Canonical parabolic subgroups of Arakelov group schemes: The function field case

ULRICH STUHLER In [Ha-Na] G. Harder and M. S. Narasimhan and independently D. Quillen in [Gray] showed the existence of a canonical filtration of a vector bundle E on a smooth complete algebraic curve X, which codifies the instability properties of the bundle E in an optimal form. This result can be also interpreted as a statement about the existence of a canonical parabolic subgroup for the reductive group scheme GL(E) of linear groups associated with E. In this form the result has been generalized by K. Behrend in [Be] to the case of arbitrary reductive group schemes G over X and it is shown again, that there always exists a canonical parabolic subgroup which keeps track of the various instability phenomena of G. The condition for G to be reductive over X however, is rather restrictive and for example in the case of a finite field of constants for X (resp. the associated field of functions F) forces the group to be generically split. In the present paper at least for the case of a finite field of constants a more general result is shown, which applies to all reductive algebraic groups over the globald field F of functions of X. The integer structures replacing the group scheme G, which are considered here, are not given in the context of algebraic geometry, but using the theory of Bruhat-Tits buildings. So the integer structure locally at a place v of F is given by a point x_{v} in the realisation of the Bruhat-Tits building of G over the completion F_{v} of F at v, which we denote in this paper as |BT (G; F_{v})|. We associate with such a point x_{v} a real valued norm ∥ ∥_{x_{v}} on the Lie algebra Lie(G) ⊗_{F} F_{v}. In the case, that x_{v} is a vertex of the polysimplicial complex BT (G; F_{v}), the norm ∥ ∥_{x_{v}} has also the following description. It is the norm associated to the o_{v}-lattice Lie(G(x_{v})) in Lie(G) ⊗_{F} F_{v}, where Lie(G(x_{v})) is the Lie algebra of the smooth group scheme G(x_{v}) given by x_{v} and defined over the valuation ring o_{v} of F_{v}. In general, the norm ∥ ∥_{x_{v}} can be obtained by an interpolation procedure, but we construct it here from the valuated root system given by x_{v} in a direct way, which is much easier than to construct the group schemes involved. The global integer structure G, called an Arakelov group scheme, is just a set

{x_{v} : x_{v} ∈ |BT (G; F_{v})|for v ∈ places (F)} ,

where for almost all v x_{v} = x_{v}^{(0)}, where the x_{v}^{(0)} are standard special vertices in BT (G; F_{v}).

Now, given such a structure of an Arakelov group scheme G over X (or F),it can be shown again, that there is a canonical parabolic subgroup with the usual properties. This is at least general enough to apply to all reductive algebraic groups over a global function field F and should be flexible enough to obtain the main results of reduction theory in this context. It also fits well with the case of number fields where G. Harder, also following Behrend’s method, has shown the existence of a canonical parabolic subgroup of an Arakelov group scheme, where again and quite in the spirit of Arakelov geometry, the integer structure at the archimedean places is just given by points in the corresponding symmetric spaces (G×_{F} ℝ)/K where K are the relevant maximal compact subgroups. Again his results are restricted to reductive algebraic groups with good reduction everywhere, but combining his results with the results of this paper, it should be possible to obtain a general result for all reductive algebraic groups over number fields and even a unified approach for reductive algebraic groups over global fields. Finally, the restriction to function fields with a finite field of constants most probably can be avoided. It comes in as a technical point only in Lemma 6.1. Nevertheless, our result is still not as general as one would like to have. One would like to give the integer structure locally at v by a bounded subset of |BT (G; F_{v})|, contained in some appartment (for finitely many places v) as in the general theory ([Ti],). For the proofs of our result we have to combine the results of the paper [H-J-S] (the language and basic properties of Arakelov vector bundles or, what is called there, generalized vector bundles) with the method of complementary poyhedra, Behrend has introduced in his paper [Be] to obtain his results. It consists in studying the situation “appartmentwise”, where the method of complementary polyhedra is a particular nice way to keep track of the various types of inequalities of degrees of vector bundles coming into the picture. We conclude the introduction by a short description of the contents of the paper. The first section recalls the concept of an Arakelov vector bundle and its various properties following [H-J-S]. Section two recalls the properties of root systems needed (mainly following [Bour] and [Be]) and introduces in 2.6. the definition of complementary polyhedra. Section 3 completely follows [Be] in introducing a formal concept of stability for complementary polyhedra and the concept of a special facet of the complementary polyhedron, which will lead directly to the canonical parabolic subgroup. (see Definitions 3.3 and 3.7 and in particular Prop. 3.8 and 3.9). Section 4 treats the local situation, as has been described already above. The main point, the construction of a norm associated to a point in the building, is in 4.5, section 5 then puts together the local data for split reductive algebraic groups over a global function field. In 5.1. the concept of an integer structure we are using, is defined. Prop. 5.9. and 5.10. describe the degrees of line bundles associated with the root subgroups of Borel subgroups resp. parabolic subgroups. Theorem 5.13. constructs for a given split maximal torus the related complementary polyhedron. At the end of section 5 the independence of our concepts from several choices (choice of a split maximal torus, choice of an “épinglage” for the root system associated with the torus) is shown. In section 6 finally the main results of this paper are proved. It is only here in Lemma 6.1., that we have to assume, that the field of constants of F is finite. Theorem 6.2 constructs the canonical parabolic subgroup for the case of a split reductive algebraic group. In 6.3. the uniqueness of the canonical parabolic subgroup is shown using Behrend’s method of complementary polyhedra. Finally Theorem 6.4. shows existence and uniqueness of a canonical parabolic subgroup for arbitrary reductive algebraic groups over F, which is easy at this stage making use of the independence results at the end of section 5.

1. Arakelov bundles

We give in this section an introduction into the concept and basic properties of Arakelov bundles on a complete smooth algebraic curve as outlined in [H-J-S] under the name of a generalized vector bundle.

In the following X is a smooth projective algebraic curve over the field of constants k, F = k(X) is the function field of the curve X. |X| denotes the set of closed points of X, which is also the set of places of F. For a closed point x ∈ |X|, F_{x} is the completion of F with respect to (a chosen) discrete valuation | |_{x} on F given by x. Furthermore o_{x} is the valuation ring of F_{x} given as

o_{x} = {u ∈ F_{x} : |u|_{x} ≤ 1}

(so the valuation is written multiplicatively), O_{X,x} denotes as usual the local ring of X at x. η is the generic point of X, the local ring O_{X,η} is the function field F itself. (1.1) Definition: A normed F_{x}-vector space (V, ∥ ∥) is an F_{x}-vector space V together with a real valual norm

∥ ∥ : V → ℝ, v ↦ ∥v∥ for v ∈ V

satisfying the axioms i) ∥v∥ ≥ 0 and ∥v∥ = 0 iff v = 0, for v ∈ V. ii) ∥v_{1} + v_{2}∥ ≤ Max{∥v_{1}∥, ∥v_{2}∥} for v_{1}, v_{2} ∈ V. iii) ∥λv∥ = |λ|_{x}∥v∥ for λ ∈ F_{x}, v ∈ V.

Example. Suppose, E_{x} is a lattice in the F_{x}-vector space V_{x}, so E_{x} is a free o_{x}-module of finite rank, contained in V_{x}, such that the canonical homomorphism

F ⊗_{o_{x}} E_{x} → V_{x}

is an isomorphism. Then for v ∈ V, v ≠ 0, one defines

∥v∥ := sup{|λ|_{x} : λ ∈ F_{x}, λ^{−1}v ∈ E_{x}}

and of course ∥0∥ = 0 defines a norm on V_{x}, such that E_{x} is exactly the unit ball {v ∈ V_{x} : ∥v∥ ≤ 1} of (V_{x}, ∥ ∥). (1.2) Definition: V is a finite dimensional F -vector space. A set of norms

{∥ ∥_{x} : F_{x} ⊗_{F} V → ℝ : x ∈ |X|}

is called coherent iff there is an open affine subset X′ = Spec(O) ⊂ X, a free O-module E of rank d and an isomorphism of F -vector spaces F ⊗_{O} E →^{∼} V, such that the norms ∥ ∥_{x} for x ∈ |X′| are given under the isomorphism above by the unit balls O_{x} ⊗_{O} E ⊂ F_{x} ⊗_{F} V.

Remark. Of course X′ is in no way uniquely determined by the situation. For example it can be always replaced by a smaller open affine subset in X′.

(1.3) Definition: An Arakelov bundle E = (V; ∥ ∥_{x} for x ∈ |X|) of rank d consists of a d-dimensional F -vector space V and a coherent system of norms {∥ ∥_{x} : x ∈ |X|} on the vector spaces F_{x} ⊗_{F} V.

Remarks.

i) So this is exactly the concept of generalized vector bundles in [H-J-S: 2.1]. ii) Suppose E is a locally free sheaf (resp. vector bundle) of rank d on the

curve X. Canonically associated with E is an Arakelov bundle (E_{η}; ∥ ∥_{x} for x ∈ |X|) where V := E_{η} is the generic stalk of the sheaf E, the norm ∥ ∥_{x} is given again by the lattice o_{x} ⊗_{O_{X,x}} E_{x} ↪ F_{x} ⊗_{F} E_{η} for x ∈ |X|.

iii) A similar construction works if one is given the data (M; ∥ ∥_{x} for x ∈ S) where S is a finite subset of |X|, Spec(O) = X \S as an open affine subset of X with associated Dedekind ring and M a projective O-module of rank d. Defining V := F ⊗_{O} M and fixing the real norms ∥ ∥_{x} for the missing x ∈ |X \ S| as above via the lattices O_{x} ⊗_{O} M ⊂ F_{x} ⊗_{F} M gives the associated Arakelov bundle (V; ∥ ∥_{x} for x ∈ |X|).

iv) For a given Arakelov bundle E = (V; ∥ ∥_{x} for x ∈ X), E_{η} := V denotes the associated vector space, which can be considered also as the generic stalk for the Arakelov bundle E.

(1.4) Definition and Lemma.

i) A homomorphism φ : E_{1} → E_{2} between two Arakelov bundles E_{i} = (V_{i}; ∥ ∥_{x}^{(i)} for x ∈ |X|) (i = 1, 2) is an F -linear map φ_{η} : V_{1} → V_{2} satisfying for all x ∈ |X|, v ∈ V_{1,x} the inequalities

∥(id_{F_{x}} ⊗_{F} φ_{η})(v)∥_{x}^{(2)} ≤ ∥v∥_{x}^{(1)}

ii) If φ_{η} above is an isomorphism and the φ_{x} := id_{F_{x}} ⊗_{F} φ_{η} are normpreserving, that is we have equality in i), then φ is an isomorphism in the category of Arakelov bundles.

Proof. To show in ii) that φ is an isomorphism in the category of Arakelov bundles given by i), one checks directly that φ^{−1} : E_{2} → E_{1} induced by φ_{η}^{−1} is a homomorphism of Arakelov bundles. □

Remarks:

i) The category of Arakelov bundles is additive. ii) If E_{1}, E_{2} are vector bundles resp. locally free sheaves on X and φ :

E_{1} → E_{2} is a homomorphism of locally free sheaves, the homomorphism φ_{η} : E_{1,η} → E_{2,η} of the generic stalks induces a homomorphism of the associated Arakelov bundles.

iii) If the E_{i} are given as

(E_{i} = (M_{i}; ∥ ∥_{x}^{(i)} for x ∈ S) (i = 1, 2)

in the sense of Remarks iii) after 1.3., a homomorphism φ : E_{1} → E_{2} can be given by an O-module homomorphism φ : M_{1} → M_{2} satisfying the contraction property of 1.4. for x ∈ S.

(1.5) Definition:

i) An Arakelov bundle E = (V; ∥ ∥_{x} for x ∈ |X|) of rank one, that is, dim_{F} (V) = 1, is also called an Arakelov line bundle.

ii) Suppose, d = rank(E) = dim_{F} (V). The d-th exterior power

Λ^{d}(E) := (Λ^{d}(V) : Λ^{d}(∥ ∥_{x}) for x ∈ |X|)

is called the determinant line bundle of E. It is an Arakelov line bundle which we often denote det(E).

Remark. Similarly we have the other tensor operations in the category of Arakelov bundles, in particular the tensor product E_{1} ⊗ E_{2} and the Hom-bundle \underline{Hom}(E_{1}, E_{2}). For some more explanations see [H-J-S: 1.2. Remarks] (1.6) Definition and Proposition.

i) If E = (V; ∥ ∥_{x} for x ∈ |X|) is an Arakelov bundle on X, for any F -linear subspace V_{1} ⊂ V one has the subbundle

(V_{1}; ∥ ∥_{x} |_{F_{x}⊗_{F} V_{1}} for x ∈ |X|).

The inclusion i : V_{1} → V induces a homomorphism of Arakelov bundles. ii) Dually the quotient vector space V_{2} := V/V_{1} obtains norms on F_{x} ⊗_{F} V_{2} =:

V_{2,x} for x ∈ |X| as follows: Suppose v̄ = v + V_{1} ∈ V/V_{1}, then one defines

∥v̄∥_{x}^{(2)} := inf{∥v + v_{1}∥_{x} : v_{1} ∈ V_{1,x}} .
E/E_{1} := (V/V_{1} : ∥ ∥_{x}^{(2)} for x ∈ |X|)

is an Arakelov bundle which is called the quotient bundle of E by E_{1}. The projection p : V → V/V_{1} induces a homomorphism p : E → E/E_{1} =: E_{2} of Arakelov bundles.

(1.7) Proposition: Suppose E, E_{1}, E_{2} are Arakelov bundles as in 1.6. Then the sequence

0 → E_{1} ⟶^{i} E ⟶^{p} E/E_{1} = E_{2} → 0

is exact in the additive category of Arakelov bundles. Any exact sequence in the additive category of Arakelov bundles is of this form up to isomorphism.

Proof. See [H-J-S:1.6 and 2.5] □

(1.8) Definition: Suppose the valuations | |_{x} for x ∈ |X| are chosen such that the product formula ∏_{x∈|X|} |u|_{x} = 1 holds for alle nonzero u ∈ F.

i) Suppose L = (V; ∥ ∥_{x} for x ∈ |X|) is an Arakelov line bundle on X. If v ∈ V is any nonzero element of V, the degree of L is given as

deg(L) := − ∑_{x∈|X|} log ∥v∥_{x}

as real number. This is independent of the choice of v. ii) For an Arakelov bundle E of arbitrary rank one defines

deg(E) := deg(det(E)).

(1.9) Proposition:

i) If L_{1}, L_{2} are Arakelov line bundles, one has

deg(L_{1} ⊗ L_{2}) = deg(L_{1}) + deg(L_{2}).

ii) If

0 → E_{1} → E → E_{2} → 0

is an exact sequence of Arakelov bundles, one has a canonical isomorphism of line bundles

det(E_{1}) ⊗ det(E_{2}) →^{∼} det(E).

In particular

deg(E) = deg(E_{1}) + deg(E_{2})

holds.

Proof. See [H-J-S; 2.6]

(1.10) Definition: For E a nontrivial Arakelov bundle, one defines

μ(E) := \frac{deg(E)}{rg(E)} ,

the slope of the Arakelov bundle E.

(1.11) Definition: An Arakelov bundle E is called stable (semistable) iff for all nonzero subbundles E_{1} of E the inequality

μ(E_{1}) < ( resp. ≤) μ(E))

holds.

(1.12) Proposition: An Arakelov bundle E on X has a unique flag of subbundles

0 = E_{0} ⊂_{≠} E_{1} ⊂_{≠} … ⊂_{≠} E_{r} = E

satisfying the following properties:

i) E_{i}/E_{i−1} is semistable for each possible i.

ii) μ(E_{i}/E_{i−1}) > μ(E_{i+1}/E_{i}) holds for each possible i. Moreover this flag satisfies iii) E_{i}/E_{i−1} is the largest subbundle of E/E_{i−1} such that

μ(E_{i}/E_{i−1}) = μ_{max}(E/E_{i−1}),

where μ_{max}(...) denotes the maximal possible value of μ for a submodule of E/E_{i−1}

iv) E_{i}/E_{i−1} is the smallest quotient bundle of E_{i}, such that

μ(E_{i}/E_{i−1}) = μ_{min}(E_{i})

where μ_{min}(E_{i}) denotes the minimal possible value of a quotient of E_{i}.

Proof. See again [H-J-S: 2.11.] □

Remark. This is the analogue of the Harder-Narasimhan canonical flag (also found independently by D. Quillen) for the case of an Arakelov bundle.

(1.13) Corollary. Suppose, E′ is a subbundle of an Arakelov bundle E. Assume that E′_{1} ⊂ ⋯ ⊂ E′_{r} is the HN-filtration of E′ and E_{1}/E′ ⊂ ⋯ ⊂ E_{s−1}/E′ is the HN-filtration of the bundle E/E′. We assume furthermore that

μ_{max}(E/E′) < μ_{min}(E′)

holds. Then the HN-filtration of E is the flag

0 ⊂ E′_{1} ⊂ … ⊂ E′_{r−1} ⊂ E′ ⊂ E_{1} ⊂ … ⊂ E_{s−1} ⊂ E.

In particular E′ is part of the canonical filtration of E.

Proof. See [H-J-S; 2.12] □

2. Root systems and complementary polyhedra

In this section we remind the reader of the concept and techniques of complementary polyhedra, which will be applied in later sections. We follow the exposition of [Be]. We start with a vector space V over the reals ℝ and a reduced root system Φ ⊂ V satisfying

i) Φ is finite and generates V.

ii) For every root α ∈ Φ there is a unique α̌ ∈ V^{∗} , V^{∗} the dual vector space, such that ⟨α, α̌⟩ = 2 for the canonical pairing holds and

s_{α} : V → V, x ↦ s_{α}(x) := x − ⟨x, α̌⟩α

satisfies s_{α}(Φ) = Φ. iii) For every root α ∈ Φ we have α̌(Φ) ⊂ ℤ. iv) If α ∈ Φ, then 2α ∉ Φ (the root system is reduced).

For α ∈ Φ we denote the hyperplane

H(α) := {x ∈ V : ⟨x, α̌⟩ = 0}.

The set of hyperplanes {H(α) : α ∈ Φ} gives the usual decomposition of the vector space V into facets F, where for x, y ∈ V x, y are in the same facet called F iff for all α ∈ Φ the following statement holds:

⟨x, α̌⟩ ≤_{>} 0 iff ⟨y, α̌⟩ ≤_{>} 0.

The facets of maximal dimension are open subsets of V and are called chambers. The set F of facets has a partial order by F_{1} ≤ F_{2} iff F_{1} ⊆ F̄_{2}. F(V, Φ) denotes the partially ordered set of facets, C the subset of chambers. (Besides the set Φ of roots and the set Φ^{∨} of coroots there is additionally the lattice of weights in V,

Λ := {λ ∈ V : ⟨λ, α̌⟩ ∈ ℤ for all α̌ ∈ Φ^{∨}}

as well as the lattice of coweights in V^{∗},

Λ^{∨} := {λ̌ ∈ V^{∗} : ⟨α, λ̌⟩ ∈ ℤ for all α ∈ Φ}

For a basis {α_{1}, …, α_{n}} = Δ [Bou, VI, 1.5] of the system of roots Φ one has the subset

Φ_{+} := {α ∈ Φ : α = ∑_{i=1}^{n} r_{i}α_{i}, r_{i} ≥ 0 for all i}

of positive roots and correspondingly

Φ_{−} := {α ∈ Φ : α = ∑_{i=1}^{n} r_{i}α_{i}, r_{i} ≤ 0 for all i}

of negative roots. One has the disjoint decomposition

Φ = Φ_{+} ∪ Φ_{−}.

Corresponding to Δ, Φ_{+}, Φ_{−} one has the dual notions Δ^{∨}, Φ_{+}^{∨}, Φ_{−}^{∨} given as Δ^{∨} := {α_{1}^{∨}, …, α_{n}^{∨}} e.t.c. A weight λ ∈ Λ is called dominant (with respect to a fixed root basis {α_{1}, …, α_{n}}) iff ⟨λ, α̌_{i}⟩ ≥ 0 for all i = 1, …, n. Similarly one has the conecpt of dominant coweights. Finally a system of fundamental dominant weights (with respect to some basis {α_{1}, …, α_{n}} of Φ) is a set of weights

{λ_{1}, …, λ_{n}} ⊂ Λ,

such that ⟨λ_{i}, α̌_{j}⟩ = δ_{ij} for i ≤ i, j ≤ n. Dually one has the notion of a system of fundamental dominant coweights.

(2.1) Definition: F denotes a facet of (V, Φ), λ ∈ Λ is a dominant fundamental weight. λ is a vertex of the facet F iff λ ∈ F̄ the closure of F in V. vert(F) denotes the set of all vertices of the facet F. One has the following

(2.2) Proposition:

i) Let F be a facet. Then the set of vertices vert(F) is a finite set {λ_{1}, …, λ_{r}} of linearly independent vectors.

ii) F = {x ∈ V: There exist a_{i}, …, a_{r} > 0, such that x = ∑_{i=1}^{r} a_{i}λ_{i}}.

For the proof see [Be, Prop. 1.4] □

Besides vert(F) it is usefull to have the set of fundamental weights

vert′(F) := {λ ∈ Λ: there exists a chamber C ∈ C, C̄ ⊃ F, such that λ ∈ vert(C)}.

Again one has the corresponding dual notions vert(F^{∨}), vert′(F^{∨}).

Remark. Let F denote a facet of (V, Φ). If λ ∈ vert′(F), then there is a facet F′ of (V, Φ) such that vert(F′) = vert(F) ∪ {λ}. Similarly for λ ∈ vert(F), there is a facet of (V, Φ) with set of vertices vert(F) \ {λ}.

(2.3) Lemma:

i) Let F ∈ F(V, Φ), let

Φ(F) := {α ∈ Φ : ⟨α, λ̌⟩ ≥ 0 for all λ ∈ vert(F)}

Then Φ(F) is a parabolic set of roots in the sense of [Bou, VI, 1.7] ii) Let R ⊂ Φ be a parabolic subset of roots. Denote

vert(R) := {λ ∈ Λ : λ is a fundamental weight satisfying ⟨α, λ̌⟩ ≥ 0 for all α ∈ R}

Then there exists a unique facet F with vert(F) = vert(R), such that Φ(F) = R holds.

iii) The two maps defined in i), ii) are inverse to each other and are bijections

between the set of facets F and the set of all parabolic subsets of Φ. The following Proposition will be very usefull in chapter 5.

(2.4) Proposition: Let F ∈ F(V, Φ) be a facet, Φ(F) the corresponding parabolic set of roots. Then

∑_{α∈Φ(F)} α ∈ F .

For the proof see [Be, Prop. 1.9]. □

Associated with a facet F ∈ F(V, Φ) and the corresponding parabolic set of roots Φ(F) there is a root system connected with the reductive part of the parabolic associated with Φ(F). This is obtained as follows:

Suppose, F ∈ F(V, Φ) is a facet. We have the decompositions into direct summands

\begin{aligned}V & = & ⟨F ⟩ ⊕ (F^{∨})^{⊥} \\ V^{∗} & = & ⟨F^{∨}⟩ ⊕ F^{⊥} \end{aligned}

where ⟨ ⟩ denotes the linear span and ( )^{⊥} the orthogonal complement in the dual space. We denote V_{F} := (F^{∨})^{⊥}, V_{F}^{∗} := F^{⊥}. Φ_{F} := Φ ∩ V_{F} is a root system in V_{F}, the reduction of the root system Φ to the facet F. Φ_{F}^{∨} := Φ^{∨} ∩ V_{F}^{∗} can be seen as the dual root system under the identification of the dual space of V_{F} with V_{F}^{∗} as above. C(F) denotes the set of chambers of (V_{F} , Φ_{F}). (2.5) Lemma: The projection pr : V → V_{F} given by the direct sum decomposition V = ⟨F ⟩ ⊕ (F^{∨})^{⊥} = ⟨F ⟩ ⊕ V_{F}, induces a bijection

\begin{matrix}pr : \{F′ ∈ F(V, Φ) : \overline{F′} ⊇ F \} & → & F(V_{F} , Φ_{F}), \\ F′ & ↦ & pr(F′) \end{matrix}

One has: dim pr(F′) = dim(F′) − dim(F). In particular one obtains a bijection of the sets of chambers, also called pr,

pr : {C ∈ C(V, Φ) : C̄ ⊇ F } → C(F).

Proof. See again [Be, Lemma 1.11] □ Also, one can conclude easily that pr induces a bijection of vert′(F) \ vert(F) with the set of fundamental weights in the weight-lattice Λ(F) ⊂ V_{F}. We come to the main subject of this section, the concept of complementary polyhedra. (2.6) Definition: Let (V, Φ) be a root system. A map d : C → V^{∗}, C ↦ d(C) is a complementary polyhedron for (V, Φ), iff the following two axioms hold.

1) If λ ∈ vert(C) ∩ vert(D) for the chambers C, D ∈ C(V, Φ), then

⟨λ, d(C)⟩ = ⟨λ, d(D)⟩.

2) If for two chambers C, D ∈ C, vert(C) ∩ vert(D) = {λ_{1}, …, λ_{n−1}} and the root α ∈ Φ satisfies: α|_{C} > 0 and α|_{D} < 0, then

⟨α, d(C)⟩ ≤ ⟨α, d(D)⟩.

(2.7) Definition: For d : C → V^{∗} a complementary polyhedron as above, F ∈ F(V, Φ) a facet, the dual polyhedron d(F) of F is defined as

d(F) := convex hull{d(C) : C̄ ⊇ F }.

Remarks.

i) In particular, for F = {0} ∈ F(Φ),

d({0}) = convex hull {d(C) : C ∈ C(V, Φ)}

ii) In Definition 2.6, 2) above the reader should check that the conditions

concerning the roots α ∈ Φ, −α ∈ Φ are compatible. iii) One checks easily that given a complementary polyhedron d : C → V^{∗}

and a facet F ∈ F(V, Φ),

\begin{aligned}d_{F} : C_{F} & → & V_{F}^{∗}, \\ d_{F} (C̃) & := & p̌r d(pr^{−1}(C̃)) \end{aligned}

for C̃ ∈ C_{F}, the set of chambers of (V_{F} , Φ_{F}), is a complementary polyhedron for the root system (V_{F} , Φ_{F}). This is called the reduction of (V, Φ) to F.

iv) One has the following description of the dual polyhedron d(F), where

H(λ) := {x ∈ V^{∗} : ⟨λ, x⟩ = ⟨λ, d(C)⟩}

for λ a fundamental weight, C a chamber satisfying C̄ ⊃ F,

E_{λ}^{+} := {x ∈ V^{∗} : ⟨λ, x⟩ > ⟨λ, d(C)⟩}

and (Ē_{λ}^{+}) its closure in V^{∗}:

d(F) = ⋂_{λ∈vert(F)} H(λ) ∩ ⋂_{λ∈vert′(F)} \overline{E(λ)^{∗}}

In particular one has

d(\{0\}) = ⋂_{λ∈vert′(0)} \overline{E(λ)^{∗}},

where of course vert′(0) is the set of all fundamental weights. For the proof of this fact see again [Be, Lemma 2.5].

3. Stability and special facets

In this section we describe the concept of stability of a complementary polyhedron. If the complementary polyhedron is not stable it will be shown that there is a unique special facet violating stability. Again we follow [Be] completely. Suppose again, (V, Φ) is a root system. For a facet F ∈ F(V, Φ) we denote

R(F) := {α ∈ Φ : ⟨α, λ̌⟩ ≥ 0 for all λ ∈ vert(F)}

As remarked above this is a parabolic set of roots. Furthermore we define

U(F) := {α ∈ R(F) : there exists λ ∈ vert(F), satisfying ⟨α, λ̌⟩ > 0}

Remarks:

i) Obviously one has U(F) ⊂ R(F) and R(F) = U(F) ⋃^{·} Φ_{F}.

ii) Thinking in terms of Lie algebras or algebraic groups, R(F) can be seen as the set of roots of a parabolic subalgebra resp. parabolic subgroup, U(F) as the set of roots of its unipotent radical.

We have the following fundamental definition of a degree function: (3.1) Definition: (V, Φ) denotes a root system, d : C(V, Φ) → V^{∗} a complementary polyhedron, F ∈ F(V, Φ) a facet. Then the degree of F is given as

deg(F) := ∑_{α∈U(F)}⟨α, d(C)⟩
= ∑_{α∈R(F)}⟨α, d(C)⟩

where C ∈ C(V, Φ) is any chamber of (V, Φ) such that C̄ ⊇ F. Remarks:

i) Of course, by property i) of a complemenary polyhedron, deg(F) is independent of the choice of the chamber C, C̄ ⊇ F.

ii) Suppose F, F′ ∈ F(V, Φ) are facets such that F̄′ ⊇ F. Denote pr : V → V_{F} the reduction map. Then one has

deg(F′) = deg pr(F′) + deg F,

where deg pr(F′) is taken with respect to the root system (V_{F} , Φ_{F}) and the complementary polyhedron d_{F} : C(V_{F}; Φ_{F}) → V_{F}^{∗} ⊂ V^{∗}.

One has the following series of equivalences (3.2) Lemma:

i) For every facet F ∈ F(V, Φ) one has deg(F) ≤ 0.

ii) For every F ∈ F(V, Φ), dim(F) = 1, one has deg(F) ≤ 0.

iii) 0 ∈ d(F). Proof. i) ⟹ ii) is obvious. The equivalence ii) ⟺ iii) is just the description of d(F) at the end of section 2. Finally, if 0 ∈ d(F) for a facet F ∈ F(V, Φ), then as we have just remarked one has ⟨λ, d(C)⟩ ≤ 0 for λ ∈ vert(C). But ∑_{α∈R(F)} α ∈ F as remarked in Proposition

2.4. Therefore ∑_{α∈R(F)}⟨α, d(C)⟩ ≤ 0 for C ∈ C, C̄ ⊇ F. □

(3.3) Definition: A complementary polyhedron d : C(V, Φ) → V^{∗} is semistable iff the equivalent conditions of Lemma 3.2 hold. For later use we introduce some numerical invariants. (3.4) Definition: Suppose F ∈ F(V, Φ) is a facet, λ ∈ vert(F) a vertex. Then

Ψ(F; λ) := {α ∈ Φ : (α, λ̌⟩ = 1 and ⟨α, μ̌⟩ = 0 for all μ ∈ vert(F) \ {λ}}

Remark. Ψ(F; λ) is called the elementary set of roots associated to F and λ. (3.5) Lemma: One has

∑_{α∈Ψ(F;λ)} α ∈ ⟨λ : λ ∈ vert(F)⟩,

the linear span of the vertices of F. Proof. [Be, Lemma 3.6] □ (3.6) Definition: Given F, λ ∈ vert(F) as above, one denotes the so called numerical invariant of F with respect to λ, d

n(F; λ) := ∑_{α∈Ψ(F;λ)}⟨α, d(C)⟩,

where C is any chamber satisfying C̄ ⊇ F. Remarks. i) One sees immediately that

d(C) = ∑_{λ∈vert(C)} n(C; λ)λ̌

ii) Again the behavior of the numerical invariants with respect to reduction is very simple: Suppose F, F′ ∈ F(V, Φ), F̄′ ⊇ F and λ ∈ vert(F′) \ vert(F). Then

n(F′; λ) = n(pr(F′); pr(λ))

with respect to the projection map pr : V → V_{F}. We come now to the fundamental concept of a special facet F with respect to a complementary polyhedron d : C(V, Φ) → V^{∗}.

(3.7) Definition: A facet F as above is special with respect to the complementary polyhedron d : C(V, Φ) → V^{∗}, if the following conditions are satisfied:

i) For all λ ∈ vert(F), one has n(F; λ) > 0.

ii) The reduction (V_{F}; Φ_{F}) with complementary polyhedron

d_{F} : C(V_{F}; Φ_{F}) → V_{F}^{∗} is semistable in the sense of Definition 3.3. Next, it will be shown, that the special facet of a complementary polyhedron is uniquely determined.

Let us first remark, that for any facet F ∈ F(V, Φ) there is a unique point y(F) ∈ V^{∗} given as

{y(F)} := ⋂_{λ∈vert(F)} H(λ) ∩ ⟨F̌⟩,

where as before ⟨F̌⟩ denotes the linear span of F̌ in V^{∗}, H(λ) is the hyperplane given as

H(λ) := {x ∈ V^{∗} : ⟨λ, x⟩ = ⟨λ, d(C)⟩ for C ∈ C(V, Φ), C̄ ⊇ F }

This is well defined as can be seen as follows:

(3.8) Proposition: The facet F ∈ F(V, Φ) is special if the following conditions are satisfied:

i) y(F) ∈ F̌

ii) y(F) ∈ d(F)

Proof. Again we repeat the arguments of [Be, Lemma 3.11] Suppose {λ_{1}, …, λ_{r}} = vert(F), where furthermore {λ_{1}, …, λ_{n}} is a set of dominant fundamental weights corresponding to a chamber C, C̄ ⊇ F. The system of positive roots associated to C is {α_{1}, …, α_{n}} such that ⟨α_{i}, λ̌_{j}⟩ = δ_{ij} for i, j ∈ {1, …, n}. Consider again

Ψ(F; λ_{i}) = {α ∈ Φ : ⟨α, λ̌_{i}⟩ = 1 and ⟨α, μ̌⟩ = 0 for all μ ∈ vert(F) \ {λ_{i}}} .

We denote for i = 1, …, r:

β_{i} := \frac{1}{#Ψ(F; λ_{i})} ( ∑_{α∈Ψ(F;λ_{i})} α)

One has ⟨β_{i}, λ̌_{j}⟩ = δ_{ij} for i, j = 1 … r by definition of the β_{i}. Therefore using Lemma 3.5 we can conclude that β_{i} ∈ ⟨λ_{1}, …, λ_{r}⟩ and furthermore {β_{1}, …, β_{r}} is a basis of this linear space by the orthogonality relations above. By the definition of y(F) one has ⟨λ_{i}, y(F)⟩ = ⟨λ_{i}, d(C)⟩ for i = 1, …, r. Therefore ⟨β_{i}, y(F)⟩ = ⟨β_{i}, d(C)⟩ (i = 1, …, r). This implies because of y(F) ∈ ⟨F̌⟩,

y(F) = ∑_{i=1}^{r}⟨β_{i}, y(F)⟩λ̌_{i}
(because of ⟨β_{i}, λ̌_{j}⟩ = δ_{ij} for i, j = 1, …, r)
= ∑_{i=1}^{r}⟨β_{i}, d(C)⟩λ̌_{i}
= ∑_{i=1}^{r} \frac{n(F; λ_{i})}{#Ψ(F; λ_{i})}λ̌_{i}

Therefore one has y(F) ∈ F̌ iff all n(F; λ_{i}) > 0 for i = 1, …, r, that is n(F; λ) > 0 for all λ ∈ vert(F). We show next the equivalence of conditions ii) in Definition 3.7 resp. Lemma 3.2 above. One has (Φ_{F}; d_{F}) semistable iff 0 ∈ d(F). Because p̌r : V^{∗} → V_{F}^{∗} induces an isomorphism under restriction to d(F) ⊂ V^{∗}, one has 0 ∈ d(F) iff 0 ∈ p̌r(d(F)). This holds if and only if ⟨F̌⟩ ∩ d(F) ≠ ∅ because ker(p̌r) = ⟨F̌⟩. But this means

exactly: y(F) ∈ d(F), since d(F) ⊂ ⋂_{i=1}^{r} H(λ_{i}) and {y(F)} = ⋂_{i=1}^{r} H(λ_{i}) ∩ ⟨F̌⟩. □

Geometrically it is more intuitive to formulate this result using the more or less canonical identification of V and V^{∗} using a scalar product invariant under the Weyl group of (V, Φ). As remarked above, a coroot α̌ gets identified with a a positive scalar multiple of α such that the relation α̌ = \frac{2α}{(α,α)} holds. Furthermore for a basis \{α_{1}, …, α_{n}\} of (V, Φ) one has (α_{i}, λ̌_{j}) = ⟨α_{i}, λ̌_{j}⟩ = 0 for i ≠ j upon identification. Similarly (α̌_{i}, λ_{j}) = ⟨λ_{j}, α̌_{i}⟩ = 0 for i ≠ j. Therefore λ_{j}, λ̌_{j} for j = 1, …, n are also positive scalar multiples of each other. The corresponding factors can be computed immediately: ⟨α_{i}, λ̌_{i}⟩ = 1, similarly ⟨λ_{i}, α̌_{i}⟩ = 1, therefore (α_{i}, λ̌_{i}) = 1 and (α̌_{i}, λ_{i}) = 1, using α̌_{i} = \frac{2α_{i}}{(α_{i},α_{i})} we obtain \frac{2}{(α_{i},α_{i})}(α_{i}, λ_{i}) = 1, therefore \frac{2}{(α_{i},α_{i})}λ_{i} = λ̌_{i} for i = 1, …, n. (3.9) Proposition: Under the above invariant identification of V with V^{∗}, for a facet F ∈ F(V, Φ), y(F) is the unique point of d(F) closest to the origin 0 of V. Conversely, if y is the unique point of d(F) closest to the origin and F is the unique facet of (V, Φ) such that y ∈ F, then F is a special facet and y = y(F). Proof. Again for the convenience of the reader we repeat the argument of [Be, Prop. 3.13]. Suppose vert(F) = {λ_{1}, …, λ_{r}}, C ∈ C(V, Φ) a chamber such that C̄ ⊇ F holds. We take x ∈ d(F) arbitrary and we will show: ∥x∥ ≥ ∥y(F)∥. Because x ∈ d(F), it follows by definition that ⟨λ_{i}, x⟩ ≥ ⟨λ_{i}, d(C)⟩ for i = 1, …, r. By definition of y(F) on the other hand we have ⟨λ_{i}, y(F)⟩ = ⟨λ_{i}, d(C)⟩ (we could of course write brackets ( , ) as well!). Therefore we obtain ⟨λ_{i}, x − d(F)⟩ ≥ 0. Because y(F) ∈ F̌ = F (under identification) we have y(F) = ∑_{i=1}^{r} y_{i}λ_{i} where y_{i} > 0 for i = 1, …, r. Therefore it follows that

∥x∥^{2} = ∥y(F)∥^{2} + ∥x − y(F)∥^{2} + 2(y(F), x − y(F))
≥ ∥y(F)∥^{2} + 2(y(F), x − y(F))
= ∥y(F)∥^{2} + 2 ∑_{i=1}^{r} y_{i}(λ_{i}, x − y(F))
≥ ∥y(F)∥^{2}

This shows the first half of the proposition.

For the second half consider y as above. Consider F ∈ F(V, Φ) such that y ∈ F. We have the C^{∞}-function

f : [0, 1] → ℝ,
t ↦ ∥y + t(d(C) − y)∥^{2}

For all t ∈ [0, 1] we have y + t(d(C) − y) ∈ d(F), since d(F) is convex. By assumption on y, f assumes its minimum at t = 0. Hence f′(0) ≥ 0. Differentiation yields (y, d(C) − y) ≥ 0. Since y ∈ F, we have y = ∑_{i=1}^{r} y_{i}λ_{i} where y_{i} > 0 for i = 1, …, r. Therefore we obtain

∑_{i=1}^{r} y_{i}(λ_{i}, d(C) − y) ≥ 0.

On the other hand y ∈ d(F) implies by definition of d(F) (λ_{i}, d(C) − y) ≤ 0 for i = 1, …, r. This forces (λ_{i}, d(C) − y) = 0 for i = 1, …, r. This implies y ∈ ⋂_{λ∈vert(F)} H(λ). In particular y ∈ d(F) ∩ F, so F is special and y = y(F). □

(3.10) Corollary. A root system (V, Φ) with complementary polyhedron d : C → V^{∗} has a unique special facet.

(3.11) Corollary. Suppose F′ ∈ F(V, Φ) such that F′ ⊂ F̄ and such that for every λ ∈ vert(F) \ vert(F′) n(F; λ) ≤ 0. Then deg(F) ≤ deg(F′) holds.

Proof. Reducing to F′ as described in 3.7, one can assume that F′ = {0}. Therefore F is a facet all of whose numerical invariants satisfy n(F; λ) ≤ 0 for λ ∈ vert(F). Denote x = ∑_{α∈R(F)} α.

We obtain for C ∈ C(V, Φ), C̄ ⊇ F:

deg(F) = ⟨x, d(C)
= ∑_{λ∈vert(F)} \frac{n(F; λ)}{#Ψ(F; λ)}⟨x, λ̌⟩

By Proposition we have that ⟨x, λ̌⟩ ≤ 0 for all λ ∈ vert(F). Thus deg(F) ≤ 0.

(3.12) Corollary. The special facet F of (Φ, d) is the largest element in the set of facets of maximal degree of (Φ, d).

4. The local situation

In this section F denotes a nondiscrete locally compact topological field of arbitrary characteristic. Of course later on in this paper we are only interested in the case char(F) = p > 0. | | : F → ℝ denotes the multiplicatively written discrete valuation uniquely determined up to equivalence which is associated with the topology of F ([We], I.2., Th. 4, I.3., Th. 5) o = {x ∈ F : |x| ≤ 1} is the valuation ring in F, (π) = πo the maximal ideal of o with some choice of a uniformising element π ∈ o. G denotes a split reductive algebraic group over F, T_{0} a split maximal torus over F in G, B_{0} ⊇ T_{0} a Borel subgroup. Φ = Φ(G; T_{0}) is the system of roots of G with respect to T_{0}, Φ ⊂ X^{∗}(T_{0}), where X^{∗}(T_{0}) is the character group of T_{0}, Φ_{+} is the system of positive roots in Φ corresponding to the pair (B_{0}, T_{0}), Δ ⊂ Φ_{+} a basis of the root system Φ. X_{∗}(T_{0}) = Hom(𝔾_{m}, T_{0}) is the group of one-parameter subgroups of T_{0}, \underline{\underline{A}}_{0} := X_{∗}(T_{0}) ⊗_{ℤ} ℝ the associated real vector space. T_{0}(F) acts on \underline{\underline{A}}_{0} by translations using the homomorphism

ν : T_{0}(F) → X_{∗}(T_{0}) ⊗_{ℤ} ℝ

uniquely determined by the requirement

|χ(t)|_{F} = |π|_{F}^{⟨ν(t),χ⟩}

for any character χ ∈ X^{∗}(T_{0}), t ∈ T_{0}(F) and π ∈ o a uniformising element of F as above. This action of T_{0}(F) on A_{0} by translations via ν extends to an action of the F - rational points of the normalizer N(T_{0}) of T_{0}, N(T_{0})(F), by affine transformations on \underline{\underline{A}}_{0} again denoted by ν. ([La], Prop. 1.8, [Ti], 1.1.(5)).

We fix now a Chevalley(-Steinberg) system for (G|F; T_{0}) in the sense of ([La], p. 38, Def. 4.3, 4.4). So this is a system of F -isomorphisms

χ_{α} : 𝔾_{a}|F → U_{α}|F

for the root subgroups U_{α}, α ∈ Φ(G; T_{0}), such that the following properties hold: i) χ_{α}, χ_{−α} are associated, that is, there is an F -group homomorphism

ε_{α} : SL(2)|F → G|F

such that for all u ∈ 𝔾_{a}(F) = F the following conditions hold:

χ_{α}(u) = ε_{α} ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) and
χ_{−α}(u) = ε_{α} ( \begin{matrix}1 & 0 \\ −u & 1 \end{matrix} ) .

Denote

m_{α} := χ_{α}(1)χ_{α}(−1)χ_{α}(1)
= ε_{α} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )

for the following. ii) For all α, β ∈ Φ(G; T_{0}) there exists ε ∈ {±1} such that for all u ∈ 𝔾_{a}(F) =

F the following condition holds:

χ_{s_{α}(β)}(u) = m_{α}χ_{β}(εu)m_{α}^{−1},

where s_{α} : X^{∗}(T)_{∗} ⊗_{ℤ} ℝ → X^{∗}(T) ⊗_{ℤ} ℝ is the reflection given by α. To be able to define the Bruhat-Tits building associated to G|F, we need several subgroups of G to be introduced now. Given x ∈ \underline{\underline{A}}_{0} = X_{∗}(T_{0}) ⊗_{ℤ} ℝ, we define U_{α,x} := \{u ∈ U_{α}(F) : |χ_{α}^{−1}(u)|_{F} ≤ |π|_{F}^{−⟨x,α⟩}}, (α ∈ Φ(G; T_{0})). Furthermore

U_{x} := ⟨U_{α,x} : α ∈ Φ⟩ ⊆ G = G(F),

the group generated by all U_{α,x}, α ∈ Φ. U_{x}^{±} := U_{x} ∩ U^{±}, where U^{+} resp. U^{−} is the unipotent subgroup of B resp. of the opposite Borel subgroup B^{−}. Finally P_{x} = ⟨U_{x}, N_{x}⟩, the subgroup of G generated by U_{x} and N_{x} := {n ∈ N(T_{0})(F) : ν(n)x = x}. We can recall now the definition of the realisation of the Bruhat-Tits building BT (G; F), |BT (G; F)|:

(4.1) Definition: |BT (G; F)| := G(F) × \underline{\underline{A}}_{0}/ ∼, where the equivalence relation “∼” is given as follows: for (g, x), (h, y) ∈ G(F) × \underline{\underline{A}}_{0}, (g, x) ∼ (h, y) iff there exists an n ∈ N(T_{0})(F) such that y = ν(n)x in \underline{\underline{A}}_{0} and furthermore g^{−1}hn ∈ U_{x}.

The polysimplicial complex BT (G; F) related to the realisation |BT (G; F)| above can be described as follows: One has the affine hyperplanes in \underline{\underline{A}}_{0} given by equations

H(α, l) := \{x ∈ \underline{\underline{A}}_{0} : ⟨x, α⟩ = l\}

where α ∈ Φ(G; T_{0}), l ∈ ℤ log |π|. This induces a decomposition of \underline{\underline{A}}_{0} into facets, which can be transported to all of |BT (G; F)| using the action of G. It induces the polysimplicial structure of BT (G; F). Furthermore |BT (G; F)| itself can be equipped with a G-equivariant metric, such that in particular the appartments \underline{\underline{A}} = g\underline{\underline{A}}_{0}, g ∈ G, are isometric to Euclidean spaces ([Ti] 2.3., [La] 9.15).

(4.2) Proposition: ([La], Prop. 9.3) One has

P_{x} = {g ∈ G = G(F) : gx = x}

We will need later on the functorial behavior of the building under field extension. So suppose that F ↪ F′ ↪ F′′ are finite field extensions of local fields.

(4.3) Theorem: ([Ti] 2.6) There is a unique system of injections

∼_{F′↪F′′}: |BT (G ×_{F} F′; F′)| → |BT (G ×_{F} F′; F′′)|

satisfying the following properties:

i) If F′ ↪ F′′ is a Galois extension of F′, the image Im(∼_{F′↪F′′}) is pointwise fixed under the Galois group Gal(F′′/F′).

ii) The restriction of i_{F′↪F′′} to any appartment of |BT (G ×_{F} F′; F′)| is an

affine mapping into an appartment of |BT (G ×_{F} F′′; F′′)|. iii) ∼_{F′↪F′′} is an G(F′)-equivariant embedding. iv) For extensions F′ ↪ F′′ ↪ F′′′ the relation

∼_{F′′↪F′′′} ∘ ∼_{F′↪F′′}=∼_{F′↪F′′′}

holds. v) If F′ ↪ F′′ is an unramified (or at least tamely ramified) Galois extension

∼_{F′↪F′′}: |BT (G ×_{F} F′; F′)| → |BT (G ×_{F} F′′; F′′)|^{Gal(F′′/F′)}

is a homeomorphism and even an isometry. Suppose now, that Ω is a nonempty subset of an appartment |\underline{\underline{A}}| ⊂ |BT (G; F)| such that the projection pr_{ss}(Ω) to the semisimple part of |BT (G; F)| is bounded. Associated with Ω is a smooth group scheme G(Ω), defined over the valuation ring o ⊂ F, whose properties are described below and whose construction is given at least for the case Ω = {x} a point in |BT (G; F)|. For the general construction see ([La], 5.17,6.1)

i) One has canonical isomorphisms

G(Ω) ×_{o} F →^{∼} G|F

ii) For every unramified finite extension F′ of F with ring of integers o′ ⊂ F′, the (abstract) group G(Ω)(o′) equals the pointwise stabiliser G(F′)^{Ω′} of Ω′, where Ω′ is the image of Ω under the canonical embedding |BT (G; F)| ↪ |BT (G; F′)|.

iii) Suppose , F′, o′, Ω′ as above. Then G(Ω′) (with generic fibre G ×_{F} F′) is

canonically isomorphic to G(Ω) ×_{o} o′. iv) G(Ω) is determined up to unique isomorphism over o.

v) If G|F is a split group (the case, we are only considering in the moment) and x ∈ |BT (G; F)| is a special point, G(x) is a Chevalley group scheme over o. We indicate the construction of G(x). We assume that x ∈ |\underline{\underline{A}}_{0}|, the standard appartment. The case of arbitrary x ∈ |BT (G; F)| is reduced to this making use of the action of G(F) in the obvious way.

1) For any α ∈ Φ(G; T_{0}) there exists a unique smooth, affine group scheme U_{α}(x) over o, of finite type, with generic fibre U_{α}|F and

U_{α}(x)(o) = U_{α,x} = {χ_{α}(u) ∈ U_{α}(F) : |u|_{F} ≤ |π|_{F}^{−⟨x,α⟩}}.

U_{α}(x) is determined by these properties up to unique isomorphism. ([La], Prop. 4.12).

2) There exist smooth, affine o-group schemes U^{±}(x) of finite tpye with generic fibre U^{±}|F and the property that for any good ordering of Φ^{±}, the F -isomorphism

∏_{α∈Φ^{±}} U_{α} →^{∼} U^{±}

can be extended to an o-isomorphism

∏_{α∈Φ^{±}} U_{α}(x) →^{∼} U^{±}(x)

([La], 5.6, for the concept of a good ordering see [La], 0.15 resp. [BT I], 1.3.15).

3) For the torus T_{0}|F we denote T_{0}|o the unique (up to unique isomorphism) extension to a torus group scheme over o. Then there exists a unique rational group law on U^{−}(x) × T_{0} × U^{+}(x), which extends the rational group law on U^{−} × T_{0} × U^{+}, coming from group law on the algebraic group G|F via the open embedding

U^{−} × T_{0} × U^{+} → G

over F.

4) There is a unique smooth group scheme G(x) over o, which has generic fibre G|F and is a smooth model of the rational o-group U^{−}(x) × T_{0} × U^{+}(x). Associated with the group scheme G(x) whose construction was just indicated, is the Lie algebra o-lattice

Lie G(x) ⊂ Lie(G|F).

If x is additionally a special vertex of BT (G; F), we can associate with the lattice LieG(x) a real valued norm ∥ ∥_{x} on Lie(G|F) by the procedure of section 1, such that the unit ball of (Lie(G|F), ∥ ∥_{x}) is exactly Lie G(x). To construct such a real valued norm for arbitrary points x ∈ |BT (G; F)| we have to proceed as follows. We first assume that x ∈ |\underline{\underline{A}}_{0}| = X_{∗}(T_{0}) ⊗_{ℤ} ℝ ⊂ |BT (G; F)|. Because G|F is a split reductive algebraic group we have the decomposition of the Lie algebra Lie(G|F) with respect to the torus T_{0} as follows:

Lie(G|F) = Lie(T_{0}) ⊕ ⨁_{α∈Φ} \underline{u}_{α} ,

where the root spaces \underline{u}_{α} = Lie(U_{α}) are one-dimensional F -vector spaces on which T_{0}(F) is acting via the adjoint representation by multiplication with the character α : T_{0} → 𝔾_{m}. \underline{t}_{0} = Lie(T_{0}), as an o-lattice in Lie(T_{0}|F), gives a real valual norm ∥ ∥_{0,\underline{t}_{0}} on Lie(T_{0}|F). ∥ ∥_{0,\underline{u}_{α}} is the norm given by the lattice \underline{u}_{α} = Lie U_{α}(o) ⊂ Lie(U_{α}|F) for α ∈ Φ(G; T_{0}). For x ∈ |\underline{\underline{A}}_{0}| = X_{∗}(T_{0}) ⊗_{ℤ} ℝ arbitrary, we define

∥ ∥_{x,\underline{u}_{α}} := ∥ ∥_{0,\underline{u}_{α}} · |π|^{⟨α,x⟩}
∥ ∥_{x,\underline{t}_{0}} := ∥ ∥_{0,\underline{t}_{0}}

(4.4) Definition: The norm ∥ ∥_{x} on Lie(G|F) is the orthogonal sum

(Lie(G); ∥ ∥_{x}) = ⨁_{α∈Φ}(\underline{u}_{α}; ∥ ∥_{x,\underline{u}_{α}}) ⊕ (\underline{t}_{0}∥ ∥_{x,\underline{t}_{0}})

(4.5) Proposition: The stabilizer P_{x} fixes the norm ∥ ∥_{x} on Lie(G|F) under the adjoint action.

Proof. This is a straightforward, though somewhat lengthy computation. By definition we have P_{x} = ⟨U_{x}; N_{x}⟩, U_{x} = ⟨U_{α,x} : α ∈ Φ⟩ and N_{x} = {n ∈ N(F) : ν(n)x = x}. Therefore it is enough to show

(1) All groups U_{α,x}, α ∈ Φ, are fixing ∥ ∥_{x}.

(2) N_{x} fixes ∥ ∥_{x}. As this is a group action, it will be sufficient to show that all the groups U_{α,x}, N_{x} are norm contracting with respect to ∥ ∥_{x}. Below we will need the elements X_{α} := Lie(χ_{α})(1) ∈ Lie(U_{α}) ⊂ Lie(G) as well as H_{α} := −[X_{α}, X_{−α}] (α ∈ Φ).

Ad (1) We consider χ_{α}(u) ∈ U_{α,x}, X_{β} as above for α, β ∈ Φ. We have to show

∥Ad(χ_{α}(u))(X_{β})∥_{x} ≤ ∥X_{β}∥_{x}

By definition we have

U_{α,x} = {χ_{α}(u) : |u| ≤ |π|^{⟨x,α⟩}}

By ([BTII], 3.2.5., p. 54) we have the commutator formulas

(i) Ad(χ_{α}(u))(X_{β}) = X_{β} + ∑_{i>0} c_{i,α,β}u^{i}X_{iα+β} (α ≠ −β) resp.

(ii) Ad(χ_{α}(u))(X_{−α}) = X_{−α} − u H_{α} + u^{2}X_{α} and finally (iii) Ad(χ_{α}(u))(H) = H − α(H)uX_{α}, where in the formulas above c_{i,α,β} ∈ ℤ

(or more precisely the image of ℤ in F). Now we have in the case α ≠ −β

∥c_{i,α,β}u^{i}X_{iα+β}∥_{x} ≤ |u|^{i}∥X_{iα+β}∥_{x}
= |u|^{i}∥X_{iα+β}∥_{0}|π|^{⟨x,iα+β⟩}
= |u|^{i}|π|^{i⟨x,α⟩}|π|^{x,β⟩}
≤ |π|^{⟨x,β⟩} because |u| ≤ |π|^{−⟨x,α⟩})
= ∥X_{β}∥_{0}|π|^{⟨x,β⟩}
= ∥X_{β}∥_{x}.

Similarly, we compute

∥Ad(χ_{α}(u))(X_{−α})∥_{x} = ∥X_{−α} − uH_{α} + u^{2}X_{α}∥_{x}

We have

∥X_{−α}∥_{x} = ∥X_{−α}∥_{0}|π|^{−⟨x,α⟩}
= |π|^{−⟨x,α⟩}
∥uH_{α}∥_{x} = ∥uH_{α}∥_{0}
= |u|∥H_{α}∥_{0}
≤ |u| ≤ |π|^{−⟨x,α⟩},

because H_{α} ∈ \underline{t} = Lie(T_{0}), therefore ∥H_{α}∥_{0} ≤ 1. (In fact = 1). Finally,

∥u^{2}X_{α}∥_{x} = |u|^{2}∥X_{α}∥_{0}|π|^{⟨x,α⟩}
≤ |π|^{−⟨x,α⟩}.

By the sharp triangle inequality we obtain

∥Ad(χ_{α}(u))(X_{−α})∥_{x} ≤ |π|^{−⟨x,α⟩}.

On the other hand we have

∥X_{−α}∥_{x} = |π|^{−⟨x,α⟩},

which shows the claim in this case. To finish case 1), we have to show ∥Adχ_{α}(u)(H)∥_{x} ≤ ∥H∥_{x} for α ∈ Φ, H ∈ Lie(T). By the commutator formulas above we have

Ad(χ_{α}(u))(H) = H − ⟨H, α⟩uX_{α}.

To show our claim in this situation, it is of course enough to show:

∥⟨H, α⟩uX_{α}∥_{x} ≤ ∥H∥_{x} = ∥H∥_{0}.

But we have

∥⟨H, α⟩uX_{α}∥_{x} = |⟨H, α⟩||u||π|^{⟨x,α⟩}.

We have the character

α : T_{0}|F → 𝔾_{m}|F,

inducing

Lie(α) : Lie(T_{0}) → Lie(𝔾_{m}) = F,

canonically. Therefore,

|Lie(α)(H)| = |⟨H, α⟩| ≤ ∥H∥_{0},

as α(T_{0}) ⊆ 𝔾_{m}(o), that is, α respects the underlying integer structure. Our claim follows also in this case and we have shown 1) of Proposition 4.

Ad (2). We have to show for n ∈ N_{x} = N ∩ P_{x}

∥Ad(n)(v)∥_{x} ≤ ∥v∥_{x}

for v ∈ Lie(G|F). It is sufficient to do the cases

i) v = X_{α} ∈ Lie(G|F)

ii) v = H ∈ Lie(T_{0}|F) ⊂ Lie(G|F).

From these ii) is immediate, because quite generally N(T_{0}) is acting via the Weyl group N(T_{0})/T_{0} and respects the integer structure Lie(T_{0}) ⊂ Lie(T_{0}|F), which defines the norm on Lie(T_{0}|F). We show i): We have for n ∈ N_{x} = N ∩ P_{x}

∥Ad(n)(X_{α})∥_{x} = ∥Ad(n)(X_{α})∥_{0} · |π|^{⟨x,^{v}ν(n)(α)⟩},

where ^{v}ν(n) is the linear part of the affine map ν(n). Furthermore

∥X_{α}∥_{x} = ∥X_{α}∥_{0} · |π|^{⟨x,α⟩}

and

∥Ad(n)(X_{α})∥_{0} = |π|^{⟨ν(n^{−1})(0)−0,α⟩}

([La], Corollary 7.8 iii)), because additionally ∥X_{α}∥_{0} = ∥X_{v_{ν(n)(α)}}∥_{0} = 1. Therefore it is sufficient for showing ∥Ad(n)(X_{α})∥_{x} = ∥X_{α}∥_{x}, to have for all α ∈ Φ

⟨ν(n^{−1})(0) − 0, α⟩ + ⟨x, ^{v}ν(n)(α)⟩ = ⟨x, α⟩

But this is equivalent to showing (ν(n^{−1})(0)) + ^{v}ν(n^{−1})x = x, which is true because of n ∈ P_{x} we have

x = ν(n^{−1})(x) = ^{v}ν(n^{−1})(x) + (ν(n^{−1})(0) − 0).

This shows Proposition 4.5. □

(4.6) Definition: Suppose x ∈ |BT (G; F)| is an arbitrary point. Then the norm ∥ ∥_{x} on Lie(G|F) is given by transport of structure as g_{∗}∥ ∥_{x_{0}}, where g ∈ G(F) satisfies gx_{0} = x for x_{0} ∈ |\underline{\underline{A}}_{0}|.

Remark: The definition is meanigful by Proposition 4.5.

(4.7) Proposition: Suppose, x ∈ |BT (G; F)| is a special vertex, G(x) the associated group scheme over o, Lie G(x) its Lie algebra over o. We denote by ∥ ∥′_{x} the norm on Lie(G|F) associated with the o-lattice Lie G(x) ⊂ Lie(G|F). Then one has the equality of norms

∥ ∥_{x} = ∥ ∥′_{x}

on Lie(G|F).

Proof. Because both constructions of norms above are equivariant with respect to the action of G, we can assume that x ∈ |\underline{\underline{A}}_{0}|. By ([La], Th. 6.1, p. 61) we have the direct sum decomposition of o-modules of Lie G(x)

Lie G(x) = ⨁_{α∈Φ} \underline{u}_{α,x} ⊕ \underline{t}_{0},

where

\underline{u}_{α,x} = Lie(U_{α,x}), \underline{t}_{0} = Lie(T_{0})

We have the corresponding orthogonal decomposition of ∥ ∥_{x} by definition. Because the restriction ∥ ∥_{x} to Lie(T |F) is given by definition by the o-lattice \underline{t}_{0} ⊂ Lie(T |F), it is sufficient to show the equality of norms

∥ ∥_{x,\underline{u}_{α}} = ∥ ∥′_{x,\underline{u}_{α}} (α ∈ Φ)

on \underline{u}_{α} = Lie(U_{α}|F).

∥ ∥_{x,\underline{u}_{α}} = ∥ ∥_{0,\underline{u}_{α}} · |π|^{⟨x,α⟩}

and ∥ ∥_{0,\underline{u}_{α}} by definition is given by \underline{u}_{α,0} ⊂ \underline{U}_{α} as a sublattice. \underline{u}_{α,x} ⊂ \underline{u}_{α} as a sublattice is given by ([La], Prop. 4.2.4. resp. [BT2], 4.3.5) as φ_{α}^{−1}([−⟨x, α⟩, ∞]), where φ_{α} = ω ∘ χ_{α}^{−1} : U_{α}(F) → ℝ ∪ {∞}, ω = (log ∘| |)(log |π|)^{−1}. Because x is a special point, we can find u ∈ F, such that |u| = |π|^{−⟨x,α⟩}. Then \underline{u}_{α,x} = u·\underline{u}_{α,0} ⊂ \underline{u}_{α} and also \{v ∈ \underline{u}_{α} : ∥v∥_{x} ≤ 1\} = u·u_{α,0}.□

5. The global situation

In this section we return to our original notations from sections 1 – 3. So in particular F is again a global function field with field of constants k = 𝔽_{q}, X denotes the associated smooth projective curve over k, G is a split reductive algebraic group over F, T_{0} denotes a fixed split maximal torus over F. For v ∈ |X|, the set of closed points of X, F_{v} is the completion of F at v. The set of valuations {| |_{v} : v ∈ |X|} on the set of F_{v} is chosen in such a way that the product formula holds: for a ∈ F, a ≠ 0, one has:

∏_{v∈|X|} |a|_{v} = 1

We consider the various Bruhat-Tits buildings BT (G ×_{F} F_{v}; F_{v}) (v ∈ |X|) constructed in section 4 for G ⊗_{F} F_{v} with respect to the split torus T_{0} ⊗_{F} F_{v}. From now on for simplicity we write BT (G; F_{v}) instead of BT (G ×_{F} F_{v}; F_{v}). Spec(O) denotes an open affine part of X, X \ Spec(O) =: S the complementary finite set of closed points in X.

(5.1) Definition: An integer structure G = (G|O; x_{v} for v ∈ S) for G|F consists of the following data:

i) a reductive group scheme G|O over Spec(O); ii) for any v ∈ S a point

x_{v} ∈ |BT (G; F_{v})|

Remark. For any closed point w ∈ Spec(O), the smooth reductive group scheme G|O determines by restriction a smooth reductive group scheme G|O_{w} = (G|O)⊗_{O} O_{w} over the valuation ring O_{w} ⊂ F_{w}. This determines a special point x_{w} ∈ |BT (G ×_{F} F_{w}; F_{w})|. Conversely, a special point x_{w} ∈ |BT (G ×_{F} F_{w}; F_{w})| determines a reductive group scheme G|O_{w}. In this sense the integer structures on G|F, namely

(G|O; x_{v} for v ∈ S)

and

(G|Spec(O) \ {w}; {x_{v} : v ∈ S} ∪ {x_{w}})

are considered to be equivalent as are all the integer structures which are obtained by a repeated application of the procedure above. (5.2) Definition and Proposition. An integer structure (G|O; x_{v} for v ∈ S) determines the structure of an Arakelov bundle on Lie(G|F), given as Lie(G|O)|; ∥ ∥_{x_{v}} for v ∈ S), which is compatible with the equivalence relation described above. Proof. (Lie(G|O) ↪ Lie(G|F) gives an O-lattice in Lie(G|F), the ∥ ∥_{x_{v}} are norms on Lie(G ×_{F} F_{v}) for v ∈ |X|, as described in section 4. The compatibility with the equivalence relation for integer structures is explained in Proposition 4.6. □ Suppose now, P ⊂ G is a parabolic subgroup, defined over F. (G|O, x_{v} for v ∈ S) denotes an integer structure on G|F. By [De-Gr], the parabolic subgroup P extends in a unique way to a parabolic subgroup scheme P|O ↪ G|O. In particular this has as unipotent radical the subgroup scheme Rad_{u}(P) ⊂ P and associated reductive group scheme P/Rad_{u}(P) =: H.

Lie(Rad_{u}P) =: rad_{u}(P) ⊂ rad_{u}(P) = Lie(Rad_{u}(P))

Furthermore the real valued norms ∥ ∥_{x_{v}} on Lie(G⊗_{F} F_{v}) (v ∈ S) can be restricted to rad_{u}(P) ⊗_{F} F_{v}. They are denoted for simplicity again by ∥ ∥_{x_{v}}. In this way we obtain the Arakelov bundle

(rad_{u}(P); ∥ ∥_{x_{v}} for v ∈ S))

with underlying F -vector space rad_{u}(P). We have obtained (5.3) Proposition: Given an integer structure

G = (G|O; x_{v} ∈ |BT (G; F_{v})| for v ∈ S)

and an F -rational parabolic subgroup P ⊂ G. Then there is associated an Arakelov bundle \underline{\underline{rad}}_{u}(P) = (rad_{u}(P); ∥ ∥_{x_{v}} for v ∈ S) with underlying F -rational vector space rad_{u}(P). This is an Arakelov subbundle of (Lie(G)|O); ∥ ∥_{x_{v}} for v ∈ S) with underlying F -vector space Lie(G|F) By ([De-Gr]; exp. XXV, 2.1) we have a filtration of Rad_{u}(P) by normal unipotent subgroup schemes

Rad_{u}(P) = U_{0} ⊃ U_{1} ⊃ … ⊃ U_{r} = (1)

which extends the corresponding filtration of Rad_{u}(P) by unipotent subgroups U_{i} over F, i = 0, …, r. This filtration can be described as follows: suppose, T ⊂ P is a split maximal torus, B a Borel group such that T ⊂ B ⊂ P holds, furthermore all three groups are supposed to be defined over F. {α_{1}, …, α_{n}} denotes a basis of the root system Φ(G, T), associated with T and B. The Dynkin diagram Dyn(G; T) is given by the set {α_{1}, …, α_{n}} together with the usual extra structure. ([Bo], IV,1.4.7). H := P/Rad_{u}(P) is a split reductive group over F with split maximal torus T̄ isomorphic to T under the projection homomorphism P → P/Rad_{u}(P). We assume that the Dynkin diagram of (P/Rad_{u}(P); T̄) is given by the set of roots {α_{r+1}, …, α_{n}}.

(5.4) Definition: The type of the parabolic subgroup P is

t(P) := {α_{1}, …, α_{r}}

Remarks. Of course for this definition we have to specify also the torus T. The type gives the roots not occuring in the Dynkin diagram of H. In particular for the case P = B of a Borel subgroup we have t(D) = {α_{1}, …, α_{n}}. {λ̌_{1} …, λ̌_{r}} is the set of coweights corresponding to the set of roots t(P) = {α_{1}, …, α_{r}}. We consider the free abelian group ⨁_{i=1}^{r} ℤλ̌_{i}.

(5.5) Definition:

i) A vector

v = ∑_{i=1}^{r} a_{i}λ̌_{i} ∈ ⨁_{i=1}^{r} ℤλ̌_{i}

is called positive iff a_{i} ≥ 0 for i = 1, …, r. ii) For a positive vector v we denote

Ω(v) := {α ∈ Φ_{+}(G; T) : ⟨α, λ̌_{i}⟩ ≥ a_{i} for i = 1, …, r}

iii) U(v_{;}G; T) is the unipotent algebraic subgroup generated by the root sub-

groups U_{α}, α ∈ Ω(v). iv) W (P; v) := U(v)/⟨U(v′) : v′ > v⟩ is the vector space group, given as the

algebraic quotient group of U(v) by the subgroup generated by all U(v′)

for all vectors v′ > v that is v′ = ∑_{i=1}^{r} a′_{i}λ̌_{i}, a′_{i} ≥ a_{i} for i = 1, …, r and

v′ ≠ v.

(5.6) Definition:

i) For α ∈ Φ(G; T), the height is defined as ht(α) := ∑_{i=1}^{r}⟨α, λ̌_{i}⟩.

ii) U_{j} := ⟨U_{α} ⊂ Rad_{u}(P) : ht(α) ≥ j⟩, the algebraic subgroup of Rad_{u}(P) generated by all root subgroups U_{α}, ht(α) ≥ j.

Remarks:

i) One has the decomposition

U_{j}/U_{j+1} = ∏_{v∈T_{j+1}} W (P; v)

where

T_{j+1} := {v = ∑_{i=1}^{r} a_{i}λ̌_{i} : ∑_{i=1}^{r} a_{i} = j + 1}

ii) The definition of all these unipotent groups depends only upon the pair

B ⊂ P and is independent of the choice of the split torus T ⊂ B. iii) All these groups can be extended over Spec(O) for a given reductive group.

The Arakelov bundle structure (rad_{u}(P); ∥ ∥_{x_{v}} for v ∈ S) on rad_{u}(P) induces Arakelov bundles on Lie(U_{j}) = \underline{u}_{j}, Lie(U_{j}/U_{j+1}) = \underline{u}_{j}/\underline{u}_{j+1} and also on W = w(P; v) := LieW (P; v) which we denote \underline{\underline{u}}_{j}, \underline{\underline{u}}_{j}/\underline{\underline{u}}_{j+1} and \underline{\underline{w}}(P; v). We have an orthogonal sum decomposition

\underline{\underline{u}}_{j}/\underline{\underline{u}}_{j+1} = ⨁_{v∈T_{j+1}} \underline{\underline{w}}(P; v)

(5.7) Definition: For a positive v ∈ ⨁_{i=1}^{r} ℤλ̌_{i} one denotes

n(P; v) := deg(\underline{\underline{w}}((P; v)).

In particular for v = λ̌_{i} one writes

n(P; v) =: n(P; λ̌_{i})

Of course the degree depends on the integer structure G under consideration.

We give now a different construction of the invariants n(B; v) for the case of a Borel group B. Again we start with the integer structure G, additionally we fix a Borel group B and a maximal split torus T ⊂ B, all defined over F. (5.8) Definition: For a root α ∈ Φ(G; T) one denotes

i) in case α > 0:

V (α) := ⨁_{β≥α} \underline{u}_{β}; V′(α) := ⨁_{β>α} \underline{u}_{β}

ii) in case α < 0:

V (α) := Lie(T) ⊕ ⨁_{β≮α} \underline{u}_{β}
V′(α) := Lie(T) ⊕ ⨁_{β≰α} \underline{u}_{β}

In the usual way we extend the linear subspaces of Lie(G) above to an Arakelov bundle V(α) and its subbundle V′(α), both subbundles of the Arakelov bundle Lie(G). The quotient bundle V(α)/V′(α) obviously is an Arakelov line bundle over X. (5.9) Proposition and Definition: For a root α > 0 there is a canonical isomorphism

\underline{\underline{u}}_{α} →^{∼} V(α)/V′(α)

For α < 0, this will serve as a definition of \underline{\underline{u}}_{α}. Proof. Denoting j := ht(α) there are injective homomorphisms

V (α) → Lie(U_{j})
V′(α) → Lie(U_{j+1})

inducing corresponding embeddings of the associated Arakelov bundles. This induces a corresponding homomorphism

\underline{\underline{u}}_{α} → ⨁_{v∈T_{j+1}} \underline{\underline{w}}(P; v) (orthogonal sum!)

and an isomorphism of line bundles

\underline{\underline{u}}_{α} →^{∼} \underline{\underline{w}}(P; v)

for the v corresponding to α. □

(5.10) Proposition: Suppose, G = (G|O; x_{v} for v ∈ S) is an integer structure on G|F, B ⊂ G a Borel subgroup over F. Then one has canonical isomorphisms as Arakelov line bundles

i) \underline{\underline{u}}_{α} ⊗ \underline{\underline{u}}_{β} →^{∼} \underline{\underline{u}}_{α+β} (α, β and α + β ∈ Φ(G; T))

ii) \underline{\underline{u}}_{α} ⊗ \underline{\underline{u}}_{−α} →^{∼} \underline{O}, the trivial rank one bundle.

Proof. 1) We assume first, that the characteristic char(F) = p > 5, avoiding the characteristics 2, 3, 5. As is well known from the usual commutator formulas in Lie theory, there is an isomorphism for the case α, β, α + β ∈ Φ(G; T) induced by forming commutators

(X̄_{α}, Ȳ_{β},) ↦ [\overline{X_{α}, Y_{β}}]

This shows i) under the assumption p > 5. To show ii), it suffices by i) to do this for a simple positive root α ∈ Φ_{+}(G; T). One considers the minimal parabolic subgroup P_{α} ⊃_{≠} B, such that one has

P_{α} ⊃_{≠} B ⊃_{≠} Rad_{u}(P) ⊃_{≠} Rad_{u}(P_{α}).

P_{α}/Rad_{u}(P_{α}) is a reductive group over F, the quotient of this by its center is a group isomorphic to P GL_{2}|F. The integer structure G induces in the usual way an integer structure on the group P GL_{2}|F. As can be checked easily, this integer structure is given by an Arakelov bundle E of rank two on X. Also the Arakelov bundle E is uniquely determined up to similarity. The Borel group B above induces a subbundle L ↪ E and we obtain an exact sequence of bundles

0 → L → E → E/L → 0.

It is easy to see that the Arakelov line bundle \underline{\underline{u}}_{α} is given as Hom(E/L, L), similarly \underline{\underline{u}}_{−α} as Hom(E/L, L). From this and the canonical isomorphism Hom(E/L, L)^{∗} →^{∼} Hom(E/L, L) the result follows.

2) With a different argument we show the proposition now for arbitrary characteristic p > 0. So, we assume given an integer structure G = (G|O; x_{v} ∈ |BT (G; F_{v})| for v ∈ S) and a Borel subgroup B ⊂ G, defined over F. We will compare this situation with a corresponding integer structure G^{(0)} = (G^{(0)}|O); x_{v}^{(0)} ∈ |BT (G; F_{v})| for v ∈ S), B^{0} ⊃ T_{0} fixed, directly related to the standard appartment which will be constructed below.

As a first step, we can choose g ∈ G(F) such that gBg^{−1} = B_{0}. Additionally, by enlarging the set S in the sense of the definition of integer structures, we can assume that G|O = G^{(0)}|O. Next, as a purely local problem, we construct for v ∈ S, x_{v}^{(0)} ∈ ∥\underline{\underline{A}}_{0}∥, b_{v} ∈ B_{0}(F_{v}), such that b_{v}x_{v}^{(0)} = x_{v} holds. To do this, we can find a maximal split torus T^{(v)} ⊂ G|F_{v} over F_{v}, such that B_{0} ⊃ T^{(v)} holds and additionally x_{v} ∈ X_{∗}(T^{(v)}) ⊗_{ℤ} ℝ =: |\underline{\underline{A}}|, the appartment corresponding to the torus T^{(v)}. This is possible by ([B-T1], 2.9.4 and 4.2.5.)

As the maximal split tori T^{(v)}, T_{0} ⊂ B_{0} are conjugate to each other under an element b_{v} ∈ G(F_{v}), such that b_{v}T_{0}b_{v}^{−1} = T^{(v)} holds, we pose x_{v}^{(0)} := b_{v}^{−1}x_{v} ∈ |\underline{\underline{A}}_{0}|. We define an adele (b_{v}) ∈ B_{0}(𝔸_{F}), given by b_{v} = 1 for v ∉ S, b_{v} as above for v ∈ S. In an obvious sense we have obtained

Ad((b_{v}))(G^{(0)}) = G

Because we have G^{(0)}|O = G|O and B = B_{0}, we can describe for α ∈ Φ(G; T_{0}) the Arakelov line bundles \underline{\underline{u}}_{α}^{(0)} resp. \underline{\underline{u}}_{α} by an invertible O-module L_{α} and real norms ∥ ∥_{x_{v}^{(0)}} resp. ∥ ∥_{x_{v}} on L_{α} ⊗_{O} F_{v} for v ∈ S. It is immediate to check then, that ∥ ∥_{x_{v}} = |α(b_{v})|_{v}∥ ∥_{x_{v}^{(0)}}. Here α(b_{v}) is given as the composition

B_{0}(F_{v}) → B_{0}/Rad_{u}(B_{0})(F_{v}) = T_{0}(F_{v}) →^{α} F_{v}^{×}.

But now, Proposition 5.10. obviously holds true for the integer structure G^{(0)} with respect to B = B_{0}. From this and the formula above the proposition holds true in general. □

(5.11) Definition: G denotes a split reductive algebraic group over the global function field F, S = {v_{1}, …, v_{t}} a finite set of places of F, Spec(O) = X \ S the corresponding open affine part of the smooth projective curve X associated to F, G := (G|O; x_{v} ∈ |BT (G; F_{v})| for v ∈ S) an integer structure on G|F. G is (semi)stable iff for all F -rational proper parabolic subgroups P of G, one has

deg(\underline{\underline{rad}}_{u}(P)) < 0 (resp. ≤ 0),

where \underline{\underline{rad}}_{u}(P)) is the Arakelov bundle associated to rad_{u}(P) and G. T ⊂ G|F is a split maximal torus of G, defined over F. We will associate a complementary polyhedron with the root system Φ(G; T), using the given integer structure G. Suppose C is a chamber in C(X^{∗}(T) ⊗_{ℤ} ℝ; Φ(G; T)) with corresponding Borel subgroup B ⊂ G, {α_{1}, …, α_{n}} the basis of Φ(G; T) corresponding to B. We have the orthogonal decomposition

\underline{\underline{u}}_{0}/\underline{\underline{u}}_{1} = \underline{\underline{rad}}_{u}(P) = ⨁_{i=1}^{n} \underline{\underline{u}}_{α_{i}},

where \underline{\underline{u}}_{α_{i}} = \underline{\underline{w}}(B; λ̌_{i}).

(5.12) Definition: One denotes

d(C) := ∑_{i=1}^{n} n(B; λ̌_{i})λ̌_{i}
:= ∑_{i=1}^{n} deg(\underline{\underline{u}}_{α_{i}})λ̌_{i} ∈ X_{∗}(T) ⊗_{ℤ} ℝ

(5.13) Theorem: The map d : C(V, Φ) → V^{∗}, where V := X^{∗}(T) ⊗_{ℤ} ℝ, d as above, is a complementary polyhedron for the root system Φ(G, T) in the sense of 2.6.

Proof. We have to check the two properties 1) and 2) of 2.6. Ad 1): Suppose, λ is a fundamental weight for Φ(G; T), P its associated maximal parabolic subgroup, such that P ⊃ T, B is a Borel subgroup with T ⊂ B ⊂ P. C = C(B) is the chamber in V = X^{∗}(T) ⊗_{ℤ} ℝ associated to B. Suppose, {α_{1}, …, α_{n}} is again the basis of Φ(G; T) corresponding to B such that t(P) = {α_{1}}. {λ̌_{1}, …, λ̌_{n}} denotes the set of coweights. We obtain

deg(\underline{\underline{rad}}_{u}(P))
= ∑_{\underline{u}_{α}⊂rad_{u}(P)} deg(\underline{\underline{u}}_{α})
= ∑_{\underline{u}_{α}⊂rad_{u}(P)} ∑_{i=1}^{n}⟨α, λ̌_{i}⟩ deg(\underline{\underline{u}}_{α_{i}})
= ⟨ ∑_{\underline{u}_{α}⊂rad_{u}(P)} α, ∑_{i=1}^{n} deg(\underline{\underline{u}}_{α_{i}})λ̌_{i}⟩
= ⟨ ∑_{\underline{u}_{α}⊂rad_{u}(P)} α, d(C)⟩

But now, Lemma 3.5 shows that

∑_{\underline{u}_{α}⊂rad_{u}(P)} α = c · λ

for some positive real number c, independent of the choice of B satisfying T ⊂ B ⊂ P. Therefore we obtain the relation

c · ⟨λ, d(C)⟩ = deg(\underline{\underline{rad}}_{u}(P)).

It follows that ⟨λ, d(c)⟩ is independent of the choice of B satisfying T ⊂ B ⊂ P. This proves part 1) of Theorem 5.10. Ad 2): Again, we can follow [Be]. Let C, C′ be chambers having precisely (n − 1) vertices in common. We denote {α_{1}, …, α_{n}} the simple roots defined by C, {λ_{1}, …, λ_{n}} = vert(C) the corresponding set of weights which are the vertices of C. Similarly {α′_{1}, …, α′_{n}} resp. {λ′_{1}, …, λ′_{n}} are the roots and weights with respect to C′. B = B(C) and B′ = B(C′) denote the Borel subgroups corresponding to the chambers C, C′. We assume without restriction that C′ = s_{1}(C) for s_{1} the reflection corresponding to α_{1}. So we have s_{1}(x) = x − ⟨x, α̌_{1}⟩α_{1} and therefore the weights s_{1}(λ_{1}), …, s_{1}(λ_{n}) are given as λ_{1} − α_{1}, λ_{2}, …, λ_{n}. Therefore vert(C′) = {λ_{1} − α_{1}, λ_{2}, …, λ_{n}}. s_{1} induces canonically a reflection on V^{∗} which we denote again by s_{1}, and which is just the reflection corresponding to α̌_{1}. Again we have s_{1}(λ̌_{1}) = λ̌_{1} − α̌_{1}, s_{1}(λ̌_{j}) = λ̌_{j} for j = 2, …, n.

By definition we have

d(C′) = ∑_{i=1}^{n} n(C′; λ′_{i})λ̌′_{i}
= n(C′; λ_{1} − α_{1})(λ̌_{1} − α̌_{1})
+ ∑_{i=2}^{n} n(C′; λ_{i})λ̌_{i}.

Therefore we obtain

⟨α_{1}, d(C′)⟩ = n(C′; λ_{1} − α_{1})⟨α_{1}, λ̌_{1} − α̌_{1}⟩
+ ∑_{i=2}^{n} n(C′; λ_{i})⟨α_{1}, λ̌_{i}⟩

Because ⟨α_{1}, λ̌_{j}⟩ = δ_{1,j} holds, we obtain

⟨α, d(C′)⟩ = −n(C′; λ_{1} − α_{1})

By definition again we have

n(C′; λ_{1} − α_{1}) = deg(\underline{\underline{u}}(−α_{1})) = − deg(\underline{\underline{u}}(α_{1}))
= deg(V_{C′}(α_{1})/V′_{C′}(α_{1})).

V_{C′}(α_{1}) is given by the set of roots {α ∈ Φ : α ≮_{C′} α_{1}}, similarly V′_{C′}(α_{1}) is given by the set of roots

{α ∈ Φ : α ≰_{C′} α_{1}}.

Next we show the following claims:

i) {α ∈ Φ : α_{1} ≤_{C} α} ⊂ {α ∈ Φ : α ≮_{C′} α_{1}} ii) {α ∈ Φ : α_{1} <_{C} α} ⊂ {α ∈ Φ : α ≰_{C′} α_{1}}

(Here the added C or C′ denotes which Borel subgroup resp. corresponding order we consider).

Obviously i) implies ii). We show i): We assume α_{1} ≤_{C} α. This holds iff δ_{1,i} ≤ ⟨α_{1}, λ_{i}⟩ for i = 1, …, n. We assume that α ∉ {β ∈ Φ : β ≮_{C′} α_{1}}. That is, we assume that α <_{C′} α_{1}. This implies ⟨α, λ̌_{i}⟩ ≤ ⟨α_{1}, λ̌_{i}⟩ for i = 2, …, n (because λ_{2}, …, λ_{n} ∈ vert(C′)). But then we have ⟨α, λ̌_{i}⟩ ≤ 0 and ⟨α, λ̌_{i}⟩ ≥ 0 for i = 2, …, n, implying ⟨α, λ̌_{i}⟩ = 0 for i = 2, …, n. Therefore α is a scalar multiple of α_{1}, that is, we have α = α_{1} or α = −α_{1}. But α ≠ α_{1}, because α <_{C′} α_{1}. Therefore we have α = −α_{1}. But this contradicts α_{1} ≤_{C} α. The statements i) and ii) above follow. Using this we obtain inclusions

V_{C}(α_{1}) ↪ V_{C′}(α_{1}),
V′_{C}(α_{1}) ↪ V′_{C′}(α_{1}),

This induces a homomorphism of Arakelov line bundles

V_{C}(α_{1})/V′_{C′}(α_{1}) → V_{C′}(α_{1})/V′_{C′}(α_{1}).

This homomorphism is not zero, as it is the identity on the generic fibre. But then we obtain the inequality of degrees

deg(\underline{\underline{u}}_{α_{1}}(B)) = n(C; λ_{1}) = ⟨α_{1}, d(C)⟩
= deg(V_{C}(α_{1})/V′_{C}(α_{1})) ≤ deg(V_{C′}(α_{1})/V′_{C′}(α′))
= deg(\underline{\underline{u}}_{α_{1}}(B′)) = ⟨α_{1}, d(C′)⟩.

This is exactly ii) of Theorem 5. □

We end this chapter with some remarks concerning the functorial behavior of the complementary polyhedra constructed. Assume first, that G = (G|O; x_{v} ∈ BT (G; F_{v}) for v ∈ S) is again an integer structure on G|F. Suppose P ⊂ G is a parabolic subgroup over F, T ⊂ P a maximal split torus of P, which then is also a maximal split torus of G. Forming the associated reductive group H|F, H := P/Rad_{u}(P), T induces a maximal split torus of H. The integer structure G|O induces an integer structure P|O of P |F and also an integer structure H|O as explained earlier. Similarly, the x_{v} ∈ |BT (G; F_{v})| induce a structure of normed vector spaces on Lie(G ⊗_{F} F_{v}) for v ∈ S and thereby on Lie(P ⊗_{F} F_{v}) and also, by forming the quotient, on Lie(H ⊗_{F} F_{v}). Call these norms ∥ ∥x̄_{v} for v ∈ S. Forming our construction of a complementary polyhedron above for (H|F; T) with respect to the integer structure (H|O; ∥ ∥_{x_{v}} for v ∈ S) it is still possible to obtain the complementary polyhedron d_{P} also in this situation, although we have not checked here, that the norms ∥ ∥x̄_{v} actually come from points x̄_{v} ∈ |BT (H; F_{v})| in the sense of the relevant construction of §3.

It is easy to obtain (see [Be], Prop. 6.9):

(5.14) Proposition: The complementary polyhedron d_{P} for (H|F; T) with respect to the integer structure (in the sense explained above) (H|O; ∥ ∥x̄_{v} for v ∈ S) is given also as the reduction in the sense of 2.7., Remarks of the complementary polyhedron d for (G|F; T) with respect to the integer structure G = (G|O; x_{v} ∈ |BT (G; F_{v})| for v ∈ S) constructed in Theorem 5.10.

Finally we remark that the constructions explained in this section, in particular the construction of the complementary polyhedron, are functorial for finite field extension F ↪ F′. Namely given again (G|F; T) with a maximal split torus T, (G|O; x_{v} ∈ BT (G; F_{v}) for v ∈ S) an integer structure. Then we define O′ ⊂ F′ as the integral closure of O in F′. For v ∈ S the valuation | |_{v} on F_{v} can be extended to the finitely many different v′|v of F′ dividing v. Thereby one obtains completions F′_{v′} for v′ ∈ S′ (S′ the set of prolongations of S). By the functoriality of buildings one obtains an embedding

|BT (G; F_{v})| ↪ |BT (G ×_{F} F′; F′_{v′})|

for v ∈ S, v′ ∈ S′. Denote x_{v′} the image of x_{v} under this map.

Finally one has canonical identifications

X_{∗}(T) ⊗_{ℤ} ℝ˜⟶X_{∗}(T′) ⊗_{ℤ} ℝ

for T′ = T ⊗_{F} F′.

(5.15) Proposition: The complementary polyhedra d, d′ formed with respect to (G|F; T) and the integer structure G = (G|O; x_{v} ∈ BT (G; F_{v}) for v ∈ S) resp. (G′|F′; T′) and the extended integer structure above are equal under identification of the vector spaces X_{∗}(T) ⊗_{ℤ} ℝ resp. X_{∗}(T′) ⊗_{ℤ} ℝ.

At the end of this section we discuss the dependence or better independence of our constructions from the choices made for the group G|F. These consisted in the choice of the maximal split torus T_{0}|F and the choice of an “épinglage” that is a Chevalley-Steinberg system of F -rational isomorphisms χ_{α} : 𝔾_{a}|F ˜→U_{α}|F for α ∈ Φ = Φ(T_{0}) ⊂ X^{∗}(T_{0}).

We show first the independance of the invariants from the chosen Chevalley-Steinberg system, which will come down to a simple application of the product formula for F.

(5.16) Proposition: Suppose

G = (G|O; x_{v} ∈ |BT (G; F_{v})| for v ∈ S)

is an integer structure on G|F, B ⊂ G is a Borel subgroup of G defined over F, g ∈ G(F) an element. ^{g}B denotes gBg^{−1}, ^{g}G := (g(G|O)g^{−1}; gx_{v} for v ∈ S) the transformed integer structure. Then for the degrees of the Arakelov line bundles \underline{\underline{u}}_{α} = \underline{\underline{u}}_{α}(B; G) and ^{g}\underline{\underline{u}}_{α} := \underline{\underline{u}}_{gα}(^{g}B,^{g} G),

deg(\underline{\underline{u}}_{α}) = deg(^{g}\underline{\underline{u}}_{α})

holds for all α ∈ Φ(T_{0}).

Proof. Immediate, because g induces an isomorphism of the Arakelov line bundles \underline{\underline{u}}_{α} and ^{g}\underline{\underline{u}}_{α}. □

Remark. The same holds for the invariants associated with F -rational parabolic subgroups of G.

(5.17) Theorem: Suppose, B and G are as above in 5.16. Then the degrees deg \underline{\underline{u}}_{α}(B, \underline{\underline{G}}) for α ∈ Φ(T_{0}) are independent of the choice of the Chevalley-Steinberg system χ_{α} : 𝔾_{a}|F ˜→U_{α}|F (α ∈ Φ(T_{0}))

Proof. Suppose χ_{α}, χ′_{α} : 𝔾_{a}|F ˜→U_{α}|F (α ∈ Φ(T_{0})) are two Chevalley-Steinberg systems. Then there is a system of elements c_{α} ∈ F^{×} (α ∈ Φ(T_{0})), such that c_{α} · χ′_{α} = χ_{α}. For the x_{v} ∈ |BT (G; F_{v})|, (v ∈ S), given by the integer structure G, one can find elements b_{v} ∈ B(F_{v}), such that

b_{v}x_{v} = x_{v}^{(0)}

holds, where

x_{v}^{(0)} ∈ Im(X_{∗}(T_{0}) ⊗_{ℤ} ℝ ↪ |BT (G ×_{F} F_{v}; F_{v})|)

for v ∈ S.

By enlarging the set S in the sense of equivalence of integer structures as described in the remark after 5.1., we can assume that the reductive group scheme G|O is the Chevalley form of G associated with T_{0}, which is additionally independent of the Chevalley-Steinberg systems chosen, when we assume |c_{α}|_{v} = 1 for all v ∉ S, α ∈ Φ(T_{0}). G^{(0)} denotes the integer structure G^{(0)} := (G|O : x_{v}^{(0)} for v ∈ S) on G|F. One has

deg \underline{\underline{u}}_{α}(B; G) = (∏_{v∈S} |α(b_{v})|_{v}) · deg \underline{\underline{u}}_{α}(B; G^{(0)})

and

deg′ \underline{\underline{u}}_{α}(B; G) = (∏_{v∈S} |α(b_{v})|_{v}) · deg′ \underline{\underline{u}}_{α}(B; G^{(0)}))

where deg (resp. deg′( )) is taken with respect to the Chevalley-Steinberg system (χ_{α}) resp. (χ′_{α}). Furthermore one has

deg′(\underline{\underline{u}}_{α}(B; G^{(0)})) = (∏_{v∈S} |c_{α}|_{v}) deg(\underline{\underline{u}}_{α}(B; G))

for all α ∈ Φ(T_{0}). Because

∏_{v∈S} |c_{α}|_{v} = ∏_{v∈|X|} |c_{α}|_{v} = 1,

5.16. follows. □

(5.18) Corollary. Given B, G as above, the degree of the Arakelov line bundle \underline{\underline{u}}_{α}(B; G) is independent of the choice of the maximal split torus T_{0}|F in G|F.

Proof. Suppose, T̃_{0} is another maximal split torus of G defined over F. We can find g ∈ G(F), such that

g T_{0}g^{−1} = T̃_{0}.

{χ_{α} : (α ∈ Φ(T_{0})} denotes again the Chevalley-Steinberg system with respect to T_{0}. Then Int(g) ∘ χ_{α} =: χ̃_{α} (α ∈ Φ(T_{0})) is a Chevalley-Steinberg system for T̃_{0} and it follows from the definition, that deg \underline{\underline{u}}_{α}(B; G) is the same with respect to (T_{0}, (χ_{α})) or with respect to (T̃_{0}; (χ̃_{α})). But as we have seen in 5.16, that deg \underline{\underline{u}}_{α}(B; G) for a fixed split maximal torus does not depend upon the “épinglage” chosen, it follows from the consideration above, that it does not depend upon the split maximal torus chosen. □

Remark. The results 5.16, 5.17 and 5.18 hold equally true for the invariants associated with parabolic subgroup P ⊂ G, defined over F.

6. The canonical parabolic subgroup

The notations of section 5 are used again. We fix an integer structure G = (G|O; x_{v} for v ∈ S) for the split reductive group G over F.

(6.1) Lemma: The set of real numbers \{deg(\underline{\underline{rad}}_{u}(P)) : P ⊂ G a parabolic subgroup} is bounded above and attains its maximum.

Proof. We consider the Arakelov bundle (Lie(G|O); ∥ ∥_{x_{v}} on Lie(G ⊗_{F} F_{v}) for v ∈ S). As the field of constants of F is finite, it follows from [H-J-S], that there are only finitely many subbundles of a given Arakelov bundle with degree larger than a fixed given real number c. □

Remark. It is only here, that it is not immediately possible to generalize our arguments from function fields over a finite field of constants to function fields over an arbitrary field of constants. In [H-J-S], it is shown that the statement of Lemma 6.1 remains true for the case G = GL(d)/F.

(6.2) Theorem: Suppose G|F, G are as above. Assume, G is not semistable with respect to this integer structure. If P ⊂ G is a parabolic subgroup of G over F, such that deg(\underline{\underline{rad}}_{u}(P)) is maximal and additionally P is maximal with this property, then P is canonical.

Proof. Again we follow [Be], Prop. 6.2. If P/Rad_{u}(P) with the induced integer structure is not semistable, there exists a parabolic subgroup Q ⊂ P, such that Q̄ := Q/Rad_{u}(P) is violating the semistability of P/Rad_{u}(P). This means that

deg(\underline{\underline{rad}}_{u}(Q̄)) = deg(\underline{\underline{rad}}_{u}(Q)/\underline{\underline{rad}}_{u}(P)) > 0.

But then

deg(\underline{\underline{rad}}_{u}(Q)) = deg(\underline{\underline{rad}}_{u}(Q̄)) + deg(\underline{\underline{rad}}_{u}(P)) > deg(\underline{\underline{rad}}_{u}(P)),

a contradiction to the choice of P. Next, we consider the unique parabolic subgroup Q ⊃ P, such that the type t(Q) = t(P) \ {α}, α some simple root with respect to an appropriate split maximal torus T contained in P.

We fix a split maximal torus T and a Borel subgroup B, both over F, such that

T ⊂ B ⊂ P ⊂ Q

holds. We denote {α_{1}, …, α_{n}} a basis for Φ(G; T) with respect to T, such that

vert(P) := vert(F (P)) = {λ_{1}, …, λ_{r}}
vert(Q) := vert(F (Q)) = {λ_{2}, …, λ_{r}}

and α = α_{1} above. We have ⟨α_{i}, λ̌_{j}⟩ = δ_{ij} for i, j ∈ {1, …, n} as before. We denote under the decomposition

V = X^{∗}(T) ⊗_{ℤ} ℝ = ⟨F (Q)⟩ ⊕ (F (Q)^{∨})^{⊥}
V (Q) := (F (Q)^{∨})^{⊥} and
pr : V → V (Q)

From the assumptions on P, Q we have

deg(\underline{\underline{rad}}_{u}(P)) > deg(\underline{\underline{rad}}_{u}(Q)),

which is equivalent to deg(\underline{\underline{rad}}_{u}(P̄)) > 0. Using Theorem 5.13 for the root system and complementary polyhedra reduced to Q (that is, everything projected to V (Q)), we have

deg(\underline{\underline{rad}}_{u}(P̄)) = ⟨v̄ := ∑_{\underline{u}_{α}⊂rad_{u}(P̄)} ᾱ, pr(d(C))⟩
= ∑_{λ̄∈vert(pr(F (P)))} \frac{n(P̄; λ̄)}{#Ψ(P̄; λ̄)}⟨v̄, λ̌⟩.

But we have vert(pr(F (P_{0})) = {λ̄_{1}} = {λ_{1}} and ⟨v̄, λ̌_{1}⟩ > 0, therefore

n(P̄; λ̄_{1}) = n(P; λ_{1}) > 0.

This was to be shown. □

(6.3) Theorem: Suppose, G is a reductive group over F, G = (G|O; x_{v} ∈ |BJ (G; F_{v})| for v ∈ S) an integer structure. Assume that the Arakelov group scheme G obtained in this way is not semistable. Then G contains a unique canonical parabolic subgroup P. It is the largest element in the set of parabolic subgroups over F, of maximal degree in G with respect to G.

Proof. Again see ([Be], Theorem 7.3). Suppose P, Q ⊂ G are both canonical parabolic subgroups, defined over F. We find a split maximal torus T over F, such that T ⊂ P, Q. By assumption P, Q are canonical, which implies that the associated facets F (P), F (Q) with respect to Φ(G; T) in X^{∗}(T) ⊗_{ℤ} ℝ are both special in the sense of Definition 3.7. But then Propositions 3.8, 3.9 imply, that F (P) = F (Q) which of course implies P = Q. □

We can generalize our results now immediately to the case of arbitrary reductive group G over F.

Suppose, G = (G|O; x_{v} ∈ |BT (G; F_{v})| for v ∈ S) is an integer structure for G|F. Here we assume in particular, that (G|O) is a reductive group scheme over O. Suppose, F′ ⊇ F is a Galois field extension, over which G is split reductive. Then there is a unique extension of the integer structure G to G′ for G ×_{F} F′ over F′. This is obtained as follows. Denoting O′ ⊇ O the integral closure of O in F′, we can assume by enlarging S, that O′ is not ramified over O. G|O ×_{O} O′ is a reductive split group scheme over O′. Denoting v′|v an extension of the valuation v on F to F′, by functoriality of the building we have maps

\begin{aligned}|BT (G ×_{F} F_{v}; F_{v})| & → & |BT (G ×_{F} F′_{w} : F′_{w})| \\ |BT (G; F_{v})| & → & |BT (G ×_{F} F′; F′_{v′})| \end{aligned}

for v ∈ S, v′|v. Denoting x_{v′|v} the image of x_{v} under this map we obtain the integer structure

G ×_{F} F′ := (G|O ×_{O} O′; x_{v′|v} ∈ |BT (G ×_{F} F′; F′_{v′})| for v ∈ S, v′|v)

Then we obtain

(6.4) Theorem: With the notations above, G ×_{F} F′ is either a semistable Arakelov group scheme with respect to the integer structure G ×_{F} F′ or the canonical parabolic subgroup P ⊂ G ×_{F} F′ from Theorem 6.3. is already defined over F.

Proof. Suppose, T_{0} ⊂ G is a maximal torus of G over F, which splits over F′. If σ ∈ Gal(F′/F) is an element of the Galois group, we obtain

deg(\underline{\underline{rad}}_{u}(P)) = deg(\underline{\underline{rad}}_{u}(^{σ}(P))),

where the degrees are determined by specifying a Chevalley -Steinberg system (χ_{α}) for T_{0} ×_{F} F′ on the left resp. the Chevalley-Steinberg system (^{σ}χ_{α}) on the right. By the functoriality of the construction the equality follows. However, 5.14 gives

deg(\underline{\underline{rad}}_{u}(^{σ}P; ^{σ}(χ_{α}))) = deg(\underline{\underline{rad}}_{u}(^{σ}P; (χ_{α})))

because of the independence of the Chevalley-Steinberg system chosen. As the same result holds true for the other numerical invariants associated with P resp. ^{σ}P, by the uniqueness of the canonical parabolic subgroup, we obtain P = ^{σ}P for all σ ∈ Gal(F′/F). From this follows that the canonical parabolic subgroup P is already defined over F. □

Remark: Theorem 6.4. shows, that the canonical parabolic subgroup P ⊂ G, which is defined over F, can be characterized already over F without going over to a splitting field F′ ⊃ F. P is the unique parabolic subgroup, such that the Arakelov group scheme P/Rad_{u}P (given by P and the integer structure G given)is semistable and P is the largest parabolic such that deg(\underline{\underline{rad}}_{u}(P)) is maximal.

References

[Be] K. Behrend, Semistability of reductive group schemes over curves. Math. Ann. 301, 281-305 (1995).

[Bo] A. Borel, Linear algebraic groups (Second enlarged edition)

Graduate texts in Mathematics 126, Springer-Verlag (1997).

[Bou] N. Bourbaki, Groupes et algébres de Lie, Chapitres 4, 5 etc.

Éléments de Mathematique, Fascicule XXXIV, Hermann (1968).

[B-T1] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. No. 41, IHES, (1972).

[B-T2] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’un donnée radicielle valuée. Publ. Math. No. 60, IHES, (1984).

[De-Gr] M. Demazure, A. Grothendieck, Schémas en groupes SGA 3. Lect. Notes Math., Vols. 151, 152, 153. Springer-Verlag (1970).

[Gray] D. R. Grayson, Finite generation of K-groups of a curve over a finite field. Springer Lect. Notes Math. vol. 966, 69-90. Springer-Verlag (1982).

[Ha] G. Harder, Halbeinfache Gruppenschemata über vollständigen Kurven. Inv. Math. 6, 107-149, (1968).

[Ha-Na] G. Harder, M.S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann. 212, 215-248, (1975).

[H-J-S] N. Hoffmann, J. Jahnel, U. Stuhler, Generalized vector bundles on curves. J. reine angew. Math. 495, 35-60 (1998).

[La] E. Landvogt, A compactification of the Bruhat-Tits building.

Springer Lect. Notes Math. vol. 1619. Springer-Verlag (1996).

[La2] E. Landvogt, Some functorial properties of the Bruhat-Tits building. J. reine angew. Math. 518, 213-241 (2000).

[Ti] J. Tits, Reductive groups over local fields. In: Automorphic forms, representations and L-functions Proc. Symp. Pure Math. vol. XXXIII, part 1. 29-70, (1979).

[We] A. Weil, Basic number theory. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Band 144. Springer-Verlag (1967).

U. Stuhler Mathematisches Institut der Universität Göttingen Bunsenstr. 3/5 37073 Göttingen e-mail: stuhler@uni-math.gwdg.de