On structure constants of Iwahori-Hecke algebras for Kac-Moody groups

We consider the Iwahori-Hecke algebra associated to an almost split Kac-Moody group $G$ (affine or not) over a nonarchimedean local field $K$. It has a canonical double-coset basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ indexed by a sub-semigroup $W^+$ of the affine Weyl group $W$. The multiplication is given by structure constants $a^ {\mathbf u}_{\mathbf w,\mathbf v}\in N=Z_{\geq0}$ : $T_{\mathbf w}*T_{\mathbf v}=\sum_{\mathbf u\in P_{\mathbf w,\mathbf v}} a^ {\mathbf u}_{\mathbf w,\mathbf v} T_{\mathbf u}$. A conjecture, by Bravermann, Kazhdan, Patnaik, Gaussent and the authors, tells that $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ is a polynomial, with coefficients in $N$, in the parameters $q_{i}-1,q'_{i}-1$ of $G$ over $K$. We prove this conjecture when $\mathbf w$ and $\mathbf v$ are spherical or, more generally, when they are said generic: this includes all cases of $\mathbf w,\mathbf v\in W^+$ if $G$ is of affine or strictly hyperbolic type. In the split affine case (where $q_i=q'_i=q$, $\forall i$) we get a universal Iwahori-Hecke algebra with the same basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ over a polynomial ring $Z[Q]$; it specializes to our Iwahori-Hecke algebra when one sets $Q=q$.


Introduction
Let G be a split, semi-simple, simply connected algebraic group over a non archimedean local field K. So K is complete for a discrete, non trivial valuation with a finite residue field κ. We write O ⊂ K for the ring of integers and q for the cardinality of κ. Then G is locally compact. In this situation, Nagayoshi Iwahori and Hideya Matsumoto in [22], introduced an open compact subgroup K I of G, now known as an Iwahori subgroup. If N is the normalizer of a suitable split maximal torus T (K * ) n , then (K I , N ) is a BN pair. The Iwahori-Hecke algebra of G is the algebra I H R = I H R (G, K I ) of locally constant, compactly supported functions on G, with values in a ring R, that are bi-invariant by the left and right actions of K I . The multiplication is given by the convolution product.
If H (O * ) n is the maximal compact subgroup of T , then H ⊂ K I and W = N/H is the affine Weyl group. One has the Bruhat decomposition G = K I · W · K I = one has T w = T ri 1 * · · · * T ri s . In a Coxeter group one knows the rules to get (using the Coxeter relations between the r i ) a reduced expression from a non reduced expression (e.g. the product of two reduced expressions w = r i1 · . . . · r is and v = r j1 · . . . · r jt ). So one deduces easily (using the above relations between the T ri ) that each structure constant a u w,v (for u, v, w ∈ W ) is in Z[q]. More precisely it is a polynomial in q − 1 with coefficients in N = Z 0 . This polynomial depends only on u, v, w and W .
So one has a universal description of I H Z as a Z[q]−algebra, depending only on W .
There are various generalizations of the above situation. First one may replace G by a general reductive group over K, isotropic but potentially non split. Then one has to consider the relative affine Weyl group W , which is a Coxeter group. One may still define a compact, open Iwahori subgroup K I and there is a Bruhat decomposition G = K I · W · K I . Now the description of I H R involves parameters q i (satisfying T 2 ri = q i · 1 + (q i − 1) · T ri ) which are potentially different from q. This gives the Iwahori-Hecke algebra with unequal parameters. There is a pleasant description of I w,v are Laurent polynomials in the q i , q i [l.c. 6.7]; they are true polynomials if w, v ∈ W v (Y ∩ T • ) and v is "regular" [l.c. 3.8].
In this article, we prove the conjecture 1 when w and v are in W +g (see 3.4). We remark also that W + = W +g in the affine case (twisted or not) or the strictly hyperbolic case, even if G is not split. This is a first step towards the description of an abstract algebra I H Z[Q] (resp. I H g Z[Q] ) over Z[Q] in the affine (or strictly hyperbolic) case (resp. in the general case).
One should mention here that one may give a more precise description of the Iwahori-Hecke algebra using a Bernstein-Lusztig presentation (see [17], [8] and [3]). But this description is given in a new basis and the coefficients of the change of basis matrix are Laurent polynomials in the parameters q i , q i . So this description is not sufficient to prove the conjecture.
Actually this article is written in a more general framework explained in Section 1: as in [3], we work with an abstract masure I and we take G to be a strongly transitive group of vectorially-Weyl automorphisms of I . In Section 2 we gather the additional technical tools (e.g. decorated Hecke paths) needed to improve the results of [3,Section 3]. We get our main results about a u w,v in Section 3: we deal with the cases w, v spherical. In Section 4 we deal with the remaining cases where w, v are in W +g , i.e. when w, v are said generic. 1. General framework 1.1. 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 free family in V and (α i ) i∈I a free family in the dual V * . We ask these data to satisfy the conditions of [31, 1.1]. In particular, the formula 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 The set Φ is an (abstract, reduced) real root system in the sense of [26], [27] or [1]. We shall sometimes also use the set [23]. It is an (abstract, reduced) root system in the sense of [1].
The fundamental positive chamber is 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 support of such a face is the vector space it generates. The set J or the face w · F v (J) or an element of this face is called spherical if the group W v (J) generated by {r i | i ∈ J} (which is the fixator or stabilizer in W v of F v (J)) is finite. An element of a vectorial chamber ±w · C v f is called regular. The Tits cone T (resp. its interior T • ) is the (disjoint) union of the positive (resp. and spherical) vectorial faces. It is a W v −stable convex cone in V . One has T = T • = V (resp. V 0 ⊂ T T • ) in the classical (resp. non classical) case, i.e. when W v is finite (resp. infinite). By the above characterization of spherical faces, T • is the set of x ∈ T whose fixator in W v is finite.
We say that 1.2. The model apartment. As in [31, 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. More precisely, they may be written We ask this apartment to be semi-discrete and the origin 0 to be special. This means that these walls are the hyperplanes 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 [19, Lemma 1.3] (i.e. replacing Φ by another system Φ 1 ) we may (and shall) assume that Λ α = Z, ∀α ∈ Φ.
For α = w(α i ) ∈ Φ, k ∈ 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 points 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. We know that W a = W v Q ∨ and we write W a R = W v V ; here Q ∨ and V have to be understood as groups of translations. An automorphism of A is an affine bijection ϕ : A → A stabilizing the set of pairs (M, α ∨ ) of a wall M and the coroot associated with α ∈ Φ such that M = M (α, k), k ∈ Z. The group Aut(A) of these automorphisms contains W a and normalizes it. We consider also the group Aut W The Tits cone T and its interior T o are convex and W v −stable cones, therefore, we can define three W v −invariant preorder relations on A: If W v has no fixed point in V {0} (i.e. V 0 = {0}) and no finite factor, then they are orders; but, in general, they are not.
1.3. Faces, sectors. The faces in A are associated to the above systems of walls and half-apartments. As in [9], 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 [9] or [18].
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 contain intersections of F Algebraic Combinatorics, Vol. 4 #3 (2021)

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On structure constants of Iwahori-Hecke algebras for Kac-Moody groups and neighbourhoods of x. In particular, if x = y ∈ A, we denote the germ in x of the segment [x, y] (resp. of the interval ]x, y]) by [x, y) (resp. ]x, y)).
For y =x, the segment germ [x, y) is called of sign ± if y − x ∈ ±T . The segment [x, y] (or the segment germ [x, y) or the ray with origin x containing y) is called preordered if x y or y x and generic if x Given F a filter of subsets of A, its strict enclosure cl A (F ) (resp. closure F ) 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 ∪ {∞} (resp. containing the closure S of some S ∈ F ). One considers also the (larger) enclosure cl # A (F ) of [33, 3.6.1] (introduced in [10,11,12] and well studied in [21], see also [20]). It is the filter made of the subsets of A containing an element of F of the shape ∩ α∈Ψ D(α, k α ), with Ψ ⊂ Φ finite and k α ∈ Z (i.e. a finite intersection of half apartments).
A local face F in the apartment A is associated to a point x ∈ A, its vertex, and a vectorial face There is an order on the local 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 local face F is the smallest dimension of an affine space generated by some S ∈ F . The (unique) is spherical if the direction of its support meets the open Tits cone (i.e. when F v is spherical), then its pointwise stabilizer W F in W a or W a R is finite and fixes x. We shall actually here speak only of local faces, and sometimes forget the word local or write F = F (x, F v ).
A local chamber is a maximal local face, i.e. a local face F (x, ±w·C v f ) for x ∈ A and w ∈ W v . The fundamental local positive (resp. negative) chamber is . A (local) panel is a spherical local face maximal among local faces which are not chambers, or, equivalently, a spherical face of dimension n − 1. Its support is a hyperplane parallel to a wall.
The point 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.
The sign of f and F is the sign of F v .
1.4. The Masure. In this section, we recall the definition and some properties of a masure given by Guy Rousseau in [31] and simplified by Auguste Hébert [21].
and only if, there exists w ∈ W a satisfying f = f 0 • w. An isomorphism (resp. a Weylisomorphism, a vectorially-Weyl isomorphism) between two apartments ϕ : ; the group of these isomorphisms is written )). As the filters in A defined in 1.3 above (e.g. local faces, sectors, walls,...) are permuted by Aut(A), they are well defined in any apartment of type A and exchanged by any isomorphism.
A masure (formerly called an ordered affine hovel) of type A is a set I endowed with a covering A of subsets called apartments, each endowed with some structure of an apartment of type A. We recall here the simplification and improvement of the original definition given by Auguste Hébert in [21]: these data have to satisfy the following two axioms: If R is the germ of a splayed chimney and if F is a local face or a germ of a chimney, then there exists an apartment containing R and F .
Actually a filter or subset in I is called a preordered (or generic) segment (or segment germ), a local face, a spherical sector face or a spherical sector face germ if it is included in some apartment A and is called like that in A. We do not recall here what is (a germ of) a (splayed) chimney; it contains (the germ of) a (spherical) sector face. We shall actually use (MA iii) uniquely through its consequence (b) below.
In the affine case the hypothesis "A ∩ A contains a generic ray" may be omitted in (MA ii).
We list now some of the properties of masures we shall use.
(a) If F is a point, a preordered segment, a local face or a spherical sector face 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 a Weyl-isomorphism from A onto A fixing cl # A (F ), see [21, 5.11] or [20, 4.4.10]. Hence any isomorphism from A onto A fixing F fixes F (and even cl # A (F ) ∩ supp(F )). More generally the intersection of two apartments A, A is always closed (in A and A ), see [21, 3.9] is an isomorphism (resp. a Weyl-isomorphism, a vectorially-Weyl isomorphism). We write Aut(I ) (resp. Aut W (I ), Aut W R (I )) the group of these automorphisms.
for y > x (resp. y < x) may be considered as a building, the positive (resp. negative) tangent building. The corresponding faces are the local faces of positive (resp. negative) direction and vertex x. For such a local face F , we write sometimes [ The buildings T + Similarly two segment germs η ∈ T + x I and ζ ∈ T − x I are said opposite if they are in a same apartment A and opposite in this apartment (i.e. in the same line, with opposite directions). 1.4.4. We assume that I has a strongly transitive group of automorphisms G, i.e. 1.4.1(a) and (c) above (after replacing cl # A by cl A ) are satisfied by isomorphisms induced by elements of G, cf. [33, 4.10] and [15, 4.7].
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 W = ν(N ) ⊂ Aut(A) of affine automorphisms of A which permutes the walls, local faces, sectors, sector-faces... and contains the affine Weyl We denote the stabilizer of 0 ∈ A in G by K and the pointwise stabilizer (or fixator) of C + 0 (resp. C − 0 ) by K I = K + I (resp. K − I ). This group K I is called the Iwahori subgroup.
1.4.5. We ask W = ν(N ) to be vectorially-Weyl for its action on the vectorial faces. This means that the associated linear map − → w of any contains W a and stabilizes M, 1.4.6. Note that there is only a finite number of constants q M as in the definition of thickness. Indeed, we must have with two orbits. So, Q ∨ ⊂ W a has at most two orbits in the set of the constants q M (αi,k) : one containing the q i = q M (αi,0) and the other containing the q i = q M (αi,±1) . Hence, the number of (possibly) different q M is at most 2 · |I|. We denote this set of parameters by Q = {q i , q i | i ∈ I}.
In [3, 1.4.5] one proves the following further equalities: Algebraic Combinatorics, Vol. 4 #3 (2021) We consider also the polynomial algebra Z[Q], where Q is the set Q = {Q i , Q i | i ∈ I} of indeterminates, satisfying the same equalities: The main examples of all the above situation are provided by the Kac-Moody theory, as already indicated in the introduction. More precisely let G be an almost split Kac-Moody group over a non archimedean complete field K. We suppose moreover the valuation of K discrete and its residue field κ perfect. Then there is a masure I on which G acts strongly transitively by vectorially Weyl automorphisms. If K is a local field (i.e. κ is finite), then we are in the situation described above. This is the main result of [10], [11], [12] and [33]. When G is actually split, this result was known previously by [19] and [32]. And in this case all the constants q M , q i , q i are equal to the cardinality q of the residue field κ.
We gave in [3, 6.7] a proof of conjecture 1 for this split case; see also [28]. Actually these proofs are proofs of conjecture 2, as the polynomials a u w,v are Laurent polynomials inherited from the description of I H as a specialization of the associative Bernstein-Lusztig algebra over Z[Q]: the algebra I H Z[Q] over Z[Q] defined by these structure constants on the basis (T w ) w∈W + is associative.
1.4.8. Remark. All isomorphisms in [31] are Weyl-isomorphisms, and, when G is strongly transitive, all isomorphisms constructed in l.c. are induced by an element of G.
1.5. Type 0 vertices. The elements of Y , through the identification Y = N · 0 ⊂ A, are called vertices of type 0 in A; they are special vertices. We note We know that I is endowed with a G−invariant preorder which induces the known one on A. Moreover, if x y, then x and y are in the same apartment.
One has G + = K(N ∩ G + )K; more precisely the map It does not depend on the choices we made (by 1.8(b) below).
For (x, y) ∈ I 0 × I 0 = {(x, y) ∈ I 0 × I 0 | x y}, the vectorial distance d v (x, y) takes values in Y ++ . Actually, as I 0 = G·0, K is the stabilizer 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 )/G of G−orbits in I 0 × I 0 and Y ++ .
Further, d v gives the inverse of the map Y ++ → K\G + /K, as any g ∈ G + is in Algebraic Combinatorics, Vol. 4 #3 (2021) 1.7. Paths and retractions. We consider piecewise linear continuous paths π : [0, 1] → A such that each (existing) tangent vector π (t) belongs to an orbit W v · λ for some λ ∈ C v f . Such a path is called a λ−path; it is increasing with respect to the preorder relation on A.
Let C z (resp. S) be a local chamber with vertex z (resp. a sector germ) in an apartment A of I . For all x ∈ I z = {y ∈ I | y z} (resp. x ∈ I ) there is an apartment A containing x and C z (resp. S). And this apartment is conjugated to A by an element of G fixing C z (resp. S) (cf. 1.4.1(a) and 1.4.4). So, by the usual arguments we can define the retraction ρ = ρ A,Cz from I z (resp. ρ = ρ A,S from I ) onto A z = A ∩ I z (resp. onto the apartment A) with center C z (resp. S).
For any such retraction ρ, the image of any segment [x, y] with (x, y) ∈ I × I and [18, 4.4]. In particular, ρ(x) ρ(y). By definition, if A is another apartment containing S (resp. C z ), then ρ induces an isomorphism from A onto A. As we assume the existence of the strongly transitive group G, this isomorphism is the restriction of an automorphism of I . 1.8. Preordered convexity. Let C ± (resp. C ± 0 ) be the set of all local chambers of direction ± (resp. with moreover vertices of type 0). A positive (resp. negative) local chamber of vertex x ∈ I will often be written C x (resp. C − x ) and its direction Proposition. Let x, y ∈ I with x y. We consider two local faces F x , F y with respective vertices x, y. Then (a) F x and F y are contained in a common apartment.  [20, 4.4.16, 4.4.17]. In (b) the case of {x, y} is proved in [31, 5.4] as, by [21, 5.1] or [20, 4.4.1], one may replace cl by cl # . This property is called the preordered convexity of intersections of apartments.
Consequence. We define W + = W v Y + (resp. W +g = W v Y +g ) which is a subsemigroup of W , and call it the Tits-Weyl (resp. generic Tits-Weyl) semigroup. An element w ∈ W +g is called generic (in a large sense) and spherical if, moreover, Let ε, η ∈ {+, −}. If C ε x ∈ C ε 0 and 0 x, we know by (b) above, that there is an apartment A containing C η 0 and C ε x . But all apartments containing C η 0 are conjugated We have proved the Bruhat decompositions G + = K ± I · W + · K ± I and the Birkhoff decompositions G + = K ∓ I · W + · K ± I . For uniqueness, see 1.10 below. Similarly we also have G +g = K ± I · W +g · K ± I and G +g = K ∓ I · W +g · K ± I . 1.9. Remark. If the generalized Cartan matrix M is of affine or strictly hyperbolic type (in the sense of [23, 4.3

or Ex. 4.1]), then any non spherical vectorial face is
Then f −1 (y) 0 and there is w ∈ W + such that f −1 (C y ) = w · C + 0 . By 1.8(b), w does not depend on the choice of A. We define the W −distance between the two local chambers C x and C y to be this unique element:

and only if there is a minimal gallery (of local chambers in T +
x I ) from C x to C y of type (i 1 , . . . , i r ), in particular x = y. When x = y, this definition coincides with the one in 1.4.2.
Let us consider an apartment A and local chambers C x , C y , actually C x * w depends on A, but not on an identification of A with A. For x y z, we have (in A) the Chasles relation: When C x = C + 0 and C y = g · C + 0 (with g ∈ G + ), d W (C x , C y ) is the only w ∈ W + such that g ∈ K I · w · K I . This is the uniqueness result in Bruhat decomposition: The W −distance classifies the orbits of K I on {C y ∈ C + 0 | y 0}, hence also the orbits of G on C + 0 × C + 0 . 1.11. Iwahori-Hecke Algebras. We consider any commutative ring with unity R. The Iwahori-Hecke algebra I H R associated to I with coefficients in R introduced in [3] is as follows: To each w ∈ W + , we associate a function T w from C + 0 × C + 0 to R defined by The Iwahori-Hecke algebra I H R is the free R−module w∈W + a w T w | a w ∈ R, a w = 0 except for a finite number of w , endowed with the convolution product: where C z ∈ C + 0 is such that x z y.

On structure constants of Iwahori-Hecke algebras for Kac-Moody groups
Actually, I H R can be identified with the natural convolution algebra of the functions G + → R, bi-invariant under K I and with finite support (in K I \G + /K I ); this is the definition given in the introduction.
More precisely I H R is the space of functions ϕ : C + 0 × C + 0 → R, that are left G−invariant and with support a finite union of orbits (see the last two lines of 1.10). To a ϕ ∈ I H R is associated ϕ G : we get the convolution product (in the classical case, we take a Haar measure on G with K I of measure 1). One also considers the subspace I H g R = w∈W +g R · T w . From 4.2 and Remark 3.3(2) one sees that it is a subalgebra of I H R . We call it the generic Iwahori-Hecke algebra associated to I with coefficients in R. From 1.9 one has I H R = I H g R in the affine or strictly hyperbolic cases.
We now recall some useful results of [3] in order to introduce the structure constants and a way to compute them.
. Let us fix two local chambers C x and C y in C + 0 with x y and d W (C x , C y ) = u ∈ W + . We consider w and v in W + . Then the number a u w,v of C z ∈ C + 0 with x z y, d W (C x , C z ) = w and d W (C z , C y ) = v is finite (i.e. in N).

Theorem 1.2 ([3, 2.4]). For any ring R, I H R is an algebra with identity element
where P w,v is a finite subset of W + , such that a u w,v = 0 for u / ∈ P w,v .

Projections and retractions
In this section we introduce the new tools that we shall use in the next section to compute the structure constants of the Iwahori-Hecke algebra.

Projections of chambers.
2.1.1. Projection of a chamber C y on a point x. Let x ∈ I , C y ∈ C + with x y, x = y. We consider an apartment A containing x and C y (by 1.8(a) above) and write C y = F (y, C v y ) in A. For y ∈ y + C v y sufficiently near to y, α(y − x) = 0 for any root α and y − x ∈ T • . So ]x, y ) is in a unique positive local chamber pr x (C y ) of vertex x; this chamber satisfies [x, y) ⊂ pr x (C y ) ⊂ cl A ({x, y }) and does not depend on the choice of y . Moreover, if A is another apartment containing x and C y , we may suppose y ∈ A ∩ A and ]x, y ), cl A ({x, y }), pr x (C y ) are the same in A . The local chamber pr x (C y ) is well determined by x and C y , it is the projection of C y in T + x I . The same things may be done changing + to − or to . But, in the above situation, if C y ∈ C − , we have to assume x o < y to define pr x (C y ) ∈ C + : otherwise ]x, y ) might be outside x + T .
When x = y, we write pr x (C y ) = C y .
Algebraic Combinatorics, Vol. 4 #3 (2021) 2.1.2. Projection of a chamber C y on a generic segment germ. Let x ∈ I , δ = [x, x ) a generic segment-germ and C y ∈ C with x y. By 2.1.1 we can consider pr x (C y ) ∈ C + (with the hypothesis x o < y if C y ∈ C − ). We consider now an apartment A containing [x, x ) and pr x (C y ) (by 1.8(a) above).
We consider inside A the prism denoted by prism δ (C y ) obtained as the intersection of all half-spaces D(α, k) (for α ∈ Φ and k ∈ R) that contain pr x (C y ) and such that δ ⊂ M (α, k). We can see that if δ = [x, x ) is regular, prism δ (C y ) = A. If the apartment A contains δ and C y (hence also pr x (C y )) we may replace pr x (C y ) by C y in the above definition of prism δ (C y ).
Lemma 2.1. In prism δ (C y ), there is a unique local chamber of vertex x that contains δ in its closure. This chamber is independent of the choice of A.
N.B. This local chamber is, by definition, the projection pr δ (C y ) of the chamber C y on the segment-germ δ. It is the local chamber containing δ in its closure which is the nearest from pr x (C y ): either d W (pr x (C y ), pr δ (C y )) is minimum or d * W (pr x (C y ), pr δ (C y )) is maximum.
The same things may be done when one supposes y x and C y ∈ C − or y o < x and C y ∈ C + .
Proof. In the apartment A, we consider δ + the segment-germ δ if δ is in T + x I (where op A (δ) denotes the opposite segment-germ in A). By 1.4.2, we can consider in the building T + x I the minimal galleries from pr x (C y ) to δ + (more exactly to a chamber C such that δ + ∈C). The last chamber of each of these galleries is the same (as it has to be on the same side as pr x (C y ) of any hyperplane of A, containing δ + and parallel to a wall); we denote it C ++ x . This chamber is associated to a positive system of roots Φ + and a root basis (α 1 , . . . , α ), satisfying α i (δ) = 0 ⇐⇒ i r, where 0 r < (we identify x and 0). Then, we have the characterization of the prism : p ∈ prism δ (C y ) ⇐⇒ (α i (p) 0 for 1 i r). We consider w r the element of highest length in the finite Weyl group (r αi ) i r .
The local chamber C ++ if not) is the unique chamber with vertex x of prism δ (C y ) that contains δ in its closure. Indeed, if C is such a chamber, then if ]x, p) ⊂ C, we have α i (p) > 0 for all i r (because C ⊂ prism δ (C y )) and α i (p) of the same sign as α i (δ) if i > r (because δ ⊂C). So In the case δ ∈ T + x I , the characterization of C ++ x in the building T + x I proves that it does not depend on the choice of A.
The chamber op A (w r (C ++ x )) also only depends on δ and C y if δ ∈ T − x I . It is sufficient to prove that it intersects conv A (δ ∪ pr x (C y )). Indeed, let us choose ξ and y such that [x, ξ) = δ and ]x, y) ⊂ pr x (C y ). We have α i (ξ) = 0 for i r, α i (ξ) < 0 for i > r and α i (y) > 0 for i r. So for t near 1 enough, α i (tξ + (1 − t)y) > 0 for i r and < 0 for i > r, so ]x, tξ + (1 − t)y) ⊂ op A (w r (C ++ x ). By Proposition 1.8, the local chamber op A (w r (C ++ x )) is included in all apartments containing δ and pr x (C y ), so is independent of the choice of A.

2.2.
Centrifugally folded galleries of chambers. Let z be a point in the standard apartment A. We have twinned buildings T + z I (resp. T − z I ). As in 1.4.2, 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 z (z + C v ) (resp. local closed chambers C = germ z (z + C v )), where C v is a vectorial chamber, cf. [18, 4.5] or [31, § 5]. The distances (resp. codistances) between these chambers Algebraic Combinatorics, Vol. 4 #3 (2021) 2.4. Hecke paths. The Hecke paths we consider here are slight modifications of those used in [19]. They were defined in [3], or in [2] (for the classical case).
Let us fix a local chamber C x ∈ C 0 ∩ A.

Definition.
A Hecke path of shape λ ∈ Y ++ with respect to C x in A is a λ−path in A that satisfies the following assumptions. For all p = π(t), we ask x o < p, so we can consider the local negative chamber C − p = pr p (C x ) by 2.1.1. Then we assume moreover that for all t ∈ [0, 1] {0, 1}, there exist 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 j = 1, . . . , s: . One says then that these two sequences are a (W v π(t) , C − π(t ))−chain from π − (t) to π + (t). Actually W v π(t) is the subgroup of W v generated by the r β such that M (β, −β(π(t))) is a wall.
Lemma 2.2. Let π ⊂ A be a Hecke path with respect to C x as above. Then, (a) For t varying in [0, 1] and p = π(t), the set of vectorial rays R + (x − π(t)) is contained in a finite set of closures of (negative) vectorial chambers. (b) There is only a finite number of pairs (M, t) with a wall M containing a point p = π(t) for t > 0, such that π − (t) is not in M and x is not in the same side of M as π − (t) (but may be x ∈ M ). (c) One writes p 0 = π(t 0 ), p 1 = π(t 1 ), . . . , p π = π(t π ) with 0 = t 0 < t 1 < · · · < t π −1 < 1 = t π the points p = π(t) satisfying to (b) above (or t = 0, t = 1). Then any point t where the path is (centrifugally) folded with respect to C x at π(t) appears in the set {t k | 1 k π − 1}. Proof. (a) The λ−path π is a union of line segments [p 0 , p 1 ] ∪ [p 1 , p 2 ] ∪ · · · ∪ [p n−1 , p n ]. By hypothesis on Hecke paths, for each point p = π(t), x − p is in the open negative Tits cone −T • (in particular only in a finite number of closures of negative vectorial chambers). Let p ∈ [p i , p i+1 ], then x − p = x − p i − (p − p i ) and R + (x − p) ⊂ conv(R + (x − p i ), −R + (p − p i )) and this convex hull is independent of p and only in a finite number of closures of (negative) vectorial chambers (as (x − p i ) ∈ −T • and (p − p i ) ∈ R + (p i+1 − p i ) ⊂ T ). So (a) is proved.
(b) There is only a finite number of vectorial walls separating (strictly) a chamber in the set of (a) above and a vector p i − p i+1 . And, for each such vectorial wall, there is only a finite number of walls with this direction meeting the compact set π ([0, 1]). Moreover such a wall meets a segment ]p i , p i+1 ] at most once or contains (c) The folding points are among {p 1 , . . . , p π −1 } by (iv) and (ii) above for j = 1.

2.5.
Retractions and liftings of line segments.

Local study.
In tangent buildings, the centrifugally folded galleries are related with retractions of opposite segment germs, by the following lemma proved in [19,Lemma 4.6].
We consider a point z ∈ A and a negative local chamber C − z in A − z . Let ξ and η be two segment germs in A + z = A ∩ T + z I . Let −η and −ξ opposite respectively η and ξ Algebraic Combinatorics, Vol. 4 #3 (2021) in A − z . Let i be the type of a minimal gallery between C − z and C −ξ , where C −ξ is the negative (local) chamber containing −ξ such that d W (C − z , C −ξ ) is of minimal length. Let Q be a chamber of A + z containing η. We suppose ξ and η conjugated by W v z .
Lemma. The following conditions are equivalent: (a) There exists an opposite ζ to η in Moreover the possible ζ are in one-to-one correspondence with the disjoint union of the sets C m

2.5.2.
Consequence. Let C x be a positive local chamber in A and z ∈ A a point such that x o < z. We consider C − z = pr z (C x ). Then one knows that the restriction of the retraction ρ = ρ A,Cx to the tangent twin building T z I is the retraction ρ Az,C − z . We consider two points y, z 0 in I such that x o < z 0 y, with d v (z 0 , y) = λ ∈ Y ++ . By 1.7, the image ρ([z 0 , y]) is a λ−path π from ρ(z 0 ) to ρ(y). For z ∈ [z 0 , y[, we consider an apartment A containing [z, y) and C x , hence also C − z . We write p = ρ(z). The restriction ρ| A is the restriction to A of an automorphism ϕ of I fixing C x (and an isomorphism from A to A); ϕ induces an isomorphism ϕ| TzI from T z I onto T z I . One has ρ| TzI = ρ Ap,C − p • ϕ| TzI = ϕ| Az • ρ Az,C − z . So one may use the above Lemma, more precisely the implication (a) =⇒ (c): we get a (W v p , C − p )−chain from π − (t) to π + (t) (if p = π(t)).
We have proved that π = ρ([z 0 , y]) is a Hecke path of shape λ with respect to C x in A. This result is a part of [3,Theorem 3.4]. It is also a consequence of the proof of [2, Th. 3.8] which deals with the classical case of buildings.
2.5.3. Liftings of Hecke paths. One considers in A a positive local chamber C x , a Hecke path π of shape λ ∈ Y ++ with respect to C x and the retraction ρ = ρ A,Cx . Given a point y ∈ I with ρ(y) = π(1), we consider the set S Cx (π, y) of all segment germs [z, y] in I such that ρ([z, y]) = π. The above Lemma (essentially (b)) is used in [3] to compute the cardinality of S Cx (π, y).
We consider the notation of 1.7 and the numbers t k of Lemma 2.2. Then p k = π(t k ), ξ k = −π − (t k ), η k = π + (t k ) and i k is the type of a minimal gallery between C − p k and C −ξ k , where C −ξ k is the negative (local) chamber such that −ξ k ⊂ C −ξ k and d W (C − p k , C −ξ k ) is of minimal length. Let Q k be a fixed chamber in A + z k containing η k in its closure and Γ + Q k (C − p k , i k , −η k ) be the set of all the galleries (C − z k , C 1 , . . . , C r ) of type i k in A − z k , centrifugally folded with respect to Q k and with −η k ∈ C r . The following result is Theorem 3.4 in [3]. One uses the notation of 2.2 and 2.3. One considers paths π more general than Hecke paths. The idea is to lift the path π step by step starting from its end by using the above Lemma. We shall generalize it in Theorem 3.3 by lifting decorated Hecke paths (see just below). Theorem 2.3. The set S Cx (π, y) is non empty if, and only if, π is a Hecke path with respect to C x . Then, we have a bijection S Cx (π, y) In particular, the number of elements in this set is a polynomial in the numbers q ∈ Q with coefficients in Z depending only on A.
Algebraic Combinatorics, Vol. 4 #3 (2021) Lemma. Let us consider in a masure I two preordered line segments or rays δ 1 , δ 2 in apartments A 1 , A 2 , sharing the same origin x. One supposes the segments germs germ x (δ 1 ) and germ x (δ 2 ) opposite (in any apartment containing them both). Then there is a line in an apartment A of I containing δ 1 and δ 2 . In particular, if δ 1 , δ 2 are line segments (resp. rays), then δ 1 ∪ δ 2 is also a line segment (resp. a line).
Proof. The case of line segments is Lemma 4.9 in [19]. The case of rays may be deduced from the fact stated in part 2 of the proof of [31,Prop. 5.4]. As we shall not use it, we omit the details.
3.3. The main formula. Let us fix two local chambers C x and C y in C + 0 with x y and d W (C x , C y ) = u = ν · u ∈ W + . We consider w = λ · w and v = µ · v in W + . Then we know that the structure constant a u w,v is the number of C z0 ∈ C + 0 with x z 0 y, d W (C x , C z0 ) = w and d W (C z0 , C y ) = v; moreover this number is finite, see Proposition 1.1. In Lemmas 3.1 and 3.2 we gave conditions equivalent to these W −distance conditions. We choose the standard apartment A containing C x and C y , and we identify C x with the fundamental local chamber C + 0 . The datum of z 0 is equivalent to the datum of the segment [z 0 , y] or of the decorated segment [z 0 , y] associated, as in 2.5, to [z 0 , y] and C y . We consider then the decorated Hecke path π image of [z 0 , y] by the retraction ρ A,Cx .
To the Hecke path π underlying a decorated Hecke path π are associated π ∈ N and numbers t 0 = 0 < t 1 < t 2 < · · · < t π = 1 as in Lemma 2.2 and Definition 2.6. We write p k = π(t k ). We write C + p (resp. C * p instead of C p ) the decorations of π at a point p of π. We write C + z (resp. C z ) the decorations of a decorated segment at one of its points z.
We use freely the notations from 2.1, 2.2 and 2.3.