A new axiomatics for masures II

Let $A$ be a Kac-Moody matrix (also known as generalized Cartan matrix) i.e a square matrix $A=(a_{i,j})_{i,j\in I}$ with integers coefficients, indexed by a finite set $I$ and satisfying:

1. 1.

   $\forall i\in I,\ a_{i,i}=2$

2. 2.

   $\forall(i,j)\in I^{2}|i\neq j,\ a_{i,j}\leq 0$

3. 3.

   $\forall(i,j)\in I^{2},\ a_{i,j}=0\Leftrightarrow a_{j,i}=0$.

A root generating system of type $A$ is a $5$-tuple $\mathcal{S}=(A,X,Y,(\alpha_{i})_{i\in I},(\alpha_{i}^{\vee})_{i\in I})$ made of a Kac-Moody matrix $A$ indexed by $I$, of two dual free $\mathbb{Z}$-modules $X$ (of characters) and $Y$ (of cocharacters) of finite rank $\mathrm{rk}(X)$, a family $(\alpha_{i})_{i\in I}$ (of simple roots) in $X$ and a family $(\alpha_{i}^{\vee})_{i\in I}$ (of simple coroots) in $Y$. They have to satisfy the following compatibility condition: $a_{i,j}=\alpha_{j}(\alpha_{i}^{\vee})$ for all $i,j\in I$.

Let $\mathbb{A}=Y\otimes\mathbb{R}$. Every element of $X$ induces a linear form on $\mathbb{A}$. We consider $X$ as a subset of the dual $\mathbb{A}^{*}$ of $\mathbb{A}$: the $\alpha_{i}$, $i\in I$ are viewed as linear forms on $\mathbb{A}$. For $i\in I$, we define an involution $r_{i}$ of $\mathbb{A}$ by $r_{i}(v)=v-\alpha_{i}(v)\alpha_{i}^{\vee}$ for all $v\in\mathbb{A}$. Its space of fixed points is $\ker\alpha_{i}$. The subgroup of $\mathrm{GL}(\mathbb{A})$ generated by the $\alpha_{i}$ for $i\in I$ is denoted by $W^{v}$ and is called the vectorial Weyl group of $\mathcal{S}$. Then $(W^{v},\{r_{i}|i\in I\})$ is a Coxeter system.

One defines an action of the group $W^{v}$ on $\mathbb{A}^{*}$ as follows: if $x\in\mathbb{A}$, $w\in W^{v}$ and $\alpha\in\mathbb{A}^{*}$, then $(w.\alpha)(x)=\alpha(w^{-1}.x)$. Let $\Phi=\{w.\alpha_{i}|(w,i)\in W^{v}\times I\}$ be the set of (real) roots. Then $\Phi\subset Q$, where $Q=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}$. Let $Q^{+}=\bigoplus_{i\in I}\mathbb{N}\alpha_{i}$, $\Phi^{+}=Q^{+}\cap\Phi$ and $\Phi^{-}=(-Q^{+})\cap\Phi$. Then $\Phi=\Phi^{+}\sqcup\Phi^{-}$.

We set $Q^{\vee}_{\mathbb{R}}=\bigoplus_{i\in I}\mathbb{R}\alpha_{i}^{\vee}$ and $Q^{\vee}_{\mathbb{R}_{+}}=\bigoplus_{i\in I}\mathbb{R}_{+}\alpha_{i}^{\vee}$. For $x,y\in\mathbb{A}$, we write $x\leq_{Q^{\vee}_{\mathbb{R}}}y$ if $y-x\in Q^{\vee}_{\mathbb{R}_{+}}$.

Define $C_{f}^{v}=\{v\in\mathbb{A}|\ \alpha_{i}(v)>0,\ \forall i\in I\}$. We call it the fundamental vectorial chamber. For $J\subset I$, one sets $F^{v}(J)=\{v\in\mathbb{A}|\ \alpha_{i}(v)=0\ \forall i\in J,\alpha_{i}(v)>0\ \forall i\in J\backslash I\}$. Then the closure $\overline{C_{f}^{v}}$ of $C_{f}^{v}$ is the union of the $F^{v}(J)$ for $J\subset I$. The positive (resp. negative) vectorial faces are the sets $w.F^{v}(J)$ (resp. $-w.F^{v}(J)$) for $w\in W^{v}$ and $J\subset I$. A vectorial face is either a positive vectorial face or a negative vectorial face. We call positive chamber (resp. negative) every cone of the form $w.C_{f}^{v}$ for some $w\in W^{v}$ (resp. $-w.C_{f}^{v}$). For all $x\in C_{f}^{v}$ and for all $w\in W^{v}$, $w.x=x$ implies that $w=1$. In particular the action of $w$ on the positive chambers is simply transitive. The Tits cone $\mathcal{T}$ is defined by $\mathcal{T}=\bigcup_{w\in W^{v}}w.\overline{C^{v}_{f}}$. One defines a $W^{v}$-invariant preorder $\leq$ on $\mathbb{A}$, the Tits preorder by:

We now define the Weyl group $W$ of $\mathbb{A}$. If $\mathcal{X}$ is an affine subspace of $\mathbb{A}$, one denotes by $\vec{\mathcal{X}}$ its direction. One equips $\mathbb{A}$ with a family $\mathcal{M}$ of affine hyperplanes called real walls such that:

1. 1.

   For all $M\in\mathcal{M}$, there exists $\alpha_{M}\in\Phi$ such that $\vec{M}=\ker(\alpha_{M})$.

2. 2.

   For all $\alpha\in\Phi$, there exists an infinite number of hyperplanes $M\in\mathcal{M}$ such that $\alpha=\alpha_{M}$.

3. 3.

   If $M\in\mathcal{M}$, we denote by $r_{M}$ the reflection with respect to the hyperplane $M$ whose associated linear map is $r_{\alpha_{M}}$. We assume that the group $W$ generated by the $r_{M}$ for $M\in\mathcal{M}$ stabilizes $\mathcal{M}$.

The group $W$ is the Weyl group of $\mathbb{A}$. We assume that $0$ is special (i.e we assume that $\ker\alpha\in\mathcal{M}$ for every $\alpha\in\Phi$) and thus $W\supset W^{v}$.

For $\alpha\in\mathbb{A}^{*}$ and $k\in\mathbb{R}$, set $M(\alpha,k)=\{v\in\mathbb{A}|\alpha(v)+k=0\}$. Then for all $M\in\mathcal{M}$, there exists $\alpha\in\Phi$ and $k_{M}\in\mathbb{R}$ such that $M=M(\alpha,k_{M})$. For $\alpha\in\Phi$, set $\Lambda_{\alpha}=\{k_{M}|\ M\in\mathcal{M}\mathrm{\ and\ }\vec{M}=\ker(\alpha)\}$. Then $\Lambda_{w.\alpha}=\Lambda_{\alpha}$ for all $w\in W^{v}$ and $\alpha\in\Phi$.

If $\alpha\in\Phi$, one denotes by $\tilde{\Lambda}_{\alpha}$ the subgroup of $\mathbb{R}$ generated by $\Lambda_{\alpha}$. By (3), $\Lambda_{\alpha}=\Lambda_{\alpha}+2\tilde{\Lambda}_{\alpha}$ for all $\alpha\in\Phi$. In particular, $\Lambda_{\alpha}=-\Lambda_{\alpha}$ and when $\Lambda_{\alpha}$ is discrete, $\tilde{\Lambda}_{\alpha}=\Lambda_{\alpha}$ is isomorphic to $\mathbb{Z}$.

One sets $Q^{\vee}=\bigoplus_{\alpha\in\Phi}\tilde{\Lambda}_{\alpha}\alpha^{\vee}$. This is a subgroup of $\mathbb{A}$ stable under the action of $W^{v}$. Then one has $W=W^{v}\ltimes Q^{\vee}$.

If $F$ is a filter on a set $E$, and $E^{\prime}$ is a subset of $E$, one says that $F$ contains $E^{\prime}$ if every element of $F$ contains $E^{\prime}$. If $E^{\prime}$ is nonempty, the set $F_{E^{\prime}}$ of subsets of $E$ containing $E^{\prime}$ is a filter. We will sometimes regard $E^{\prime}$ as a filter on $E$ by identifying $F_{E^{\prime}}$ and $E^{\prime}$. If $F$ is a filter on $E$, its closure $\overline{F}$ (resp. its convex hull) is the filter of subsets of $E$ containing the closure (resp. the convex envelope) of some element of $F$. A filter $F$ is said to be contained in an other filter $F^{\prime}$: $F\subset F^{\prime}$ (resp. in a subset $Z$ in $E$: $F\subset Z$) if and only if any set in $F^{\prime}$ (resp. if $Z$) is in $F$.

If $x\in\mathbb{A}$ and $\Omega$ is a subset of $\mathbb{A}$ containing $x$ in its closure, then the germ of $\Omega$ in $x$ is the filter $germ_{x}(\Omega)$ of subsets of $\mathbb{A}$ containing a neighborhood of $x$ in $\Omega$.

A sector in $\mathbb{A}$ is a set of the form $\mathfrak{s}=x+C^{v}$ with $C^{v}=\pm w.C_{f}^{v}$ for some $x\in\mathbb{A}$ and $w\in W^{v}$. A point $u$ such that $\mathfrak{s}=u+C^{v}$ is called a base point of $\mathfrak{s}$ and $C^{v}$ is its direction. The intersection of two sectors of the same direction is a sector of the same direction.

The sector-germ of a sector $\mathfrak{s}=x+C^{v}$ is the filter $\mathfrak{S}$ of subsets of $\mathbb{A}$ containing an $\mathbb{A}$-translate of $\mathfrak{s}$. It only depends on the direction $C^{v}$. We denote by $+\infty$ (resp. $-\infty$) the sector-germ of $C_{f}^{v}$ (resp. of $-C_{f}^{v}$).

A ray $\delta$ with base point $x$ and containing $y\neq x$ (or the interval $]x,y]=[x,y]\backslash\{x\}$ or $[x,y]$ or the line containing $x$ and $y$) is called preordered if $x\leq y$ or $y\leq x$ and generic if $y-x\in\pm\mathring{\mathcal{T}}$, the interior of $\pm\mathcal{T}$.

For $\alpha\in\Phi$, and $k\in\mathbb{R}\cup\{+\infty\}$, let $D(\alpha,k)=\{v\in\mathbb{A}|\alpha(v)+k\geq 0\}$ (and $D(\alpha,+\infty)=\mathbb{A}\mathrm{}$) and $D^{\circ}(\alpha,k)=\{v\in\mathbb{A}|\ \alpha(v)+k>0\}$ (for $\alpha\in\Phi$ and $k\in\mathbb{R}\cup\{+\infty\}$).

Let $\mathscr{L}$ be the set of families $(\Lambda^{\prime}_{\alpha})_{\alpha\in\Phi}$ such that $\Lambda_{\alpha}\subset\Lambda^{\prime}_{\alpha}\subset\mathbb{R}$ and $\Lambda_{\alpha}^{\prime}=-\Lambda_{-\alpha}^{\prime}$, for $\alpha\in\Phi$.

An apartment is a root generating system equipped with a Weyl group $W$ (i.e with a set $\mathcal{M}$ of real walls, see 2.1.3) and a family $\Lambda^{\prime}\in\mathscr{L}$. Let $\underline{\mathbb{A}}=(\mathcal{S},W,\Lambda^{\prime})$ be an apartment and $\mathbb{A}$ be the underlying affine space. A set of the form $M(\alpha,k)$, with $\alpha\in\Phi$ and $k\in\Lambda^{\prime}_{\alpha}$ is called a wall of $\mathbb{A}$ and a set of the form $D(\alpha,k)$, with $\alpha\in\Phi$ and $k\in\Lambda^{\prime}_{\alpha}$ is called a half-apartment of $\mathbb{A}$. A subset $X$ of $\mathbb{A}$ is said to be enclosed if there exist $k\in\mathbb{N}$, $\beta_{1},\ldots,\beta_{k}\in\Phi$ and $(\lambda_{1},\ldots,\lambda_{k})\in\prod_{i=1}^{k}\Lambda^{\prime}_{\beta_{i}}$ such that $X=\bigcap_{i=1}^{k}D(\beta_{i},\lambda_{i})$.

Let $\underline{\mathbb{A}}=(\mathcal{S},W,\Lambda^{\prime})$ be an apartment and $\mathbb{A}$ be the underlying affine space.

Let $x\in\mathbb{A}$ and $F^{v}$ be a vectorial face. The local-face $F^{\ell}(x,F^{v})=germ_{x}(x+F^{v})$ is the filter defined as the intersection of $x+F^{v}$ with the the filter of neighborhoods of $x$ in $\mathbb{A}$. The face $F(x,F^{v})$ is the filter consisting of the subsets containing a finite intersection of half-spaces $D(\alpha,\lambda_{\alpha})$ or $D^{\circ}(\alpha,\lambda_{\alpha})$, with $\lambda_{\alpha}\in\Lambda^{\prime}_{\alpha}\cup\{+\infty\}$ for all $\alpha\in\Phi$ (at most one $\lambda_{\alpha}\in\Lambda_{\alpha}$ for each $\alpha\in\Phi$). We say that $F$ is positive (or negative) if $F^{v}$ is.

Let $F,F^{\prime}$ be two faces. We say that $F^{\prime}$ dominates $F$ if $F\subset\overline{F^{\prime}}$. The dimension of a face $F$ is the smallest dimension of an affine space generated by some $S\in F$. Such an affine space is unique and is called its support. A face is said to be spherical if the direction of its support meets the open Tits cone $\mathring{\mathcal{T}}$; then its pointwise stabilizer $W_{F}$ in $W^{v}$ is finite.

A chamber (or alcove) is a face of the form $F(x,C^{v})$ where $x\in\mathbb{A}$ and $C^{v}$ is a vectorial chamber of $\mathbb{A}$.

A panel is a face which is maximal (for the domination relation) among the faces contained in at least one wall.

Let $F=F(x,F^{v})$ be a face ($x\in\mathbb{A}$ and $F^{v}$ is a vectorial face). The chimney $\mathfrak{r}(F,F^{v})$ is the filter consisting of the sets containing an enclosed set containing $F+F^{v}$. The face $F$ is the basis of the chimney and the vectorial face $F^{v}$ its direction. A chimney is splayed if $F^{v}$ is spherical.

A shortening of a chimney $\mathfrak{r}(F,F^{v})$, with $F=F(x,F_{0}^{v})$ is a chimney of the form $\mathfrak{r}(F(x+\xi,F_{0}^{v}),F^{v})$ for some $\xi\in\overline{F^{v}}$. The germ of a chimney $\mathfrak{r}$ is the filter of subsets of $\mathbb{A}$ containing a shortening of $\mathfrak{r}$ (this definition of shortening is slightly different from the one of [Rou11] 1.12 but follows [Rou17] 3.6) and we obtain the same germs with these two definitions).

An apartment of type $\mathbb{A}$ is a set $\underline{A}$ with a nonempty set $\mathrm{Isom}(\mathbb{A},A)$ of bijections (called Weyl-isomorphisms) such that if $f_{0}\in\mathrm{Isom}(\mathbb{A},A)$ then $f\in\mathrm{Isom}(\mathbb{A},A)$ if and only if there exists $w\in W$ satisfying $f=f_{0}\circ w$. We will say isomorphism instead of Weyl-isomorphism in the sequel. An isomorphism between two apartments $\phi:A\rightarrow A^{\prime}$ is a bijection such that ($f\in\mathrm{Isom}(\mathbb{A},A)$ if, and only if, $\phi\circ f\in\mathrm{Isom}(\mathbb{A},A^{\prime})$). We extend all the notions that are preserved by $W$ to each apartment. Thus sectors, enclosures, faces and chimneys are well defined in any apartment of type $\mathbb{A}$. If $A$ and $B$ are two apartments, and $\phi:A\rightarrow B$ is an apartment isomorphism fixing some set $X$, we write $\phi:A\overset{X}{\rightarrow}B$.

In this definition, we say that an apartment contains a germ of a filter if it contains at least one element of this germ. We say that a map fixes a germ if it fixes at least one element of this germ.

The main examples of masures are the masures associated with (almost) split Kac-Moody group over valued fields constructed in [Rou16] and [Rou17].

By [Héb20, Theorem 5.1], definition 2.3 is equivalent to [Rou11, Définition 2.1] (at least under Assumption 2.1).

Let $A$ be a Kac-Moody matrix and $\mathcal{S}$ be a root generating system of type $A$. We consider the group functor $\mathbf{G}$ associated with the root generating system $\mathcal{S}$ in [Tit87] and in [Rém02, Chapitre 8]. This functor is a functor from the category of rings to the category of groups satisfying axioms (KMG1) to (KMG 9) of [Tit87]. When $R$ is a field, $\mathbf{G}(R)$ is uniquely determined by these axioms by Theorem 1’ of [Tit87]. This functor contains a toric functor $\mathbf{T}$, from the category of rings to the category of commutative groups (denoted $\mathcal{T}$ in [Rém02]) and two functors $\mathbf{U^{+}}$ and $\mathbf{U^{-}}$ from the category of rings to the category of groups.

Let $\mathcal{K}$ be a field equipped with a non-trivial valuation $\omega:\mathcal{K}\rightarrow\mathbb{R}\cup\{+\infty\}$, $\mathcal{O}$ be its ring of integers and $G=\mathbf{G}(\mathcal{K})$ (and $U^{+}=\mathbf{U^{+}}(\mathcal{K})$, …). For all $\epsilon\in\{-,+\}$, and all $\alpha\in\Phi_{\epsilon}$, we have an isomorphism $x_{\alpha}$ from $\mathcal{K}$ to a group $U_{\alpha}$. For all $k\in\mathbb{R}$, one defines a subgroup $U_{\alpha,k}:=x_{\alpha}(\{u\in\mathcal{K}|\ \omega(u)\geq k\})$. Let $\mathcal{I}$ be the masure associated with $G$ constructed in [Rou16]. Then for all $\alpha\in\Phi$, $\Lambda_{\alpha}=\Lambda^{\prime}_{\alpha}=\omega(\mathcal{K}^{*})$. If moreover $\omega(\mathcal{K}^{*})$ is discrete, one has (up to renormalization) $\Lambda_{\alpha}=\mathbb{Z}$ for all $\alpha\in\Phi$. Moreover, we have:

• -

  the fixator of $\mathbb{A}$ in $G$ is $H=\mathbf{T}(\mathcal{O})$ (by remark 3.2 of [GR08])

• -

  for all $\alpha\in\Phi$ and $k\in\mathbb{Z}$, the fixator of $D(\alpha,k)$ in $G$ is $H.U_{\alpha,k}$ (by 4.2 7) of [GR08])

• -

  for all $\epsilon\in\{-,+\}$, $H.U^{\epsilon}$ is the fixator of $\epsilon\infty$ (by 4.2 4) of [GR08])

• -

  when moreover the residue field of $(\mathcal{K},\omega)$ contains $\mathbb{C}$, the fixator of $\{0\}$ in $G$ is $K_{s}=\mathbf{G}(\mathcal{O})$ (by example 3.14 of [GR08]).

If moreover, $\mathcal{K}$ is local, with residue cardinal $q$, each panel is contained in $1+q$ chambers.

The group $G$ is reductive if and only if $W^{v}$ is finite. In this case, $\mathcal{I}$ is the usual Bruhat-Tits building of $G$ and one has $\mathcal{T}=\mathbb{A}$.

As the Tits preorder $\leq$ on $\mathbb{A}$ is invariant under the action of $W^{v}$, one can equip each apartment $A$ with $\leq_{A}$. Let $A$ be an apartment of $\mathcal{I}$ and $x,y\in A$ be such that $x\leq_{A}y$. Then by [Rou11, Proposition 5.4], if $B$ is an apartment containing $x$ and $y$, $[x,y]_{A}=[x,y]_{B}$ and there exists an apartment isomorphism $\psi:A\overset{[x,y]}{\rightarrow}B$. In particular, $x\leq_{B}y$. This defines a relation $\leq$ on $\mathcal{I}$. By Théorème 5.9 of [Rou11], $\leq$ is a preorder on $\mathcal{I}$. It is invariant by apartment isomorphisms: if $A,B$ are apartments, $\phi:A\rightarrow B$ is an apartment isomorphism and $x,y\in A$ are such that $x\leq y$, then $\phi(x)\leq\phi(y)$. We call it the Tits preorder on $\mathcal{I}$.

Let $\mathfrak{s}$ be a sector-germ of $\mathcal{I}$ and $A$ be an apartment containing it. Let $x\in\mathcal{I}$. By (MA iii), there exists an apartment $A_{x}$ of $\mathcal{I}$ containing $x$ and $\mathfrak{s}$. By (MA ii), there exists an apartment isomorphism $\phi:A_{x}\rightarrow A$ fixing $\mathfrak{s}$. By [Rou11, 2.6], $\phi(x)$ does not depend on the choices we made and thus we can set $\rho_{A,\mathfrak{s}}(x)=\phi(x)$.

The map $\rho_{A,\mathfrak{s}}:\mathcal{I}\rightarrow A$ is the retraction onto $A$ centered at $\mathfrak{s}$.

We denote by $\rho_{+\infty}:\mathcal{I}\twoheadrightarrow\mathbb{A}$ (resp. $\rho_{-\infty}$) the retraction onto $\mathbb{A}$ centered at $+\infty$ (resp. $-\infty$).