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%%%%% Auteur
%%%1
\author{\firstname{Nicole} \lastname{Bardy-Panse}}
\address{Universit{\'e} de Lorraine\\
CNRS, IECL\\
Nancy\\
F-54000, France}
\email{Nicole.Panse@univ-lorraine.fr}
%%%2
\author{\firstname{Guy} \lastname{Rousseau}}
\address{Universit{\'e} de Lorraine\\
CNRS, IECL\\
Nancy\\
F-54000, France}
\email{Guy.Rousseau@univ-lorraine.fr}
%%%%% Sujet
\keywords{Building, Hecke algebra, Kac--Moody group, masure, local field.}
\subjclass{20G44, 20C08, 20G25, 20E42, 51E24}
%%%%% Gestion
\DOI{10.5802/alco.163}
\datereceived{2020-03-09}
\daterevised{2020-11-25}
\dateaccepted{2020-12-04}
%%%%% Titre et résumé
\title{On structure constants of Iwahori--Hecke algebras for Kac--Moody groups}
\begin{abstract}
We consider the Iwahori--Hecke algebra $^I\!\mathscr{H}$ associated to an almost split Kac--Moody group $G$ (affine or not) over a nonarchimedean local field $\mathcal{K}$.
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 $a^ {\mathbf u}_{\mathbf w,\mathbf v}\in\mathbb{N}=\mathbb{Z}_{\geq 0}$: $T_{\mathbf w}*T_{\mathbf v}=\sum_{\mathbf u\in P_{\mathbf w,\mathbf v}} a^ {\mathbf u}_{\mathbf w,\mathbf v} T_{\mathbf u}$.
A conjecture, by {Braverman}, Kazhdan, Patnaik, Gaussent and the authors, tells that $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ is a polynomial, with coefficients in $\mathbb{N}$, in the parameters $q_{i}-1,q'_{i}-1$ of $G$ over $\mathcal{K}$.
We prove this conjecture when $\mathbf w$ and $\mathbf v$ are spherical
or, more generally, when they are said {to be} 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 $\mathbb{Z}[Q]$; it specializes to $^I\!\mathscr{H}$ when one sets $Q=q$.
\end{abstract}
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\begin{document}
\maketitle
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\section*{Introduction}\label{seIntro}
Let $G$ be a split, semi-simple, simply connected algebraic group over a non archimedean local field $\shk$.
So $\shk$ is complete for a discrete, non trivial valuation with a finite residue field $\qk$.
We write $\sho\subset\shk$ {for} the ring of integers and $q$ {for} the cardinality of $\qk$.
Then $G$ is locally compact. In this situation, Nagayoshi Iwahori and Hideya Matsumoto in~\cite{IM65}, 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 $T\simeq (\shk^*)^n$, then $(K_{I},N)$ is a BN pair.
The Iwahori--Hecke algebra of $G$ is the algebra ${^I\!}\SHH_{R}={^I\!}\SHH_{R}(G,K_{I})$ 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 $H\simeq(\sho^*)^n$ is the maximal compact subgroup of $T$, then $H\subset K_{I}$ and $W=N/H$ is the affine Weyl group.
One has the Bruhat decomposition $G=K_{I}\mycdot W\mycdot K_{I}=\sqcup_{\mathbf w\in W} K_{I}\mycdot \mathbf w\mycdot K_{I}$.
If one considers the characteristic function $T_{\mathbf w}$ of $K_{I}\mycdot \mathbf w\mycdot K_{I}$, we get a basis of ${^I\!}\SHH_{R}$: ${^I\!}\SHH_{R}=\oplus_{\mathbf w\in W} R\mycdot T_{\mathbf w}$.
The convolution product is given by $T_{\mathbf w}*T_{\mathbf v}=\sum_{\mathbf u\in P_{\mathbf w,\mathbf v}} a^ {\mathbf u}_{\mathbf w,\mathbf v} T_{\mathbf u}$, with $P_{\mathbf w,\mathbf v}$ a finite subset of $W$.
The numbers $a^ {\mathbf u}_{\mathbf w,\mathbf v}\in R$ are the structure constants of ${^I\!}\SHH_{R}$. The unit is $1=T_{e}$.
Iwahori and Matsumoto gave a precise (and now classical) definition of {$^I\!\SHH_{R}$} by generators and relations.
The group $W$ is an infinite Coxeter group generated by $\{r_{0},\ldots,r_{n}\}$.
Then {$^I\!\SHH_{R}$} is generated by $\{T_{r_{0}},\ldots,T_{r_{n}}\}$ with relations $T_{r_{i}}^2=q\mycdot 1+(q-1)\mycdot T_{r_{i}}$ and $T_{r_{i}}*T_{r_{j}}*T_{r_{i}}*\cdots=T_{r_{j}}*T_{r_{i}}*T_{r_{j}}*\cdots$ (with $m_{i,j}$ factors on each side) for $i\neq j$, if $m_{i,j}$ is the finite order of $r_{i}r_{j}$.
For $\mathbf w=r_{i_{1}}\mycdot \ldots\mycdot r_{i_{s}}$ a reduced expression in $W$, one has $T_{\mathbf w}=T_{r_{i_{1}}}*\cdots*T_{r_{i_{s}}}$.
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 (\eg the product of two reduced expressions $\mathbf w=r_{i_{1}}\mycdot \ldots\mycdot r_{i_{s}}$ and $\mathbf v=r_{j_{1}}\mycdot \ldots\mycdot r_{j_{t}}$).
So one deduces easily (using the above relations between the $T_{r_{i}}$) that each structure constant $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ (for $\mathbf u, \mathbf v,\mathbf w \in W$) is in $\Z[q]$.
More precisely it is a polynomial in $q-1$ with coefficients in $\N=\Z_{\geq0}$.
This polynomial depends only on $\mathbf u, \mathbf v,\mathbf w$ and $W$.
So one has a universal description of ${^I\!}\SHH_{\Z}$ 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 $\shk$, 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 $G=K_{I}\mycdot W\mycdot K_{I}$.
Now the description of ${^I\!}\SHH_{R}$ involves parameters $q_{i}$ (satisfying $T_{r_{i}}^2=q_{i}\mycdot 1+(q_{i}-1)\mycdot T_{r_{i}}$) which are {potentially} different from $q$.
This gives the Iwahori--Hecke algebra with unequal parameters.
There is a pleasant description of ${^I\!}\SHH_{R}$ using the Bruhat--Tits building associated to the BN pair $(K_{I},N)$, see \eg~\cite{P06}.
For now more than twenty years, there is an increasing interest in the study of Kac--Moody groups over local fields, see the works of {Braverman}, Garland, Kapranov, Kazhdan, Patnaik, Gaussent and the authors: \eg~\cite{BPGR16, BPGR17, BrGKP14, BrK11, BrK14, BrKP14, Ga95, GaG95, GR13, Kap01}.
It has been possible to define and study for Kac--Moody groups (supposed at first affine) the spherical Hecke algebra, the Iwahori--Hecke algebra, the Satake isomorphism, \ldots.
This is also closely related to more abstract works on Hecke algebras by Cherednik and Macdonald, \eg~\cite{Che92, Che95, Ma03}.
We are mainly interested in Iwahori--Hecke algebras for Kac--Moody groups over local fields.
They were introduced and described by {Braverman}, Kazhdan and Patnaik in the affine case~\cite{BrKP14} and then in general by Gaussent and the authors~\cite{BPGR16}.
So let us consider a Kac--Moody group $G$ (affine or not) over the local field $\shk$.
We suppose it split (as defined by Tits~\cite{T87}) or more generally almost split~\cite{Re02}. Let us choose also a maximal split subtorus.
To this situation are %is
associated an affine (relative) Weyl group $W$ and an Iwahori subgroup $K_{I}$ (defined up to conjugacy by $W$), see~\ref{sect1.4.5} %\ref{1.3} (5)
and~\ref{sect1.4.7} %(7)
below.
This group $W$ is not a Coxeter group but may be described as a semi-direct product $W=W^v\ltimes Y$, where $W^v$ is a Coxeter group, the relative Weyl group, and $Y$ is (essentially) the cocharacter group of the torus.
\looseness-1
Unfortunately the Bruhat decomposition ``$G=K_{I}\mycdot W\mycdot K_{I}$'' fails to be true (even in the untwisted affine case, \ie for loop groups).
One has to consider the sub-semigroup $W^+=W^v\ltimes Y^+$ (\resp $W^{+g}=W^v\ltimes Y^{+g}$) of $W$, where $Y^+$ (\resp $Y^{+g}$) is the intersection of $Y$ with the Tits cone $\sht$ (\resp with a cone $\sht^\circ\cup V_{0}\subset\sht$, where $\sht^\circ$ is the open Tits cone) in $V=Y\otimes_{\Z}\R$ (see~\ref{1.2},~\ref{1.4}, and~\ref{1.11} below).
Then $G^+=K_{I}\mycdot W^+\mycdot K_{I}$ (\resp $G^{+g}=K_{I}\mycdot W^{+g}\mycdot K_{I}\subset G^+$) is a sub-semigroup of $G$: the Kac--Moody--Tits semigroup (\resp the generic Kac--Moody--Tits semigroup).
We may consider the characteristic functions $T_{\mathbf w}$ of the double cosets $K_{I}\mycdot \mathbf w\mycdot K_{I}$ and one proves in~\cite{BPGR16} that:
%\medskip
The space ${^I\!}\SHH_{R}$ (\resp ${^I\!}\SHH^g_{R}$) of $R-$valued functions with finite support on $K_{I}\backslash G^ {+}/K_{I}$ (\resp $K_{I}\backslash G^ {+g}/K_{I}$) is naturally endowed with a structure of algebra (see~\ref{s2}).
We get thus the Iwahori--Hecke algebra ${^I\!}\SHH_{R}=\oplus_{\mathbf w\in W^+} R\mycdot T_{\mathbf w}$ (\resp the generic Iwahori--Hecke algebra ${^I\!}\SHH^g_{R}=\oplus_{\mathbf w\in W^{+g}} R\mycdot T_{\mathbf w}$).
The product is given by structure constants $a^ {\mathbf u}_{\mathbf w,\mathbf v}\in\N=\Z_{\geq0}$: $T_{\mathbf w}*T_{\mathbf v}=\sum_{\mathbf u\in P_{\mathbf w,\mathbf v}} a^ {\mathbf u}_{\mathbf w,\mathbf v} T_{\mathbf u}$.
\begingroup
\renewcommand\thecdrthm{\arabic{cdrthm}}
%\begin{enonce*}[plain]{Conjecture 1}
\begin{conj}[{\cite[2.5]{BPGR16}}]\label{conj1} Each $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ is a polynomial, with coefficients in $\N=\Z_{\geq0}$, in the parameters $q_{i}-1,q'_{i}-1$ of the situation, see~\ref{sect1.4.6} %\ref{1.3}.6
below.
This polynomial depends only on the affine Weyl group $W$ acting on the apartment $\A$ and on $\mathbf w,\mathbf v,\mathbf u\in W^+$.
\end{conj}
One may consider that this is a translation of the following question of Braverman, Kazhdan and Patnaik:
%\begin{enonce*}[plain]{Question}
\begin{ques*}[{\cite[end of~1.2.4]{BrKP14}}] Has the algebra ${^I\!}\SHH_{\C}$ a purely algebraic or combinatorial description with respect to the coset basis $(T_{\mathbf w})_{\mathbf w\in W^+}$?
\end{ques*}
%\end{enonce*}
But a more precise formulation of this question is as follows:
%\begin{enonce*}[plain]{Conjecture 2}
\begin{conj}\label{conj2}
The algebra ${^I\!}\SHH_{\Z}$ (or ${^I\!}\SHH^g_{\Z}$) is the specialization of an algebra ${^I\!}\SHH_{\Z[\SHQ]}$ (or ${^I\!}\SHH^g_{\Z[\SHQ]}$) with the same basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ (or $(T_{\mathbf w})_{\mathbf w\in W^{+g}}$) over $\Z[\SHQ]$.
Here $\SHQ$ is a set of indeterminates $Q_{i},Q'_{i}$ (with some equalities between them, see~\ref{sect1.4.6} %\ref{1.3}.6
below) and the specialization is given by $Q_{i}\mapsto q_{i},Q'_{i}\mapsto q'_{i}, \forall i\in I$.
The algebra ${^I\!}\SHH_{\Z[\SHQ]}$ (or ${^I\!}\SHH^g_{\Z[\SHQ]}$) depends only on the affine Weyl group $W$ acting on the apartment $\A$.
\end{conj}
Let us consider the split case: $G$ is a split Kac--Moody group, all parameters $q_{i},q'_{i}$ are equal to $q=\vert\qk\vert$ and all indeterminates $Q_{i},Q'_{i}$ are equal to a single indeterminate $Q$.
Then the conjecture~\ref{conj1} has already been proved by Gaussent and the authors~\cite[6.7]{BPGR16} and independently by Muthiah~\cite{Mu15} if, moreover, $G$ is untwisted affine.
Actually the same proof gives also conjecture~\ref{conj2}, see~\ref{sect1.4.7} %~\ref{1.3}.7
below.
In the general (non split) case, weakened versions were obtained in~\cite{BPGR16}: the $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ are Laurent polynomials in the $q_{i},q'_{i}$ [\lc 6.7]; they are true polynomials if $\mathbf w,\mathbf v \in W^v\ltimes(Y\cap\sht^\circ)$ and $\mathbf v$ is ``regular'' [\lc 3.8].
In this article, we prove the conjecture~\ref{conj1} when $\mathbf w$ and $\mathbf v$ are in $W^ {+g}$ (see~\ref{sc8}).
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 ${^I\!}\SHH_{\Z[\SHQ]}$ (\resp ${^I\!}\SHH^g_{\Z[\SHQ]}$) over $\Z[\SHQ]$ 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~\cite{GaG95},~\cite{BrKP14} and~\cite{BPGR16}).
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~\ref{s1}: as in~\cite{BPGR16}, we work with an abstract masure $\SHI$ and we take $G$ to be a strongly transitive group of vectorially-Weyl automorphisms of $\SHI$.
In Section~\ref{pr} we gather the additional technical tools (\eg decorated Hecke paths) needed to improve the results of~\cite[Section~3]{BPGR16}.
We get our main results about $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ in Section~\ref{sc}: we deal with the cases $\mathbf w,\mathbf v$ spherical.
In Section~\ref{s4} we deal with the remaining cases where $\mathbf w,\mathbf v$ are in $W^ {+g}$, \ie when $\mathbf w,\mathbf v$ are said generic.
\endgroup
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\section{General framework}\label{s1}
\subsection{Vectorial data}\label{1.1} We consider a quadruple $(V,W^v,(\qa_i)_{i\in I}, (\qa^\vee_i)_{i\in I})$ where $V$ is a finite dimensional real vector space, $W^v$ a subgroup of $\GL (V)$ (the vectorial Weyl group), $I$ a finite set, $(\qa^\vee_i)_{i\in I}$ a {free} family in $V$ and $(\qa_i)_{i\in I}$ a free family in the dual $V^*$.
We ask these data to satisfy the conditions of~\cite[1.1]{R11}.
In particular, the formula $r_i(v)=v-\qa_i(v)\qa_i^\vee$ defines a linear involution in $V$ which is an element in $W^v$ and $(W^v,\{r_i\mid i\in I\})$ is a Coxeter system.
To be more concrete, we consider the Kac--Moody case of [\lc; 1.2]: the matrix $\M=(\qa_j(\qa_i^\vee))_{i,j\in I}$ is a generalized Cartan matrix.
Then $W^v$ is the Weyl group of the corresponding Kac--Moody Lie algebra $\g g_\M$ and the associated real root system is
\[
\QF=\{w(\qa_i)\mid w\in W^v,i\in I\}\subset Q=\bigoplus_{i\in I}\,\Z\mycdot \qa_i.
\]
We set $\QF^\pm{}=\QF\cap Q^\pm{}$ where $Q^\pm{}=\pm{}(\bigoplus_{i\in I}\,(\Z_{\geq 0})\mycdot\qa_i)$ and $Q^\vee=(\bigoplus_{i\in I}\,\Z\mycdot \qa_i^\vee)$, $Q^\vee_\pm{}=\pm{}(\bigoplus_{i\in I}\,(\Z_{\geq 0})\mycdot \qa_i^\vee)$.
We have $\QF=\QF^+\cup\QF^-$ and, for $\qa=w(\qa_i)\in\QF$, $r_\qa=w\mycdot r_i\mycdot w^{-1}$ and $r_\qa(v)=v-\qa(v)\qa^\vee$, where the coroot $\qa^\vee=w(\qa_i^\vee)$ depends only on $\qa$.
The set $\QF$ is an (abstract, reduced) real root system in the sense of~\cite{MP89},~\cite{MP95} or~\cite{Ba96}.
We shall sometimes also use the set $\QD=\QF\cup\QD^+_{im}\cup\QD^-_{im}$ of all roots (with $-\QD^-_{im}=\QD^+_{im}\subset Q^+$, $W^v-$stable) defined in~\cite{K90}.
It is an (abstract, reduced) root system in the sense of~\cite{Ba96}.
The \emph{fundamental positive chamber} is $C^v_f=\{v\in V\mid\qa_i(v)>0,\forall i\in I\}$.
Its closure $\overline{C^v_f}$ is the disjoint union of the vectorial faces $F^v(J)=\{v\in V\mid\qa_i(v)=0,\forall i\in J$, $\qa_i(v)>0,\forall i\in I\setminus J\}$ for $J\subset I$.
We set $V_0 = F^v(I)$.
The positive (\resp negative) vectorial faces are the sets $w\mycdot F^v(J)$ (\resp $-w\mycdot F^v(J)$) for $w\in W^v$ and $J\subset I$.
The support of such a face is the vector space it generates.
The set $J$ or the face $w\mycdot F^v(J)$ or an element of this face is called \emph{spherical} if the group $W^v(J)$ generated by $\{r_i\mid i\in J\}$ {(which is the fixator or stabilizer in $W^v$ of $F^v(J)$)} is finite.
An element of a vectorial chamber $\pm w\mycdot C^v_f$ is called \emph{regular}.
The \emph{Tits cone} $\sht$ (\resp its interior $\sht^\circ$) is the (disjoint) union of the positive (\resp and spherical) vectorial faces. It is a $W^v-$stable convex cone in $V$.
One has $\sht=\sht^\circ=V$ (\resp $V_{0} \subset \sht \setminus \sht^\circ$) in the classical (\resp non classical) case, \ie when $W^v$ is finite (\resp infinite).
{By the above characterization of spherical faces, $\sht^\circ$ is the set of $x\in\sht$ whose fixator in $W^v$ is finite.}
We say that $\A^v=(V,W^v)$ is a \emph{vectorial apartment}.
\subsection{The model apartment}\label{1.2} As in~\cite[1.4]{R11} the model apartment $\A$ is $V$ considered as an affine space and endowed with a family $\shm$ of walls.
These walls are affine hyperplanes directed by $\ker(\qa)$ for $\qa\in\QF$.
More precisely, they may be written $M(\qa,k)=\{v\in V\mid\qa(v)+k=0\}$, for $\qa\in\QF$ and $k\in\R$.
We ask this apartment to be \motgras{semi-discrete} and the origin $0$ to be \motgras{special}.
This means that these walls are the hyperplanes $M(\qa,k)=\{v\in V\mid\qa(v)+k=0\}$ for $\qa\in\QF$ and $k\in\QL_\qa,$ with $\QL_\qa=k_\qa\mycdot \Z$ a non trivial discrete subgroup of $\R$.
Using~\cite[Lemma~1.3]{GR13} (\ie replacing $\QF$ by another system $\QF_1$) we may (and shall) assume that $\QL_\qa=\Z, \forall\qa\in\QF$.
For $\qa=w(\qa_i)\in\QF$, $k\in\Z$ and $M=M(\qa,k)$, the reflection $r_{\qa,k}=r_M$ with respect to $M$ is the affine involution of $\A$ with fixed points the wall $M$ and associated linear involution $r_\qa$.
The affine Weyl group $W^a$ is the group generated by the reflections $r_M$ for $M\in \shm$; we assume that $W^a$ stabilizes $\shm$.
We know that $W^a=W^v\ltimes Q^\vee$ and we write $W^a_\R=W^v\ltimes V$; here $Q^\vee$ and $V$ have to be understood as groups of translations.
An automorphism of $\A$ is an affine bijection $\qf:\A\to\A$ stabilizing the set of pairs $(M,\qa^\vee)$ of a wall $M$ and the coroot associated with $\qa\in\QF$ such that $M=M(\qa,k)$, $k\in\Z$. The group $\Aut (\A)$ of these automorphisms contains $W^a$ and normalizes it.
We consider also the group $\Aut ^W_\R(\A)=\{\qf\in \Aut (\A)\mid\vect{\qf}\in W^v\}=\Aut (\A)\cap W^a_\R$.
For $\qa\in\QF$ and $k\in\R$, $D(\qa,k)=\{v\in V\mid\qa(v)+k\geq 0\}$ is a half-space, it is called a \emph{half-apartment} if $k\in\Z$. We write $D(\alpha,\infty) = \mathbb A$.
The Tits cone $\mathcal T$
and its interior $\mathcal T^o$ are convex and $W^v-$stable cones, therefore, we can define three $W^v-$invariant preorder relations on $\mathbb A$:
\[
x\leq y\;\Leftrightarrow\; y-x\in\mathcal T
; \quad x\stackrel{o}{<} y\;\Leftrightarrow\; y-x\in\mathcal T^o
; \quad x\stackrel{o}{\leq} y\;\Leftrightarrow\; y-x\in\mathcal T^o\cup V_{0}.
\]
If $W^v$ has no fixed point in $V\setminus\{0\}$ (\ie $V_{0}=\{0\}$) and no finite factor, then they are orders; but, in general, they are not.
\subsection{Faces, sectors}
\label{suse:Faces}
The faces in $\mathbb A$ are associated to the above systems of walls
and half-apartments. As in~\cite{BrT72}, they
are no longer subsets of $\mathbb A$, but filters of subsets of $\mathbb A$. For the definition of that notion and its properties, we refer to~\cite{BrT72} or~\cite{GR08}.
If $F$ is a subset of $\mathbb A$ containing an element $x$ in its closure,
the germ of $F$ in $x$ is the filter $\germ_x(F)$ consisting of all subsets of $\mathbb A$ which contain intersections of $F$ and neighbourhoods of $x$. In particular, if $x\neq y\in \mathbb A$, we denote the germ in $x$ of the segment $[x,y]$ (\resp of the interval $]x,y]$) by $[x,y)$ (\resp $]x,y)$).
For $y {\neq} x$, the segment germ $[x,y)$ is called of sign $\pm$ if $y-x\in\pm\sht$.
The segment $[x,y]$ (or the segment germ $[x,y)$ or the {ray} with origin $x$ containing $y$) is called \emph{preordered} if $x\leq y$ or $y\leq x$ and \emph{generic} if $x\stackrel{o}{<} y$ or $y\stackrel{o}{<} x$.
Given $F$ a filter of subsets of $\mathbb A$, its \emph{strict enclosure} $\cl _{\mathbb A}(F)$ (\resp \emph{closure} $\overline F$) is the filter made of the subsets of $\mathbb A$ containing an element of $F$ of the shape $\cap_{\alpha\in\Delta}D(\alpha,k_\alpha)$, where $k_\alpha\in {\Z}\cup\{\infty\}$ (\resp containing the closure $\overline S$ of some $S\in F$).
One considers also the (larger) \emph{enclosure} $\cl _\A^\#(F)$ of~\cite[3.6.1]{R13} (introduced in~\cite{Ch12, Ch10, Ch11} and well studied in~\cite{He17a}, see also~\cite{Heb18}).
It is the filter made of the subsets of $\mathbb A$ containing an element of $F$ of the shape $\cap_{\alpha\in\Psi}D(\alpha,k_\alpha)$, with $\Psi\subset\QF$ finite and $k_\alpha\in {\Z}$ (\ie a finite intersection of half apartments).
\medskip
A \emph{local face} $F$ in the apartment $\mathbb A$ is associated
to a point $x\in \mathbb A$, its vertex, and a vectorial face $F^v$ in $V$, its direction. It is defined as $F=\germ_x(x+F^v)$ and we denote it by $F=F^\ell(x,F^v)$.
Its closure is $\overline{F^\ell}(x,F^v)=\germ_x(x+\overline{F^v})$ {.}
There is an order on the local faces: the assertions ``$F$ is a face of $F'$'',
``$F'$ covers $F$'' and ``$F\leq F'$'' are by definition equivalent to
$F\subset\overline{F'}$.
The dimension of a local face $F$ is the smallest dimension of an affine space generated by some $S\in F$.
The (unique) such affine space $E$ of minimal dimension is the support of $F$; if $F=F^\ell(x,F^v)$, $\supp (F)=x+\supp (F^v)$.
A local face $F=F^\ell(x,F^v)$ is spherical if the direction of its support meets the open Tits cone (\ie 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 $F=F(x,F^v)$.
\medskip
A \emph{local chamber} is a maximal local face, \ie a local face $F^\ell(x,\pm w\mycdot C^v_f)$ for $x\in\A$ and $w\in W^v$.
The \emph{fundamental local positive (\resp negative)} chamber is $C_0^+=\germ_0(C^v_f)$ (\resp $C_0^-=\germ_0(-C^v_f)$).
A \emph{(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.
\medskip
A \emph{sector} in $\mathbb A$ is a $V-$translate $\mathfrak s=x+C^v$ of a vectorial chamber
$C^v=\pm w\mycdot C^v_f$, $w \in W^v$. The point $x$ is its \emph{base point} and $C^v$ its \emph{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 \emph{sector-germ} of a sector $\mathfrak s=x+C^v$ in $\mathbb A$ is the filter $\mathfrak S$ of
subsets of~$\mathbb A$ consisting of the sets containing a $V-$translate of $\mathfrak s$, it is well
determined by the direction $C^v$. So, the set of
translation classes of sectors in $\mathbb A$, the set of vectorial chambers in $V$ and
the set of sector-germs in $\mathbb A$ are in canonical bijection.
A \emph{sector-face} in $\mathbb A$ is a $V-$translate $\mathfrak f=x+F^v$ of a vectorial face
$F^v=\pm w\mycdot F^v(J)$. The sector-face-germ of $\mathfrak f$ is the filter $\mathfrak F$ of
subsets containing a translate $\mathfrak f'$ of $\mathfrak f$ by an element of $F^v$ (\ie $\mathfrak
f'\subset \mathfrak f$). If $F^v$ is spherical, then $\mathfrak f$ and $\mathfrak F$ are also called
spherical. The sign of $\mathfrak f$ and $\mathfrak F$ is the sign of $F^v$.
\subsection{The Masure}\label{1.3}
In this section, we recall the definition and some properties of a masure given by Guy Rousseau in~\cite{R11} and simplified by Auguste H{\'e}bert~\cite{He17a}.
%\medskip
%\parni{\bf 1)}
%\begin{enumerate}
%\item
\subsubsection{}\label{sect1.4.1} An apartment of type $\mathbb A$ is a set $A$ endowed with a set $\Isom ^W\!(\mathbb A,A)$ of bijections (called Weyl-isomorphisms) such that, if $f_0\in \Isom ^W\!(\mathbb A,A)$, then $f\in \Isom ^W\!(\mathbb A,A)$ if, and only if, there exists $w\in W^a$ satisfying $f = f_0\circ w$.
An isomorphism (\resp a Weyl-isomorphism, a vectorially-Weyl isomorphism) between two apartments $\varphi:A\to A'$ is a bijection such that, for any $f\in \Isom ^W\!(\mathbb A,A)$, $f'\in \Isom ^W\!(\mathbb A,A')$, $f'^{-1}\circ\qf\circ f \in \Aut (\A)$ (\resp $\in W^a$, $\in \Aut ^W_\R(\A)$); the group of these isomorphisms is written $\Isom (A,A')$ (\resp $\Isom ^W(A,A')$, $\Isom ^W_\R(A,A')$).
As the filters in $\A$ defined in~\ref{suse:Faces} above (\eg local faces, sectors, walls,...) are permuted by $\Aut (\A)$, they are well defined in any apartment of type $\A$ and exchanged by any isomorphism.
%\end{enumerate}
A \emph{masure} (formerly called an \emph{ordered affine hovel}) of type $\mathbb A$ is a set $\SHI$ endowed with a covering $\mathcal A$ 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 H\'ebert in~\cite{He17a}: these data have to satisfy the following two axioms:
\begingroup
\advance\leftmargini by 3mm
\begin{enumerate}[label=(MA~\roman{enumi})]\setcounter{enumi}{1}
%(MA ii)
\item\label{MAii} If two apartments $A,A'$ are such that $A\cap A'$ contains a generic ray, then $A\cap A'$ is a finite intersection of half-apartments (\ie $A\cap A'=\cl _A^\#(A\cap A')$) and there exists a Weyl isomorphism $\qf:A\to A'$ fixing $A\cap A'$.
\item\label{MAiii} %(MA iii)
If $\g R$ 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 $\g R$ and $F$.
\end{enumerate}
\endgroup
\medskip
Actually a filter or subset in $\SHI$ 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~\ref{MAiii} uniquely through its consequence~\ref{sect1.4.1b} below.
In the affine case the hypothesis ``$A\cap A'$ contains a generic ray'' {may be omitted} in~\ref{MAii}.
%\medskip
We list now some of the properties of masures we shall use.
\begin{enumerate}[label=(\alph*)]
\item\label{sect1.4.1a}
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 $A\cap A'$ contains the
enclosure {$\cl _A^\#(F)$} of $F$ and there exists a Weyl-isomorphism from $A$ onto $A'$ fixing $\cl _A^\#(F)$, see~\cite[5.11]{He17a} or~\cite[4.4.10]{Heb18}.
Hence any isomorphism from $A$ onto $A'$ fixing $F$ fixes $\overline F$ (and even $\cl_A^\#(F)\cap \supp (F)$).
More generally the intersection of two apartments $A,A'$ is always closed (in $A$ and $A'$), see~\cite[3.9]{He17a} or~\cite[4.2.17]{Heb18}.
\item\label{sect1.4.1b}
%{\bf b)}
If $\mathfrak F$ is the germ of a spherical sector face and if $F$ is a {local} face or a germ of a sector face, then there exists an apartment that contains $\mathfrak F$ and $F$.
\item\label{sect1.4.1c}
%{\bf c)}
If two apartments $A,A'$ contain $\mathfrak F$ and $F$ as in~\ref{sect1.4.1b}, then their intersection contains {$\cl _A^\#(\mathfrak F\cup F)$} and there exists a Weyl-isomorphism from $A$ onto $A'$ fixing {$\cl _A^\#(\mathfrak F\cup F)$}.
\item\label{sect1.4.1d}
%{\bf d)}
We consider the relations, %$²$,
$\stackrel{o}{<}$ and $\stackrel{o}{\leq}$ on $\SHI$ defined as follows:
\begin{multline*}
\hspace*{10mm}x \leq y \text{ (\resp } x\stackrel{o}{<}y,x\stackrel{o}{\leq}y) \\
\iff \exists A\in\sha \text{ such that }x,y\in A\text{ and } x {\leq}_A y \text{ (\resp } x\stackrel{o}{<}_Ay,x\stackrel{o}{\leq}_Ay).
\end{multline*}
Then $\leq$ (\resp $\stackrel{o}{<}$, $\stackrel{o}{\leq}$) is a well defined preorder relation, in particular transitive; it is called the \emph{Tits preorder} (\resp \emph{Tits open preorder, large Tits open preorder}), see~\cite{He17a}.
\item\label{sect1.4.1e}
%{\bf e)}
We ask here $\SHI$ to be thick of \motgras{finite thickness}: the number of local chambers covering a given (local) panel in a wall has to be finite $\geq 3$.
This number is the same for any panel $F$ in a given wall $M$~\cite[2.9]{R11}; we denote it by $1+q_M=1+q_{F}$.
\item\label{sect1.4.1f}
%{\bf f)}
An automorphism (\resp a Weyl-automorphism, a vectorially-Weyl automorphism) of $\SHI$ is a bijection $\qf:\SHI\to\SHI$ such that $A\in\sha\iff \qf(A)\in\sha$ and then $\qf\vert_A:A\to\qf(A)$ is an isomorphism (\resp a Weyl-isomorphism, a vectorially-Weyl isomorphism).
{We write $\Aut (\SHI)$ (\resp $\Aut ^W(\SHI)$, $\Aut ^W_\R(\SHI)$) the group of these automorphisms.}
\end{enumerate}
%\medskip
%\parni{\bf 2)}
\subsubsection{}\label{sect1.4.2}
For $x\in\SHI$, the set $\sht^+_x\SHI$ (\resp $\sht^-_x\SHI$) of segment germs $[x,y)$ for $y>x$ (\resp $y0$ small) the negative (\resp positive) segment-germ of $\pi$ at $t$, for $00 $ for all $i\leq r$ (because $C\subset \prism _\qd (C_y)$) and $\qa_i(p) $ of the same sign as $\qa_i(\qd) $ if $i>r$ (because $\qd\subset \bar C$). So $C=C_x^{++}$ if $\qd\in \sht^+_x\SHI$ (\resp $C=\op _A(w_r(C_x^{++}))$ if $\qd\in \sht^-_x\SHI$).
In the case $\qd\in \sht^+_x\SHI $, the characterization of $C_x^{++}$ in the building $ \sht^+_x\SHI$ proves that it does not depend {on} the choice of $A$.
The chamber $\op_A(w_r(C_x^{++}))$ also only depends on $\qd$ and $C_y$ if $\qd\in \sht^-_x\SHI$. It is sufficient to prove that it intersects {$\conv _A(\qd \cup \pr _x(C_y))$}.
Indeed, let us choose $\qx$ and $y$ such that $[x,\qx)= \qd$ and $]x,y)\subset \pr _x(C_y)$.
We have $\qa_i(\qx) =0 $ for $i\leq r$, $\qa_i(\qx) <0 $ for $i> r$ and $\qa_i(y) >0 $ for $i\leq r$.
So for $t$ near $1$ enough, $\qa_i(t\qx+(1-t)y) >0 $ for $i\leq r$ and $<0$ for $i>r$, so $]x, t\qx+(1-t)y)\subset \op _A(w_r(C_x^{++})$.
By Proposition~\ref{1.11}, the local chamber $\op _A(w_r(C_x^{++}))$ is included in all apartments containing $\qd$ and $\pr _x(C_y)$, so is independent of the choice of $A$.
\end{proof}
\subsection{Centrifugally folded galleries of chambers}
\label{sc1}
Let $z$ be a point in the standard apartment $\mathbb A$. We have twinned buildings $\mathcal T_z^+\SHI$ (\resp $\mathcal T_z^-\SHI$).
As in~\ref{sect1.4.2}, %\ref{1.3}.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 $\overline{C}=\germ_z(z+\overline{C^v})$), where $C^v$ is a vectorial chamber, \cf~\cite[4.5]{GR08} or~\cite[\S~5]{R11}.
The distances (\resp codistances) between these chambers are written $d^W$ (\resp $d^{*W}$).
To $\A$ is associated a twin system of apartments $\A_z = (\A_z^-,\A_z^+)$.
Let $\mathbf i = (i_1,\ldots , i_r)$ be the type of a minimal gallery. We choose in $\A^-_z$ a negative (local) chamber $C^-_z$ and denote by $C^+_z$ its opposite in $\A^+_z$.
We consider now galleries of (local) chambers $\mathbf c = (C_z^-,C_1,\ldots ,C_r)$ in the
apartment $\mathbb A_z^-$ starting at $C_z^-$ and of type $\mathbf i$.
Their set is written $\Gamma (C^-_{z},\mathbf i)$.
We consider the root $\beta_j$ corresponding to the common
limit hyperplane $M_j = M({\beta_j},-\qb_j(z))$ of type $i_j$ of $C_{j-1}$ and $C_j $ satisfying moreover $\qb_j(C_j)\geq{}\qb_j(z)$.
We consider the system of positive roots $\QF^+$ associated to $C^+_z$. Actually, $\QF^+=w\mycdot \QF^+_f$, if $\QF^+_f$ is the system $\QF^+$ defined in~\ref{1.1} and $C^+_z=\germ_z(z+w\mycdot C^v_f)$.
We denote by $(\qa_i)_{i\in I}$ the corresponding basis of $\QF$ and by $(r_i)_{i\in I}$ the corresponding generators of $W^v$. Note that this change of notation for $\QF^+$ and $r_{i}$ is limited to subsection~\ref{sc1}.
The set $\Gamma (C^-_{z},\mathbf i)$ of galleries is in bijection with the set $\Gamma (\mathbf i) = \{1,r_{i_1}\}\times\cdots\times \{1,r_{i_r}\}$ via the map $(c_1,\ldots ,c_r)\mapsto (C_z^-, c_1 C_z^-,\ldots ,c_1\cdots c_r C_z^-)$.
Moreover $\beta_j = -c_1\cdots c_j (\alpha_{i_j})$.
\begin{defi*}
Let $\mathfrak Q$ be a chamber in $\mathbb A_z$.
A gallery $\mathbf c = (C_z^-,C_1,\ldots ,C_r)\in\Gamma (C^-_{z},\mathbf i)$ is said to be \emph{centrifugally folded} with respect to $\mathfrak Q$ if $C_j = C_{j-1}$ implies that $M_j $ is a wall and separates $\mathfrak Q$ from $C_j = C_{j-1}$.
We denote this set of centrifugally folded galleries by $\Gamma^+_{\mathfrak Q} (C^-_{z},\mathbf i)$.
We write $\Gamma^+_{\mathfrak Q} (C^-_{z},\mathbf i,C)$ the subset of galleries in $\Gamma_{\mathfrak Q} (C^-_{z},\mathbf i)$ such that $C_{r}$ is a given chamber $C$.
\end{defi*}
\subsection{Liftings of galleries}
\label{sc2}
Next, let $\qr_{\mathfrak Q}: \sht_{z}\SHI \to \mathbb A_z$ be the retraction centered at $\mathfrak Q$. To a gallery of chambers $\mathbf c = (C_z^-,C_1,\ldots ,C_r)$ in $\Gamma(C_z^-,\mathbf i)$,
one can associate the set of all galleries of type $\mathbf i$ starting at $C_z^-$ in $\sht_{z}^-\SHI$ that retract onto $\mathbf c$, we denote this set by $\mathcal C_{\mathfrak Q}(C_z^-,\mathbf c)$.
We denote the set of galleries $\mathbf c' = (C_z^-,C'_1,\ldots ,C'_r)$ in $\mathcal C_{\mathfrak Q}(C_z^-,\mathbf c)$ that are minimal (\ie satisfy $C'_{j-1}\ne C'_j$ for any $j$) by $\mathcal C_{\mathfrak Q}^m(C_z^-,\mathbf c)$.
Recall from~\cite[Proposition~4.4]{GR13}, that the set $\mathcal C_{\mathfrak Q}^m(C_z^-,\mathbf c)$ is nonempty if, and only if, the gallery $\mathbf c$ is centrifugally folded with respect to $\mathfrak Q$. Recall also from loc. cit., Corollary 4.5, that if $\mathbf c \in \Gamma_{\mathfrak Q}^+(C_z^-,\mathbf i)$, then the number of elements in $\mathcal C^m_{\mathfrak Q}(C_z^-,\mathbf c)$ is:
\[
\sharp \mathcal C^m_{\mathfrak Q}(C_z^-,\mathbf c) = \prod_{j\in J_1} (q_{j} - 1) \times \prod_{j\in J_2} q_{j}
\]
where $q_j=q_{M_j}\in\shq$,
\[
J_1\Mk =\{j\in\{1,\cdots,r\}\mid c_j=1\}=\{j\in\{1,\cdots,r\}\mid C_{j-1}=C_{j}\}\]
and
\[
J_2\Mk =\Mk \{j\Mk \in\Mk \{1,\cdots,r\}\Mk \mid\Mk C_{j-1}\Mk \neq\Mk C_{j} \text{ and $M_j$ is a wall separating $ \mathfrak Q$ (and $C_{j-1}$) from $C_j$}\}.
\]
One may remark that $\{1,\cdots,r\}$ contains the disjoint union $J_{1}\sqcup J_{2}$, but may be different from it.
The missing $j$ are precisely those $j$ such that $M_{j}$ is not a wall (hence $q_{M_j}$ is not defined)
or that $\mathfrak Q$ (and $C_{j}$) are separated from $C_{j-1}$ by $M_{j}$.
\medskip
%\bigskip
More generally let $\mathbf c^m = (C_z^-,C_1^m,\ldots ,C_r^m)$ be the minimal gallery in $\A^-_{z}$ of type $\mathbf i$.
We write $\mathcal C^m(C^ {-}_{z},\mathbf i)$ the set of all minimal galleries in $\SHI$ of type $\mathbf i$ starting from $C_z^-$.
Its cardinality is $\prod_{j\in J_2} q_{j}$, where $J_2$ is the set of $1\leq j\leq r$ such that the hyperplane $M_{j}$ separating $C^m_{j-1}$ from $C^m_{j}$ is a wall.
\begin{NB} The $q_{j}=q_{M_{j}}$ in the above formulas are in the set $\shq$ of parameters.
More precisely, by~\ref{sect1.4.6}, %\ref{1.3}.6,
if $M_{j}=M(\qb_{j},k_{j})$ with $\qb_{j}=w\mycdot \qa_{i}$ (for some $w\in W^v$, $i\in I$ and $k_{j}\in\Z$), then one has $q_{j}=q_{i}$ if $k_{j}$ is even and $q_{j}=q'_{i}$ if $k_{j}$ is odd.
\end{NB}
\subsection{Hecke paths}
\label{sc3}
The Hecke paths we consider here are slight modifications of those used in~\cite{GR13}.
They were defined in~\cite{BPGR16}, or in~\cite{BCGR13} (for the classical case).
Let us fix a local chamber $C_x\in \mathscr C_0\cap \mathbb A$.
\begin{defi*} A Hecke path of shape $\ql\in Y^{++}$ with respect to $C_x$ in $\A$ is a $\ql-$path in $\mathbb A$ that satisfies the following assumptions.
For all $p=\pi(t)$, we ask $x \stackrel{o}{<} p$, so we can consider the local negative chamber $C^-_p=\pr _{p}(C_x)$ by~\ref{sect2.1.1}. %\ref{sc0}.1.
Then we assume moreover
that for all $t\in [0,1]\setminus\{0,1\}$, there exist finite sequences
$(\xi_0=\pi_-'(t),\xi_1,\dots,\xi_s=\pi_+'(t))$ of vectors in $V$ and
$(\beta_1,\dots,\beta_s)$ of real roots such that, for all $j=1,\dots,s$:
\begin{enumerate}[label=(\roman*)]
\item\label{sect2.4_defi_i} $r_{\beta_j}(\xi_{j-1})=\xi_j$,
\item\label{sect2.4_defi_ii} $\beta_j(\xi_{j-1})<0$,
\item\label{sect2.4_defi_iii} $\beta_j(\pi(t))\in\mathbb Z$, \ie $\qp(t)$ is in a wall of direction $\ker\qb_{j}$,
\item\label{sect2.4_defi_iv} $\qb_j(C^-_{\pi(t)}) <\qb_j(\pi(t))$.
\end{enumerate}
One says then that these two sequences are a $(W^v_{\pi(t)}, C^-_{\pi(t}))-$chain from $\pi'_-(t)$ to $\pi'_+(t)$.
Actually $W^v_{\pi(t)}$ is the subgroup of $W^v$ generated by the $r_{\qb}$ such that $M(\qb,-\qb(\qp(t)))$ is a wall.
\end{defi*}
When $t\in {}]0,1[$ is such that $s\not=0$, one has $\pi'_{-}(t)\not=\pi'_{+}(t)$, the path is centrifugally folded with respect to $C_x$ at $\pi(t)$.
\begin{lemm}\label{sc3a} Let $\qp\subset\A$ be a Hecke path with respect to $C_{x}$ as above. Then,
\begin{enumerate}[label=(\alph*)]
\item\label{lemma2.2_a}
%(a)
For $t$ varying in $[0,1]$ and $p=\qp(t)$, the set of vectorial rays $\R_{+}(x-\qp(t))$ is contained in a finite set of closures of (negative) vectorial chambers.
\item\label{lemma2.2_b}
%(b)
There is only a finite number of pairs $(M,t)$ with a wall $M$ containing a point $p=\qp(t)$ for $t>0$, such that $\qp_{-}(t)$ is not in $M$ and $x$ is not in the same side of $M$ as $\qp_{-}(t)$ (but may be $x\in M$).
\item\label{lemma2.2_c}
%(c)
One writes $p_{0}=\qp(t_{0}), p_{1}=\qp(t_{1}), \ldots, p_{\ell_\qp{}}=\qp(t_{\ell_{\qp}})$ with $0=t_{0}