Stéphane Gaussent

Guy Rousseau

Spherical Hecke algebras for Kac-Moody groups over local fields

Keywords:

We define the spherical Hecke algebra H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The stabilizer K of a special point on the standard apartment plays the role of a maximal open compact subgroup. We can define H as the algebra of K-bi-invariant functions on G with almost finite support. As two points in the hovel are not always in a same apartment, this support has to be in some large subsemigroup G + of G. We prove that the structure constants of H are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We also prove the Satake isomorphism between H and the algebra of Weyl invariant elements in some completion of a Laurent polynomial algebra. In particular, H is always commutative. Actually, our results apply to abstract "locally finite" hovels, so that we can define the spherical algebra with unequal parameters.

Introduction

Let G be a connected reductive group over a local non-archimedean field K and let K be an open compact subgroup. The space H of complex functions on G, bi-invariant by K and with compact support is an algebra for the natural convolution product. Ichiro Satake [START_REF] Satake | Theory of spherical functions on reductive algebraic groups over p-adic fields[END_REF] studied this algebra H to define the spherical functions and proved, in particular, that H is commutative for good choices of K. We know now that one of the good choices for K is the fixator of some special vertex for the action of G on its Bruhat-Tits building I , whose structure is explained in [START_REF] Bruhat | Groupes réductifs sur un corps local I, Données radicielles valuées[END_REF]. Moreover H, now called the spherical Hecke algebra, may be entirely defined using I , see e.g. [P06].

Stéphane Gaussent & Guy Rousseau

Kac-Moody groups are interesting generalizations of reductive groups and it is natural to try to generalize the spherical Hecke algebra to the case of a Kac-Moody group. But there is now no good topology on G and no good compact subgroup, so the "convolution product" has to be defined only with algebraic means. Alexander Braverman and David Kazhdan [BrK10] succeeded in defining such a spherical Hecke algebra, when G is split and untwisted affine. For a well chosen subgroup K, they define H as an algebra of K-bi-invariant complex functions with "almost finite" support. There are two new features: the support has to be in a subsemigroup G + of G and it is an infinite union of double classes. Hence, H is naturally a module over the ring of complex formal power series.

Our idea is to define this spherical Hecke algebra using the hovel associated to the almost split Kac-Moody group G that we built in [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF], [START_REF] Rousseau | Groupes de Kac-Moody déployés sur un corps local, 2 Masures ordonnées[END_REF] and [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF]. This hovel I is a set with an action of G and a covering by subsets called apartments. They are in one-toone correspondence with the maximal split subtori, hence permuted transitively by G. Each apartment A is a finite dimensional real affine space and its stabilizer N in G acts on it via a generalized affine Weyl group W = W v ⋉ Y (where Y ⊂ -→ A is a discrete subgroup of translations) which stabilizes a set M of affine hyperplanes called walls. So, I looks much like the Bruhat-Tits building of a reductive group, but M is not a locally finite system of hyperplanes (as the root system Φ is infinite) and two points in I are not always in a same apartment (this is why I is called a hovel). There is on I a G-invariant preorder ≤ which induces on each apartment A the preorder given by the Tits cone T ⊂ -→ A . Now, we consider the fixator K in G of a special point 0 in a chosen standard apartment A. The spherical Hecke algebra H R is a space of K-bi-invariant functions on G with values in a ring R. In other words, it is the space H I R of G-invariant functions on I 0 × I 0 where I 0 = G/K is the orbit of 0 in I . The convolution product is easy to guess from this point of view: (ϕ * ψ)(x, y) = z∈I 0 ϕ(x, z)ψ(z, y) (if this sum means something). As two points x, y in I are not always in a same apartment (i.e. the Cartan decomposition fails: G = KN K), we have to consider pairs (x, y) ∈ I 0 × I 0 , with x ≤ y (this implies that x, y are in a same apartment). For H R , this means that the support of ϕ ∈ H R has to be in K\G + /K where G + = {g ∈ G | 0 ≤ g.0} is a semigroup. In addition, K\G + /K is in one-to-one correspondence with the subsemigroup Y ++ = Y ∩ C v f of Y (where C v f is the fundamental Weyl chamber). Now, to get a well defined convolution product, we have to ask (as in [BrK10]) the support of a ϕ ∈ H R to be almost finite: supp(ϕ) ⊂ n i=1 (λ i -Q ∨ + ) ∩ Y ++ , where λ i ∈ Y ++ and Q ∨ + is the subsemigroup of Y generated by the fundamental coroots. Note that (λ -

Q ∨ + ) ∩ Y ++ is infinite except when G is reductive.
With this definition we are able to prove that H R is really an algebra, which generalizes the known spherical Hecke algebras in the finite or affine split case ( §2). In the split case, we describe the hovel I and give a direct proof that H R is commutative ( §3).

The structure constants of H R are the non-negative integers m λ,µ (ν) (for λ, µ, ν ∈ Y ++ ) such that c λ * c µ = ν∈Y ++ m λ,µ (ν)c ν , where c λ is the characteristic function of KλK. Each chamber (= alcove) in I has only a finite number of adjacent chambers along a given panel. These numbers are called parameters of I and they form a finite set Q. In the split case, there is only one parameter q: the number of elements of the residue field κ of K. In §4 we show that the structure constants are polynomials in these parameters with integral coefficients depending only on the geometry of an apartment.

In §5 we build an action of H R on the module of functions from A ∩ I 0 to R. This gives an injective homomorphism from H R into a suitable completion R[[Y ]] of the group algebra R[Y ]; hence H R is abelian (5.3). After modification by a character this homomorphism gives the Satake isomorphism from H R onto the subalgebra R

[[Y ]] W v of W v -invariant elements in R[[Y ]].
The proof involves a parabolic retraction of I onto an extended tree inside it.

Actually, this article is written in a more general framework (explained in §1): we ask I to be an abstract ordered hovel (as defined in [START_REF] Rousseau | Masures affines[END_REF]) and G a strongly transitive group of (positive, type-preserving) automorphisms.

The general definition and study of Hecke algebras for split Kac-Moody groups over local fields was also undertaken by Alexander Braverman, David Kazhdan and Manish Patnaik (as we knew from [P10]). A preliminary draft appeared recently [START_REF] Braverman | Hecke algebras for p-adic loop groups[END_REF]. Their arguments are algebraic without use of a geometric object as a hovel, and the proofs seem complete (temporarily?) only for the untwisted affine case. In addition to the construction of the spherical Hecke algebra and the Satake isomorphism (as here), they give a formula for spherical functions and they build the Iwahori-Hecke algebra. We hope to generalize, in a near future, these results to our general framework.

One should notice that these authors use, instead of our group K, a smaller K 1 , a priori slightly different, see Remark in Section 3.4.

1 General framework

Vectorial data

We consider a quadruple (V, W v , (α i ) i∈I , (α ∨ i ) i∈I ) where V is a finite dimensional real vector space, W v a subgroup of GL(V ) (the vectorial Weyl group), I a finite set, (α ∨ i ) i∈I a family in V and (α i ) i∈I a free family in the dual V * . We ask these data to verify the conditions of [Ro11, 1.1]. In particular, the formula

r i (v) = v -α i (v)α ∨ i defines a linear involution in V which is an element in W v and (W v , {r i | i ∈ I}) is a Coxeter system.
To be more concrete we consider the Kac-Moody case of [l.c. ; 1.2]: the matrix M = (α j (α ∨ i )) i,j∈I is a generalized Cartan matrix. Then W v is the Weyl group of the corresponding Kac-Moody Lie algebra g M and the associated real root system is

Φ = {w(α i ) | w ∈ W v , i ∈ I} ⊂ Q = i∈I Z.α i . We set Φ ± = Φ ∩ Q ± where Q ± = ±( i∈I (Z ≥0 ).α i ) and Q ∨ = ( i∈I Z.α ∨ i ), Q ∨ ± = ±( i∈I (Z ≥0 ).α ∨ i ). We have Φ = Φ + ∪ Φ -and, for α = w(α i ) ∈ Φ, r α = w.r i .w -1 and α ∨ = w(α ∨ i ) depend only on α, and r α (v) = v -α(v)α ∨ .
The set Φ is an (abstract reduced) real root system in the sense of [START_REF] Moody | On infinite root systems[END_REF], [START_REF] Moody | Lie algebras with triangular decompositions[END_REF] or [START_REF] Bardy-Panse | Systèmes de racines infinis[END_REF]. We shall sometimes also use the set

∆ = Φ ∪ ∆ + im ∪ ∆ - im of all roots (with -∆ - im = ∆ + im ⊂ Q + , W v -stable) defined in [Ka90].
It is an (abstract reduced) root system in the sense of [START_REF] Bardy-Panse | Systèmes de racines infinis[END_REF].

The fundamental positive chamber is

C v f = {v ∈ V | α i (v) > 0, ∀i ∈ I}. Its closure C v f is the disjoint union of the vectorial faces F v (J) = {v ∈ V | α i (v) = 0, ∀i ∈ J, α i (v) > 0, ∀i ∈ I \ J} for J ⊂ I.
The positive (resp. negative) vectorial faces are the sets w.F v (J) (resp. -w.F v (J)) for w ∈ W v and J ⊂ I. The set J or the face w.

F v (J) is called spherical if the group W v (J) generated by {r i | i ∈ J} is finite.
The Tits cone T is the (disjoint) union of the positive vectorial faces. It is a W v -stable convex cone in V .

The model apartment

As in [START_REF] Rousseau | Masures affines[END_REF]1.4] the model apartment A is V considered as an affine space and endowed with a family M of walls. These walls are affine hyperplanes directed by Ker(α) for α ∈ Φ.

We ask this apartment to be semi-discrete and the origin 0 to be special. This means that these walls are the hyperplanes defined as follows:

M (α, k) = {v ∈ V | α(v) + k = 0} for α ∈ Φ and k ∈ Λ α (with Λ α = k α .Z a non trivial discrete subgroup of R).
Using the following lemma (i.e. replacing Φ by Φ) we shall assume that

Λ α = Z, ∀α ∈ Φ. For α = w(α i ) ∈ Φ, k ∈ Λ α (= Z) and M = M (α, k)
, the reflection r α,k = r M with respect to M is the affine involution of A with fixed point set the wall M and associated linear involution r α . The affine Weyl group W a is the group generated by the reflections r M for M ∈ M; we assume that W a stabilizes M.

For

α ∈ Φ and k ∈ R, D(α, k) = {v ∈ V | α(v) + k ≥ 0} is an half-space, it is called an half-apartment if k ∈ Λ α (= Z).
The Tits cone T and its interior T o are convex and W v -stable cones, therefore, we can define two W v -invariant preorder relations on A:

x ≤ y ⇔ y -x ∈ T ; x o ≤ y ⇔ y -x ∈ T o .
If W v has no fixed point in V \ {0} and no finite factor, then they are orders; but they are not in general.

Lemma 1.3. For all α ∈ Φ we choose k α > 0 and define α = α/k α , α ∨ = k α .α ∨ . Then Φ = { α | α ∈ Φ} is the (abstract reduced) real root system (in the sense of [START_REF] Moody | On infinite root systems[END_REF], [START_REF] Moody | Lie algebras with triangular decompositions[END_REF] or [START_REF] Bardy-Panse | Systèmes de racines infinis[END_REF]) associated to (V, W v , (k -1

α i .α i ) i∈I , (k α i .α ∨ i ) i∈I ) hence to the generalized Cartan matrix M = (k -1 α j .α j (k α i .α ∨ i )) i,j∈I
. Moreover with Φ, the walls are described using the subgroups Λ α = Z.

Proof. For α, β ∈ Φ, the group W a contains the translation τ by k α .α ∨ and τ (M (β, 0

)) = M (β, -β(k α .α ∨ )). So k α .β(α ∨ ) ∈ Λ β i.e. β( α ∨ ) = k -1 β .k α .β(α ∨ ) ∈ Z. Hence M = (k -1 α j .α j (k α i .α ∨ i )) i,
j∈I is a generalized Cartan matrix and the lemma is clear, as k wα = k α .

1.4 Faces, sectors, chimneys...

The faces in A are associated to the above systems of walls and halfapartments (i.e.

D(α, k) = {v ∈ A | α(v) + k ≥ 0}).
As in [START_REF] Bruhat | Groupes réductifs sur un corps local I, Données radicielles valuées[END_REF], they are no longer subsets of A, but filters of subsets of A. For the definition of that notion and its properties, we refer to [START_REF] Bruhat | Groupes réductifs sur un corps local I, Données radicielles valuées[END_REF] or [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF].

If F is a subset of A containing an element x in its closure, the germ of F in x is the filter germ x (F ) consisting of all subsets of A which are intersections of F and neighbourhoods of x. In particular, if x = y ∈ E, we denote the germ in x of the segment [x, y] (resp. of the interval ]x, y]) by [x, y) (resp. ]x, y)).

The enclosure cl A (F ) of a filter F of subsets of A is the filter made of the subsets of A containing an element of F of the shape

∩ α∈∆ D(α, k α ), where k α ∈ Z ∪ {∞} (here, D(α, ∞) = A).
A face F in the apartment A is associated to a point x ∈ A and a vectorial face F v in V ; it is called spherical according to the nature of F v . More precisely, a subset S of A is an element of the face F (x, F v ) if and only if it contains an intersection of half-spaces

D(α, k) or open halfspaces D • (α, k) (for α ∈ ∆ and k ∈ Z ⊔ {∞}) which contains Ω ∩ (x + F v ), where Ω is an open neighborhood of x in A. The enclosure of a face F = F (x, F v ) is its closure: the closed-face F . It is the enclosure of the local-face in x, germ x (x + F v ).
There is an order on the faces: the assertions "F is a face of F ′ ", "F ′ covers F " and "F ≤ F ′ " are by definition equivalent to F ⊂ F ′ . The dimension of a face F is the smallest dimension of an affine space generated by some S ∈ F . The (unique) such affine space E of minimal dimension is the support of F . Any S ∈ F contains a non empty open subset of E. A face F is spherical if the direction of its support meets the open Tits cone, then its fixator

W F in W is finite. Any point x ∈ A is contained in a unique face F (x, V 0 ) which is minimal (but seldom spherical); x is a vertex if, and only if, F (x, V 0 ) = {x}.
A chamber (or alcove) is a maximal face, or, equivalently, a face such that all its elements contain a nonempty open subset of A.

A panel is a spherical face maximal among faces which are not chambers, or, equivalently, a spherical face of dimension n -1. Its support is a wall. So, the set of spherical faces of A and the Tits cone completely determine the set M of walls.

A sector in A is a V -translate s = x + C v of a vectorial chamber C v = ±w.C v f (w ∈ W v ),
x is its base point and C v its direction. Two sectors have the same direction if, and only if, they are conjugate by V -translation, and if, and only if, their intersection contains another sector.

The sector-germ of a sector s = x + C v in A is the filter S of subsets of A consisting of the sets containing a V -translate of s, it is well determined by the direction C v . So the set of translation classes of sectors in A, the set of vectorial chambers in V and the set of sector-germs in A are in canonical bijection. We write S -∞ the sector-germ associated to the negative fundamental vectorial chamber -C v f . A sector-face in A is a V -translate f = x + F v of a vectorial face F v = ±wF v (J). The sector-face-germ of f is the filter F of subsets containing a translate f ′ of f by an element of F v (i.e. f ′ ⊂ f). If F v is spherical, then f and F are also called spherical. The sign of f and F is the sign of F v .

A chimney in A is associated to a face F = F (x, F v 0 ), its basis, and to a vectorial face F v , its direction, it is the filter r(F,

F v ) = cl A (F + F v ). A chimney r = r(F, F v ) is splayed if F v is
spherical, it is solid if its support (as a filter, i.e. the smallest affine subspace containing r) has a finite fixator in W v . A splayed chimney is therefore solid. The enclosure of a sector-face f = x + F v is a chimney. A halfline δ with origin in x and containing y = x (or the interval ]x, y], the segment

[x, y]) is called preordered if x ≤ y or y ≤ x and generic if x o ≤ y or y o ≤ x.
With these new notions, a chimney can be defined as the enclosure of a preordered halfline and a preordered segment-germ sharing the same origin. The chimney is splayed if, and only if, the halfline is generic.

The hovel

In this section, we recall the definition of an ordered affine hovel given by Guy Rousseau in [START_REF] Rousseau | Masures affines[END_REF].

An apartment of type A is a set A endowed with a set Isom(A, A) of bijections (called isomorphisms) such that if f 0 ∈ Isom(A, A), then f ∈ Isom(A, A) if, and only if, there exists w ∈ W a satisfying f = f 0 • w. An isomorphism between two apartments φ : A → A ′ is a bijection such that f ∈ Isom(A, A) if, and only if, φ • f ∈ Isom(A, A ′ ). As the filters in A defined in 1.4 above (e.g. faces, sectors, walls,..) are permuted by W a , they are well defined in any apartment of type A.

Definition. An ordered affine hovel of type A is a set I endowed with a covering A of subsets called apartments such that:

(MA1) any A ∈ A admits a structure of an apartment of type A;

(MA2) if F is a point, a germ of a preordered interval, a generic halfline or a solid chimney in an apartment A and if A ′ is another apartment containing F , then A ∩ A ′ contains the enclosure cl A (F ) of F and there exists an isomorphism from A onto A ′ fixing cl A (F );

(MA3) if R is a germ of a splayed chimney and if F is a face or a germ of a solid chimney, then there exists an apartment that contains R and F ;

(MA4) if two apartments A, A ′ contain R and F as in (MA3), then their intersection contains cl A (R ∪ F ) and there exists an isomorphism from A onto A ′ fixing cl A (R ∪ F );

(MAO) if x, y are two points contained in two apartments A and A ′ , and if x ≤ A y then the two segments [x, y] A and [x, y] A ′ are equal.

We ask here I to be thick of finite thickness: the number of chambers (=alcoves) containing a given panel has to be finite ≥ 3. This number is the same for any panel in a given wall M [Ro11, 2.9]; we denote it by 1 + q M . We assume that I has a strongly transitive group of automorphisms G (i.e. all isomorphisms involved in the above axioms are induced by elements of G, cf. [Ro13, 4.10]). We choose in I a fundamental apartment which we identify with A. As G is strongly transitive, the apartments of I are the sets g.A for g ∈ G. The stabilizer N of A in G induces a group ν(N ) of affine automorphisms of A which permutes the walls, sectors, sector-faces... and contains the affine Weyl group W a [Ro13, 4.13.1]. We denote the fixator of 0 ∈ A in G by K.

We ask ν(N ) to be positive and type-preserving for its action on the vectorial faces. This means that the associated linear map -→ w of any w ∈ ν(N ) is in W v . As ν(N ) contains W a and stabilizes M, we have ν(N ) = W v ⋉ Y , where W v fixes the origin 0 of A and Y is a group of translations such that:

Q ∨ ⊂ Y ⊂ P ∨ = {v ∈ V | α(v) ∈ Z, ∀α ∈ Φ}. We ask Y to be discrete in V . This is clearly satisfied if Φ generates V * i.e. (α i ) i∈I is a basis of V * .
Examples. The main examples of all the above situation are provided by the hovels of almost split Kac-Moody groups over fields complete for a discrete valuation and with a finite residue field, see [START_REF] Rousseau | Groupes de Kac-Moody déployés sur un corps local, 2 Masures ordonnées[END_REF], [Ch10], [START_REF] Charignon | Immeubles affines et groupes de Kac-Moody, masures bordées[END_REF] or [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF]. Some details in the split case can be found in Section 3.

Remarks. a) In the following, we often refer to [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF] which deals with split Kac-Moody groups and residue fields containing C. But the results cited are easily generalized to our present framework, using the above references. b) For an almost split Kac-Moody group over a local field K, the set of roots Φ is

K Φ red = { K α ∈ K Φ | 1 2 . K α ∈ K Φ}
where the relative root system K Φ describes well the commuting relations between the root subgroups. Unfortunately Φ gives a worst description of these relations.

Type 0 vertices

The elements of Y considered as the subset Y = N.0 of V = A are called vertices of type 0 in A; they are special vertices. We note

Y + = Y ∩ T and Y ++ = Y ∩ C v
f . The type 0 vertices in I are the points on the orbit I 0 of 0 by G. This set I 0 is often called the affine Grassmannian as it is equal to G/K.

In general, G is not equal to KY K = KN K [GR08, 6.10] i.e. I 0 = K.Y . We know that I is endowed with a G-invariant preorder ≤ which induces the known one on A [Ro11, 5.9]. We set

I + = {x ∈ I | 0 ≤ x} , I + 0 = I 0 ∩I + and G + = {g ∈ G | 0 ≤ g.0}; so I + 0 = G + .0 = G + /K. As ≤ is a G-invariant preorder, G + is a semigroup. If x ∈ I +
0 there is an apartment A containing 0 and x (by definition of ≤ ) and all apartments containing 0 are conjugated to A by K (axiom (MA2)); so x ∈ K.Y + as

I + 0 ∩ A = Y + . But ν(N ∩ K) = W v and Y + = W v .Y ++ (with uniqueness of the element in Y ++ ); so I + 0 = K.Y ++ , more precisely I + 0 = G + /K is the disjoint union of the KyK/K for y ∈ Y ++ .
Hence, we have proved that the map Y ++ → K\G + /K is one-to-one and onto.

1.7 Vectorial distance and Q ∨ -order

For x ∈ T , we note x ++ the unique element in C v f conjugated by W v to x. Let I × ≤ I = {(x, y) ∈ I × I | x ≤ y} be the set of increasing pairs in I . Such a pair (x, y) is always in a same apartment g.A; so g -1 yg -1 x ∈ T and we define the vectorial distance

d v (x, y) ∈ C v f by d v (x, y) = (g -1 y -g -1 x) ++ .
It does not depend on the choices we made.

For (x, y) y) takes values in Y ++ . Actually, as I 0 = G.0, K is the fixator of 0 and I + 0 = K.Y ++ (with uniqueness of the element in Y ++ ), the map d v induces a bijection between the set

∈ I 0 × ≤ I 0 = {(x, y) ∈ I 0 × I 0 | x ≤ y}, the vectorial distance d v (x,
I 0 × ≤ I 0 /G of orbits of G in I 0 × ≤ I 0 and Y ++ . Any g ∈ G + is in K.d v (0, g0).K. For x, y ∈ A, we say that x ≤ Q ∨ y (resp. x ≤ Q ∨ R y) when y -x ∈ Q ∨ + (resp. y -x ∈ Q ∨ R+ = i∈I R ≥0 .α ∨ i ).
We get thus a preorder which is an order at least when

(α ∨ i ) i∈I is free or R + -free (i.e. a i α ∨ i = 0, a i ≥ 0 ⇒ a i = 0, ∀i).

Paths

We consider piecewise linear continuous paths π : [0, 1] → A such that each (existing) tangent vector π ′ (t) is in an orbit W v .λ of some λ ∈ C v f under the vectorial Weyl group W v . Such a path is called a λ-path; it is increasing with respect to the preorder relation ≤ on A.

For any t = 0 (resp. t = 1), we let π ′ -(t) (resp. π ′ + (t)) denote the derivative of π at t from the left (resp. from the right). Further, we define w ± (t) ∈ W v to be the smallest element in its

(W v ) λ -class such that π ′ ± (t) = w ± (t).λ (where (W v ) λ is the fixator in W v of λ). Moreover, we denote by π -(t) = π(t) -[0, 1)π ′ -(t) = [π(t), π(t -ε) ) (resp. π + (t) = π(t) + [0, 1)π ′ + (t) = [π(t), π(t + ε) ) (for ε > 0 small) the positive (resp. negative) segment-germ of π at t.
The reverse path π defined by π = π(1t) has symmetric properties, it is a (-λ)-path.

For any choices of λ ∈ C v f , π 0 ∈ A, r ∈ N \ {0} and sequences τ = (τ 1 , τ 2 , . . . , τ r ) of elements in W v /(W v ) λ and a = (a 0 = 0 < a 1 < a 2 < • • • < a r = 1)
of elements in R, we define a λ-path π = π(λ, π 0 , τ , a) by the formula:

π(t) = π 0 + j-1 i=1 (a i -a i-1 )τ i (λ) + (t -a j-1 )τ j (λ) for a j-1 ≤ t ≤ a j .
Any λ-path may be defined in this way (and we may assume τ j = τ j+1 ).

Definition. [KM08, 3.27] A Hecke path of shape λ with respect to -C v f is a λ-path such that, for all t ∈ [0, 1] \ {0, 1}, π ′ + (t) ≤ W v π(t) π ′ -(t), which means that there exists a W v π(t) -chain from π ′ -(t) to π ′ + (t), i.e. finite sequences (ξ 0 = π ′ -(t), ξ 1 , . . . , ξ s = π ′ + (t))
of vectors in V and (β 1 , . . . , β s ) of real roots such that, for all i = 1, . . . , s:

i) r β i (ξ i-1 ) = ξ i , ii) β i (ξ i-1 ) < 0, iii) r β i ∈ W v π(t) i.e. β i (π(t)) ∈ Z: π(t) is in a wall of direction Ker(β i ). iv) each β i is positive with respect to -C v f i.e. β i (C v f ) > 0.
Remarks. 1) The path is folded at π(t) by applying successive reflections along the walls M (β i , -β i (π(t)) ). Moreover conditions ii) and iv) tell us that the path is "positively folded" (cf. [START_REF] Gaussent | LS-galleries, the path model, and MVcycles[END_REF]) i.e. centrifugally folded with respect to the sector germ

S -∞ = germ ∞ (-C v f ). 2) Let c -= germ 0 (-C v f
) be the negative fundamental chamber (= alcove). A Hecke path of shape λ with respect to c - [START_REF] Bardy-Panse | Une preuve plus immobilière du théorème de saturation de Kapovich-Leeb-Millson[END_REF] is a λ-path in the Tits cone T satisfying the above conditions except that we replace iv) by : iv') each β i is positive with respect to c -i.e. β i (π(t)c -) > 0.

Then ii) and iv') tell us that the path is centrifugally folded with respect to the center c -.

2 Convolution algebras

Wanted

We consider the space

H I R = H R (I , G) = {ϕ I : I 0 × ≤ I 0 → R | ϕ I (gx, gy) = ϕ I (x, y), ∀g ∈ G}
of G-invariant functions on I 0 × ≤ I 0 with values in a ring R (essentially C or Z). We want to make H I R (or some large subspace) an algebra for the following convolution product:

(ϕ I * ψ I )(x, y) = x≤z≤y ϕ I (x, z)ψ I (z, y).
It is clear that this product is associative and R-bilinear if it exists.

Via d v , H I R is linearly isomorphic to the space H R = {ϕ G : Y ++ = K\G + /K → R}, which can be interpreted as the space of K-bi-invariant functions on G + . The correspondence ϕ I ↔ ϕ G between H I R and H R is given by: ϕ G (g) = ϕ I (0, g.0) and ϕ

I (x, y) = ϕ G (d v (x, y)).
In this setting, the convolution product should be:

(ϕ G * ψ G )(g) = h∈G + /K ϕ G (h)ψ G (h -1 g),
where we consider ϕ G and ψ G as trivial on G \ G + . In the following we shall often make no difference between ϕ I or ϕ G and forget the exponents I and G . We consider the subspace H f R of functions with finite support in Y ++ = K\G + /K; its natural basis is (c λ ) λ∈Y ++ where c λ sends λ to 1 and µ = λ to 0. Clearly c 0 is a unit for * . In

H I R , (c λ * c µ ) I (x, y) is the number of triangles [x, z, y] with d v (x, z) = λ and d v (z, y) = µ.
As suggested by [BrK10] and lemma 2.4, we consider also the subspace

H R of H R of functions ϕ with almost finite support i.e. supp(ϕ) ⊂ ∪ n i=1 (λ i -Q ∨ + ) ∩ Y ++ where λ i ∈ Y ++ .

Retractions onto Y +

For all x ∈ I + there is an apartment containing x and c -[Ro11, 5.1] and this apartment is conjugated to A by an element of K fixing c -(axiom (MA2) ). So, by the usual arguments and [l.c. , 5.5] we can define a retraction ρ c -of

I + into A with center c -; its image is ρ c -(I + ) = T = I + ∩ A and ρ c -(I + 0 ) = Y + . There is also a retraction ρ -∞ of I onto A with center the sector-germ S -∞ [GR08, 4.4]. For ρ = ρ c -or ρ -∞ the image of a segment [x, y] with (x, y) ∈ I × ≤ I and d v (x, y) = λ ∈ C v f is a λ-path [GR08, 4.4].
In particular, ρ(x) ≤ ρ(y).

Convolution product

The convolution product in H R should be defined (for y ∈ Y ++ ) by

(ϕ * ψ)(y) = ϕ(z)ψ(d v (z, y))
where the sum runs over the z ∈ I + 0 such that 0 ≤ z ≤ y and ϕ(z

) = ϕ I (0, z) = ϕ G (d v (0, z)). 1) Using ρ c -we have, for λ, µ, y ∈ Y ++ , (c λ * c µ )(y) = w∈W v /(W v ) λ N c -(µ, w.λ, y) where N c -(µ, w.λ, y) is the number of z ∈ I + 0 with d v (z, y) = µ and ρ c -(z) = w.λ ∈ Y + . Note that, if N c -(µ, wλ, y) > 0, there exists a µ-path from wλ to y, hence y ∈ wλ + Y + . So c λ * c µ is the formal sum c λ * c µ = ν∈Y ++ m λ,µ (ν)c ν where the structure constant m λ,µ (ν) = w∈W v /(W v ) λ N c -(µ, w.λ, ν) ∈ Z ≥0 ∪{+∞} is also equal to the number of triangles [x, z, y] with d v (x, z) = λ and d v (z, y) = µ, for any fixed pair (x, y) ∈ I 0 × ≤ I 0 with d v (x, y) = ν (e.g. (x, y) = (0, ν)). 2) Using ρ -∞ we have m λ,µ (ν) = z ′ N -∞ (µ, z ′ , ν) where the sum runs over the z ′ in Y + (λ) = ρ -∞ ({z ∈ I + 0 | d v (0, z) = λ}) and N -∞ (µ, z ′ , ν) ∈ Z ≥0 ∪ {+∞} is the number of z ∈ I + 0 with d v (0, z) = λ, d v (z, y) = µ (for any y ∈ I + 0 with d v (0, y) = ν e.g. y = ν) and ρ -∞ (z) = z ′ . But ρ -∞ ([0, z]) is a λ-path hence increasing with respect to ≤ , so Y + (λ) ⊂ Y + . Moreover, ρ -∞ ([z, ν]) is a µ-path, so z ′ has to be in ν -Y + . Hence, z ′ has to run over the set Y + (λ) ∩ (ν -Y + ) ⊂ Y + ∩ (ν -Y + ).
Actually, the image by ρ -∞ of a segment [x, y] with (x, y) ∈ I × ≤ I and d v (x, y) = λ ∈ Y ++ is a Hecke path of shape λ with respect to -C v f [GR08, th. 6.2]. Hence the following results: [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF]5.3.3] that any Hecke path of shape λ starting in y 0 ∈ Y can be transformed in the path π λ (t) = y 0 + λt by applying successively the operators e α i or e α i for i ∈ I; moreover e α i (π)(1) = π(1)+α ∨ i and e α i (π

Lemma 2.4. a) For λ ∈ Y ++ and w ∈ W v , wλ ∈ λ -Q ∨ + , i.e. wλ ≤ Q ∨ λ. b) Let π be a Hecke path of shape λ ∈ Y ++ with respect to -C v f , from y 0 ∈ Y to y 1 ∈ Y . Then λ = π ′ (0) ++ = π ′ (1) ++ , π ′ (0) ≤ Q ∨ λ, π ′ (0) ≤ Q ∨ R (y 1 -y 0 ) ≤ Q ∨ R π ′ (1) ≤ Q ∨ λ and y 1 -y 0 ≤ Q ∨ λ. c) If moreover (α ∨ i ) i∈I is free, we may replace above ≤ Q ∨ R by ≤ Q ∨ . d) For λ, µ, ν ∈ Y ++ , if m λ,µ (ν) > 0, then ν ∈ λ + µ -Q ∨ + i.e. ν ≤ Q ∨ λ + µ. N.B. By d) above, if x ≤ z ≤ y in I 0 , then d v (x, y) ≤ Q ∨ d v (x, z) + d v (z, y). Proof. a) By definition, for λ ∈ Y , wλ ∈ λ + Q ∨ , hence a) follows from [Ka90, 3.12d] used in a realization where (α ∨ i ) i∈I is free. b) By definition of Hecke paths in 1.8, λ = π ′ (0) ++ = π ′ (1) ++ . Moreover, ∀t ∈ [0, 1], λ = π ′ -(t) ++ = π ′ + (t
(0) ≤ Q ∨ R (y 1 -y 0 ) ≤ Q ∨ R π ′ (1) ≤ Q ∨ R λ. It is proved (but not stated) in
)(1) = π(1), hence y 1 -y 0 ≤ Q ∨ λ. c) By b) y 1 -y 0 -π ′ (0) ∈ Q ∨ R+ ∩ Q ∨ = Q ∨ + , so π ′ (0) ≤ Q ∨ (y 1 -y 0 ). Idem for y 1 -y 0 ≤ Q ∨ π ′ (1). d) If m λ,µ (ν) > 0 we have an Hecke path of shape λ (resp. µ) from 0 to z ′ (resp. from z ′ to ν). So d) follows from b). Proposition 2.5. Suppose (α ∨ i ) i∈I free in V . Then for all λ, µ, ν ∈ Y ++ , m λ,µ (ν) is finite.
N.B. Actually we may replace the condition (α ∨ i ) i∈I free by (α ∨ i ) i∈I R + -free.

Proof. We have to count the z ∈

I + 0 such that d v (0, z) = λ and d v (z, ν) = µ. We set z ′ = ρ -∞ (z). By lemma 2.4b, z ′ ∈ λ-Q ∨ + and ν ∈ z ′ +µ-Q ∨ + , hence z ′ is in (λ-Q ∨ + )∩(ν -µ+Q ∨ + ) which is finite as (α ∨
i ) i∈I is free or R + -free. So, we fix now z ′ . By [GR08, cor. 5.9] there is a finite number of Hecke paths π ′ of shape µ from z ′ to ν. So, we fix now π ′ . And by [l.c. th. 6.3] (see also 4.10, 4.11) there is a finite number of segments [z, ν] retracting to π ′ ; hence the number of z is finite.

Theorem 2.6. Suppose (α ∨ i ) i∈I free or R + -free, then H R is an algebra.

Proof. We saw that for λ, µ, ν ∈ Y ++ , m λ,µ (ν) is finite; hence c λ * c µ is well defined (eventually as an infinite formal sum). Let us consider ϕ, ψ

∈ H R : supp(ϕ) ⊂ ∪ m i=1 (λ i -Q ∨ + ), supp(ψ) ⊂ ∪ n j=1 (µ j -Q ∨ + ). Let ν ∈ Y ++ . If m λ,µ (ν) > 0 with λ ∈ supp(ϕ), µ ∈ supp(ψ) (hence λ ∈ λ i -Q ∨ + , µ ∈ µ j -Q ∨ + for some i, j), we have λ + µ ∈ ν + Q ∨ + by lemma 2.4d. So λ ∈ (ν -µ + Q ∨ + ) ∩ (λ i -Q ∨ + ) ⊂ (ν -µ j + Q ∨ + ) ∩ (λ i -Q ∨ + )
, a finite set. For the same reasons µ is in a finite set, so ϕ * ψ is well defined.

With the above notations ν

∈ (λ + µ -Q ∨ + ) ⊂ ∪ i,j (λ i + µ j -Q ∨ + ), so ϕ * ψ ∈ H R . Definition 2.7. H R = H R (I , G)
is the spherical Hecke algebra (with coefficients in R) associated to the hovel I and its strongly transitive automorphism group G.

Remark. We shall now investigate H R and some other possible convolution algebras in H R by separating the cases: finite, indefinite and affine.

Finite case

In this case Φ and W v are finite, (α ∨ i ) i∈I is free, T = V and the relation ≤ is trivial. The hovel I = I + is a locally finite Bruhat-Tits building.

Let ρ be the half sum of positive roots. As 2ρ ∈ Q and ρ(α ∨ i ) = 1, ∀i ∈ I, we see that an almost finite set in Y ++ is always finite. So H R and H f R are equal. The algebra H C was already studied by I. Satake in [START_REF] Satake | Theory of spherical functions on reductive algebraic groups over p-adic fields[END_REF]. Its close link with buildings is explained in [P06]. The algebra H Z is the spherical Hecke ring of [START_REF] Kapovich | Polygons in symmetric spaces and buildings with applications to algebra[END_REF], where the interpretation of m λ,µ (ν) as a number of triangles in I is already given.

H R is not an algebra as e.g. m λ,(-w 0 )λ (0) = 0 ∀λ ∈ Y ++ (where w 0 is the greatest element in W v ).

Indefinite case

Lemma. Suppose now Φ associated to an indefinite indecomposable generalized Cartan matrix. Then there is in ∆ + im an element δ (of support I) such that δ(α ∨ i ) < 0, ∀i ∈ I and a basis (δ i ) i∈I of the real vector space Q R spanned by Φ such that δ i (T ) ≥ 0, ∀i ∈ I.

Proof. Any δ ∈ ∆ + im takes positive values on T [Ka90, 5.8]. Now, in the indefinite case, there is δ ∈ ∆ + im ∩ (⊕ i∈I R >0 .α i ) such that δ(α ∨ i ) < 0, ∀i ∈ I [l.c. 4.3], hence δ + α i ∈ ∆ + , ∀i ∈ I. Replacing eventually δ by 3δ [l.c. 5.5], we have (δ + α i )(α ∨ j ) < 0, ∀i, j ∈ I, hence δ + α i ∈ ∆ + im . The wanted basis is inside {δ} ∪ {δ 0 + α i | i ∈ I}.
The existence of δ ∈ ∆ + im as in the lemma proves that (α ∨ i ) i∈I is R + -free. So H R is an algebra. The following example 2.10 proves that H f R is in general not a subalgebra. If (α i ) i∈I generates (i.e. is a basis of) V * , H R is also an algebra (the formal spherical Hecke algebra): Let ν ∈ Y ++ , we have to prove that there is only a finite number of pairs (λ, µ) ∈ (Y ++ ) 2 such that m λ,µ (ν) > 0. Let z ′ be as in the proof of 2.5. We saw in 2.3 that

z ′ ∈ Y + ∩ (ν -Y + ) = Y ∩ T ∩ (ν -T ). By the lemma, T ∩ (ν -T ) is bounded, hence Y ∩ T ∩ (ν -T ) is finite. So we may fix z ′ . Now λ ∈ z ′ + Q ∨
+ hence (for δ as in the lemma) δ(λ) ≤ δ(z ′ ); as α i (λ) ∈ Z >0 ∀i ∈ I and δ ∈ ⊕ i∈I R >0 .α i this gives only a finite number of possibilities for λ. Similarly µ ∈ νz ′ + Q ∨ + has to be in a finite set. Actually H R is often equal to H R when (α ∨ i ) i∈I is free and (α i ) i∈I generates V * (hence the matrix M = (α j (α ∨ i )) is invertible), see the following example 2.10.

An indefinite rank 2 example

Let us consider the Kac-

Moody matrix M = 2 -3 -3 2 . The basis of Φ and V * is {α 1 , α 2 }
and we consider the dual basis

(̟ ∨ 1 , ̟ ∨ 2 ) of V . In this basis α ∨ 1 = 2 -3 , α ∨ 2 = -3
2 and the matrices of r 1 , r 2 , r 2 r 1 and r 1 r 2 are respectively

-1 0 3 1 , 1 3 0 -1 , M = 8 3 -3 -1 and M -1 = -1 -3 3 8
. The eigenvalues of M or M -1 are a ± = (7 ± √ 45)/2. In a basis diagonalizing M and M -1 we see easily that (r 2 r 1 ) n + (r 1 r 2 ) n = a n .Id V where a n = a n + + a n is in N and increasing up to infinity (a 0 = 2, a 1 = 7, a 2 = 47, a 3 = 322,...).

Consider now λ

= µ = -α ∨ 1 -α ∨ 2 = 1 1 in Y ++ ⊂ Z ≥0 .̟ ∨ 1 ⊕ Z ≥0 .̟ ∨ 2 .
We have (r 2 r 1 ) n .λ + (r 1 r 2 ) n .λ = a n .λ. This means that m λ,λ (a n .λ) ≥ N c -(λ, (r 2 r 1 ) n λ, a n .λ) ≥ 1, for all positive n (and the same thing for N -∞ ). So c λ * c λ is an infinite formal sum.

Actually

(-Q ∨ + ) ∩ Y ++ ⊃ Z ≥0 .5̟ ∨ 1 ⊕ Z ≥0 .5̟ ∨ 2 , hence Y ++ itself is almost finite!

An affine rank 2 example

Let us consider the Kac-Moody matrix

M = 2 -2 -2 2 . The basis of Φ is {α 1 , α 2 } but
we consider a realization V of dimension 3 for which {α ∨ 1 , α ∨ 2 } is free and with basis of

V * {α o = -ρ, α 1 , α 2 }. More precisely, if (̟ ∨ 0 , ̟ ∨ 1 , ̟ ∨ 2 )
is the dual basis of V , we have

α ∨ 1 =   -1 2 -2   , α ∨ 2 =   -1 -2 2 
 and the matrices of r 1 , r 2 , r 1 r 2 and r 2 r 1 are respectively

  1 1 0 0 -1 0 0 2 1   ,   1 0 1 0 1 2 0 0 -1   , M =   1 1 3 0 -1 -2 0 2 3   and M -1 =   1 3 1 0 3 2 0 -2 -1   . A classical calculus using triangulation tells us that (r 2 r 1 ) n + (r 1 r 2 ) n =   2 4n 2 4n 2 0 1 0 0 0 1   . Actually c = α ∨ 1 + α ∨ 2 = -2̟ ∨ 0 ∈ Q ∨ + is the canonical central element [Ka90, § 6
.2] and the above calculations are peculiar cases of [l.c. § 6.5].

Let's consider now

λ = µ = 2 i=1 a i ̟ ∨ i ∈ Y ++ ⊂ ⊕ 2 i=1 Z ≥0 ̟ ∨ i .
We have (r 2 r 1 ) n (λ) + (r 1 r 2 ) n (λ) = λ -2n 2 |λ|c with |λ| = a 1 + a 2 . This means that m λ,λ (λ -2n 2 |λ|c) ≥ N c -(λ, (r 2 r 1 ) n (λ), λ -2n 2 |λ|c) ≥ 1, ∀n ∈ Z (and the same thing for N -∞ ). So c λ * c λ is an infinite formal sum.

Moreover as c is fixed by r 1 and r 2 , (r 2 r 1 ) n (λ+2n 2 |λ|c)+(r 1 r 2 ) n (λ) = λ, so m λ+2n 2 |λ|c,λ (λ) ≥ 1, ∀n ∈ Z, and H R is not an algebra. Remark also that, if we consider the essential quotient V e = V /Rc, the above calculus tells that m λ,λ (λ)

≥ n∈Z N c -(λ, (r 2 r 1 ) n (λ), λ) is infinite if |λ| > 0.

Affine indecomposable case

We saw in the example 2.11 above that m λ,λ (λ) may be infinite, ∀λ ∈ Y ++ when (α ∨ i ) i∈I is not free. So, in this case, H R seems to contain no algebra except R.c 0 .

Remark also that (α ∨ i ) i∈I free is equivalent to (α ∨ i ) i∈I R + -free in the affine indecomposable case as the only possible relation between the

α ∨ i is c = 0 where c = i∈I a ∨ i .α ∨ i (with a ∨ i ∈ Z >0 ∀i ∈ I) is the canonical central element. An almost finite subset in Y ++ is a finite union of subsets like Y λ = (λ -Q ∨ + ) ∩ Y ++ . Let δ be the smallest positive imaginary root in ∆. Then δ(Q ∨ + ) = 0 so Y λ ⊂ {y ∈ Y ++ | δ(y) = δ(λ)} = Y ′ λ . But δ = i∈I a i .α i with a i ∈ Z >0 ∀i ∈ I, so the image of Y ′ λ in V e = V /Rc (where Rc = ∩ i∈I Ker(α i )) is finite. It is now clear that Y λ is a finite union of sets like µ -Z ≥0 .c with µ ∈ Y ++ .
Hence an almost finite subset as defined above is the same as an almost finite union (of double cosets) as defined in [BrK10].

The algebra H C is the one introduced by A. Braverman and D. Kazhdan in [BrK10]. We gave above a combinatorial proof that it is an algebra, without algebraic geometry.

3 The split Kac-Moody case

Situation

As in [START_REF] Rousseau | Groupes de Kac-Moody déployés sur un corps local, 2 Masures ordonnées[END_REF] or [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF], we consider a split Kac-Moody group G associated to a root generating system (RGS) S = (M, Y S , (α i ) i∈I , (α ∨ i ) i∈I ) over a field K endowed with a discrete valuation ω (with value group Λ = Z and ring of integers O = ω -1 ([0, +∞])) whose residue field κ = F q is finite . So, M = (a i,j ) i,j∈I is a Kac-Moody matrix, Y S a free Z-module,

(α ∨ i ) i∈I a family in Y S , (α i ) i∈I a family in the dual X = Y * S of Y S and α j (α ∨ i ) = a i,j . If (α i ) i∈I is free in X, we consider V = V Y = Y S ⊗ Z R and the clear quadruple (V, W v , (α i = α i ) i∈I , (α ∨ i ) i∈I ).
In general, we may define

Q = Z I with canonical basis (α i ) i∈I , then V = V Q = Hom Z (Q, R)
is also in a quadruple as in 1.1. A third example V xl of choice for V is explained in [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF]. We always denote by bar : Q → X the linear map sending α i to α i .

With these vectorial data we may define what was considered in 1.1 and 1.2 (we choose

Λ α = Λ = Z, ∀α ∈ Φ).
Now the hovel I in 1.5 is as defined in [START_REF] Rousseau | Groupes de Kac-Moody déployés sur un corps local, 2 Masures ordonnées[END_REF] or [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF] and the strongly transitive group is G = G(K). By [START_REF] Rousseau | Masures affines[END_REF]6.11] or [Ro12, 5.16] we have q M = q for any wall M .

When G is a split reductive group, I is its extended Bruhat-Tits building.

Generators for G

The Kac-Moody group G contains a split maximal torus T with character group X and cocharacter group Y S . We note T = T(K). For each α ∈ Φ ⊂ Q there is a group homomorphism x α : K → G which is one-to-one; its image is the subgroup U α . Now G is generated by T and the subgroups U α for α ∈ Φ, submitted to some relations given by Tits [T87], also available in [Re02] or [START_REF] Rousseau | Groupes de Kac-Moody déployés sur un corps local, 2 Masures ordonnées[END_REF]. We set U ± the subgroup generated by the subgroups U α for α ∈ Φ ± . We shall explain now only a few of the relations. For u ∈ K, t ∈ T and α ∈ Φ one has: (KMT4) t.x α (u).t -1 = x α (α(t).u) (where α = bar(α)) For u = 0, we note s α (u) = x α (u).x -α (u -1 ).x α (u) and s α = s α (1).

(KMT5) s α (u).t. s α (u) -1 = r α (t) (W v acts on V , Y S , X hence on T )

Weyl groups

Actually the stabilizer

N of A ⊂ I is the normalizer of T in G. The image ν(N ) of N in Aut(A) is a semi-direct product ν(N ) = ν(N 0 ) ⋉ ν(T ) with:
N 0 is the fixator of 0 in N and ν(N 0 ) is isomorphic to W v acting linearly on A = V . Actually ν(N 0 ) is generated by the elements ν( s α ) which act as r α (for α ∈ Φ).

t ∈ T acts on A by a translation of vector

ν(t) ∈ V such that χ(ν(t)) = -ω(χ(t)) for any χ ∈ X = Y * S and χ ∈ X or Q which are related by χ = χ if V = V Y or χ = bar(χ) if V = V Q . So, ν(N ) = W v ⋉ Y where Y is closely related to Y S ≃ T /T(O): as Λ = ω(K) = Z, they are equal if V = V Y and, if V = V Q , Y = bar * (Y S ) is the image of Y S by the map bar * : Y S → Hom Z (Q, Z) dual to bar. So, the choice V = V Y is more pleasant. The choice V = V Q is made e.g. in [Ch10], [Ch11] or [Re02]
and has good properties in the indefinite case, cf. 2.9. They coincide both when

(α i ) i∈I is a basis of X ⊗ R = V * Y .
This assumption generalizes semi-simplicity, in particular the center of G is then finite [Re02, 9.6.2].

The group K

The group K = G 0 should be equal to G(O) for some integral structure of G over O cf. [GR08, 3.14]. But the appropriate integral structure is difficult to define in general. So we define K by its generators:

The group N 0 is generated by T 0 = T(O) = T ∩ K and the elements s α for α ∈ Φ (this is clear by 3.3). The group U 0 , generated by the groups U α,0 = x α (O) for α ∈ Φ, is in K. We note U ± 0 = U 0 ∩ U ± . In general U ± 0 is not generated by the groups U α,0 for α ∈ Φ ± [Ro12, 4.12.3a].

It is likely that K may be greater than the group generated by N 0 and U 0 (i.e. by U 0 and T 0 ). We have to define groups U pm+ 0 ⊃ U + 0 and U nm- 0 ⊃ U - 0 as follows. In a formal positive completion G + of G, we can define a subgroup U ma+ 0 = α∈∆ + U α,0 of the subgroup U ma+ = α∈∆ + U α of G + , with U + ⊂ U ma+ (where U α,0 and U α are suitably defined for α imaginary). Then

U pm+ 0 = U ma+ 0 ∩ G = U ma+ 0 ∩ U + . The group U nm- 0 is defined similarly with ∆ -using a group U ma- 0 ⊂ U ma-in a formal negative completion G -of G. Now K = G 0 = U nm- 0 .U + 0 .N 0 = U pm+ 0 .U - 0 .N 0 [Ro12, 4.14, 5.1]
Remark. Let us denote by K 1 the group used by A. Braverman, D. Kazhdan and M. Patnaik in their definition of the spherical Hecke algebra. With the notation above, K 1 is generated by T 0 and U 0 , i.e. by T 0 , U + 0 and U - 0 , hence

K = U nm- 0 .K 1 = U pm+ 0 .K 1 , with U - 0 ⊂ U nm- 0 ⊂ U -and U + 0 ⊂ U pm+ 0 ⊂ U + . But they prove, at least in the untwisted affine case, that U -∩U + .K 1 ⊂ K 1 [BrKP12, proof of 6.4.3]; so U nm- 0 ⊂ U -∩K ⊂ U -∩U + .K 1 ⊂ K 1 and K = K 1 .
This result answers positively a question in [Ro13, 5.4], at least for points of type 0 and in the untwisted affine split case. Proposition 3.5. There is an involution θ (called Chevalley involution) of the group G such that θ(t) = t -1 for all t ∈ T and θ(x α (u)) = x -α (u) for all α ∈ Φ and u ∈ K. Moreover K is θ-stable and θ induces the identity on W v = N/T . Proof. This involution is well known on the corresponding complex Lie algebra, see [START_REF] Victor | Infinite dimensional Lie algebras[END_REF]1.3.4] where one uses for the generators e α a convention different from ours ([e α , e -α ] = -α ∨ as in [T87] or [Re02]). Hence the proposition follows when κ contains C or is at least of characteristic 0. But here we have to use the definition of G by generators and relations.

We see in [Ro12, 1.5, 1.7.5] that s α (-u) = s α (u) -1 and s α (u) = s -α (u -1 ). So for the wanted involution θ we have θ( s α (u)) = s -α (u) = s α (u -1 ). We have now to verify the relations between the θ(x α (u)) = x -α (u), θ(t) = t -1 and θ( s α (u)) = s α (u -1 ). This is clear for (KMT4) and (KMT5) (as r α = r -α ). The three other relations are:

(KMT3) (x α (u), x β (v)) =
x γ (C α,β p,q .u p v q ) for (α, β) ∈ Φ 2 prenilpotent and, for the product, γ = pα+ qβ runs in (Z >0 α+ Z >0 β)∩ Φ. But the integers C α,β p,q are picked up from the corresponding formula between exponentials in the automorphism group of the corresponding complex Lie algebra. As we know that θ is defined in this Lie algebra, we have C -α,-β p,q = C α,β p,q and (KMT3) is still true for the images by θ.

(KMT6) s α (u -1 ) = s α .α ∨ (u) for α simple and u ∈ K \ {0}. This is still true after a change by θ as θ( s α (u -1 )) = s α (u) and (-α) ∨ (u) = α ∨ (u -1 ).

(KMT7) s α .x β (u). s -1 α = x γ (ε.u) if γ = r α (β) and s α (e β ) = ε.e γ in the Lie algebra (with ε = ±1). This is still true after a change by θ because s α (e β ) = ε.e γ ⇒ s α (e -β ) = ε.e -γ (as

r α (β ∨ ) = γ ∨ ).
So, θ is a well defined involution of G, θ(U 0 ) = U 0 , θ(N 0 ) = N 0 and θ(U ± 0 ) = U ∓ 0 . But the isomorphism θ of U + onto U -can clearly be extended to an isomorphism θ from U ma+ onto U ma-sending U ma+

0 onto U ma- 0 . So θ(U pm+ 0 ) = U nm- 0 and θ(K) = K. As θ( s α ) = s α , θ induces the identity on W v = N/T .
Theorem 3.6. The algebra H R or H R is commutative, when it exists.

Notation: To be clearer we shall sometimes write

H R (G, K) or H R (G, K) instead of H R or H R .
Proof. The formula θ # (g) = θ(g -1 ) defines an anti-involution (θ # (gh) = θ # (h).θ # (g) ) of G which induces the identity on T and stabilizes K.

In particular θ # (G + ) = θ # (KY ++ K) = G + and θ # (KλK) = KλK, ∀λ ∈ Y ++ . For ϕ, ψ ∈ H R and g ∈ G + , one has: (ϕ * ψ)(g) = (ϕ * ψ)(θ # (g)) = h∈G + /K ϕ(h)ψ(h -1 θ # (g)). The map h → h ′ = θ # (h -1 θ # (g)) = gθ # (h -1 ) is one-to-one from G + /K onto G + /K. So, (ϕ * ψ)(g) = h ′ ∈G + /K ϕ(θ # (h ′-1 g))ψ(θ # (h ′ )) = h ′ ∈G + /K ϕ(h ′-1 g)ψ(h ′ ) = (ψ * ϕ)(g).
Remarks 3.7. 1) This commutativity will be below proved in general as a consequence of the Satake isomorphism. The above proof generalizes well known proofs in the reductive case, e.g. for G = GL n , θ # is the transposition.

2) When G is an almost split Kac-Moody group over the field K (supposed complete or henselian) it splits over a finite Galois extension L, the hovel K I over K exists and embeds in the hovel L I over L [Ro13, § 6]. After enlarging eventually L one may suppose that 0 is a special point in K I and L I , more precisely in the fundamental apartments K A ⊂ L A = A associated respectively to a maximal K-split torus K S and a L-split maximal torus T ⊃ K S. If we make a good choice of the homomorphisms x α : L → G(L), the associated involution θ of G(L) should commute with the action of the Galois group Γ = Gal(L/K) hence induce an involution K θ and an anti-involution

K θ # of G(K) = G(L) Γ such that K θ(K) = K θ # (K) = K and K θ # induces the identity in Y ( K S) = K S(K)/ K S(O). The commutativity of H R (G, K) or H R (G, K) would follow.
This strategy works well when G is quasi split over K; unfortunately it seems to fail in the general case.

3) The commutativity of H R or H R is linked to the choice of a special vertex for the origin 0. Even in the semi-simple case, other choices may give non commutative convolution algebras, see [START_REF] Satake | Theory of spherical functions on reductive algebraic groups over p-adic fields[END_REF] and [KeR07].

Structure constants

We come back to the general framework of § 1. We shall compute the structure constants of H R or H R by formulas depending on A and the numbers q M of 1.5. Note that there are only a finite number of them: as q wM = q M , ∀w ∈ ν(N ) and wM (α, k) = M (wα, k), ∀w ∈ W v , we may suppose

M = M (α i , k) with i ∈ I and k ∈ Z. Now α ∨ i ∈ Q ∨ ⊂ Y ; as α i (α ∨ i ) = 2 the translation by α ∨ i permutes the walls M = M (α i , k) (for k ∈ Z)
with two orbits. So Y has at most two orbits in the set of the constants q M (α i ,k) , those of q i = q M (α i ,0) and q ′ i = q M (α i ,±1) . Hence the number of (possibly) different parameters is at most 2.|I|. We denote by

Q = {q 1 , • • • , q l , q ′ 1 = q l+1 , • • • , q ′ l = q 2l }
this set of parameters.

Centrifugally folded galleries of chambers

Let x be a point in the standard apartment A. Let Φ x be the set of all roots α such that α(x) ∈ Z. It is a closed subsystem of roots. Its associated Weyl group W v x is a Coxeter group. We have twinned buildings I +

x (resp. I - x ) whose elements are segment germs [x, y) = germ x ([x, y]) for y ∈ I , y = x, y ≥ x (resp. y ≤ x). We consider their unrestricted structure, so the associated Weyl group is W v and the chambers (resp. closed chambers) are the local chambers

C = germ x (x + C v ) (resp. local closed chambers C = germ x (x + C v )), where C v is a vectorial chamber, cf. [GR08, 4.5] or [Ro11, § 5]. To A is associated a twin system of apartments A x = (A - x , A + x ). We choose in A -
x a negative (local) chamber C - x and denote C + x its opposite in A + x . We consider the system of positive roots Φ + associated to C + x (i.e.

Φ + = wΦ + f , if Φ + f is the system Φ + defined in 1.1 and C + x = germ x (x + wC v f )).
We note (α i ) i∈I the corresponding basis of Φ and (r i ) i∈I the corresponding generators of W v .

Fix a reduced decomposition of an element w ∈ W v , w = r i 1 • • • r ir and let i = (i 1 , ..., i r ) be the type of the decomposition. We consider now galleries of (local) chambers c = (C - x , C 1 , ..., C r ) in the apartment A - x starting at C - x and of type i. The set of all these galleries is in bijection with the set

Γ(i) = {1, r i 1 } × • • • × {1, r ir } via the map (c 1 , ..., c r ) → (C - x , c 1 C - x , ..., c 1 • • • c r C - x ). Let β j = -c 1 • • • c j (α i j )
, then β j is the root corresponding to the common limit hyperplane

M j = M β j of C j-1 = c 1 • • • c j-1 C - x and C j = c 1 • • • c j C -
x and satisfying to β j (C j ) ≥ β j (x) (actually M j is a wall ⇐⇒ β j ∈ Φ x ). In the following, we shall identify a sequence (c 1 , ..., c r ) and the corresponding gallery. Definition 4.2. Let Q be a chamber in A +

x . A gallery c = (c 1 , ..., c r ) ∈ Γ(i) is said to be centrifugally folded with respect to Q if c j = 1 implies β j ∈ Φ x and w -1 Q β j < 0, where

w Q = w(C + x , Q) ∈ W v (i.e. Q = w Q C + x ).
We denote this set of centrifugally folded galleries by Γ + Q (i). Proposition 4.3. A gallery c = (C - x , C 1 , ..., C r ) ∈ Γ(i) belongs to Γ + Q (i) if, and only if, C j = C j-1 implies that M j = M β j is a wall and separates Q from C j = C j-1 .

Proof. We saw that M j is a wall ⇐⇒ β j ∈ Φ x . We have the following equivalences:

(M j separates Q from C j = C j-1 ) ⇐⇒ (w -1 Q M j separates C + x from w -1 Q C j = w -1 Q C j-1 ) ⇐⇒ (w -1 Q β j is a negative root).
The group G x = G x /G Ix acts strongly transitively on I + x and I - x . For any root α ∈ Φ x with α(x) = k ∈ Z, the group U α = U α,k /U α,k+1 is a finite subgroup of G x of cardinality q x,α = q M (α,-α(x)) ∈ Q. We denote by u α the elements of this group.

Next, let ρ Q : I x → A x be the retraction centered at Q. To a gallery of chambers c = (c 1 , ..., c r ) = (C - x , C 1 , ..., C r ) in Γ(i), one can associate the set of all galleries of type i starting at C - x in I - x that retract onto c, we denote this set by C Q (c). We denote the set of minimal galleries in

C Q (c) by C m Q (c). Set g j = c j if w -1 Q β j > 0 or β j ∈ Φ x u c j (α i j ) c j if w -1 Q β j < 0 and β j ∈ Φ x .
(1)

Proposition 4.4. C Q (c) is the non empty set of all galleries (C - x = C ′ 0 , C ′ 1 , ..., C ′ r ) where ∀j : C ′ j = g 1 • • • g j C -
x with each g j chosen as in (1) above. For all j local chambers Q and C ′ j are in the apartment

g 1 • • • g j A x .
The set C m Q (c) is empty if, and only if, the gallery c is not centrifugally folded with respect to Q. The gallery (C - x = C ′ 0 , C ′ 1 , ..., C ′ r ) is minimal if, and only if, c j = 1 for any j with w -1 Q β j > 0 or β j ∈ Φ x and u c j (α i j ) = 1 for any j with c j = 1 and w -1 Q β j < 0.

Remark. For g j as in equation ( 1) we may write g j = u c j (α i j ) c j (with u c j (α i j ) = 1 if

w -1 Q β j > 0 or β j ∈ Φ x ).
Then in the product g 1 • • • g j we may gather the c k on the right and, as

c 1 • • • c k (α i k ) = -β k , we may write g 1 • • • g j = u -β 1 • • • u -β j .c 1 • • • c j . Hence C ′ j := g 1 • • • g j C - x = u -β 1 • • • u -β j C j . When u -β k = 1 we have β k ∈ Φ x and w -1 Q β k < 0; so it is clear that ρ Q (C ′ j ) = C j . The gallery (C - x = C ′ 0 , C ′ 1 , ..., C ′ r ) (of type i) is minimal if,
and only if, we may also write (uniquely)

C ′ j = u -α i 1 .u r i 1 (-α i 2 ) • • • u r i 1 •••r i j-1 (-α i j ) .r i 1 • • • r i j (C - x ) = h 1 • • • h j .r i 1 • • • r i j (C - x ) with h k u r i 1 •••r i k-1 (-α i k ) ∈ U r i 1 •••r i k-1 (-α i k ) (which fixes C - x ). In particular, C ′ j ∈ h 1 • • • h j A x .
But this formula gives no way to know when ρ Q (C ′ j ) = C j . We know only that, when

β k ∈ Φ x i.e. r i 1 • • • r i k-1 (-α i k ) ∈ Φ x , we have necessarily h k = 1. Proof. As the type i of (C - x = C ′ 0 , C ′ 1 , ..., C ′ r )
is the type of a minimal decomposition, this gallery is minimal if, and only if, two consecutive chambers are different. So the last assertion is a consequence of the first ones. We prove these properties for (C - x = C ′ 0 , C ′ 1 , ..., C ′ j ) by induction on j. We write in the following just H j for the common limit hyperplane H β j of C j-1 and C j of type i j .

There are five possible relative positions of Q, C - x and C 1 with respect to H 1 and we seek

C ′ 1 with ρ Q (C ′ 1 ) = C 1 and C ′ 1 ⊃ C - x ∩ H 1 . 0) β 1 = -c 1 α i 1 ∈ Φ x , then H 1 is not a wall, each C ′ 1 with C ′ 1 ⊃ C - x ∩ H 1 is equal to C - x or r i 1 C - x and C ′ 1 or C - x are contained in the same apartments. So C ′ 1 = C 1 = c 1 C - x ; C 1 and Q are in g 1 A x = A x with g 1 = c 1 . When C ′ 1 = C - x ,
we have c 1 = 1 and c is not centrifugally folded.

We suppose now β 1 ∈ Φ x , so H 1 is a wall. 1) C - x is on the same side of H 1 as Q and C 1 not, then

c 1 = r i 1 , β 1 = α i 1 , w -1 Q β 1 < 0, C ′ 1 = g 1 C - x = u -α i 1 r i 1 C - x = u -α i 1 C 1 . But u -α i 1 pointwise stabilizes the halfspace bounded by H 1 containing C - x , hence u -α i 1 (Q) = Q and C ′ 1 are in the apartment g 1 A x . 2) Q and C - x = C 1 are separated by H 1 , then c 1 = 1, β 1 = -α i 1 , w -1 Q β 1 < 0, C ′ 1 = g 1 C - x = u α i 1 C - x but u α i 1 pointwise stabilizes the halfspace bounded by H 1 not containing C - x , hence Q and C ′ 1 are in the apartment g 1 A x . 3) C 1 is on the same side of H 1 as Q and C - x not, then c 1 = r i 1 , β 1 = α i 1 , w -1 Q β 1 > 0 and C ′ 1 has to be C 1 so g 1 = c 1 = r i 1 , w -1 Q (α i 1 ) > 0, moreover Q and C ′ 1 = r i 1 C - x = C 1 are in the apartment g 1 A x . 4) Q and C - x = C 1 are on the same side of H 1 . Then c 1 = 1 and w -1 Q β 1 > 0; the gallery c is not centrifugally folded. So ρ Q (C ′ 1 ) = C 1 implies C ′ 1 = C - x = g 1 C - x with g 1 = c 1 = 1 as in (1). But the gallery (C - x = C ′ 0 , C ′ 1 , ..., C ′ j
) cannot be minimal. By induction we assume now that the chambers Q and

C ′ j-1 = g 1 • • • g j-1 C - x are in the apartment A j-1 = g 1 • • • g j-1 A x .
Again, we have five possible relative positions for Q, C j-1 and C j with respect to H j . We seek

C ′ j with ρ Q (C ′ j ) = C j and C ′ j ⊃ C ′ j-1 ∩ g 1 • • • g j-1 H α i j . 0) β j = -c 1 • • • c j α i j ∈ Φ x , then H j is not a wall, each C ′ j with C ′ j ⊃ C ′ j-1 ∩ g 1 • • • g j-1 H α i j is equal to C ′ j-1 = g 1 • • • g j-1 C - x or g 1 • • • g j-1 r i j C - x ; moreover C ′ j or C ′ j-1 are contained in the same apartments. So C ′ j = g 1 • • • g j-1 c j C - x and Q are in g 1 • • • g j A x = g 1 • • • g j-1 A x with g j = c j . When C ′ j = C ′ j-1
, we have c j = 1 and c is not centrifugally folded.

We suppose now β j ∈ Φ x , so H j is a wall. 1) C j-1 is on the same side of

H j = c 1 • • • c j-1 H α i j as Q and C j not, then c j = r i j , β j = c 1 • • • c j-1 α i j , w -1 Q β j < 0. Moreover Q and C ′ j-1 are on the same side of g 1 • • • g j-1 H α i j in A j-1 , and
C ′ j = g 1 • • • g j-1 u -α i j r i j C - x = g 1 • • • g j-1 u -α i j r i j (g 1 • • • g j-1 ) -1 C ′ j-1 = g 1 • • • g j-1 u -α i j (g 1 • • • g j-1 ) -1 g 1 • • • g j-1 r i j (g 1 • • • g j-1 ) -1 C ′ j-1 ,
where

g 1 • • • g j-1 r i j (g 1 • • • g j-1 ) -1 C ′ j-1 is the chamber adjacent to C ′ j along g 1 • • • g j-1 H α i j in A j-1 . Moreover, g 1 • • • g j-1 u -α i j (g 1 • • • g j-1
) -1 pointwise stabilizes the halfspace bounded by

g 1 • • • g j-1 H α i j containing C ′ j-1 and Q. So Q and C ′ j are in the apartment g 1 • • • g j A x . 2) C j-1 = C j and Q are separated by H j , then c j = 1, β j = -c 1 • • • c j-1 α i j , w -1 Q β j < 0. Moreover C ′ j-1 and Q are separated by g 1 • • • g j-1 H α i j in A j-1
, and Q and the chamber

g 1 • • • g j-1 r i j (g 1 • • • g j-1 ) -1 C ′ j-1
are on the same side of this wall. For u α i j = 1

C ′ j = g 1 • • • g j-1 u α i j C - x = g 1 • • • g j-1 u α i j (g 1 • • • g j-1 ) -1 C ′ j-1 is a chamber adjacent (or equal) to C ′ j-1 along g 1 • • • g j-1 H α i j = g 1 • • • g j-1 u α i j H α i j in g 1 • • • g j A x (with g j = u α i j ).
The root-subgroup

g 1 • • • g j-1 U α i j (g 1 • • • g j-1
) -1 pointwise stabilizes the halfspace bounded by g 1 • • • g j-1 H α i j and containing the chamber

g 1 • • • g j-1 r i j (g 1 • • • g j-1 ) -1 C ′ j-1 . So Q and C ′ j are in the apartment g 1 • • • g j A x .
3) C j is on the same side of

H j = c 1 • • • c j-1 H α i j as Q and C j-1 not, then c j = r i j , β j = c 1 • • • c j-1 α i j , w -1 Q β j > 0. and so C ′ j = g 1 • • • g j-1 r i j C - x . Whence Q and C ′ j are in the apartment g 1 • • • g j A x .
4) C j-1 = C j and Q are on the same side of

H j = c 1 • • • c j-1 H α i j , then c j = 1, β j = -c 1 • • • c j-1 α i j and w -1 Q β j > 0. The gallery c is not centrifugally folded. So ρ Q (C ′ j ) = C j implies C ′ j = C ′ j-1 = g 1 • • • g j C - x with g j = c j = 1 as in (1). But the gallery (C - x = C ′ 0 , C ′ 1 , ..., C ′ j ) cannot be minimal. Corollary 4.5. If c ∈ Γ + Q (i), then the number of elements in C m Q (c) is: ♯C m Q (c) = t(c) k=1 q j k × r(c) l=1 (q j l -1)
where

q j = q x,β j = q x,α i j ∈ Q, t(c) = ♯{j | c j = r i j , β j ∈ Φ x and w -1 Q β j < 0} and r(c) = ♯{j | c j = 1, β j ∈ Φ x and w -1 Q β j < 0}.

Galleries and opposite segment germs

Suppose now x ∈ A ∩ I + . Let ξ and η be two segment germs in A + x . Let -η and -ξ opposite respectively η and ξ in A -

x . Let i be the type of a minimal gallery between C - x and C -ξ , where C -ξ is the negative (local) chamber containing -ξ such that w(C - x , C -ξ ) is of minimal length. Let Q be a chamber of A +

x containing η. We suppose ξ and η conjugated by W v x .

Lemma. The following conditions are equivalent: (i) There exists an opposite ζ to η in

I - x such that ρ Ax,C - x (ζ) = -ξ. (ii) There exists a gallery c ∈ Γ + Q (i) ending in -η. (iii) ξ ≤ W v
x η (in the sense of 1.8, with Φ + defined as in 4.1 using C - x ). Moreover the possible ζ are in one-to-one correspondence with the disjoint union of the sets

C m Q (c) for c in the set Γ + Q (i, -η) of galleries in Γ + Q (i) ending in -η. More precisely, if m ∈ C Q (c) is associated to (h 1 , • • • , h r ) as in remark 4.4, then ζ = h 1 • • • h r (-ξ). Proof. If ζ ∈ I - x opposites η and if ρ Ax,C - x (ζ) = -ξ, then any minimal gallery m = (C - x , M 1 , ..., M r ∋ ζ)
retracts onto a minimal gallery between C - x and C -ξ . So we can as well assume that m has type i = (i 1 , ..., i r ) and then ζ determines m. Now, if we retract m from Q, we get a gallery c = ρ Ax,Q (m) in A -

x starting at C - x , ending in -η and centrifugally folded with respect to Q.

Reciprocally, let c = (C - x , C 1 , ..., C r ) ∈ Γ + Q (i), such that -η ∈ C r .
According to proposition and remark 4.4, there exists a minimal gallery m = (C - x , C ′ 1 , ..., C ′ r ) in the set C Q (c), and the chambers C ′ j can be described by

C ′ j = g 1 • • • g j C - x = h 1 • • • h j .r i 1 • • • r i j C - x where each h k fixes C -
x , hence ρ Ax,C - x restricts on C ′ j to the action of (h x . So we proved the equivalence (i) ⇐⇒ (ii) and the last two assertions. Now the equivalence (i) ⇐⇒ (iii) is proved in [GR08, Prop. 6.1 and Th. 6.3]: in this reference we speak of Hecke paths with respect to -C v f , but the essential part is a local discussion in I x (using only C -

1 • • • h j ) -1 . Let ζ ∈ C ′ r opposite
x and the twin building structure of I ± x ) which gives this equivalence.

Liftings of Hecke paths

Let π be a λ-path from z ′ ∈ Y + to y ∈ Y + entirely contained in the Tits cone T , hence in a finite union of closed sectors wC v f with w ∈ W v . By [GR08, 5.2.1], for each w ∈ W v there is only a finite number of s ∈]0, 1] such that the reverse path π(t) = π(1t) leaves, in π(s), a wall positively with respect to -wC v f , i.e. this wall separates π -(s) from -wC v f . Therefore, we are able to define ℓ ∈ N and 0 < t 1 < t 2 < • • • < t ℓ ≤ 1 such that the z k = π(t k ), k ∈ {1, ..., ℓ} are the only points in the path where at least one wall containing z k separates π -(t k ) and the local chamber c -of 1.8.2.

For each k ∈ {1, ..., ℓ} we choose for C - z k (as in 4.1) the germ in z k of the sector of vertex z k containing c -. Let i k be a fixed reduced decomposition of the element w -(t k ) ∈ W v and let Q k be a fixed chamber in

I + z k containing η k = π + (t k ). We note -ξ k = π -(t k ).
When π is a Hecke path (or a billiard path as in [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF]), ξ k and η k are conjugated by W v z k . Remark 4.10. The same things as above may be done for the retraction ρ -∞ instead of ρ c -: for all x we choose C - x = germ x (x -C v f ). For a λ-path π in A from z ′ to y, [GR08, 5.2.1] tells that we have a finite number of points z k = π(t k ) where at least a wall is left positively by the path π(t) = π(1t). We define as above i

k , Q k , η k and ξ k . Now S -∞ (π, y) is the set of all segments [z, y] such that ρ -∞ ([z, y]) = π.
In [GR08, Theorems 6.2 and 6.3], we have proven that S -∞ (π, y) is nonempty if, and only if, π is a Hecke path with respect to -C v f . Moreover, we have shown that, for I associated to a split Kac-Moody group over C((t)), S -∞ (π, y) is isomorphic to a quasi-affine toric complex variety. The arguments above prove that, with our choice for I , S -∞ (π, y) is finite, with the following precision (which generalizes to the Kac-Moody case some formulae of [GL11]): Proposition 4.11. Let π be a Hecke path with respect to -C v f from z ′ to y. Then we have a bijection:

S -∞ (π, y) ≃ ℓ k=1 c∈Γ + Q k (i k ,-η k ) C m Q k (c)
In particular the number of elements in this set is a polynomial in the numbers q ∈ Q with coefficients in Z ≥0 depending only on A.

Theorem 4.12. Let λ, µ, ν ∈ Y ++ , c -the negative fundamental alcove and suppose (α ∨ i ) i∈I R + -free. Then a) The number of Hecke paths of shape µ with respect to c -starting in z ′ = wλ (for some w ∈ W v fixing 0) and ending in y = ν is finite.

b) The structure constant m λ,µ (ν) i.e. the number of triangles [0, z, ν] in I with d v (0, z) = λ and d v (z, ν) = µ is equal to:

m λ,µ (ν) = w∈W v /(W v ) λ π ℓπ k=1 c∈Γ + Q k (i k ,-η k ) ♯C m Q k (c) (2)
where π runs over the set of Hecke paths of shape µ with respect to c -from wλ to ν and ℓ π ,

Γ + Q k (i k , -η k ) and C m Q k (c
) are defined as above for each such π. c) In particular the structure constants of the Hecke algebra H R are polynomials in the numbers q ∈ Q with coefficients in Z ≥0 depending only on A.

Proof. We saw in 2.3.1 that m λ,µ (ν) is the number of z ∈ I + 0 such that d v (0, z) = λ and d v (z, ν) = µ. Such a z determines uniquely a Hecke path π = ρ c -([z, ν]) of shape µ with respect to c -from z ′ = ρ c -(z) to ν. But d v (0, z) = λ and 0 ∈ c -, so d v (0, z ′ ) = λ i.e. z ′ = wλ with w ∈ W v . So the formula (2) follows from theorem 4.8.

We know already that m λ,µ (ν) is finite (2.5) and S c -(π, y) = ∅ (theorem 4.8), hence a) is clear. Now c) follows from corollary 4.5

Satake isomorphism

In this section, we prove the Satake isomorphism. From now on, we assume that the α ∨ i 's are free.

We denote by U -the fixator in G of the sector germ S -∞ , i.e. any u ∈ U -has to fix pointwise a sector x -C v f ⊂ A. By definition, for z ∈ I , ρ -∞ (z) is the only point of the orbit U -.z in A.

The module of functions on the type 0 vertices in A

Let A 0 = ν(N ) • 0 = Y • 0 be the set of vertices of type 0 in A. Note that A 0 can be identified to the set of horocycles of U -in I 0 , i.e. to I 0 /U -, via the retraction ρ -∞ . We consider first

F = F R = F (A 0 , R)
, the set of functions on A 0 with values in a ring R. Equivalently, F can be identified with the set of U --invariant functions on I 0 .

For µ ∈ Y , we define χ µ ∈ F as the characteristic function of U -.µ in I 0 (or {µ} in Y ). Then, any χ ∈ F R may be written χ = µ∈Y a µ χ µ with a µ ∈ R. We set supp(χ) = {µ | a µ = 0}. Now, let

F = F R = {χ ∈ F | supp(χ) ⊂ ∪ n j=1 (µ j -Q ∨ + ) for some µ j ∈ A 0 }
be the set of functions on I 0 with almost finite support. We define also the following completion of the group algebra R[Y ]: Proof. It is relatively clear that χ * ϕ is a function on I 0 /U -and that the map indeed defines an action. Let us check that this action commutes with the one of Z. Let t ∈ Z and x ∈ I 0 , then The difficult point is to show that the support condition is satisfied. For any λ ∈ Y ++ , and any ν ∈ Y ,

R[[Y ]] = {f = y∈Y a y e y | supp(f ) = {y ∈ Y | a y = 0} ⊂ ∪ n j=1 (µ j -Q ∨ + ) for some µ j ∈ A 0 } it is
(χ * ϕ)(tx) = y∈I 0 χ(y)ϕ I (y, tx) = y ′ ∈I 0 χ(ty ′ )ϕ I (ty ′ , tx) (y = ty ′ ) = y ′ ∈I 0 χ(ty ′ )ϕ I (y ′ , x) = ((χ • t) * ϕ)(x). So, (χ • t) * ϕ = (χ * ϕ) • t.
(χ µ * c λ )(ν) = y∈I 0 χ µ (y)c I λ (y, ν) = ♯{y ∈ I 0 | ρ -∞ (y) = µ and d v (y, ν) = λ}
The latest is also the cardinality of the set of all segments [y, ν] in I (y ≤ ν) of "length" λ such that y ∈ U -• µ. In addition, since the action of H commutes with the one of Z, we set

n λ (ν -µ) = (χ µ * c λ )(ν). Then n λ (ν -µ) = π ♯S -∞ (π, ν)
where the sum runs over the set of Hecke λ-paths with respect to -C v f from µ to ν (see 4.10 for the definition of

S -∞ (π, ν)). Now, Lemma 2.4 b) shows that n λ (ν -µ) = 0 implies ν -µ ≤ Q + λ. Moreover, if ν = λ + µ, then n λ (λ) = 1. Therefore, we get χ µ * c λ = ν≤ Q ∨ λ+µ n λ (ν -µ)χ ν = χ λ+µ + ν< Q ∨ λ+µ n λ (ν -µ)χ ν . (3)
This formula shows that, for any ϕ ∈ H with supp(ϕ

) ⊂ ∪ n i=1 (λ i -Q ∨ + ) and any ξ ∈ F with supp(χ) ⊂ ∪ n j=1 (µ j -Q ∨ + ), the support of χ * ϕ is contained in ∪ i,j (λ i + µ j -Q ∨ + ). More precisely, for any ν ∈ ∪ i,j (λ i + µ j -Q ∨ + )
there exists a finite number of λ ∈ supp(ϕ) and µ ∈ supp(χ) such that ν ≤ Q + λ + µ. Hence, χ * ϕ is well defined.

The Satake isomorphism

S * (c λ ) = ν≤ Q ∨ λ n λ (ν)e ν = e λ + ν< Q ∨ λ n λ (ν)e ν
We shall modify S * by some character to get the Satake isomorphism.

The module δ

We define a map δ :

Q ∨ → R * + , i∈I a i α ∨ i → i∈I (q i q ′ i ) a i
, where q i , q ′ i ∈ Q ⊂ N are as in the beginning of Section 4. We extend this homomorphism and its square root to Y (as R * + is uniquely divisible). So, we get homomorphisms δ, δ 1/2 : Y

→ R * + and δ = δ • ν, δ 1/2 = δ 1/2 • ν : Z → R * + .
We made a choice for δ. But we shall see in theorem 5.4 that the expected properties depend only on δ Q ∨ .

In the classical case, where G is a split semi-simple group and I its Bruhat-Tits building, we have q i = q ′ i = q for any i ∈ I. Hence, if we set µ = i∈I a i α ∨ i , δ 1/2 (µ) = q a i = q ρ(µ) where ρ is the half sum of positive roots.

The Satake isomorphism

From now on, we suppose that the algebra R contains the image of δ 1/2 in R * + . We define

S(c λ ) = µ≤ Q ∨ λ δ 1/2 (µ)n λ (µ)e µ = δ 1/2 (λ)e λ + µ< Q ∨ λ δ 1/2 (µ)n λ (µ)e µ
and extend it to formal combinations of the c λ with almost finite support. We get thus an algebra homomorphism S :

H → R[[Y ]] called the Satake isomorphism, as it is one to one: For ϕ = λ a λ c λ ∈ H, we have S(ϕ) = λ a λ δ 1/2 (λ)e λ + µ< Q ∨ λ δ 1/2 (µ)n λ (µ)e µ .
If ϕ = 0 and λ 0 is a maximum element in supp(ϕ), then λ 0 is also a maximum element in supp(S(ϕ)) and S(ϕ) = 0.

Remarks. a) So we already know that H is commutative. b) In the classical case where G is a split semi-simple group, S(c λ ) is defined as an integral over a maximal unipotent subgroup, we choose here U -. The Haar measure du on U -is chosen to give volume 1 to K ∩ U -, and, for an element t in the torus Z, the formula for changing variables is given by d(tut -1 ) = δ(t) -1 du. So the classical formula for the Satake isomorphism given e.g. in [START_REF] Cartier | Representations of p-adic groups: a survey, in Automorphic forms, representations and L-functions[END_REF](19) p 146] when ν(t) = µ, is:

S(c λ )(t) = δ(t) 1/2 U -c G λ (ut)du = δ(t) 1/2 U -c I λ (0, ut.0)du = δ(t) 1/2 U -c I λ (u -1 .0, t.0)du = δ(t) 1/2 y∈U -.0 c I λ (y, µ) = δ(t) 1/2 y∈I 0 χ 0 (y).c I λ (y, µ) = δ(t) 1/2 (χ 0 * c λ ) (µ
) This is the same formula as ours.

W v -invariance

There is an action of W v on Y , hence on R[Y ] by setting w.e λ = e wλ for w ∈ W v and λ ∈ Y . This action does not extend to R

[[Y ]], but we define R[[Y ]] W v = {f = a λ e λ ∈ R[[Y ]] | a λ = a wλ , ∀λ ∈ Y, ∀w ∈ W v }. This is a subalgebra of R[[Y ]
] and actually the image of the Satake isomorphism (see Theorem 5.4).

Remark. Let

C ∨ = {π ∈ V * | α ∨ i (
π) ≥ 0, ∀i ∈ I} and T ∨ = ∪ w∈W v wC ∨ be the fundamental dual chamber and the dual Tits cone in V * . By definition, for f

∈ R[[Y ]] and π ∈ C ∨ , π(supp(f )) is bounded above. Hence, for f ∈ R[[Y ]] W v , π(supp(f )
) is also bounded above for any π ∈ T ∨ . We know that the dual cone of T ∨ is the closed convex hull Γ of the set Proof. As S(c λ ) = µ≤ Q ∨ λ δ 1/2 (µ)n λ (µ)e µ we only have to prove that, for w ∈ W v , δ 1/2 (µ)n λ (µ) = δ 1/2 (wµ)n λ (wµ) or n λ (wµ) = n λ (µ)δ 1/2 (µwµ). It is sufficient to prove this for w = r i a fundamental reflection, hence to prove that n λ (r

∆ ∨im + ∪{0}, where ∆ ∨im + ⊂ Q ∨ + is
i µ) = n λ (µ)δ 1/2 (µ -r i µ) = n λ (µ)δ 1/2 (α i (µ)α ∨ i ).
By the given definition of δ, the wanted formula is:

n λ (r i µ) = n λ (µ) q i q ′ i α i (µ) (4)
The proof of this formula is postponed to the following subsections, starting with 5.5. One can already notice that α i (µ) is an integer. If it is odd, since any t ∈ Z with ν(t) = µ exchanges the walls M (α i , 0) and M (α i , α i (µ)), hence q i = q ′ i . So, in any case

q i q ′ i |α i (µ)| is an integer. Once the formula (4) is proved we know that S(H) ⊂ R[[Y ]] W v . For f = a µ e µ ∈ R[[Y ]] W v with supp(f ) ⊂ ∪ r j=1 (λ j -Q ∨ + )
, we shall build a sequence ϕ n in H such that supp(f -S(ϕ n )) ⊂ ∪ r j=1 (λ j -Q ∨ +n ) and supp(ϕ n+1ϕ n ) ⊂ Y ++ ∩ (∪ r j=1 (λ j -Q ∨ +n )), where

Q ∨ +n = { i∈I n i α ∨ i ∈ Q ∨ + | n i ≥ n}.
Then, the limit ϕ of this sequence exists in H and S(ϕ) = . So, S is onto.

We build the sequence by induction. We set ϕ 0 = 0. If ϕ 0 , • • • , ϕ n are given as above, we set {µ 1 , • • • , µ s } = supp(f -S(ϕ n )) \ ∪ r j=1 (λ j -Q ∨ +(n+1) ). For any w ∈ W v , wµ k ∈ supp(f -S(ϕ n )) ⊂ ∪ r j=1 (λ j -Q ∨ +n ), so wµ k cannot be strictly greater than µ k for ≤ Q ∨ ; this proves that µ k ∈ Y ++ . So we define ϕ n+1 = ϕ n -s k=1 a µ k (f -S(ϕ n ))δ(µ k ) -1/2 c µ k . As S(c λ ) = δ 1/2 (λ)e λ + µ< Q ∨ λ δ 1/2 (µ)n λ (µ)e µ , this ϕ n+1 is suitable.

Remark. Suppose G is a split Kac-Moody group as in Section 3. And consider the complex Kac-Moody algebra g ∨ associated with G ∨ , the Langlands dual of G. Let h ∨ = C ⊗ Z Y be the Cartan subalgebra of g ∨ . Let Rep(g ∨ ) be the category of g ∨ -modules V such that V is h ∨ -diagonalizable, the weight spaces V λ are finite dimensional and the set P(V ) of weights of V satisfies P(V ) ⊂ ∪ r j=1 (λ j -Q ∨ + ), for some λ j . One can check that Rep(g ∨ ) is stable by tensoring, hence, we can consider its Grothendieck ring K(g ∨ ). Now, the map [V ] → λ (dim V λ )e λ is an isomorphism from K(g ∨ ) onto C[[Y ]] W v . Therefore, by composing it with S, we get an isomorphism between H C and K(g ∨ ).

Extended tree associated to (A, α i )

We consider the vectorial panel -F v ({i}) in -C v f and its support the vectorial wall Ker(α i ). Their respective directions are a panel F ∞ in a wall M ∞ , in the twin buildings I ±∞ at infinite of I [Ro11, 3.3, 3.4, 3.7].

The germs of the sector panels in I of direction F ∞ are the points of an (essential) affine building I (F ∞ ), which is of rank 1 i.e. a tree [START_REF] Rousseau | Masures affines[END_REF]4.6].

The union I (M ∞ ) of the apartments in I containing a wall of direction M ∞ is an inessential affine building whose essential quotient is I (F ∞ ) [Ro11, 4.9]. More precisely I (M ∞ ) may be identified with the product of the tree I (F ∞ ) and an affine space quotient of A.

The canonical apartment of I (M ∞ ) is A endowed with a smaller set of walls: uniquely the walls of direction Ker(α i ). As we chose I semi-discrete (1.2), this is a locally finite set of hyperplanes; hence I (M ∞ ) is discrete and I (F ∞ ) a discrete tree (not an R-tree). By [Ro11, 2.9] the valences of these walls are the same in I (M ∞ ) and in I , i.e. 1 + q i and 1 + q ′ i ; hence I (F ∞ ) is a semi-homogeneous tree of valences 1 + q i and 1 + q ′ i . By definition, 0 ∈ A is in a wall of valence 1 + q i . We asked that the stabilizer N of A in G is positive and type preserving (1.5) i.e. acts on V = -→ A via W v . So, the stabilizer in W v of M ∞ is {1, r i } and M ∞ determines in V a supplementary vectorial subspace of dimension one : M ⊥ ∞ = Ker(1 + r i ). The affine space A decomposes as the product of the affine space E = A/M ⊥ ∞ with associated vector space Ker(α i ) and an affine line (= A/Ker(α i )). This decomposition is canonical i.e.

Parabolic retraction

Let x be a point in I . There is a unique sector-panel x + F ∞ of vertex x and direction F ∞ [Ro11, 4.7.1]. The germ of this sector-panel is a point in I (F ∞ ), the projection pr F∞ (x) of x onto I (F ∞ ), cf.

[Ch10], [START_REF] Charignon | Immeubles affines et groupes de Kac-Moody, masures bordées[END_REF] or [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF]4.3.5] in the Kac-Moody case. Let A x be an apartment in I containing x and F ∞ , hence x + F ∞ and germ ∞ (x + F ∞ ). But this germ is in an apartment B x of I (M ∞ ) (axiom (MA3) applied to germ ∞ (x+F ∞ ) and a sector of direction C v f ) and there exists an isomorphism ψ x of A x onto B x fixing this germ (axiom (MA2)). One writes ρ(x) = ψ x (x) ∈ I (M ∞ ). We have thus defined the retraction ρ = ρ F∞,M∞ of I onto I (M ∞ ) with center F ∞ . We shall now verify that ρ(x) does not depend on the choices made.

By definition, ρ(x) is in the hyperplane H x of B x containing germ ∞ (x + F ∞ ) and of direction M ∞ , this H x does not depend on the choice of B x . Moreover for two choices ψ x : A x → B x and ψ ′

x :

A ′ x → B x , ψ ′ x • ψ -1
x is the identity on germ ∞ (x + F ∞ ) hence on H x . It is now clear that ψ x (x) = ψ ′ x (x). Actually ρ(x) may also be defined in the following simple way: there exist y, z ∈ (x + F ∞ ) ∩ B x such that y is the middle of [x, z] in A x , then ρ(x) is the point of H x ⊂ B x such that y is the middle of [ρ(x), z] in B x .

Remark. It is possible to prove that the image by ρ of a preordered segment is a polygonal line and, in some generalized sense, a Hecke path.

Factorization of ρ -∞

The panel F ∞ is in the closure of the chamber C -∞ of I -∞ associated to -C v f . So this chamber or the associated sector-germ S -∞ determines an end of the tree I (F ∞ ) [Ro11, 4.6] i.e. a sector-germ S ′ in I (M ∞ ): S ′ is one of the two sector-germs in A (considered as an apartment of I (M ∞ ) with its small set of walls), each element in S ′ contains an half apartment of equation α i (y) ≤ k with k ∈ Z. We write ρ ′ -∞ the retraction of I (M ∞ ) onto A with center S ′ .

Lemma. The retraction ρ -∞ factorizes through ρ : ρ -∞ = ρ ′ -∞ • ρ. Proof. For x ∈ I , one chooses an apartment A x containing x and C -∞ , hence the sector x + C -∞ , its sector-germ S -∞ and its panel x + F ∞ . One chooses also an apartment

  η in any apartment containing those two. The minimality of the gallery m = (C - x , C ′ 1 , ..., C ′ r ) ensures that ρ Ax,Cx (ζ) ∈ C -ξ ; hence ρ Ax,Cx (ζ) = -ξ as they are both opposite η up to conjugation by W v

Proof.

  For any u ∈ [y, z], we have x ≤ y ≤ u ≤ z, hence x and [u, y) or [u, z) are in a same apartment[START_REF] Rousseau | Masures affines[END_REF] 5.1]. As [y, z] is compact we deduce that there are points u 0 = y, u 1 , • • • , u ℓ = z such that x and [u i-1 , u i ] are in a same apartment A i , for 1 ≤ i ≤ ℓ. Now A 1 contains x and [y, u 1 ], hence also [x, y] (axiom (MAO) of 1.5). But [y, x) and [y, u 1 ) = [y, z) are opposite, so [x, y] ∪ [y, u 1 ] = [x, u 1 ]. The lemma follows by induction.

  clearly a commutative algebra (with e y .e z = e y+z ). Actually, it is the Looijenga's coweight algebra, see Section 4.1 in [Loo]. The formula (f.χ)(µ) = y∈Y a y χ(µy), for f = a y e y ∈ R[[Y ]], χ ∈ F and µ ∈ Y , defines an element f.χ ∈ F ; in particular e y .χ µ = χ µ+y . Clearly, the map R[[Y ]] × F → F , (f, χ) → f.χ makes F into a free R[[Y ]]-module of rank 1, with any χ µ as basis element. Definition-Proposition 5.2. The map F × H → F (χ, ϕ) → χ * ϕ, where, for x ∈ I 0 , (χ * ϕ)(x) = y∈I 0 χ(y)ϕ I (y, x), defines a right action of H on F that commutes with the actions of Z = {n ∈ N | ν(n) ∈ Y } and (more generally) R[[Y ]].

  For ν(t) = µ ∈ Y and χ ∈ F , we have clearly χ • t = e -µ .χ. As a formal consequence, the right action of H commutes with the left action of R[[Y ]].

5. 3 . 1

 31 The morphism S * As F is a free R[[Y ]]-module of rank one, we have End R[[Y ]] (F ) = R[[Y ]]. So the right action of H on the R[[Y ]]-module F gives an algebra homomorphism S * : H → R[[Y ]] such that χ * ϕ = S * (ϕ).χ for any ϕ ∈ H and any χ ∈ F . As e ν .χ µ = χ µ+ν , equation (3) gives

  the set of positive imaginary roots in the dual system of roots ∆ ∨ , [Ka90, 5.8]. So, the only directions along which points in supp(f ) (for f ∈ R[[Y ]] W v ) may go to infinity are the directions in -Γ. Theorem 5.4. The Hecke algebra H R is isomorphic via S to the commutative algebra ]] W v of Weyl invariant elements in R[[Y ]].

  invariant by the stabilizer N (M ∞ ) of M ∞ in N . As a consequence we get the decompositionI (M ∞ ) = E × I (F ∞ ) which is canonical i.e. invariant by the stabilizer G(M ∞ ) of M ∞ in G. Moreover G(M ∞ )acts on E by translations only.Remark. Suppose G is an almost split Kac-Moody group over a local field K and I its associated hovel as in[START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF]. Then the stabilizerG(F ∞ ) of F ∞ in G is a parabolic subgroup, endowed with a Levi decomposition G(F ∞ ) = G(M ∞ ) ⋉ U (F ∞ ) (with U (F ∞ ) ⊂ U -) and I (M ∞ ) (resp. I (F ∞ ))is the extended (resp. essential) Bruhat-Tits building associated to the reductive group of rank one G(M ∞ ), embedded in I [Ro13, 6.12.2]. Any orbit of U (F ∞ ) in I meets I (M ∞ ) in one and only one point.The tree I (F ∞ ) is a piece of the polyhedral "compactification" of I (a true compactification when G is reductive). With the notation of[START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF], I (M ∞ ) (resp. I (F ∞ )) is the façade I (G, K, A) F∞ (resp. I (G, K, A e ) F∞ ).

  ) ++ and we know how to get π ′ + (t) from π ′ -(t) by successive reflections; this proves that π ′

+ (t) ∈ π ′ -(t) + Q ∨ R+ .

By integrating the locally constant function π ′ (t), we get π ′

Acknowledgement:

The first author acknowledges support of the ANR grants ANR-09-JCJC-0102-01 and ANR-2010-BLAN-110-02.

Stéphane Gaussent & Guy Rousseau

When π is a Hecke path with respect to c -, {z 1 , • • • , z ℓ } includes all points where the piecewise linear path π is folded and, in the other points, all galleries in Γ + Q k (i k , -η k ) are unfolded.

Let S c -(π, y) be the set of all segments [z, y] such that ρ c -([z, y]) = π.

Theorem 4.8. S c -(π, y) is non empty if, and only if, π is a Hecke path with respect to c -. Then, we have a bijection

In particular the number of elements in this set is a polynomial in the numbers q ∈ Q with coefficients in Z ≥0 depending only on A.

N.B. So the image by ρ c -of a segment in I + is a Hecke path with respect to c -.

Proof. The restriction of ρ

; so the lemma 4.6 tells that π is a Hecke path with respect to c -if, and only if, each

) is non empty. We set t 0 = 0 and t ℓ+1 = 1. We shall build a bijection from

) is a minimal gallery given by a sequence of elements (h k 1 , ..., h k r k ) ∈ (G z k ) r k , as in remark 4.4 and, for t ∈ [t n , t n+1 ], we have

is the h k j defined above; in particular each h k j fixes c -. We note g = (h ℓ 1 ...

and retracting to π -(t n ) by ρ c -. By lemma 4.6 and the above remark, this segment germ determines uniquely a minimal gallery

As above we replace each g n j by a chosen element of

is a segment ∀k ∈ {1, ..., ℓ + 1}. Moreover, for k ∈ {1, ..., ℓ}, the segment germs [s(t k ), s(t k+1 )) and [s(t k ), s(t k-1 )) are opposite. By the following lemma this proves that s itself is a segment. Lemma 4.9. Let x, y, z be three points in an ordered hovel I , with x ≤ y ≤ z and suppose the segment germs [y, z) , [y, x) opposite in the twin buildings

and S -∞ . Hence, A x and B x contain both germ ∞ (x + F ∞ ) and S -∞ ; by axiom (MA4) there exists an isomorphism ψ x of A x onto B x fixing these two germs. By the definition of the parabolic retraction, in 5.6, ρ(x) = ψ x (x). Now the apartments A x and B x of I (M ∞ ) contain both S -∞ , hence S ′ . So there is an isomorphism θ :

Counting

We want to prove equation (4): n λ (r i µ) = n λ (µ) q i q ′ i α i (µ) for λ ∈ Y ++ and µ ∈ Y , where n λ (µ) is the number of points y ∈ I 0 such that ρ -∞ (y) = -µ and d v (y, 0) = λ, cf. 5.2. For z ∈ I (M ∞ ) one writes p λ (z) ∈ Z ≥0 ∪ {∞} for the number of points y ∈ I 0 such that ρ(y) = z and d v (y, 0) = λ. By lemma 5.7, n λ (µ) is the sum of p λ (z)

) acts transitively on the apartments of I or I (M ∞ ) containing it (by axiom (MA4), as M 0 is the enclosure of two sector panel germs). Moreover

, where the sum runs over the orbits Ω of G(M 0 ) in I (M ∞ ) ∩ I 0 and n Ω (ν) is the number of points in the orbit Ω such that ρ ′ -∞ (z) = ν. To prove formula (4), it is sufficient to prove for any orbit Ω as above and any ν ∈ Y that:

We saw, in 5.5, that G(M ∞ ) leaves the decomposition I (M ∞ ) = I (F ∞ ) × E invariant and acts on E by translations. But G(M 0 ) fixes M 0 ∋ 0, so it acts trivially on E. As G(M 0 ) is transitive on the apartments containing M 0 , an orbit Ω is a set S r × {e} where S r is the sphere of radius r ∈ Z ≥0 and center 0 in the tree I (F ∞ ). The apartment A (with its small set of walls) is the product (R, Z) × E, where α i is the projection of A onto the one dimensional apartment R with vertex set Z.

So, the above formula, hence Formula (4) and Theorem 5.4 are consequences of the following proposition. The fact that q i = q ′ i when m = α i (ν) is odd, was explained in the proof of 5.4.

The tree case

Let T be a (discrete) semi-homogeneous tree. Let A ≃ R be an apartment in T whose vertices are identified with Z. The valence of the vertex s ∈ Z is 1 + q (resp. 1 + q ′ ) if s is even (resp. odd). Let -∞ be the end of A corresponding to integers converging towards -∞. Let ρ ′ be the retraction of T onto A with center -∞. For m ∈ Z ⊂ A and r ∈ Z ≥0 we write n r (m) the number of vertices in the sphere S r of center 0 and radius r in T such that ρ ′ (z) = m.

If m is odd we ask that q = q ′ .

Remark. This formula is equivalent to the W v (T)-invariance of the image of the Satake isomorphism for the Bruhat-Tits tree T. As this invariance is known, the following proof is not necessary; we give it for the convenience of the reader.

Stéphane Gaussent & Guy Rousseau

For a Bruhat-Tits tree I = T, there are two choices for I 0 (and Y ): the set of vertices at even distance from 0 or the full set of vertices. In this last case, we have to allow m to be odd and we see below that the hypothesis q = q ′ is necessary to get the formula. So, even for classical Bruhat-Tits buildings, to get the good image for the Satake isomorphism, I 0 cannot be any G-stable set of special vertices (we chose I 0 to be a G-orbit).

Proof. For z ∈ S r , let s z ∈ Z be the vertex of A such that [0, s z ] = [0, z] ∩ A. Then ρ ′ (z) = s z + (r -|s z |) ∈ Z.

We can calculate the number n r (m) of vertices z ∈ S r such that ρ ′ (z) = m: First case: s z ≥ 0 ⇐⇒ ρ ′ (z) = r. So n r (r) = qq ′ qq ′ • • • (r factors).

Second case: -r ≤ s z < 0 ⇐⇒ ρ ′ (z) < r and then ρ ′ (z) = r + 2s z i.e. s z = (ρ ′ (z)r)/2. The number n r (m) is then:

It is now easy to compare n r (m) and n r (-m). We get the wanted formula, using that q = q ′ when m is odd.