissible}

is an open subset. Especially it has a canonical structure of an adic space.

Proof. This is [He, Theorem 4.1]. □

We can define a substack 𝔇_{ν}^{wa} ⊂ 𝔇_{ν} consisting of the weakly admissible filtered isocrystals. More precisely, for an adic space X the groupoid 𝔇_{ν}^{wa}(X) consists of those triples (D, Φ, F^{•}) such that (D ⊗ k(x), Φ ⊗ id, F^{•} ⊗ k(x)) is weakly admissible for all x ∈ X. Thanks to Corollary 2.22 it is clear that this is again an fpqc-stack. The following result is now an obvious consequence of Theorem 3.1.

Corollary 3.2. The stack 𝔇_{ν}^{wa} on the category of adic spaces locally of finite type over the reflex field of ν is an open substack of 𝔇_{ν}.

4. The fibers over the adjoint quotient

We now come to the main results of this paper. We want to link the weakly admissible locus in

(Res_{K_{0}/ℚ_{p}} GL_{d} × Gr_{K,ν})^{ad}

as considered in the previous section to the adjoint quotient of the group GL_{d}. This relation was studied by Breuil and Schneider in [BS]. In this section we show that the fibers over the adjoint quotient are (base changes of) analytifications of schemes over ℚ_{p} and hence the period stacks considered here have a much more algebraic nature than the period spaces considered by Rapoport and Zink in [RZ]. In the next section we determine the image of the weakly admissible locus in the adjoint quotient and identify it with a closed Newtonstratum in the sense of Kottwitz [Ko]. First we need to recall some notations and facts about the adjoint quotient from [Ko]. We write GL_{d} = GL(V) for the general linear group over ℚ_{p}, where V = ℚ_{p}^{d}, and B ⊂ GL_{d} for the Borel subgroup of upper triangular matrices. Further we denote by A ⊂ B the diagonal torus and identify X_{∗}(A) and X^{∗}(A) with ℤ^{d} in the usual way, i.e. (m_{1}, …, m_{d}) ∈ ℤ^{d} defines

(t ↦ diag(t^{m_{1}}, …, t^{m_{d}})) ∈ X^{∗}(A)

resp. (diag(t_{1}, …, t_{d}) ↦ t_{1}^{m_{1}} … t_{d}^{m_{d}}) ∈ X_{∗}(A).

Let Δ = {α_{1}, …, α_{d−1}} be the simple roots defined by B, i.e. ⟨α_{i}, ν⟩ = ν_{i}−ν_{i+1} for all ν ∈ X_{∗}(A). We also choose lifts

ω_{i} = (1^{(i)}, 0^{(d−i)}) ∈ ℤ^{d} = X^{∗}(A)

of the dual basis ϖ_{1}, … ϖ_{d−1} ∈ X^{∗}(A ∩ SL_{d}) of the coroots. Finally W = S_{d} denotes the Weyl group of (GL_{d}, A). There is a map

c : A ⟶ 𝔸^{d−1} × 𝔾_{m}

which maps an element of A to the coefficients of its characteristic polynomial. This morphism identifies A/W with 𝔸^{d−1} × 𝔾_{m}. Now we will define a map

(4.1)  G = Res_{K_{0}/ℚ_{p}}(GL_{d})_{K_{0}} ⟶ A/W

that is invariant under φ-conjugation on the left side. Recall that we have identifications GL_{d} = GL(V) and (GL_{d})_{K_{0}} = GL(V_{0}), where V_{0} = V ⊗_{ℚ_{p}} K_{0}. For an ℚ_{p}-algebra R and g ∈ G(R) we have the R⊗_{ℚ_{p}} K_{0}-linear automorphism Φ_{g}^{f} = (g(id ⊗φ))^{f} of R ⊗_{ℚ_{p}} V_{0}. Its characteristic polynomial is an element of (R ⊗_{ℚ_{p}} K_{0})[T]. Now this polynomial is invariant under id ⊗φ and hence it already lies in R[T] which can be seen as follows: First we may assume that K_{0} embeds into R, as R = R′ ∩ (R ⊗_{ℚ_{p}} K_{0}) ⊂ R′ ⊗_{ℚ_{p}} K_{0} for any extension R′ of R. As in the proof of Proposition 2.6, we choose decompositions R ⊗_{ℚ_{p}} K_{0} = ∏_{i=1}^{f} R and D = ∏_{i=1}^{f} V_{i} such that Φ_{g} maps V_{i} to V_{i+1}, where V_{f+1} = V_{1}. Then Φ_{g}^{f} induces automorphisms Φ_{i} on V_{i}. It follows that

charpoly(Φ_{g}^{f}) = (charpoly(Φ_{1}), …, charpoly(Φ_{f})).

However, Φ_{i+1} = Φ|_{V_{i}} ∘Φ_{i}∘(Φ|_{V_{i}})^{−1} and hence charpoly(Φ_{i}) = charpoly(Φ_{i+1}). We define the morphism in (4.1) by mapping g ∈ G(R) to the coefficients of this polynomial. It is easy to check that this morphism is invariant under φ-conjugation on G and hence we get morphisms

(4.2)  \xymatrix{G × Gr_{K,ν} \ar[d] \ar[r]^{α̃} & (A/W)_{E} \ar[d]^{=} \\ D_{ν} \ar[r]^{α} & (A/W)_{E},}

where D_{ν} is the stack-quotient

D_{ν} = (G_{E} × Gr_{K,ν})/G

on the category of E-schemes, where the action of G on G_{E} × Gr_{K,ν} is the same as in (3.4). Here ν is a cocharacter as in (3.1) and E is the reflex field of ν. We also write α and α̃ for the analytification of these morphisms.

Theorem 4.1. Let x ∈ (A/W)_{E}^{ad} and ν be a cocharacter as in (3.1). Then there exists a quasi-projective k(x)-scheme X which is an open subscheme of α^{−1}(x) such that the weakly admissible locus in the fiber over x is given by

α^{−1}(x)^{wa} = X^{ad}.

Proof. The proof will be similar to the proof of [He, Theorem 4.1]. Let x = (c_{1}, …, c_{d}) ∈ k(x)^{d−1} × k(x)^{×} and let v_{x} denote the (multiplicative) valuation on k(x). First note that

c_{d} = det_{k(x)⊗_{ℚp} K_{0}}(Φ^{f}) = det_{k(x)}(Φ^{f})^{1/f}

and hence α^{−1}(x)^{wa} = ∅ unless

v_{x}(c_{d})^{1/f} = v_{x}(p)^{∑_{j∈ℤ} j \frac{1}{ef} dim_{k(x)} gr_{j} F^{•}},

where F^{•} is the universal filtration on Gr_{K,ν}. In the following we will assume that this condition is satisfied. For i ∈ {0, …, d}, consider the following functor on the category of ℚ_{p}-schemes,

S ⟼ \{ \begin{matrix}E ⊂ O_{S} ⊗_{ℚ_{p}} V_{0} locally free O_{S} ⊗_{ℚ_{p}} K_{0}-submodule \\ of rank i that is locally on S a direct summand \end{matrix} \} .

Using the theory of Quot-schemes (see [FGA, Thm 3.1] for example) this functor is easily seen to be representable by a projective ℚ_{p}-scheme Gr_{K_{0},i}. We let G = Res_{K_{0}/ℚ_{p}} GL_{d} act on Gr_{K_{0},i} in the following way: for a ℚ_{p}-scheme S, let A ∈ G(S) and E ∈ Gr_{K_{0},i}(S). We get a linear endomorphism A of O_{S} ⊗_{ℚ_{p}} V_{0} and define the action of A on E by

A · E = A((id ⊗φ)(E)).

Write

a : G × Gr_{K_{0},i} ⟶ Gr_{K_{0},i}

for this action and consider the subscheme Z_{i} ⊂ G × Gr_{K_{0},i} defined by the following fiber product:

\xymatrix{Z_{i} \ar[d] \ar[r] & G × Gr_{K_{0},i} \ar[d]^{a×id} \\ _{Gr_{K_{0},i}} \ar[r]^{Δ} & Gr_{K_{0},i} × Gr_{K_{0},i} .}

An S-valued point x of the scheme Z_{i} is a pair (g_{x}, U_{x}), where g_{x} ∈ G(S) is a linear automorphism of O_{S} ⊗_{ℚ_{p}} V_{0} and U_{x} is an O_{S} ⊗_{ℚ_{p}} K_{0}-submodule of rank i stable under Φ_{x} = g_{x}(id ⊗φ). The scheme Z_{i} is projective over G via the first projection

pr_{i} : Z_{i} ⟶ G.

Further we denote by f_{i} ∈ Γ(Z_{i}, O_{Z_{i}}) the global section defined by

f_{i}(g_{x}, U_{x}) = det((g_{x}(id ⊗φ))^{f}|_{U_{x}})

(recall f = [K_{0} : ℚ_{p}]), where the determinant is the determinant as O_{Z_{i}}- modules. We also write f_{i} for the global section on the associated adic space Z_{i}^{ad}. We write E for the pullback of the universal bundle on Z_{i} to Z_{i} ×Gr_{K,ν} and F^{•} for the pullback of the universal filtration on Gr_{K,ν}. Then the fiber product

G^{•} = (E ⊗_{K_{0}} K) ∩ F^{•}

is a filtration of E ⊗_{K_{0}} K by coherent sheaves. By the semi-continuity theorem the function

h_{i} : x ⟼ ∑_{j∈ℤ} j \frac{1}{ef} dim_{κ(x)} gr_{j} G^{•}

is upper semi-continuous on Z_{i} × Gr_{K,ν} and hence so is

h_{i}^{ad} : x ⟼ ∑_{j∈ℤ} j \frac{1}{ef} dim_{k(x)} gr_{j} (G^{•})^{ad}.

For m ∈ ℤ we write Y_{i,m} ⊂ Z_{i} × Gr_{K,ν} (resp. Y_{i,m}^{ad} ⊂ Z_{i}^{ad} × Gr_{K,ν}^{ad}) for the closed subscheme (resp. the closed adic subspace)

Y_{i,m} = {y ∈ Z_{i} × Gr_{K,ν} | h_{i}(y) ≥ m},
Y_{i,m}^{ad} = {y ∈ Z_{i}^{ad} × Gr_{K,ν}^{ad} | h_{i}^{ad}(y) ≥ m}.

Then the definitions imply that

pr_{i,m} : Y_{i,m} ⟶ G × Gr_{K,ν}
pr_{i,m} : Y_{i,m}^{ad} ⟶ (G × Gr_{K,ν})^{ad}

are proper morphism. Now

S_{i,m} = {y = (g_{y}, U_{y}, F_{y}^{•}) ∈ Y_{i,m}×_{(G×Gr_{K,ν})}α^{−1}(x) | v_{y}(f_{i}(g_{y}, U_{y})) > v_{y}(p)^{f^{2}m}}

is a union of connected components of

Y_{i,m} ×_{(G×Gr_{K,ν})} α^{−1}(x),

which can be seen as follows: Let λ_{1}, …, λ_{d} denote the zeros of the polynomial

X^{d} + c_{1}X^{d−1} + ⋯ + c_{d−1}X + c_{d}.

Then every possible value of the f_{i} is a product of some of the λ_{i} and hence f_{i} can take only finitely many values. We conclude that the subset ⋃_{i,m} pr_{i,m}(S_{i,m}) is closed and claim that

α^{−1}(x)^{wa} = (α^{−1}(x)\ ⋃_{i,m} pr_{i,m}(S_{i,m}))^{ad}.

Indeed, let z = (g_{z}, F_{z}^{•}) ∈ α^{−1}(x) ⊂ G^{ad} × Gr_{K,ν}^{ad}. Then the object

(k(z) ⊗ V_{0}, g_{z}(id ⊗φ), F_{z}^{•})

is not weakly admissible if and only if there exists a g_{z}(id ⊗φ)-stable subspace U_{z} ⊂ k(z) ⊗ V_{0} of some rank, violating the weak admissibility condition. This means z ∈ ⋃_{i,m} pr_{i,m}(S_{i,m})^{ad}. Here we implicitly use that fact that weak admissibility is stable under extension of scalars (see Corollary 2.22).

□

Remark 4.2. In view of the period domains considered in [RZ] it can be surprising that this weakly admissible locus is indeed the adification of a scheme, not just an analytic space. The main reason is the following: In [RZ] the isocrystal is fixed and the counter examples one has to exclude for the weak admissibility condition are parametrized by the ℚ_{p}-valued points of an algebraic variety. In our setting the isocrystal is not fixed and the Frobenius Φ may vary. Hence the set of counter examples is the algebraic variety itself rather than its ℚ_{p}-valued points.

Example 4.3. This example illustrates the difference with period spaces in the sense of Rapoport-Zink. Let K = ℚ_{p} and d = 2. We consider the Frobenius Φ = diag(p, p) and fix the filtration F^{•} such that

dim F^{i} = \begin{cases}2 & i ≤ 0 \\ 1 & i = 1, 2 \\ 0 & i ≥ 3. \end{cases}

In this case the flag variety Gr_{K,ν} is the projective line ℙ^{1} and the period space of [RZ] (or rather [DOR]) is the Drinfeld upper halfplane ℙ^{1}\ℙ^{1}(ℚ_{p}) which is not a scheme. On the other hand the weakly admissible set in the sense discussed above is obviously empty.

Corollary 4.4. Let x ∈ (A/W)_{E}^{ad} and consider the 2-fiber product

\xymatrix{α^{−1}(x)^{wa} \ar[r] \ar[d] & 𝔇_{ν}^{wa} \ar[d] \\ x \ar[r] & (A/W)_{E}^{ad}.}

Then there exists an Artin stack in schemes 𝔄 over the field k(x) such that

α^{−1}(x)^{wa} ≅ 𝔄^{ad}

Proof. This is an immediate consequence of Theorem 4.1 □

We end the discussion of the fibers over the adjoint quotient by discussing three examples.

Example 4.5. Let K = ℚ_{p} and d = 3. We take Φ = diag(1, p, p^{2}) and fix the type of the filtration F^{•} such that

dim F^{i} = \begin{cases}3 & i ≤ 0 \\ 2 & i = 1 \\ 1 & i = 2 \\ 0 & i ≥ 3. \end{cases}

We write G = GL_{3} and B ⊂ G for the Borel subgroup of upper triangular matrices. Further X = G/B is the full flag variety, and we are interested in the weakly admissible locus in X. One easily checks that

X^{wa} = {F^{•} ∈ X | F^{1} ∩ V_{1} = 0, and F^{2} ⊄ V_{12}},

where 0 ⊂ V_{1} ⊂ V_{12} ⊂ ℚ_{p}^{3} is the standard flag fixed by the Borel B. The subset X^{wa} is obviously stable under B and, in fact,

X^{wa} = Bw_{0}B/B,

where w_{0} is the longest Weyl group element. If x denotes the image of Φ in the adjoint quotient, then Φ is a representative of the unique φ-conjugacy class in G mapping to x and we further have α^{−1}(x) = A\G/B and

α^{−1}(x)^{wa} = A\Bw_{0}B/B.

Example 4.6. We use the same notations as in the example above, but this time Φ = diag(1, 1, p^{3}). Then

X^{wa} = {F^{•} ∈ X | F^{1} ∩ V_{12} = 0}.

As dim V_{12} = dim F^{1} = 2 it follows that X^{wa} = ∅. In this case there is a second φ-conjugacy class in G mapping to the same point in the adjoint quotient as Φ. A representative of this second φ-conjugacy class is given by

Φ′ = (\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & p^{3} \end{matrix}) .

Example 4.7. In this example let d = 2 and Φ = diag(1, p). Let K be a ramified extension of ℚ_{p} of degree e and consider flags of the type (1, …, 1), i.e. the cocharacter is defined over ℚ_{p}, the only non-trivial filtration step is F^{1} = (F_{i}^{1})_{i=1...,e} and the base change of the flag variety X = Gr_{K,ν} to K is

X_{K} = ℙ_{K}^{1} × ⋯ × ℙ_{K}^{1}.

The weakly admissible locus in X_{K} is given by

X_{K}^{wa} = {F^{•} = (F^{•})_{i} | F_{i}^{1} ≠ ∞ for all i ∈ {1, …, e}}.

Let G = Res_{K/ℚ_{p}} GL_{2} and B ⊂ G the Weil-restriction of the Borel subgroup of upper triangular matrices. Again we write w_{0} for the longest Weyl group element of G. Then

X^{wa} = Bw_{0}B/B ⊂ X = G/B.

If again x denotes the image of Φ in the adjoint quotient, then α^{−1}(x) = A\X and

α^{−1}(x)^{wa} = A\Bw_{0}B/B.

5. Newton strata and weak admissibility

The proof of Theorem 4.1 suggests that the weakly admissible locus in the fibers over a point in A/W does only depend on the valuation of the zeros of the characteristic polynomial associated to the points of the adjoint quotient. Hence we want to extend the result that the fibers over the adjoint quotient are nice spaces to the pre-image of the Newton strata in the adjoint quotient. Here we work in the category of analytic spaces in the sense of Berkovich (see [Be]), as it is not obvious how to generalize the notion of Newton strata (as defined in [Ko]) to adic spaces. Though the weakly admissible locus is not a Berkovich space in general [He, Example 4.4], we show that it becomes a Berkovich space, if we restrict ourselves to the pre-images of the Newton strata. Further we want to identify the image of the weakly admissible locus in the adjoint quotient with a (closed) Newton-stratum. As usual we will write ℋ (x) for the residue field at a point x in an analytic space and X^{an} for the analytic space associated to a scheme X.

5.1. Newton strata. We first need to recall more notations from [Ko]. We write 𝔞 = X_{∗}(A) ⊗_{ℤ} ℝ, and 𝔞_{dom} ⊂ 𝔞 for the subset of dominant elements, i.e the elements μ ∈ 𝔞 such that ⟨α_{i}, μ⟩ ≥ 0 for all i ∈ {1, …, d − 1}. For c = (c_{1}, …, c_{d}) ∈ (A/W)^{an} we write

(5.1)  d_{c} = (−v_{c}(c_{1}), …, −v_{c}(c_{d})) ∈ ℝ̃^{d−1} × ℝ,

where v_{c} denotes the (additive) valuation on ℋ (c) normalized by v_{c}(p) = 1, and ℝ̃ = ℝ ∪ {−∞}. Note that there is a sign in (5.1), as Kottwitz uses a different sign convention. For a ∈ A^{an} define ν_{a} ∈ 𝔞 by requiring

⟨λ, ν_{a}⟩ = −v_{a}(λ(a))

for all λ ∈ X^{∗}(A), where we write v_{a} for the (additive) valuation on ℋ (a). By [Ko, Proposition 1.4.1] there is a continuous map r : 𝔞 → 𝔞_{dom} mapping x ∈ 𝔞 to the dominant element with the smallest distance to x, and this map extends in a continuous way to ℝ̃^{d−1} × ℝ. Here 𝔞 ⊂ ℝ̃^{d−1} × ℝ via the chosen identification X_{∗}(A) = ℤ^{d}. Then we find that r(d_{c(a)}) is the unique dominant element in the W -orbit of ν_{a}. This follows from [Ko, Theorem 1.5.1] for all a ∈ A(ℚ̄_{p}) and, for an arbitrary point, from the fact that A(ℚ̄_{p}) is dense in A^{an} and the continuity of the construction. Note that r(d_{c}) = (−v_{c}(λ_{1}) ≥ ⋯ ≥ −v_{c}(λ_{d})) if the λ_{i} are the roots of the characteristic polynomial associated to c and Kottwitz shows that this does only depend on d_{c} and not on c itself.

Definition 5.1. For μ ∈ 𝔞_{dom} we define

(A/W)_{μ} = {c ∈ A/W | r(d_{c}) = μ}
(A/W)_{≤μ} = {c ∈ A/W | r(d_{c}) ≤ μ}.

Here ” ≤ ” is the usual dominance order on dominant coweights. We will call the first of these subsets the Newton stratum defined by μ and the second the closed Newton stratum defined by μ.

We need another description of these sets to identify them as analytic subspaces of the adjoint quotient.

Proposition 5.2. Let μ ∈ 𝔞_{dom} and I_{μ} = {i ∈ {1, …, d − 1} | ⟨α_{i}, μ⟩ = 0}. Then

(A/W)_{μ} = \{c = (c_{1} …, c_{d}) ∈ (𝔸^{d−1} × 𝔾_{m})^{an} | \begin{matrix}v_{c}(c_{i}) ≥ −⟨ω_{i}, μ⟩ , i ∈ I_{μ} \\ v_{c}(c_{i}) = −⟨ω_{i}, μ⟩ , i ∉ I_{μ} \end{matrix} \}

(A/W)_{≤μ} = \{c = (c_{1} …, c_{d}) ∈ (𝔸^{d−1} × 𝔾_{m})^{an} | \begin{matrix}v_{c}(c_{i}) ≥ −⟨ω_{i}, μ⟩ , i ≠ d \\ v_{c}(c_{d}) = −⟨ω_{d}, μ⟩. \end{matrix} \}

Proof. For all points in (A/W)^{an}(ℚ̄_{p}) this follows from [Ko, Theorem 1.5.2]. Again the proposition follows from continuity, and the fact that the points in (A/W)^{an}(ℚ̄_{p}) are dense in (A/W)^{an}. □

The category of (strict) analytic spaces is a full subcategory of the category of adic spaces locally of finite type, see [Hu3, 8.3] and the references cited there. Hence we can restrict the stacks 𝔇_{ν} and 𝔇_{ν}^{wa} to the category of analytic spaces. We write again 𝔇_{ν} and 𝔇_{ν}^{wa} for these restrictions. Further we write α^{an} (resp. α^{an}) for the analytifications of the morphisms defined in (4.2).

Theorem 5.3. Let ν be a cocharacter as in (3.1) and μ ∈ 𝔞_{dom}. Then the weakly admissible locus in the inverse image (α^{an})^{−1}((A/W)_{μ}^{an}) is an analytic space.

Proof. The proof is almost identical with the proof of Theorem 4.1. If we replace Z_{i} by

Z̃_{i} = (Z_{i})^{an} ×_{(G×Gr_{K,ν})^{an}} (α^{an})^{−1}((A/W)_{μ}^{an})

and Y_{i,m} by

Ỹ_{i,m} = Y_{i,m} ×_{(G×Gr_{K,ν})^{an}} (α^{an})^{−1}((A/W)_{μ}^{an}).

Here the functions f_{i} are not locally constant on Z̃_{i} and Ỹ_{i,m}, but their valuations (or absolute values) are. As in the proof of Theorem 4.1^{1} it follows that

S̃_{i,m} = {y = (g_{y}, U_{y}, F_{y}^{•}) ∈ Ỹ_{i,m} | |f_{i}(g_{y}, U_{y})|_{y} > |p|^{f^{2}m}}

is a union of connected components of Ỹ_{i,m} and we use the properness of the projection pr_{i,m} : Ỹ_{i,m} → (α^{an})^{−1}((A/W)_{μ}^{an}) to conclude that

(α^{an})^{−1}((A/W)_{μ}^{an})^{wa} = (α^{an})^{−1}((A/W)_{μ}^{an})\ ⋃ pr_{i,m}(S̃_{i,m})

is an open subspace. □

5.2. The image of the weakly admissible locus. In this section we determine the image of the weakly admissible locus under the map defined in (4.1). In the case of a regular cocharacter ν it was shown by Breuil and Schneider that the set of points a ∈ A such that there exists a weakly admissible filtered φ-module (D, Φ, F^{•}) with (Φ^{f})^{ss} = a is an affinoid domain, see [BS, Proposition 3.2]. Here we extend this result to the general case and give a description of this image purely in terms of the adjoint quotient A/W. The difference with the description in [BS] is that we do not need to fix an order of the eigenvalues corresponding to the order of their valuations. We fix a coweight ν as in (3.1). This coweight determines the jumps of the filtration F^{•} on Gr_{K,ν}. After passing to ℚ̄_{p} the filtration is given by

F^{•} = ∏_{ψ} F_{ψ}^{•}

where the product runs over all embeddings ψ : K ↪ ℚ̄_{p}. We write {x_{ψ,1} > x_{ψ,2} > ⋯ > x_{ψ,r}} for the jumps of the filtration F_{ψ}^{•}, i.e.

gr_{i} F_{ψ}^{•} ≠ 0 ⇔ i ∈ {x_{ψ,1}, …, x_{ψ,r}}.

1Note that in this case this does not show that the weakly admissible locus is a scheme, as our base is not a point but the Newton-stratum (A/W)_{μ}^{an} which is not a scheme.

Further denote by n_{ψ,i} the rank of F_{ψ}^{x_{ψ,i}} and write

m_{ψ,j}(i) = max(0, n_{ψ,j} + i − d).

This is the minimal dimension of the intersection of F_{ψ}^{x_{ψ,j}} with a subspace of dimension i. For i ∈ {0, …, d} define

(5.2)  l_{i} = ∑_{ψ} \frac{1}{ef} (∑_{j=1}^{r−1}(x_{ψ,j} − x_{ψ,j+1})m_{ψ,j}(i) + x_{ψ,r}m_{ψ,r}(i)) .

Definition 5.4. For a cocharacter ν and i ∈ {1, …, d} define l_{i} as in (5.2). Define a rational dominant coweight μ(ν) ∈ 𝔞_{dom} by requiring that

\frac{1}{f} ⟨ω_{i}, μ(ν)⟩ = −l_{i} for all i ∈ \{1, …, d\}.

The following result generalizes [BS, Proposition 3.2].

Theorem 5.5. Let ν be a cocharacter as in (3.1) and define μ(ν) as in Definition 5.4. Let x ∈ (A/W)^{an}, then (α^{an})^{−1}(x)^{wa} ≠ ∅ if and only if x ∈ (A/W)_{≤μ(ν)}^{an}.

Proof. Let c = (c_{1}, …, c_{d}) ∈ (A/W)_{≤μ(ν)}^{an} and denote by λ_{1}, …, λ_{t} the roots of

X^{d} + c_{1}X^{d−1} + ⋯ + c_{d−1}X + c_{d}

with multiplicities m_{i} in some finite extension L of ℋ (c) containing K_{0}. Let D = L ⊗_{ℚ_{p}} V_{0} ≅ ∏_{i=1}^{d} L^{d} and

g = (id, …, id, A) ∈ ∏_{i=1}^{f} GL_{d}(L) ≅ G(L),

where A is a matrix consisting of t Jordan blocks of size m_{i} with diagonal entries λ_{i}. Now the pair (D, Φ) = (D, g(id ⊗φ)) ∈ Isoc(k)_{L} has the property that there are only finitely many Φ-stable subobjects D′ ⊂ D. If D′ ⊂ D is a rank i subobject then

t_{N}(D′) := \frac{1}{f^{2}} v_{p}(det Φ^{f}|_{D′}) = \frac{1}{f} ∑_{j=1}^{t} m′_{j}v_{p}(λ_{j})

for some multiplicities m′_{j}, where we write v_{p} for the additive valuation. Write a = (λ_{1}^{(m_{1})}, …, λ_{t}^{(m_{t})}) ∈ A^{an}, then c(a) = c and r(d_{c(a)}) ≤ μ(ν) by assumption. It follows that

t_{N}(D′) = \frac{1}{f} v_{p}(wω_{i}(a))  = −\frac{1}{f} ⟨wω_{i}, ν_{a}⟩
= −\frac{1}{f} ⟨w′ω_{i}, r(d_{c(a)})⟩ ≥ −\frac{1}{f} ⟨ω_{i}, r(d_{c(a)})⟩
≥ −\frac{1}{f} ⟨ω_{i}, μ(ν)⟩  = l_{i}.

for some w, w′ ∈ W. Now for all Φ-stable D′ ⊂ D consider the open subset

U_{D′} ⊂ Gr_{K,ν} ⊗_{ℚ_{p}}K_{0}

of all filtrations F^{•} such that dim(F_{ψ}^{x_{ψ,j}} ∩ D′_{K}) = max(0, n_{ψ,j} + i − d) for all embeddings ψ. This is open as the right hand side is the minimal possible dimension of such an intersection. Since Gr_{K,ν} is geometrically irreducible we find that the intersection ⋂_{D′⊂D} U_{D′} is non-empty and hence there exists an F -valued point F^{•} in this intersection, where F is some extension of L. Now we have (D ⊗_{K_{0}} F, Φ ⊗ id, F^{•}) ∈ Fil Isoc(k)_{F}^{K} and this object is weakly admissible since for all Φ-stable D′ ⊂ D we have

deg(D′) = l_{i} − t_{N}(D′) ≤ 0

where i is the rank of the subobject D′ (and here we write the degree additively). Further, by the definition of g, we find that g maps to c under the map α̃. Conversely assume that c ∈ (A/W)^{an} such that ∅ ≠ α^{−1}(c). Let (D, Φ, F^{•}) be an F -valued point of this fiber for some field F containing K_{0}. Then D decomposes into D_{1} ×⋯×D_{f} and we denote by μ_{1}, …, μ_{t} the distinct eigenvalues of Φ^{f}|_{D_{1}} and by d_{i} their multiplicities (as zeros of the characteristic polynomial). We write (λ_{1}, …, λ_{d}) = (μ_{1}^{(d_{1})}, …, μ_{t}^{(d_{t})}). Then c = c(λ_{1}, … λ_{d}) and we claim that

\frac{1}{f} ∑_{j∈I} v_{p}(λ_{j}) ≥ l_{i}

for all I ⊂ {1, …, d} with ♯I = i. This claim clearly implies c ∈ (A/W)_{≤μ(ν)}^{an}.

Let I ⊂ {1, …, d} and write (λ_{j})_{j∈I} = (λ′_{1}^{(m_{1})}, …, λ′_{t}^{(m_{t})}), where we assume that the λ′_{j} are pairwise distinct. Then ∑_{j=1}^{t} m_{j} = i = ♯I. Using the Jordan canonical form on easily sees that there exists a subobject D′ ⊂ D such that

(Φ^{f}|_{D′∩D_{1}})^{ss} = diag(λ′_{1}^{(m_{1})}, … λ′_{t}^{(m_{t})})

and hence

∑_{j≥1} m_{j} \frac{1}{f} v_{p}(λ′_{j}) ≥ ∑_{j∈ℤ} j \frac{1}{ef} dim gr_{j} D′_{K} ≥ l_{i},

which yields the claim. □

We end by giving two examples of closed Newton strata in the adjoint quotient.

Example 5.6. Let K = ℚ_{p} and d = 3. We fix the cocharacter ν as in Example 4.5 and Example 4.6, i.e.

dim F^{i} = \begin{cases}3 & i ≤ 0 \\ 2 & i = 1 \\ 1 & i = 2 \\ 0 & i ≥ 3. \end{cases}

One easily checks that l_{1} = 0, l_{2} = 1 and l_{3} = 3, i.e.

μ(ν) : t ⟼ diag(1, t^{−1}, t^{−2}).

The image of the weakly admissible locus in the adjoint quotient is given by

(A/W)_{≤μ(ν)}^{an} = \{c = (c_{1}, c_{2}, c_{3}) ∈ 𝔸^{2} × 𝔾_{m} | \begin{matrix}v_{c}(c_{1}) ≥ 0, \\ v_{c}(c_{2}) ≥ 1, \\ v_{c}(c_{3}) = 3. \end{matrix} \}

If a = (a_{1}, a_{2}, a_{3}) ∈ A with v_{a}(a_{1}) ≤ v_{a}(a_{2}) ≤ v_{a}(a_{3}), then [BS, Proposition 3.2] says that there exists a weakly admissible filtered φ-module (D, Φ, F^{•}) with filtration of type ν such that Φ^{ss} = a if and only if

\begin{cases}0 ≤ v_{a}(a_{1}) \\ 1 ≤ v_{a}(a_{1}) + v_{a}(a_{2}) \\ 3 = v_{a}(a_{1}) + v_{a}(a_{2}) + v_{a}(a_{3}). \end{cases}

This is clearly equivalent to our condition in the adjoint quotient. This result also explains Example 4.5 and Example 4.6.

Example 5.7. Again we let K = ℚ_{p} and d = 3. Fix a cocharacter ν such that

dim F^{i} = \begin{cases}3 & i ≤ 0 \\ 2 & i = 1 \\ 0 & i ≥ 2. \end{cases}

One easily checks that l_{1} = 0, l_{2} = 1 and l_{3} = 2, and the image of the weakly admissible locus is

(A/W)_{≤μ(ν)}^{an} = \{c = (c_{1}, c_{2}, c_{3}) ∈ 𝔸^{2} × 𝔾_{m} | \begin{matrix}v_{c}(c_{1}) ≥ 0, \\ v_{c}(c_{2}) ≥ 1, \\ v_{c}(c_{3}) = 2. \end{matrix} \}

References

[Be] V. Berkovich, Spectral theory and analytic geometry over nonarchimedian fields, Math. surveys and monographs 33, American Math. Soc. 1990.

[BGR] S. Bosch, U. Güntzer, R. Remmert, Non-archimedian analysis, Springer-Verlag, Berlin 1984.

[Bo] S. Bosch, Lectures on formal and rigid geometry, Preprint 378, University of Münster, SFB 478-Preprint Series, Münster 2005. Available at http://www.math.uni-muenster.de/sfb/about/publ/heft378.pdf .

[BS] C. Breuil, P. Schneider, First steps towards p-adic Langlands functoriality, J. reine angew. Math. 610 (2007), 149 - 180.

[CF] P. Colmez, J.-M. Fontaine, Constructions des représentations p-adiques semi-stables, Invent. Math. 140 (2000), 1-43.

[DOR] J-F. Dat, S. Orlik, M. Rapoport, Period domains over finite and p-adic fields, Cambridge Tracts in Math. 183, Cambridge University Press 2010.

[FGA] A. Grothendieck, Fondements de la Géometrie Algébrique IV. Les schémas de Hilbert, Séminaire Bourbaki, t. 13, 1960/61, no. 221.

[He] E. Hellmann, On arithmetic families of filtered φ-modules and crystalline representations, preprint 2011, arXiv:1010.4577v2.

[Hu1] R. Huber, Continuous valuations, Math. Z. 212 (1993), 445-447.

[Hu2] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), 513-551.

[Hu3] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math., E30, Friedr. Vieweg & Sohn, Braunschweig, 1996.

[Ko] R. E. Kottwitz, Dimensions of Newton strata in the adjoint quotient of reductive groups, Pure Appl. Math. Q. 2 (2006), 817-836.

[PR] G. Pappas, M. Rapoport, Φ-modules and coefficient spaces, Moscow Math. J. 9 no. 3 (2009), 625- 664.

[RZ] M. Rapoport, T. Zink, Period spaces for p-divisible groups, Ann. of Math. Studies, vol 141, Princeton University Press, Princeton, NJ, 1996.

[Zi] T. Zink, Cartiertheorie kommutativer formaler Gruppen, Teubner Texte zur Math. 68, Leipzig, 1984.

Eugen Hellmann

Mathematisches Institut der Universität Bonn Endenicher Allee 60 53115 Bonn Germany hellmann@math.uni-bonn.de 992

Documenta Mathematica 16 (2011)