Bruhat-Tits Theory from Berkovich's Point of View. Analytic Filtrations

Bruhat-Tits theory from Berkovich’s point of view. Analytic filtrations Arnaud Mayeux

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Arnaud Mayeux. Bruhat-Tits theory from Berkovich’s point of view. Analytic filtrations. 2020. hal-02566872v1

HAL Id: hal-02566872 https://hal.archives-ouvertes.fr/hal-02566872v1 Preprint submitted on 7 May 2020 (v1), last revised 16 Nov 2020 (v2)

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Arnaud Mayeux

Let k be a p-adic eld. We dene ltrations by k-anoid groups, in the Berkovich analytication Gan of a connected reductive group G/k, related to Moy-Prasad ltrations. They are parametrized by a cone, whose basis is the Bruhat-Tits building and whose vertex is the neutral element, in Gan via the notions of Shilov boundary and holomorphically convex envelope.

Contents

1 Schematic congruence subgroups 4

2 Berkovich k-analytic spaces 7 2.1 Analytication of k-schemes and k◦-schemes ...... 7 2.2 Analytic groups and descent ...... 10

3 Bruhat-Tits buildings and Moy-Prasad ltrations 11 3.1 Rational points ...... 11 3.2 Demazure group schemes ...... 12 3.3 Moy-Prasad ltrations ...... 12

4 Denitions and rst properties of analytic ltrations 12 4.1 Notions of potentially Demazure objects ...... 12 4.2 Filtrations of rational potentially Demazure k-anoid groups 14 4.2.1 The split rational case ...... 14 4.2.2 The general case ...... 18 4.3 Filtrations of Lie algebra ...... 22

5 Filtrations associated to points in the Bruhat-Tits building 22

5.1 Denitions and properties of Gx,r ...... 22 5.2 Denitions and properties of the map θ ...... 24 5.3 A cone ...... 28 5.4 Comparison with Moy-Prasad ltrations in the tame case . . 29 5.5 Filtrations of the Lie algebra ...... 30 5.6 Examples and pictures ...... 30 5.6.1 The split torus of rank one ...... 30

5.6.2 A computation of Gx,0 in the case of a wild torus of norm one elements in a quadratic extension ...... 33 Appendix: On notions of rational points 36

1 Acknowledgments

I thank Anne-Marie Aubert, Marco Maculan and Antoine Ducros for help and comments. This research work was done during author's thesis at IMJ- PRG.

Introduction

Let G be a connected reductive group over k. Berkovich [1, chapter 5] (split case) and Rémy-Thuillier-Werner [14] (general case) showed that there is a canonical embedding of the Bruhat-Tits building of G into the Berkovich analytication Gan of G. This embedding and related ideas form what is called Berkovich's point of view on Bruhat-Tits buildings . In this article, we observe that Berkovich's point of view allows to dene and parametrize natural k-analytic ltrations related to Moy-Prasad ltrations [11] [12]. Let x be a point in the reduced Bruhat-Tits building of G over k. The group G(k) acts on the reduced and enlarged buildings. This action is compatible with the canonical projection from the enlarged building to the reduced one. Let us consider the stabilizer stabg (x) of a preimage of x in the enlarged building, this is a compact open subgroup of G(k) independent of the preimage. The idea of Berkovich's point of view is to construct a k-anoid group Gx which realizes stabg (x). The space Gx is equipped with an ordered relation and has a maximal point: its Shilov Boundary denoted θ(x). The space Gx can be recovered from θ(x) taking the holomorphically convex envelope. The preceding constructions and results for the general case are done in [14]. Now our ltrations are certains k-anoid groups Gx,r contained in Gx (we will assume that x and r are rational) satisfying also that the Shilov boundary of Gx,r is a singleton θ(x, r), and that the holomorphically convexe envelope 0 of θ(x, r) is Gx,r. We have Gx,0 = Gx, and for r > r we have Gx,r ⊂ Gx,r0 . 6= When r goes to +∞, θ(x, r) goes to the neutral element. The set {θ(x, r) | r ≥ 0} is a segment joining θ(x) to the neutral element.

2 θ(x, 0)

θ(x, r)

This is an heuristic picture representing Gx and its ltrations. Pictorially, the group stabg (x) is the set of lower extremal points, Gx is the whole picture and Gx,r is the orange part. The k-points of Gx,r are the orange lower extremal points, they coincide with the corresponding Moy-Prasad group in many cases. The Shilov boundaries of Gx and Gx,r are the points θ(x, 0) and θ(x, r). The neutral element is the central lower point. Let us introduce some terminology. A k-anoid group H is a k-anoid Demazure group if it is the generic ber of the formal completion of a De- mazure model G of G along its special ber, it is a potentially k-anoid De- mazure group if there exists a K-anoid extension of k such that H ×M(k) an M(K) is a K-anoid Demazure group of G ×M(k) M(K); if the extension can be choosen nite then H is a rational potentially k-anoid Demazure group. We work with rational potentially k-anoid Demazure groups. Let us x such a H. Let us x a nite extension K/k such that H ×M(k) M(K) is the generic ber of the formal completion of a Demazure model G/K◦ of G ×k K along its special ber. We explain in this article that G is well- dened (unique) and Gal(K/k)-stable. Now let r be a positive rational number, we can assume r ∈ ord(K). We dene k-anoid groups Hr taking the projection an of the generic ber of the prK/k : G ×M(k) M(K) → M(k)

3 formal completion of Γe(K,k)r(G) along its special ber. Here Γe(K,k)r(G) is the schematic e(K, k)r-th congruence of G [20], [13]. We show that Hr is a an well-dened k-anoid subgroup of G and that the Shilov boundary of Hr is a singleton σr and that the holomorphically convex envelope of σr is Hr. Now each rational point x in the Bruhat-Tits building gives rise to a rational potentially k-anoid Demazure groups Gx. Applying the general construction, we get k-anoid ltrations Gx,r for any positive rational number. We show that this construction is compatible by base change in a natural sense and that in tame situations Gx,r(k) is the corresponding Moy-Prasad group for r > 0. We then study the map an θ : BT(G, k) × Q≥0 → G , where BT(G, k) denotes the dense subset of BT(G, k) made of rational points. We prove that it is continous and injective. We also dene similar ltrations for the Lie algebra of G. An appendix compare our notion of rational points with Broussous-Lemaire one [4].

Notations

p : a prime number

k/Qp : a nite extension πk : a uniformizer of k

ord : valuation on algebraic extensions of k such that ord(πk) = 1 e > 1 : a real number strictly bigger than 1 |•| = e−ord(•) norm ◦ k = {x ∈ k | |x| ≤ 1 } = ok ◦◦ k = {x ∈ k | |x| < 1 } = pk k˜ = k◦/k◦◦ residual eld G : connected reductive k-group scheme BT(G, k): reduced Bruhat-Tits building BTE(G, k): enlarged Bruhat-Tits building Gan : analytication of G (Berkovich k-analytic space)

1 Schematic congruence subgroups

In this section we recall some results about congruence subgroups for group schemes over k◦. Congruence groups are built using dilatations [3] [10]. It preserves group schemes structures. We refer the reader to [3], [20], [13] and [10] for dilatations, higher dilatations and schematic congruence groups (see also [17]). We now introduce directly congruence subgroups.

4 Denition 1.1. Let G = spec(A) be a at k◦-group scheme of nite type and n be a positive integer. Let I be the augmentation ideal of A. Put −n X −in i . Then is An = A[πk I] = A + πk I ⊂ A ⊗k◦ k Γn(G) := spec(An) i≥1 called the n-th congruence subgroup of G. It is a canonical group scheme satisfying ◦ ◦ ◦ n ◦ Γn(G)(k ) = ker (G(k ) → G(k /πk k )) . ◦ This is a at k -group scheme together with a morphism Γn(G) → G. Let X = spec(A) be an ane k-scheme of nite type. Let K/k be a nite Galois extension. Let X = spec(A) be an ane at K◦-scheme of nite type such that X×K◦ K = X ×k K. We thus have A⊗K◦ K = A⊗k K. By atness, we see A as a subset of A⊗k K. In this situation, we say that X is Gal(K/k)- stable if A is Gal(K/k)-stable in A ⊗k K. The following Proposition shows that Galois stability is preserved under the operations of taking congruence subgroups.

Lemma 1.2. Let G = spec(A) be an ane k-group scheme of nite type. Let K/k be a Galois extension and A be a at sub-Hopf K◦-algebra of nite type of the Hopf K-algebra AK = A ⊗k K such that A ⊗K◦ K = AK . Put G = spec(A) and assume it is Gal(K/k)-stable. Then for any positive integer n, the congruence subgroup Γn(G) = spec(An) is Gal(K/k)-stable. ◦ Proof. Let εA : A → K be the augmentation, J = ker(εA). Let us remark that εA is the restriction to A of the augmentation εA ⊗ Id : A ⊗k K → K of AK . So J = ker(εA ⊗ Id) ∩ A. The set ker(εA ⊗ Id) is Gal(K/k)-stable, and A is stable by hypothesis, so J is Gal(K/k)-stable as the intersection of two -stable subsets of . By 1.1 P −in i, and Gal(K/k) A ⊗k K An = A + i≥1 πK J so it is Gal(K/k)-stable.

We have a compatibility between extension of scalars and taking congruence subgroups (up to ramication index).

Lemma 1.3. Let K/k be a nite extension, let e(K, k) be the ramication ◦ index, let πK be a uniformizer of K. Let G = spec(A) be an ane at k - ◦ group scheme of nite type. Then the Hopf algebras of Γn(G) ×k◦ K and ◦ Γe(K,k)n(G ×k◦ K ) are egal in (A ⊗k◦ k) ⊗k K. ◦ Proof. Let εA : A → k be the augmentation and J = ker(εA). Let An be the Hopf ◦-algebra such that , then X −in i k Γn(G) = spec(An) An = A + πk J i≥1 by 1.1. So, we have   ◦ ◦ ◦ X −in i ◦ (1) Hopf(Γn(G)×k◦ K ) = An ⊗k◦ K = A⊗k◦ K + πk J ⊗k◦ K . i≥1

5 ◦ ◦ ◦ Let εA ⊗ Id : A ⊗k◦ K → K be the augmentation of A ⊗k◦ K and JK◦ = ◦ ◦ ker(εA ⊗ Id). By 1.1, the Hopf K -algebra of Γe(K,k)n(G ×k◦ K ) is

◦ ◦ X −ie(K,k)n i (2) Hopf(Γe(K,k)n(G ×k◦ K )) = A ⊗k◦ K + πK (JK◦ ) . i≥1

◦ ◦ ◦ n ◦ Because of K is at over k , JK◦ = J ⊗k◦ K . We claim that J ⊗k◦ K = ◦ n (J ⊗k◦ K ) , we report the proof of this claim after deducing the required result. We deduce the equality

◦ Hopf(Γn(G) ×k◦ K ) =   By equation (1) ◦ X −in i ◦ = A ⊗k◦ K +  πk J  ⊗k◦ K i≥1

By properties of sum and tensor product ◦ X −in i ◦ = A ⊗k◦ K + πk J ⊗k◦ K i≥1 ◦ X −in i ◦ = A ⊗k◦ K + πk J ⊗k◦ K i≥1

By the claim ◦ X −in ◦ i = A ⊗k◦ K + πk (J ⊗k◦ K ) i≥1 e(K,k) −e(K,k)in ◦ ◦ and see before the claim ◦ X i πkK = πK K = A ⊗k◦ K + πK (JK◦ ) i≥1 ◦ By equation (2) = Hopf(Γe(K,k)n(G ×k◦ K )) as required. Let us now prove the claim, i.e. that we have the equality of ideals ◦ n n ◦ n (J ⊗k◦ K ) = (J ⊗k◦ K ). Let us rst prove the inclusion ⊃. Since J is ◦ n made of sums of products of n elements in J and because of (J ⊗k◦ K ) is stable by addition since it is an ideal, it is enough to show that any element of n ◦ ◦ n the form x = (j1 . . . jn ⊗λ) ∈ (J ⊗k◦ K ) is contained in (J⊗k◦ K ) . This is a consequence of the identity (j1 . . . jn ⊗λ) = (j1 ⊗λ)(j2 ⊗1) ... (jn ⊗1). Now ◦ n let us prove the inclusion ⊂. Since (J ⊗k◦ K ) is made of sums of products ◦ ◦ of n elements in J ⊗k◦ K and since (J ⊗k◦ K ) is made of sums of pure n ◦ tensors and since (J ⊗k◦ K ) is stable by addition, it is enough to show that ◦ n any element of the form x = (j1 ⊗λ1) ... (jn ⊗λn) ∈ (J⊗k◦ K ) is contained n ◦ in (J ⊗k◦ K ). This is a consequence of the identity (j1 ⊗ λ1) ... (jn ⊗ λn) = (j1 . . . jn ⊗ λ1 . . . λn). This ends the proof of the claim and so the lemma is proved.

Remark 1.4. Lemma 1.3 can also be proved using [10, 2.5].

We nish this section with the following proposition.

6 Proposition 1.5. [20, 2.8] Let G be a smooth k◦-group scheme ane and ˜ of nite type. Let n ∈ N, then the special ber of Γn(G) is a vector k-group scheme. In particular it is smooth and its special ber is irreducible.

2 Berkovich k-analytic spaces

2.1 Analytication of k-schemes and k◦-schemes For general denitions and notations about Berkovich analytic spaces, we refer the reader to [2, 1] and references given there (in particluar [1]). To each scheme X of nite type over k, Berkovich [1, 3.4] associated a k- analytic space Xan such that for any non-Archimedean eld K/k, there is a bijection Xan(K) ' X(K). Let us describe the analytication explicitly in the ane case. Proposition 2.1. (Analytication of k-schemes)[1, 3.4.2] If X = spec(A), where A is a nitely generated ring over k, then the underlying topological space Xan coincides with the set of all multiplicative seminorms on A whose an restriction to k is the norm on k. A point x in X is also denoted | |x. If x ∈ Xan, we dene an Hol(x) := {y ∈ X | |f|y≤ |f|x ∀f ∈ A}. The following proposition is extracted from Rémy-Thuillier-Werner's work [14, 1.2.4] [16, 2.1.1] (see also [1, 5.3.2]). Denition/Proposition 2.2. (Analytication of k◦-schemes) Let A be a at topologically nitely presented k◦-algebra whose spectrum M(A) we denote X. Let X = spec(A ⊗k◦ k) be the generic ber of X. The map × | |A : A ⊗k◦ k → R≥0, a 7→ inf{|λ| | λ ∈ k and a ∈ λ(A ⊗ 1)}

| |A is a norm on A ⊗k◦ k. The Banach algebra A = A ⊗k◦ k obtained by completion is a strictly k-anoid algebra whose spectrum is denoted by Xbη and is called the generic ber of the formal completion of X along its special ber. This anoid space is naturally an anoid domain in Xan (whose points are multiplicative seminorms on A⊗k◦ k which are bounded with respect ˆ ˜ to the seminorm |.|A). Moreover, there is a reduction map τ : Xη → X ×k◦ k dened as follows: a point x in Xˆ η gives a sequence of ring homomorphisms: A → H(x)◦ → H](x) ˜ ˜ whose kernel τ(x) denes a prime ideal of A ⊗k◦ k, i.e a point in X ×k◦ k. If the scheme X is integrally closed in its generic ber in particular if X is smooth then τ is the reduction map of Berkovich (see [1, 2.4]). And so the Shilov Boundary of Xˆ η is in bijection with the irreducible components of ˜ the special ber X ×k◦ k. Moreover, the spectral norm ρ on A is egal to | |A if and only if the algebra A ⊗k◦ k is reduced [16, Proposition 2.1.1].

7 Corollary 2.3. Let X = spec(A) be a smooth k◦-scheme with irreducible special ber. Let X be the generic ber of X. Then

an 1. Xbη is a strictly k-anoid domain of X

2. The Shilov boundary of Xbη is a singleton egal to | |A

an 3. Xbη is the holomorphically convex enevelope of | |A ∈ X . Proof. 1. This is contained in 2.2.

2. Since the special ber of X is irreducible, the Shilov boundary of Xbη is a ˜ singleton by 2.2. The algebra A⊗k◦ k is reduced since X is smooth, thus by 2.2, is the spectral norm. This implies that | |A Shi(spec(\A)η) = | |A (see [1, page 26], see also [14, proof of 2.4(ii)]).

3. Recall that the holomorphically convex envelope of | |A is

an Hol(| |A) = {x ∈ G | |f|x ≤ |f|A ∀f ∈ Hopf(G) }.

By 2.2 the k-anoid algebra A of Xbη is the completion of Hopf(G) relatively to the norm | |A. Let i denote the natural corresponding an injective k-algebras morphism Hopf(G) → A. The inclusion Xbη ⊂ G is given by

ι : M(A) → Gan

| |x 7→ | |x ◦ i .

Since M(A) is the set of all multiplicative seminorms on A bounded by | |A, ι(M(A)) is contained in the holomorphically convex envelope of | |A. Reciprocally, let x ∈ Hol(| |A), x = | |x is a multiplicative seminorm Hopf(G) → R≥0 such that |f|x≤ |f|A ∀f ∈ Hopf(G). Since A is the completion of Hopf(G) relatively to | |A, | |x induces a multiplicative seminorm on A bounded by | |A. This ends the proof.

If K/k is an anoid extension, and X is a k-analytic space, we denote by the canonical surjective map coming from prK/k X ×M(k) M(K) → X the canonical cartesian square

X ×M(k) M(K) / M(K)

X / M(k).

8 If K/k is a nite Galois extension and X is a k-analytic space, the group acts on and induces an isomorphism Gal(K/k) X×M(k)M(K) prK/k (X×M(k) M(K))/Gal(K/k) ' X. This implies that if DK is a subset of X ×M(k) then is -stable if and only if −1 . M(K) DK Gal(K/k) prK/k(prK/k(DK )) = DK We now state a descent theorem, this is due to Rémy-Thuillier-Werner.

Theorem 2.4. [14, Appendix A] Let X be a k-anoid space. Let K be a k-anoid extension. Let D be a subset of X, then D is a k-anoid domain of if and only if the subset −1 is a -anoid domain in X prK/k(D) K X ×M(k) M(K).

Corollary 2.5. Let X be a k-anoid space. Let K/k be a nite Galois extension. Let DK be a Gal(K/k)-stable K-anoid domain of X ×M(k) , put . Then M(K) D = prK/k(DK ) 1. D is a k-anoid domain of X

2. D ×M(k) M(K) = DK . Proof. 1. Since is -stable, −1 , so DK Gal(K/k) prK/k(prK/k(DK )) = DK −1 is -anoid, so by Theorem 2.4, is -anoid. prK/k(D) K D k 2. Since D is k-anoid, this is a consequence of [14, Appendix A begin- ning of A.1 ].

We now show that being Galois stable is preserved by taking the generic ber of the formal completion along the special ber.

Proposition 2.6. Let K/k be a nite Galois extension. Let X = spec(A) be an ane k-scheme of nite type and let X = spec(A) be a smooth K◦- scheme of nite type such that X ×K◦ K = X ×k K and such that X ×K◦ K˜ is irreducible. Assume that A is a Gal(K/k)-stable subalgebra of A ⊗k K. Then the generic ber of the formal completion of X along its special ber is ˆ a Gal(K/k)-stable K-anoid domain Xη of X ×M(k) M(K).

ˆ an Proof. Let | |x ∈ Xη ⊂ X ×M(k) M(K), it is a seminorm on A ⊗k K ˆ bounded by | |A. Let γ ∈ Gal(K/k), we need to show that | |x.γ stay in Xη. Let f ∈ A ⊗k K, then (| |x.γ)(f) = |γ.f|x. By denition of | |x, we have |γ.f|x ≤ |γ.f|A. Since A is Gal(K/k) stable in A ⊗k K, we have γ.A = A for all γ ∈ Gal(K/k) and we deduce the following.

9 |γ.f|A = inf {|λ| | γ.f ∈ A ⊂ A ⊗K◦ K} λ∈K× −1 = inf {|λ| | f ∈ γ A ⊂ A ⊗K◦ K} λ∈K×

= inf {|λ| | f ∈ A ⊂ A ⊗K◦ K} λ∈K×

= |f|A

Consequently, we have (| |x.γ)(f) = |γ.f|x ≤ |γ.f|A = |f|A. Thus | |x.γ ≤ ˆ | |A, and so (| |x.γ) ∈ Xη by Corollary 2.3. The proof ends here.

2.2 Analytic groups and descent

Proposition 2.7. Let G be a k-analytic group, let K/k be an anoid extension, let HK be a K-anoid subgroup of G ×M(k) M(K), let H = , if it is a -anoid domain of then it is a -anoid subgroup prK/k(HK ) k G k of G.

Proof. Let m : G ×M(k) G → G be the multiplication map and inv : G → G the inversion map comming from the group-structure on G. We have to show that the restriction maps m : H ×M(k) H → G and inv : H → G factor through H. Consider the following diagram whose four squares are commutative. m HK × H Kt / HK p i i

' m " p GK × GK p / GK

H × H t p H q p i i ' m " G × G / G Let x be in H × H, it is enough to show that there is y in H such that m ◦ i(x) = i(y). Let z in HK × HK such that p(z) = x, then m ◦ i(x) = m ◦ p ◦ i(z) = p ◦ m ◦ i(z) = p ◦ i ◦ m(z) = i ◦ p ◦ m(z) So y = p ◦ m(z) works. The same argument works for inv.

10 3 Bruhat-Tits buildings and Moy-Prasad ltrations

3.1 Rational points

Let G be a connected reductive group over k. Rousseau [15] proved that for each extension K/k of non archimedean local elds there is a canonical injective map ιK/k : BT(G, k) → BT(G, K) which is continous and G(k)- equivariant.

Denition 3.1. A point x ∈ BT(G, k) is called rational if there exists a nite extension k0/k such that

0 1. ik0/k(x) is a special point of BT(G, k ), 2. G is split over k0.

The set of rational points is denoted BT(G, k).

Proposition 3.2. The set BT(G, k) is a dense subset of BT(G, k).

Proof. Remark rst that if G is split over k, it is obvious that BT(G, k) is dense in BT(G, k), since for any maximal split torus S over k and any nite extension K/k, the apartement A(G, S)/K is obtained from A(G, S)/k adding regularly e(K, k) times more walls. Let us now prove the proposition. It is enough to show that for any maximal split torus S of G over k, A(G, S) is dense in A(G, S). Let L be a nite Galois extension such that G is split over L. By [5, 4.1.1,4.1.2,5.1.12], there exists a torus T ⊃ S dened over k such that T ×k L is a maximal split torus of G ×k L. There exists a facet F in A(G, T )/L which is Gal(L/k)-stable. The barycentre x of F is Gal(L/k)- stable and so x ∈ A(G, S)/k (since (A(G, T )/L)Gal(L/k) = A(G, S)/k). By [6, 6.3.4, lines 8-9], the point x becomes special over a nite extension K/L. So we have proved that there exists one rational point x in A(G, S). Now the set of points {g.x | g ∈ S(k)} consists in a dense subset of A(G, S) constitued of rational points. Indeed, let us rst show that this set consists in rational points. So let g ∈ S(k), there exists a nite extension K/L such that g ∈ S(K). The point x is special in the building BT(G, K) (since G is split over L and x is special in the building BT(G, L)), so g.x is special in BT(G, K). By denition T (k) acts on A(G, T )/L by translation (the translation vector v associated to t ∈ T (k) is given by the usual formula "< v, α >= −ord(α(t)) ∀α", see [5, 4.2.3(I)]) and for any g ∈ S(k) ⊂ T (k), we have g.x ∈ A(G, S)/k, so g.x is a rational point in BT(G, k). Since ord(k) is dense in R, {g.x | g ∈ S(k)} is dense in A(G, S). The propositon follows.

In an appendix at the end of this document, we compare this notion of rational points with Broussous-Lemaire one [4].

11 3.2 Demazure group schemes

Denition 3.3. A Demazure k◦-group scheme is a connected and split reductive k◦-group scheme ([7] [1, 1.1.2]).

If x ∈ BT(G, k), we denote by stab(g x) the stabilizer in G(k) of a preimage of x in the enlarged building.

Proposition 3.4. [14, end of page 15] [5, 4.6.22] If G is split over k and x is a special point, then the canonical scheme attached to x by Bruhat- ◦ ◦ Tits Gx, satisfying Gx(k ) = stab(g x), is a Demazure k -group scheme and Gx ×k◦ k = G. Moreover Gx is smooth and its special ber is irreducible.

3.3 Moy-Prasad ltrations

To any point x ∈ BT(G, k) and any r ∈ R≥0 A. Moy and G. Prasad [11] [12] attached a compact subgroup MP , they also introduced a G(k)x,r ⊂ G(k) subgroup MP of the Lie algebra . If 0 then MP MP , g(k)x,r g(k) r ≥ r G(k)x,r0 ⊂ G(k)x,r we thus get ltrations. We refer to [18] for an other denition of these ltrations, with suitable normalizations, they are dened there only if G split over a tamely ramied extension. We will only deal with Moy-Prasad ltrations in this tame situation and we will use Yu's normalization. See also [20, 0.4] and [19] for informations about Moy-Prasad ltrations.

Fact 3.5. [18, line 36 page 588] [9, line 15 page 278] Let r > 0 and x ∈ BT(G, k) and assume G splits over a tamely ramied extension, then for any nite tamely ramied extension E/k, G(E)MP ∩ G(k) = G(k)MP . ιE/k(x),r x,r 4 Denitions and rst properties of analytic ltrations

4.1 Notions of potentially Demazure objects

Let G = spec(A) be a connected reductive k-group scheme. Let Gan be the k- analytic group associated to G by analytication. B. Rémy, A. Thuillier and A. Werner [14] have introduced the notion of potentially k-anoid Demazure subgroup of Gan. We also introduce a related notion of rational potentially k-anoid Demazure subgroup of Gan.

Denition 4.1. A k-anoid subgroup H ⊂ Gan is a k-anoid Demazure group if it is the generic ber of the formal completion of a Demazure model G of G along its special ber. It is a potentially k-anoid Demazure group if there exists a K-anoid extension of k such that H ×M(k) M(K) is a an K-anoid Demazure group of G ×M(k) M(K); if the extension can be choosen nite then H is a rational potentially k-anoid Demazure group.

12 Proposition 4.2. [14] [8] Let H be a potentially k-anoid Demazure subgroup of Gan. Then

1. The Shilov Boundary of H is reduced to a point σH . 2. The underlying k-anoid domain of H is the holomorphically convex envelope of σH .

Denition 4.1 and Proposition 4.2 give birth naturally to the following notions.

Denition 4.3. Let x be a point in Gan. It is a Demazure point if its holomorphically convex envelope in Gan is a k-anoid Demazure subgroup of Gan. It is a potentially Demazure point if its holomorphically convex envelope in Gan is a potentially k-anoid Demazure subgroup of Gan. It is a rational potentially Demazure point if its holomorphically convex envelope in Gan is a rational potentially k-anoid Demazure subgroup of Gan. We denote by Dem(G), Dem[ (G), Dem(G) the corresponding subsets of Gan, of course the following inclusions hold

Dem(G) ⊂ Dem(G) ⊂ Dem[ (G) ⊂ Gan.

Proposition 4.4. Let A, A0 be two k◦-subalgebra of Hopf(G) such that G = spec(A) and G0 = spec(A0) are two Demazure k◦-group scheme with generic 0 an 0 bers G. If Gb η = Gcη (equality in G ), then A = A . In other words, the schematic Demazure model associated to a k-anoid Demazure group is well-dened.

0 an Proof. The spaces Gb η and Gcη are two k-anoid domain in G whose Shilov 0 boundary are singletons. By Corollary 2.3, we have Shi(Gcη) = Shi(Gb η) = | |A = | |A0 . By denition, | |A is a norm on Hopf(G) given by the formula |f|A = inf {|λ| | f ∈ λ(A ⊗ 1)}. The valuation of k is discrete, so we have λ∈k×

× f ∈ A ⇔ 1 ∈ {λ ∈ k | f ∈ λ(A ⊗ 1)} ⇔ inf {|λ| | f ∈ λ(A ⊗ 1)} ≤ 1 ⇔ |f|A ≤ 1. λ∈k×

0 0 Similarly we have f ∈ A ⇔ |f|A0 ≤ 1. So nally f ∈ A ⇔ f ∈ A , as required.

Lemma 4.5. Let H be a rational potentially k-anoid Demazure subgroup of an G . Let K/k be a Galois extension such that H×M(k)M(K) is a K-anoid an ◦ Demazure subgroup of G ×M(k) M(K). Let G = spec(A) be the K - Demazure group scheme attached to H ×M(k) M(K), i.e H ×M(k) M(K) = (it is well dened by 4.4). Then is -stable. spec(\A)η A Gal(K/k)

13 Proof. Let σ ∈ Gal(K/k). On one hand, we have σ(H ×M(k) M(K)) = , so . On the other hand, we have H ×M(k) M(K) σ(spec(\A)η) = spec(\A)η . So we have . Thus by spec(\σ(A))η = σ(spec(\A)η) spec(\σ(A))η = spec(\A)η 4.4, we have σ(A) = A.

4.2 Filtrations of rational potentially Demazure k-anoid groups

We now introduce the ltration of potentially Demazure k-anoid groups. We start by a simple case.

4.2.1 The split rational case

Assume G is split and let x be a Demazure point in Gan. Let G be the ◦ unique (see 4.4) k -Demazure group scheme such that H := Hol(x) = Gb η. Let T be a maximal k◦-split torus of G and Φ be the corresponding set of roots. Let B be a Borel subgroup such that T is a Levi subgroup of B. Let Φ−, Φ+ be the corresponding sets of negative and positive roots. For each ◦ α ∈ Φ, we have a canonical k -root subgroup Uα ⊂ G. Choose an ordering on Φ−, Φ+, then the multiplication morphism of k◦-schemes Y Y Uα ×k◦ T ×k◦ Uα → G (3) α∈Φ− α∈Φ+ is an open immersion. Its image, which does not depend on the choice of the ordering, is denoted Ω and is called the grosse cellule of G. Taking generic bers, we obtain similar objects for G. The objects

T := T ×k◦ k

Uα := Uα ×k◦ k

B := B ×k◦ k are respectively a maximal split torus, a root subgroup, and a Borel subgroup of G = G×k◦ k. We can identify canonically Φ with the set of roots associated to G, T . Moreover (3) induces an open immersion Y Y Uα ×k T ×k Uα → G α∈Φ− α∈Φ+

whose image, independent of the ordering, is denoted Ω and is called the grosse cellule of G. We can identify Ω and Ω ×k◦ k. The grosse cellule Ω is ane and the open immersion Ω → G corresponds to an injective morphism of Hopf algebras from Hopf(G) to Hopf(Ω) (see [1, line 24 page 103]. We are going to construct k-anoid subgroups Hr of Hol(x) satisfying an that Shi(Hr) ∈ G is a singleton. So Shi(Hr) will appear as a function on

14 Hopf(G), the Hopf algebra of G. This leads us to study rstly Hopf algebras of various ane group schemes. The torus T is split so it is isomorphic to ◦ s (Gm/k ) for some integer s. Fix an isomorphism ◦ T ' spec(k [X1,...,Xs,Y1,...,Ys]/(XiYi = 1 for 1 ≤ i ≤ s)). Fix an integral Chevalley basis of Lie(G, k◦), it induces, for each root α ∈ Φ, ◦ ◦ a k -isomorphism Uα ' Gadd, where Gadd is the additive group over k . ◦ Thus we have xed an isomorphism Uα ' spec(k [Zα]), i.e. we have xed an ◦ isomorphism Hopf(Uα) ' k [Zα], for any root α. Let r ∈ Z≥0, and consider ◦ the r-th congruence k -group scheme Γr(G) (see 1.1). By [20] we have an open immersion

Y Y Γr(Uα) ×k◦ Γr(T) ×k◦ Γr(Uα) → Γr(G), (4) α∈Φ− α∈Φ+

its image does not depend on the ordering and is Γr(Ω).

Denition/Proposition 4.6. Using the process given in 2.2, let Hr be , the generic ber of the formal completion of along its special Γ\r(G)η Γr(G) ber. We have

1. is a -anoid subgroup of Γ\r(G)η k H 2. Its Shilov boundary is equal to . Shi(Hr) | |Hopf(Γr(G)) 3. is the holomorphically convex envelope of . Hr | |Hopf(Γr(G))

Proof. Since for r ≥ 0, Γr(G) is smooth with irreducible special ber, this is a direct consequence of Corollary 2.3.

In order to give an explicit description of Shi(Hr), we need to study the Hopf algebra of Γr(Ω). We start by studying the Hopf algebra of Ω. Since Y Y Ω = Uα ×k T ×k Uα, α∈Φ− α∈Φ+ we obtain O O Hopf(Ω) = Hopf(Uα) ⊗k Hopf(T ) ⊗k Hopf(Uα). α∈Φ− α∈Φ+

The torus T is egal to T ×k◦ k. The previously xed isomorphism ◦ T ' spec(k [X1,...,Xs,Y1,...,Ys]/(XiYi = 1 for 1 ≤ i ≤ s)) induces a similar isomorphism over k for T . The set1

1Note that the condition (k 6= 0 ⇒ l = 0) is equivalent to the condition (l 6= 0 ⇒ k = 0). So it is a symmetric condition. Remark also that k denote a eld and also an integer, this is not a problem.

15 k l {X Y | k, l ∈ N; k 6= 0 ⇒ l = 0} is a basis of the k-vector space k[X,Y ]/XY − 1. We need an other basis of Hopf(Gm), centered at unity. The set k l {(X − 1) (Y − 1) | k, l ∈ N; k 6= 0 ⇒ l = 0} is a basis of the k-vector space k[X,Y ]/XY −1. The previously xed isomor- ◦ phisms {Hopf(Uα) ' k [Zα]}α∈Φ induce isomorphisms Hopf(Uα) ' k[Zα]. mα We identify the corresponding objects. The set {Zα | mα ∈ Z≥0} is a basis of the k-vector space Hopf(Uα). These considerations allow us to x an isomorphism

  s !   O O O Hopf(Ω) '  k[Zα] ⊗k k[Xi,Yi]/XiYi − 1 ⊗k  k[Zα]) α∈Φ− i=1 α∈Φ+

' k[X1,...,Xs,Y1,...,Ys, {Zα}α∈Φ]/(XiYi − 1, 1 ≤ i ≤ s) Moreover the set s Y ki li Y mα { (Xi−1) (Yi−1) Zα | ki, li, mα ∈ N; ∀1 ≤ i ≤ s, ki 6= 0 ⇒ li = 0} i=1 α∈Φ is a k-basis of the k-vector space Hopf(Ω). So given f ∈ Hopf(Ω), f can be written uniquely as

s X Y ki li Y mα . f = ak1...ksl1...ls,mαα∈Φ (Xi − 1) (Yi − 1) Zα k1,...,ks,l1,...,ls,mαα∈Φ i=1 α∈Φ In order to simplify the notation, we denote a parameter k1, . . . , ks, l1, . . . , ls, mα, α ∈ Φ with ki, li, mα ∈ N; ki 6= 0 ⇒ li = 0 by the symbol u, and U the set of all such parameters. Moreover, the element s Y ki li Y mα is denoted by the symbol u. (Xi−1) (Yi−1) Zα ((X − 1)(Y − 1)Z) i=1 α∈Φ With these notations, an element f ∈ Hopf(Ω) is written uniquely as

X u f = au ((X − 1)(Y − 1)Z) . u∈U Since Y Y Γr(Ω) = Γr(Uα) ×k◦ Γr(T) ×k◦ Γr(Uα), α∈Φ− α∈Φ+ we obtain O O Hopf(Γr(Ω)) = Hopf(Γr(Uα)) ⊗k◦ Hopf(Γr(T)) ⊗k◦ Hopf(Γr(Uα)) α∈Φ− . α∈Φ+

16 Using 1.1 , we have

◦ −r Hopf(Γr(Uα)) = k [πk Zα] and

◦ −r −r −r −r Hopf(Γr(T)) = k [πk (X1 − 1), . . . , πk (Xs − 1), πk (Y1 − 1), . . . , πk (Ys − 1)] ⊂ Hopf(T ). Finally, we get the formula

◦ −r −r −r Hopf(Γr(Ω)) = k [{πk Zα}α∈Φ, {πk (Xi − 1), πk (Yi − 1)}1≤i≤s] ⊂ Hopf(Ω).

Proposition 4.7. With the same notations as in 4.6, Shi(Hr) is a norm on an an Hopf(G) inside G . The point Shi(Hr) belongs to Ω and corresponds to a norm on Hopf(Ω). The norm Shi(Hr) factorizes trough the canonical injective morphism of Hopf algebras Hopf(G) → Hopf(Ω). The corresponding norm on Hopf(Ω) is explicitely given, using the notations introduced previously, by the following formula

Hopf(Ω) −→ R≥0 X u −r|u| au ((X − 1)(Y − 1)Z) 7→ max |au|e u∈U u∈U X where |u| is egal to k1 + ... + ks + l1 + ... + ls + mα. α∈Φ

Proof. By 2.2, is the unique point such that the re- Shi(Γ\r(G)η) ∈ Γ\r(G)η ˜ duction map sends to the generic point of Γr(G) ×k◦ k. Let x denote the ˜ ˜ generic point of Γr(G) ×k◦ k. The closure x of x is egal to Γr(G) ×k◦ k. ˜ ˜ The special ber Γr(Ω) ×k◦ k is open in Γr(G) ×k◦ k (and non empty), con- ˜ ˜ sequently x is contained in Γr(Ω) ×k◦ k. Indeed, assume x 6∈ Γr(Ω) ×k◦ k, ˜ ˜ then x is contained in the closed subset Γr(G) ×k◦ k \ Γr(Ω) ×k◦ k, and so ˜ ˜ x 6= Γr(G) ×k◦ k, this is a contradiction. So x is contained in Γr(Ω) ×k◦ k. The commutative diagram πΩ ˜ Γ\r(Ω)η / Γr(Ω) ×k◦ k 3 x

πG ˜ Γ\r(G)η / Γr(G) ×k◦ k whose vertical arrows are inclusions shows that −1 Shi(Γ\r(G)η) = πΩ (x) ∈ . So . By 2.2 and 2.3, is the Γ\r(Ω)η Shi((Γ\r(G)η) = Shi(Γ\r(Ω)η) Shi(Γ\r(Ω)η) norm on given as follows. | |Hopf(Γr(Ω)) Hopf(Ω)

17 X u For f ∈ Hopf(Ω), write f = au ((X − 1)(Y − 1)Z) . u∈U and |f|Hopf(Γr(Ω)) = inf{|λ| | λ ∈ k f ∈ λ(Hopf(Γr(Ω) ⊗ 1)} and −r |u| ◦ = inf{|λ| | λ ∈ k au ∈ λ(πk ) k ∀u ∈ U} and −r |u| = inf{|λ| | λ ∈ k |au| ≤ |λ||πk | ∀u ∈ U} and r |u| = inf{|λ| | λ ∈ k |au||πk| ≤ |λ| ∀u ∈ U} r |u| = max|au||πk| u∈U −r|u| = max|au|e u∈U

This ends the proof.

4.2.2 The general case ¯ Lemma 4.8. Let k be an algebraic closure of k. Let r ∈ Q≥0. Let H be a rational potentially k-anoid Demazure subgroup of Gan. There exists a nite Galois extension K/k in k¯ such that: • r ∈ ord(K) an • H ×M(k) M(K) is a K−anoid Demazure subgroup of G ×M(k) M(K).

Proof. By denition, there exists a nite extension L/k, such that H ×M(k) an M(L) is a L-anoid Demazure subgroup of G ×M(k) M(L). There exists a nite extension E/k, such that r ∈ ord(E). Let K be a nite Galois extension of k such that L, E ⊂ K. Then K satises the required properties.

Denition 4.9. Let H be a rational potentially k-anoid Demazure sub- an group of G and r ∈ Q≥0. Let K be a nite extension as in the previous ◦ lemma. Let G be the Demazure K -group scheme such that H×M(k)M(K) = G . Then, we put H = pr (Γ \(G) ), the projection of the generic b η r K/k e(K,k)r η ber of the formal completion along the special ber of the e(K, k)r-th congruence subgroup of G. We have the following proposition.

Proposition 4.10. We have

1. In denition 4.9, Hr is independant of the choice of K.

2. H0 = H

an 3. Hr is a k-anoid subgroup of G , it is a k-anoid subgroup of H.

18 Proof. We rst prove (3), then (1) and (2). By 4.5, A is Gal(K/k)-stable.

So by 1.2, Γe(K,k)r(G) is Gal(K/k)-stable. Thus by 2.6, Γe(K,k\)r(G)η is -stable in an . Consequently by 2.5, Gal(K/k) G ×M(k)M(K) prK/k(Γe(K,k\)r(G)) is a -anoid domain in an. Moreover by 2.7, is a - k G prK/k(Γe(\K,k)r(G)) k anoid group. This nishes . Let us now show that (3) prK/k(Γe(\K,k)r(G)) does not depend on the choice of the extension. So let K and K0 be two extensions satisfying the conditions of Lemma 4.8. Let G /K◦ and 0 0◦ G /K be the Demazure group schemes such that H ×M(k) M(K) = Gb η 0 0 00 and H ×M(k) M(K ) = Gcη. Let K be a nite Galois extension such that 0 00 an 00 K,K ⊂ K . The following equalities hold (in (G ×M(k) M(K )))

00 00 00 H ×M(k) M(K ) = (H ×M(k) M(K)) ×M(K) M(K ) = Gb η ×M(K) M(K ) 0 00 0 00 = (H ×M(k) M(K )) ×M(K0) M(K ) = Gcη ×M(K0) M(K ) and

00 00◦ Gb η ×M(K) M(K ) = (G ×\K◦ K )η 0 00 0 00◦ Gcη ×M(K0) M(K ) = (G ×\K0◦ K )η.

00◦ 0 00◦ We thus get an equality (G ×\K◦ K )η = (G ×\K0◦ K )η. Using 4.4, 00◦ 0 00◦ 00 we deduce an equality G ×K◦ K = G ×K0◦ K , let G denote this 00◦ 00◦ K -Demazure-group scheme. By Proposition 1.3, Γe(K00,k)r(G ×K◦ K ) = 00◦ 00 00 Γ (G) × ◦ K . So Γ \00 (G ) = Γ \(G) × M(K ). We e(K,k)r K e(K ,k)r η e(K,k)r η M(K) deduce that

00 pr 00 (Γ \00 (G ) ) = Γ \(G) . K /K e(K ,k)r η e(K,k)r η

00 00 However, pr 00 (Γ \00 (G ) ) = pr (pr 00 (Γ \00 (G ) )). K /k e(K ,k)r η K/k K /K e(K ,k)r η 00 So pr 00 (Γ \00 (G ) ) = pr (Γ \(G) ). By symmetry, we get K /k e(K ,k)r η K/k e(K,k)r η 00 0 pr 00 (Γ \00 (G ) ) = pr 0 (Γ \0 (G ) ). So pr (Γ \(G) ) = K /k e(K ,k)r η K /k e(K ,k)r η K/k e(K,k)r η 0 pr 0 (Γ \0 (G ) ), and (1) is proved. Let K/k be a nite Galois exten- K /k e(K ,k)r η ◦ sion such that H ×M(k) M(K) is a Demazure K -anoid group scheme Gb η. Then

H0 = prK/k(Γ\0(G)η)

= prK/k(Gb η) = H So (2) is proved and the proof ends here.

19 We now have a fundamental result.

Proposition 4.11. Let , , and as in Denition 4.9. Let K H r K/k G Ae(K,k)r ◦ be the Hopf K -algebra of Γe(K,k)r(G). 1. The Shilov boundary of is reduced to a point . Hr σHr 2. The map

| | K :Hopf(G ×k K) → ≥0 Ae(K,k)r R K f 7→ inf {|λ| | f ∈ λ(Ae(K,k)r ⊗ 1) ⊂ Hopf(G ×k K)}. λ∈K×

is a norm on Hopf(G ×k K), moreover

| | K |Hopf(G)= Shi(Hr). Ae(K,k)r

3. The k-anoid algebra of Hr is the completion of Hopf(G) relatively to the norm | | K |Hopf(G). Ae(K,k)r 4. is the holomorphically convex envelope of . Hr σHr 5. and . Shi(Hol(σHr )) = σHr Hol(Shi(Hr)) = Hr

Proof. 1. The Shilov boundary of Γ \(G) is a singleton by the split e(K,k)r η rational case (see 4.6). The Shilov boundary of Γ \(G) surjects e(K,k)r η onto the Shilov boundary of Hr by [1, 1.4.5 proof], and so the Shilov boundary of Hr is a singleton.

2. By 4.7 the map | | K is a norm on Hopf(G×k K). By 2.3, | | K Ae(K,k)r Ae(K,k)r is the Shilov boundary of Γ \(G) . The Shilov boundary of H is e(K,k)r η r egal to pr (Shi(Γ \(G) )) and pr is realized by the restriction K/k e(K,k)r η K/k map of functions from Hopf(G ×k K) to Hopf(G). This explains both assertions.

3. We have already prove it in the split rational case. We adapt the argument given by [14, proof of 2.4 (ii)] to descent this result. Let AHr be the -anoid algebra of . Since is reduced, the norm of k Hr AHr AHr coincides with the spectral norm [1, 2.1.4] of , and so it is egal to AHr | |σ | | since σ = Shi(H ). Let Hopf(G) Hr be the completion of σHr Hr r Hopf(G) relatively to the norm | | . The injective morphism of k- σHr algebras (corresponding to an), extends to i : Hopf(G) → AHr Hr ⊂ G | |σ an isometric embedding i : Hopf(G) Hr → A . Let A Hr Hr×M(k)M(K)

20 be the K-anoid algebra of Hr ×M(k) M(K). By denition Hr ×M(k) M(K) is egal to Γ \(G) (G is the K◦-Demazure group scheme e(K,k)r η used to dene Hr). So, by the split rational case,

| |σ A = Hopf(G × K) Hr×M(k)M(K) . Hr×M(k)M(K) k

In particular Hopf(G × K) is dense in A = A . In k Hr×M(k)M(K) Hr other words is dense in ˆ . It follows that ˆ Hopf(G)⊗k K AHr ⊗kK i⊗kK : | |σ Hr ˆ ˆ is an isomorphism of Banach algebras, Hopf(G) ⊗kK → AHr ⊗kK | |σ hence Hr by [1, Lemma A.5]. Hopf(G) = AHr

4. By the previous assertion, the k-anoid algebra Ar of Hr is the completion of Hopf(G) relatively to the norm | | . Let i denote the σHr natural corresponding inclusion . The inclusion Hopf(G) → AHr Hr = M(A ) ⊂ Gan is given by σHr

ι : M(A ) → Gan σHr | |x 7→ | |x ◦ i .

Since M(A ) is the set of all multiplicative bounded seminorms on σHr A , ι(M(A )) is contained in the holomorphically convex enve- σHr σHr lope of . Reciprocally, let , is a multiplicative semi- σHr x ∈ Hol(σHr ) x norm Hopf(G) → such that |f| ≤|f| ∀f ∈ Hopf(G). Since A R≥0 x σHr r is the completion of Hopf(G), x induces a multiplicative seminorm on bounded by . This ends the proof. Ar σHr 5. These are obvious consequences of the previous assertions.

Remark 4.12. If r > s ∈ Q≥0, Hr ⊂ Hs. 6= Proof. This is an easy consequence of the denition taking the K-points for any suciently big extension K/k. Proposition 4.13. The map an, is continous. Q≥0 → G r 7→ σHr Proof. One can adapt the proof of 5.7.

21 4.3 Filtrations of Lie algebra

Let g be the k-Lie algebra of G it is a a k-scheme. In this section we an dene k-anoid groups hr ⊂ g , for any rational potentially Demazure k- anoid subgroup H and any r ∈ Q≥0. So let H be a rational potentially an Demazure k-anoid subgroup of G and r ∈ Q≥0. In 4.9, we have dened an analytic group Hr. In order to dene Hr, we have choosen a certain extension K/k (see 4.9). Let G be the K◦-Demazure group scheme such that H ×M(k) M(K) = Gb η. Let Γe(K,k)r(G) be the e(K, k)r-th congruence ◦ subgroup of Γ. Let Lie(Γe(K,k)r(G)) be its K -Lie algebra, it is in particular a K◦-group scheme, it is a smooth group scheme over K◦, its special ber is irreducible with reduced ˜ -algebra. We denote by the canonical map K prK/k an an g ×M(k) M(K) → g . Denition 4.14. With the previously introduced notations, we put h = pr Lie(Γ\ (G)) , r K/k e(K,k)r η the projection of the generic ber of the formal completion along its special

ber of Lie(Γe(K,k)r(G)).

Proposition 4.15. With the previously introduced notations, hr is a k- anoid domain of gan, it is a k-anoid group, moreover

1. The Shilov boundary of is reduced to a point and is egal to hr σhr | | | . Hopf(Lie(Γe(K,k)r(G))) Hopf(g) 2. Hol(σhr ) = hr

3. The k-anoid algebra of hr is the completion of Hopf(g) relatively to the norm | | | . Hopf(Lie(Γe(K,k)r(G))) Hopf(g) Proof. The proof is similar to that of Proposition 4.11.

5 Filtrations associated to points in the Bruhat- Tits building

5.1 Denitions and properties of Gx,r Let G be a connected reductive k-group scheme, let x ∈ BT(G, k) be a rational point in the reduced Bruhat-Tits building of G and let r be a positive rational number.

Proposition 5.1. There exists a nite Galois extension K/k such that

1. ιK/k(x) is a special point in BT(G, K), 2. G is split over K

22 3. r is in ord(K×) Proof. Since x is rational, by Denition 3.1 there is a nite Galois extension K0/k such that (1) and (2) are satied. It is obvious that there exists a nite Galois extension K00/k such that (3) is satised. Let K be a Galois extension of k containing K0 and K00. The eld K satises the three properties.

Let K be an extension of k as in Proposition 5.1. Let G = Gx be the canonical K◦-Demazure group scheme attached to x ∈ BT(G, K) characterized by the fact that its K◦-points form the stabilizer of a preimage of ιK/k(x) in the enlarged Bruhat-Tits building (see section 3). In these con- ◦ ditions, the K -Hopf algebra of G is Gal(K/k)-stable in A ⊗k K. As usual, let denote the projection an an. prK/k G ×M(k) M(K) → G Denition/Proposition 5.2. Let be , it is a rational po- • Gx prK/k(Gb η) tentially Demazure k-anoid subgroup of Gan equal to the k-anoid group dened and considered in [14, theorem 2.1]. It is characterized by the fact that for any non archimedean extension k0/k,

0 0 Gx(k ) = stab(g ιk0/k(x)) ⊂ G(k ). an • Using 4.9, we obtain a k-anoid subgroup (Gx)r of G , it is equal to pr (Γ \(G) ). We write G instead of (G ) . K/k e(K,k)r η x,r x r Proof. The fact that is a rational potentially Demazure -anoid prK/k(Gb η) k an subgroup of G equal to the k-anoid group Gx dened and considered in [14, denition 2.1] is explained during the proof of [14, 2.1]. The last part of the proposition is a direct consequence of 4.9.

The previous section 4 gives us the following properties of Gx,r. Proposition 5.3. We have:

an 1. Gx,r is a k-anoid subgroup of G .

2. The Shilov boundary of Gx,r is reduced to a point that we denote θ(x, r). The point θ(x, r) ∈ Gan is a norm on Hopf(G) egal to | | | . Hopf(Γe(K,k)r(G)) Hopf(G)

3. Gx,r is the holomorphically convex envelope of θ(x, r).

4. If r = 0, Gx,r = Gx where Gx is the k-analytic group dened in [14, 2.1].

5. The k-anoid algebra of Gx,r is the completion of Hopf(G) relatively to the norm | | | , i.e. the completion of Hopf(G) Hopf(Γe(K,k)r(G)) Hopf(G) relatively to θ(x, r). Proof. These are corollaries of 4.10 and 4.11 .

23 5.2 Denitions and properties of the map θ Proposition 5.4. Let x be a rational point in the reduced Bruhat-Tits building of G, let r be a positive rational number and let g ∈ G(k), then

−1 1. Gg.x,r = gGx,rg 2. θ(g.x, r) = gθ(x, r)g−1

Proof. By 5.3, the two assertions are equivalent. Let us prove the rst assertion. The case r = 0 is proved in [14, 2.5]. For r ≥ 0 rational, choose an ◦ extension K/k as in the denition of Gx,r and let G be the K -Demazure group scheme attached to the special point iK/k(x) ∈ BT(G, K). The point ◦ g.iK/k(x) ∈ BT(G, K) is special and the K -Demazure group scheme at- −1 tached to g.iK/k(x) ∈ BT(G, K) is Gg.x = gGxg . We then deduce the equality

Gg.x,r = (Gg.x)r = prK/k Γe(K,k\)r(Gg.x)η

−1 = prK/k Γe(K,k)\r(gGxg )η

−1 = prK/k (gΓe(K,k\)r(Gx)g )η

−1 = prK/k g Γe(K,k\)r(Gx)η g = g pr Γ \(G ) g−1 K/k e(K,k)r x η −1 = gGx,rg .

We now introduce a natural map.

Denition 5.5. Let Q≥0 denote the semi-eld of positive real rational numbers. Let BT(G, k) be the set of rational points of the reduced Bruhat-Tits building of G. Let

an θ : BT(G, k) × Q≥0 → G be the map sending (x, r) to the Shilov boundary of the previously dened k-anoid group Gx,r. Remark 5.6. Let k0/k be a nite extension of k. Let x ∈ BT(G, k0), let 0 0 r ∈ Q≥0, we dene a k -anoid group as follows. Let K/k be a nite Galois extension such that G is split over K, ιK/k0 (x) is special in BT(G, K) and r ∈

24 ◦ ord(K). Let G be the K -Demazure group scheme attached to ιK/k0 (x). We 0 0 0 an put G = pr 0 (Γ \(G) ), this is a k -anoid subgroup of (G× k ) . x,r K/k e(K,k)r η k 0 0 an Let θk0 be the corresponding map BT(G, k ) × Q≥0 → (G ×k k ) , sending to 0 . If 0 comes from , i.e. for (x, r) Shi(Gx,r) x ∈ BT(G, k ) k x = ιk0/k(x) a point , we also denote naturally the 0-anoid group 0 x ∈ BT(G, k) k Gx,r by G . Remark that we have used in the denition the ramication ιk0/k(x),r index e(K, k) and not e(K, k0), this reects the fact that k is a "reference" object in this work, indeed we work with the valuation ord which is normalized relatively to k. These choices allow us to state the following proposition.

an Proposition 5.7. 1. The map θ : BT(G, k) × Q≥0 → G is G(k)- equivariant relatively to the actions:

• g.(x, r) = (g.x, r) for all (x, r) ∈ BT(G, k) × Q≥0 • g.x = gxg−1 for all x ∈ Gan

2. For any nite extension k0/k, the diagram θ 0 k0 0 an BT(G, k ) × Q≥0 / (G ×k k ) O pr 0 ιk0/k×Id k /k

θ an BT(G, k) × Q≥0 / G

is commutative, where the map θk0 is dened in the previous remark 5.6. Moreover, for any rational point x ∈ BT(G, k) and any r ∈ Q≥0, 0 an 0 the equality of k -anoid subgroups of G ×M(k) M(k ) holds:

G = G × M(k0). ιk0/k(x),r x,r M(k)

3. For any nite extension k0/k,

G (k0) ∩ G(k) = G (k) ιk0/k(x),r x,r

an 4. The map θ : BT(G, k) × Q≥0 → G is continuous and injective. Proof. 1. We have to show that g.θ(x, r) = θ(g.(x, r)). This is a direct consequence of 5.4, indeed

θ(g.(x, r)) = θ(g.x, r) = gθ(x, r)g−1 = g.θ(x, r)

2. We use the notation of Remark 5.6. Let K/k be the extension used to dene as , we can assume that 0 . Gx,r Gx,r = prK/k(Γe(K,k\)r(G)η) k ⊂ K We have 0 , thus 0 . By Gx,r = prK/k0 (Γe(K,k\)r(G)η) Gx,r = prk0/k(Gx,r) denition θ(x, r) = Shi(Gx,r), by the previous sentence and properties

25 of Shilov boundaries, this is egal to 0 . The commu- prk0/k(Shi(Gx,r)) tativity of the diagram follows. We have pr−1 (G ) = G K/k x,r ιK/k(x),r by denition of Gx,r and since K/k is a Galois extension. We thus get pr−1 (pr−1 (G )) = G . We also have pr−1 (G0 ) = K/k0 k0/k x,r ιK/k(x),r K/k0 x,r G by denition of G0 and since K/k0 is a Galois extension. iK/k(x),r x,r We thus obtain

−1 −1 −1 0 . prK/k0 (prk0/k(Gx,r)) = prK/k0 (Gx,r)

This implies −1 0 , since is surjective. We thus prk0/k(Gx,r) = Gx,r prK/k0 have 0 0 . Gx,r = Gx,r ×M(k) M(k ) 3. This is a direct consequence of the previous assertion and the fact that an Gx,r is a k-anoid domain of G . 4. We follow [14, Proposition 2.6 (ii) and Proposition 2.8 (iii)] for the continuity. Let k0/k be a nite extension such that G is split over k0. Since the maps 0 and ιk0/k × Id : BT(G, k) × Q≥0 → BT(G, k ) × Q≥0 0 an an are continous, it is enough to show that prk0/k :(G ×k k ) → G 0 0 an BT(G, k ) × Q≥0 → (G ×k k ) is continous. In other words, we can assume G is split over k. So assume G is split over k and choose a special point x ∈ BT(G, k). Let G be the k◦-Demazure group scheme attached to x. Let T be a maximal split k◦-torus of G and let B be a k◦-Borel such that T is a Levi subgroup of B. Let Φ, Φ−, Φ+ be the corresponding set of roots. Choose a Chevalley basis of the k◦-Lie algebra of G. We are in a similar situation as in 4.2.1, and we use the same notations as 4.2.1 in the following. We can use x to identify the ∗ apartement A(T, k) with V (T ) = HomAb(X (T ), R). It is enough to an show that the restriction map A(T, k) × Q≥0 → (G) is continous. We claim that for any rational point y in A(T, k) = V (T ) and any an r ∈ Q≥0, the point θ(y, r) belong to Ω and corresponds to the norm

Hopf(Ω) → R≥0

X u −r|u| Y mα au((X − 1)(Y − 1)Z) 7→ max|au|e e u∈U u∈U α∈Φ

∗ where < ., . > is the map V (T )×X (T ) → R , (y, α) 7→< y, α >= y(α). This claim implies the continuity since this formula is continous in2 the variable (y, r) . We now prove the claim following closely [14, Proposition 2.6 (ii)]. Since y is a rational point, there exists a nite

2Be carefull that what is denoted by u in [14] is here denoted by y and u here is a parameter for a basis of Hopf(Ω) (see 4.2.1)

26 extension such that with . Let K be , K/k y = t.x t ∈ T (K) Uα Uα ×k K K K Ω be Ω ×k K and T be T ×k K. For any t ∈ T (K), and any root , the element normalizes the root group K and conjuguation α ∈ Φ t Uα by induces an automorphism of K which is just the homothety of t Uα ratio ×. If we read it through the isomorphisms K , α(t) ∈ K Gadd → Uα we have a commutative diagram K K spec(Hopf(T )[{Zα}α∈Φ]) / Ω

τ int(t) K K spec(Hopf(T )[{Zα}α∈Φ]) / Ω K ∗ K where τ is induced by the Hopf(T )-automorphism τ of Hopf(T )[{Zα}α∈Φ mapping Zα to α(t)Zα for any α ∈ Φ. It follows that, over K, θ(t.x, r) = −1 an tθ(x, r)t is the point of (G ×k K) dened by the multiplicative norm on K X u Hopf(Ω ) mapping f = au((X − 1)(Y − 1)Zα) to u∈U !

∗ X Y mα u |τ (f)|θ(x,r) = au α(t) ((X − 1)(Y − 1)Zα) θ(x,r) u∈U α∈Φ

−r|u| Y mα = max|au|e |α(t)| u∈U α∈Φ

−r|u| Y mα = max|au|e e u∈U α∈Φ

Since G is assumed to be split over k and by properties of Shilov boundaries, we get the claim by restriction from K to k. This ends the proof of the continuity.

Let us explain the injectivity. Let (x1, r1) and (x2, r2) be in BT(G, k) × Q≥0 such that θ(x1, r1) = θ(x2, r2). Let us rst explain, by the absurd, that necessarily we have r1 = r2. So assume by the absurd that there exists (x1, r1) and (x2, r2) with r1 6= r2 such that θ(x1, r1) = θ(x2, r2). Assume for example r1 > r2. Since θ(x1, r1) = θ(x2, r2), taking holomorphically convexe envelope, we have by 5.3. Since and are Gx1,r1 = Gx2,r2 x1 x2 rational points, there exists a nite extension K/k such that ∃g ∈ G(K) such that g.ιK/k(x1) = ιK/k(x2). By 5.4, we thus get gG g−1 = G = G . ιK/k(x1),r1 g.ιK/k(x1),r1 ιK/k(x2),r1 By 4.12, we thus obtain

−1 −1 gGι (x ),r g = gGι (x ),r g ⊂ Gι (x ),r . K/k 2 2 K/k 1 1 6= K/k 2 2

We have thus deduced the existence of a K-anoid group Gabsurd := G satisfying iK/k(x2),r2

27 −1 gGabsurdg ⊂ Gabsurd, 6=

this is absurd. So we have proved that θ(x1, r1) = θ(x2, r2) ⇒ r1 = r2. Let us now prove that we also necessarily have x1 = x2. Assume rst G is split over k. Let (x1, r) and (x2, r) be in BT(G, k) × Q≥0. We know, by properties of Bruhat-Tits buildings, that there exists an apartement A(T, k) such that x1 and x2 belongs to this apartement. The choice of a special point in A(T, k) induces, as in the proof of the continuity, an explicit map an an A(T, k) × Q≥0 → G which factorizes through Ω . Let (y, r) ∈ A(T, k) × Q≥0, the explicit formula for θ(y, r)

Hopf(Ω) → R≥0

X u −r|u| Y mα au((X − 1)(Y − 1)Z) 7→ max|au|e e u∈U u∈U α∈Φ claimed and proved before (during the proof of the continuity) shows that

θ(x1, r) = θ(x2, r) ⇒ x1 = x2. Indeed, the formula give us

−r θ(x1, r)(Zα) = e e for any root α −r θ(x2, r)(Zα) = e e for any root α, consequently,

θ(x1, r) = θ(x2, r) ⇒< x1, α >=< x2, α > for all roots α

⇒ x1 = x2 as required. In general, if G is not split, we prove injectivity using a nite Galois extension k0/k such that G is split over k0 and using the diagram θ 0 k0 0 an BT(G, k ) × Q≥0 / (G ×k k ) O pr 0 ιk0/k×Id k /k

θ an BT(G, k) × Q≥0 / G . By the split case, the map θk0 is injective. The map ιk0/k ×Id is injective. So it is enough to show the restriction of 0 an an to the prk0/k :(G ×k k ) → G image of θk0 ◦ ιk0/k × Id is injective. This is a consequence of the fact that is 0 -stable for any . θk0 (ιk0/k(x), r) Gal(k /k) (x, r) ∈ BT(G, k) × Q≥0

5.3 A cone

an We have dened a continous and injective map θ : BT(G, k) × Q≥0 → G . By completion, we get a continous and injective map θ : BT(G, k) × R≥0 →

28 an an G . For all x ∈ BT(G, k), we put θ(x, +∞) = eG, where eG ∈ G is the neutral element. Let R≥0 be R≥0 ∪ +∞. an Denition/Proposition 5.8. The set θ(BT(G, k), R≥0) ⊂ G is a topological cone in Gan. Its base is the reduced Bruhat-Tits building and its vertex is the neutral element. If p = θ(x, r) ∈ Gan is in this cone, the depth of p is by denition the number r. The subset θ(BT(G, k), Q≥0) ∪ eG is called the rational cone.

Proof. For any x ∈ BT(G, k), the point θ(x, r) approaches eG as r approaches +∞. This makes clear 5.8.

5.4 Comparison with Moy-Prasad ltrations in the tame case

Let G be a connected reductive k-group scheme that split over a tamely ramied extension. Recall that MP denote the normalized Moy-Prasad G(k)x,r ltration (see section 3). The following facts

• [20, 8.8] if G is split and x is special, then Moy-Prasad ltrations are obtained by taking set-theoretic congruence subgroups of the integral points of the attached integral Demazure group Gx; • Fact 3.5 of this paper. Moy-Prasad ltrations are compatible relatively to eld extensions in the tame case; together with the denitions of Gx,r imply the following proposition.

Proposition 5.9. Assume we can choose the extension K/k tamely ramied in order to dene (see Denitions 5.2 and 4.9), then MP . Gx,r Gx,r(k) = G(k)x,r Proof. Let K/k be a nite tamely ramied extension such that we can write G = pr Γ \(G) . The following equality hold. x,r K/k e(K,k)r η

Gx,r(k) = Gx,r(K) ∩ G(k) = Γ \(G) (K) ∩ G(k) e(K,k)r η ◦ = Γe(K,k)(K ) ∩ G(k) ◦ ◦ e(K,k)r ◦ = ker(G(K ) → G(K /πK K )) ∩ G(k) ◦ ◦ r ◦ = ker(G(K ) → G(K /πkK ) ∩ G(k) MP = G(K)x,r ∩ G(k) MP = G(k)x,r

This ends the proof.

29 5.5 Filtrations of the Lie algebra By section 4.3 and Denition 5.2, we obtain analytic ltrations of the Lie algebra gx,r := (gx)r, for each x ∈ BT(G, k) and each r ∈ Q≥0. We recall the formal denition in the following denition.

Denition 5.10. Let x ∈ BT(G, k) and r ∈ Q≥0. Let K/k as in 5.1, then g = pr Lie(Γ\ (G)) where G is the K◦-Demazure group scheme x,r K/k e(K,k)r η attached to ιK/k(x) ∈ BT(G, K).

5.6 Examples and pictures In this section we give some examples and pictures of the previously introduced objects.

5.6.1 The split torus of rank one

Let R be a commutative ring. The R-algebra R[X,Y ]/(XY − 1) is naturally a Hopf R-algebra. Recall that its augmentation map is

R[X,Y ]/XY − 1 → R X 7→ 1 Y 7→ 1

and its kernel is generated by X−1 and Y −1. Now let A be k[X,Y ]/XY − 1 (k is our xed p-adic eld). Let G be spec(A), it is a split torus of rank one over k. The morphism of k-algebra k[X] → k[X,Y ]/XY − 1 induces a morphism of ane scheme 1. It also induces an inclusion G → Ak an 1 an, it is injective and an 1 an . The reduced Bruhat- G ⊂ (Ak) G = (Ak) \ 0 Tits building of G is a singleton {x}. The point x is special and G is split over k. The grosse cellule of G is G. The k◦-Demazure group scheme attached to x is G = spec(k◦[X,Y ]/XY − 1). Let make explicit the denition of the k-anoid group Gx,r for r ≥ 0. If r = 0, Gx,r = Gx,0 = Gb η and Gb η = M(k{X,Y }/XY − 1). Assume now r > 0, we have to choose a nite Galois extension K/k such that r ∈ ord(K). Let G be the K◦-Demazure ◦ group scheme attached to ιK/k(x). It is egal to spec(K [X,Y ]/XY − 1). By denition G is egal to pr Γ \(G) . The K◦-scheme Γ (G) x,r K/k e(K,k)r η e(K,k)r is the e(K, k)r−th congruence subgroup of G. The ring Hopf(Γe(K,k)r(G)) is egal to ◦ −e(K,k)r −e(K,k)r , since the K [πK (X − 1), πK (Y − 1)] ⊂ K[X,Y ]/XY − 1 kernel of the augmentation is generated by X − 1 and Y − 1. The K-anoid group Γ \(G) is the Berkovich spectrum of the K-anoid algebra ob- e(K,k)r η tained completing K[X,Y ]/XY − 1 relatively to the norm | | . Hopf(Γe(K,k)r(G))

30 Writting as X k1 k2 ( is f ∈ K[X,Y ]/XY − 1 ak1k2 (X − 1) (Y − 1) U

(k1,k2)∈U the set of parameter for the basis of K[X,Y ]/XY − 1 centered at unity, see 4.2.1), the norm | | is explicitely given by the map Hopf(Γe(K,k)r(G))

K[X,Y ]/XY − 1 → R≥0 f 7→ ◦ −e(K,k)r −e(K,k)r inf {|λ| | f ∈ λ(K [πK (X − 1), πK (Y − 1)]) ⊂ K[X,Y ]/XY − 1} λ∈K×

k1 k2 ◦ −e(K,k)r −e(K,k)r = inf {|λ| | ak1k2 (X − 1) (Y − 1) ∈ λ(K [πK (X − 1), πK (Y − 1)])∀(k1, k2) ∈ U} λ∈K×

−e(K,k)r(k1+k2) ◦ = inf {|λ| | ak1k2 ∈ λπK K ∀(k1, k2) ∈ U} λ∈K×

−re(K,k)(k1+k2) = inf {|λ| | |ak1k2 | ≤ |λπK | ∀(k1, k2) ∈ U} λ∈K×

−r(k1+k2) = inf {|λ| | |ak1k2 |e ≤ |λ| ∀(k1, k2) ∈ U} λ∈K×

−r(k1+k2) = max |ak1k2 |e . (k1,k2)∈U

Completing K[X,Y ]/XY − 1, we deduce that the K-anoid algebra of Γ \(G) is re(K,k) η

X k1 k2 and −r |u| as { ak1k2 (X − 1) (Y − 1) | ak1k2 ∈ k |ak1k2 |(e ) → 0 |u|→ ∞}

(k1,k2)∈U We denote it as K{er(X − 1), er(Y − 1)}/XY − 1. The Shilov boundary of Γ \(G) is | | . The Shilov boundary θ(x, r) of e(K,k)r η Hopf(Γe(K,k)r(G)) pr (Γ \(G) ) is | | restricted to the k-algebra Hopf(G). K/k e(K,k)r η Hopf(Γe(K,k)r(G)) The point θ(x, r) ∈ Gan is thus egal to the norm on k[X,Y ]/XY − 1 which map X k1 k2 to −r(k1+k2). It corre- ak1k2 (X − 1) (Y − 1) max |ak1k2 |e (k1,k2)∈U (k1,k2)∈U sponds via the embedding an 1 an to the norm usually denoted G → (Ak) \ 0 inside 1 an. | |1,e−r (Ak) We have the picture

31 θ(x, 0) = | |1,1 Reduced Bruhat-Tits Building

θ(x, 1) = | |1,e−1

θ(x, 2) = | |1,e−2

θ(x, 3) = | |1,e−3

θ(x, 4) = | |1,e−4

θ(x, +∞) = | |1,0 ... 1 1 + π3 1 + π2 1 + π2 + π3 1 + π + π2 1 + π δ 1 + π−1

32 giving some points (of course it is not exhaustive) of an inside 1 an. Here G (Ak) δ is an element in (k◦)× \ 1 + k◦◦. The point θ(x, 0) is mapped to the so- called Gauss point, and corresponds to the reduced Bruhat-Tits building. When r ≥ 0 is increasing the point θ(x, r) is getting closer to 1, the neutral an element of G . The holomorphically convex envelope Gx,r of θ(x, r) should be thought as all the points under (attainable by going only down) θ(x, r) and the k-rational points of Gx,r as certain lower extremities. In this situation the cone is the red line, it is homeomorphic to the segment [0, 1] ( Note that r7→e−r [0, +∞] ' [1, 0] ).

5.6.2 A computation of Gx,0 in the case of a wild torus of norm one elements in a quadratic extension

2 In this section k = Q2. The polynomial X − 2 does not have any solution in k. Let πl ∈ k be a root of this polynomial and let l be the eld k(πl) ⊂ k. The extension l/k is a wildly ramied Galois extension. We have [l : k] = e(f : k) = 2. The element πl is a uniformizer of l. The k-vector space l is 2-dimensional and {1, πl} is a k-basis. So each element in l can be written as x + πly with x, y ∈ k. The norm of x + πly is egal to (x + πly)(x − πly) = x2 − 2y2. The set of norm 1 elements is an algebraic group. Let us write the Hopf algebra of the corresponding ane k-group scheme G. The Hopf k-algebra of G is k[X,Y ]/X2 − 2Y 2 − 1, moreover the comultiplication ∆, the antipode τ and the augmentation ε are

∆ : Hopf(G) → Hopf(G) ⊗ Hopf(G) X 7→ X ⊗ X + 2Y ⊗ Y Y 7→ X ⊗ Y + Y ⊗ X

τ : Hopf(G) → Hopf(G) X 7→ X Y 7→ −Y

ε : Hopf(G) → k X 7→ 1 Y 7→ 0 . The k-group G is a torus, indeed the equation 2 2 2 2 k[X,Y ]/X − 2Y − 1 ⊗k l ' l[X,Y ]/X − 2Y − 1

' l[X,Y ]/(X + πlY )(X − πlY ) − 1 ' l[U, V ]/UV − 1

33 shows that . The reduced Bruhat-Tits building G ×spec(k) spec(l) ' Gm/l BT(G, k) is a singleton {x}. The point x is a special point of BT(G, k) and ιK/k(x) ∈ BT(G, K) is special for any nite extension K/k. The group G is not split over k, it is split over l. Let us make explicit the group Gx,0. We need to nd an extension K/k such that G is split over K, ιK/k(x) is special, and r = 0 ∈ ord(K). The eld K = l works. By denition the -analytic group is egal to , where is the ◦-Demazure k Gx,0 prl/k(Gb η) G l group scheme attached to il/k(x). By the previous example 5.6.1, in the coordinate U, V , G = spec(l◦[U, V ]/UV − 1). Thus in the coordinate X,Y , Hopf(G) is egal to the l◦-subalgebra of l[X,Y ]/X2 − 2Y 2 − 1 generated by ◦ l ,X + πlY,X − πlY . By 5.3, the k-anoid algebra of Gx,0 is the completion 2 2 of Hopf(G) = k[X,Y ]/X − 2Y − 1 relatively to the norm | |Hopf(G) |Hopf(G) (recall that | |Hopf(G) is a norm on Hopf(G ×k l) ). So let us make as explicit as possible the norm | |Hopf(G) |Hopf(G). By denition, we have

| |Hopf(G) : Hopf(G ×k l) → R≥0

f 7→ inf {|λ| | f ∈ λ(Hopf(G) ⊗ 1) ⊂ Hopf(G ×k l)}. λ∈l× We deduce that

2 2 | |Hopf(G) :l[X,Y ]/X − 2Y − 1 → R≥0 ◦ f 7→ inf {|λ| | f ∈ λ(< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l)}. λ∈l× And so, by restriction

2 2 | |Hopf(G) |Hopf(G) : k[X,Y ]/X − 2Y − 1 → R≥0 ◦ f 7→ inf {|λ| | f ∈ λ(< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l)}. λ∈l× We have to complete k[X,Y ]/X2 − 2Y 2 − 1 relatively to this norm, in order to simplify notation let us put || || = | |Hopf(G) |Hopf(G). Let us compute the value ||X||. By denition it is egal to ◦ inf {|λ| | X ∈ λ(< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l)}. λ∈l× Since

◦ X 6∈ (< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l) ◦ πlX 6∈ (< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l) 2 ◦ 2X = πl X ∈ (< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l),

we deduce that ||X|| = |2−1| = e (2 is a uniformizer of k). Let us now compute the value ||Y ||. By denition it is egal to

34 ◦ inf {|λ| | Y ∈ λ(< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l)}. λ∈l× Since

◦ Y 6∈ (< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l) ◦ πlY 6∈ (< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l) 2 ◦ 2X = πl Y 6∈ (< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l) 3 ◦ 2πlX = πl Y ∈ (< l ,X − πlY,X + πlY >) ⊂ Hopf(G ×k l),

−3 3 we deduce that 2 . ||Y || = |πl | = e By completion, the k-Banach algebra of Gx,0 is egal to

−1 3 −1 2 2 k{e X, (e 2 ) Y }/X − 2Y − 1 , || ||

−1 3 −1 where k{e X, (e 2 ) Y } is the k-algebra

X k k k 3 k 1 2 1 2 2 as . { ak1k2 X Y | |ak1k2 |e (e ) → 0 k1 + k2 → ∞} ⊂ k[[X,Y ]] k1,k2

−1 3 −1 2 Let us check directly that the k-anoid algebra of Gx,0 is k{e X, (e 2 ) Y }/X − 2 −1 3 −1 2 2 2Y −1 , || ||. We need to check that (k{e X, (e 2 ) Y }/X −2Y −1)⊗ˆ kl is isomorphic to the l-anoid algebra of Gb η. In the coordinates U, V , the l- −1 3 −1 2 anoid algebra of Gb η is l{U, V }/UV −1. The l-algebra (k{e X, (e 2 ) Y }/X − 2 −1 3 −1 2 2 2Y −1)⊗ˆ kl is isomorphic to l{e X, (e 2 ) Y }/X −2Y −1). The isomorphism previously considered l[X,Y ]/X2 − 2Y 2 − 1 ' l[U, V ]/UV − 1 induces maps

−1 3 −1 2 2 l{e X, (e 2 ) Y }/X − 2Y − 1 ↔ l{U, V }/UV − 1

X + πlY ← U [ X − πlY ← V [ U + V X 7→ 2 U − V Y 7→ . 2πl These maps are mutual inverse k-Banach algebras isometries.

35 Appendix: On notions of rational points

Let k be a non archimedean local eld and G be a connected reductive k-group scheme. We have two notions of rational points in the reduced Bruhat-Tits building BT(G, k).

1. Here G = GLN in the original denition of Broussous-Lemaire [4]. A point x ∈ BT(G, k) is called barycentrically rational if it is the barycentre of vertices in a chamber with rational weights (this denition is natural after Broussous-Lemaire work, see their work on comparison of ltrations [4]). We denote by the associated subset BTratbar (G, k) of BT(G, k).

2. A point x ∈ BT(G, k) is called specially rational (or just rational in this paper) if there exists K/k nite such that

(a) ιK/k(x) ∈ BT(G, K) is a special point (ιK/k is the canonical map between buildings, this notion of rational point is introduced in this text (see section 3)) (b) G is split over K (this condition will always be satised in this appendix).

We denote (it was denoted before) the asso- BTratspe (G, k) BT(G, k) ciated subset of BT(G, k).

In this appendix we prove that they are equivalent in the case G = GLN , i.e . BTratspe (GLN , k) = BTratbar (GLN , k)

Proof that the two notions are equivalent for G = GLN

Here G = GLN , it is split /k and the reduced building is a simplicial complex. That last condition means that any facet F is a simplex. Let F be a maximal facet in an apartement A, and x it. Let S1,...,Si,...,SN be the vertex of the facet F . Put I = {1,...,N}. • Since F is a maximal simplex, for all i ∈ I, the set −−→ Ri = {SiSj | j ∈ I and i 6= j}

is a repère of A. That means that for any i ∈ I and each P ∈ A, there exists3 unique real numbers such that −−→ X −−→. x1,..., xbi, . . . xN SiP = xjSiSj j∈I j6=i The numbers are called the coordinates of in the repère x1,..., xbi, . . . , xN P Ri. 3The symbol hat over a symbol in a list of symbol means that we ommit it.

36 4 • (Since G = GLN ) The directions of the walls in A are in bijection with the vertex of the maximal simplex F as follows

{Vertex of F } ↔ {direction of the walls in A}

Si 7→Di = {direction of the wall containing S1,..., Sbi,...,SN }

• Let K/k be a nite extension, since G is split, for any maximal split torus S, the simplicial structure on the associated apartement A(G, S) satises the following: The apartement A(G, S)/K is obtained from A(G, S)/k adding regularly e times more walls for each direction. Fix a vertex Si, we thus get a direction Di , and we put:

The set of walls having direction , and coming from nite extensions W allsSi = { Di }

Let P ∈ A (think P ∈ F ). Write the coordinates of P in the repère −−→ X −−→ Ri: SiP = xjSiSj. By Thalès Theorem, the following are equivalent j∈I\i ( ): - j ∈ I \ i P ∈ W allsSj - The j-th coordinate xj of P in the repère Ri is a rational number.

Recall that a point P is special over an extension K/k if for every direction Di, there exists a wall of BT(G, K) such that P is contained in this wall. We deduce that

P ∈ BTratspe (G, k) ⇔ ∀j ∈ I,P ∈ W allSj ⇔ ∀i, j ∈ I; i 6= j; the j-th coordinate of P

in the repère Ri is rational. ⇔ ∀i ∈ I, the coordinates of P in the

repère Ri are rational numbers. Let us now prove that . • BTratspe = BTratbar We start by the inclusion . Let . Let , we write in ⊂ x ∈ BTratspe i ∈ I P the repère Ri −−→ X −−→ SiP = xjSiSj j∈I\i with xj rational numbers. We deduce the relation, using Chasles

4By the direction of a wall we mean the vectorial part.

37 −−→ X −−→ −−→ SiP = xj(SiP + PSj). j∈I\i

This allows us to write X −−→ X −−→ 0 = (( xj) − 1)SiP + xjPSj j∈I\i j∈I\i

This makes clear that by denition of barycentres. P ∈ BTratbar (G, k) Let us prove the reverse inclusion . Let . We have ⊃ P ∈ BTratbar (G, k) to show that for each i, the coordinates of P in the repère Ri are rational numbers. By denition of , there exists rational numbers BTratbar (G, k) cj such that X −−→ cjPSj = 0. j∈I We thus get

X −−→ −−→ −−→ cj(PSi + SiSj) + ciPSi = 0. j∈I\i

So we obtain X −−→ X −−→ cjSiSj + ( cj)PSi = 0. j∈I\i j∈I

X Putting k = cj (it is 6= 0), we get j∈I

−−→ X cj −−→ S P = S S . i k i j j∈I\i

This shows that the coordinates of P in the repère Ri are rational numbers. So , as required. P ∈ BTratspe (G, k)

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BICMR, PEKING UNIVERSITY [email protected]