Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields

TL;DR

This paper extends the theory of Hecke algebras to split Kac-Moody groups over local fields by defining a completion, analyzing its center, and establishing an isomorphism with the spherical Hecke algebra, using the masure structure.

Contribution

It introduces a completion of the Iwahori-Hecke algebra for Kac-Moody groups and proves its center is isomorphic to the spherical Hecke algebra, generalizing known results for reductive groups.

Findings

The completed Iwahori-Hecke algebra's center is isomorphic to the spherical Hecke algebra.

The masure I serves as an analogue of the Bruhat-Tits building for Kac-Moody groups.

Construction of Hecke algebras for special and spherical facets in the Kac-Moody setting.

Abstract

Let G be a split Kac-Moody group over a non-archimedean local field. We define a completion of the Iwahori-Hecke algebra of G. We determine its center and prove that it is isomorphic to the spherical Hecke algebra of G using the Satake isomorphism. This is thus similar to the situation of reductive groups. Our main tool is the masure I associated to this setting, which is the analogue of the Bruhat-Tits building for reductive groups. Then, for each special and spherical facet F , we associate a Hecke algebra. In the Kac-Moody setting, this construction was known only for the spherical subgroup and for the Iwahori subgroup.

Equations

$${\mathcal{H}}_s\underset{\sim }{\overset{S}{\to }}\mathbb{C}[Q^{\vee }{]}^{W^v}\stackrel{g}{\to }\mathcal{H},\text{and}\mathrm{Im}(g)=\mathcal{Z}(\mathcal{H}),$$

$${\mathcal{H}}_s\underset{\sim }{\overset{\text{leavevmode}S\text{leavevmode}}{\to }}\mathbb{C}[[Y{]]}^{W^v}\stackrel{g}{\to }\hat{\mathcal{H}},\text{and}\mathrm{Im}(g)=\mathcal{Z}(\hat{\mathcal{H}}),$$

$$\forall x\in \mathbb{A},w\in W^v,\alpha \in {\mathbb{A}}^{\ast },(w\cdot \alpha )(x):=\alpha (w^{-1}\cdot x).$$

$$D(\alpha ,k):=\{t\in \mathbb{A}\mid \alpha (t)+k\ge 0\}\text{and}{D}^{\circ }(\alpha ,k):=\{t\in \mathbb{A}\mid \alpha (t)+k>0\}.$$

$$F^v(J)=\{v\in \mathbb{A}\mid \forall j\in J,{\alpha }_j(v)=0\text{ and }\forall i\in I\setminus J,{\alpha }_i(v)>0\};$$

$$\forall (x,y)\in {\mathbb{A}}^2,x\le y⟺y-x\in \mathcal{T}.$$

$$\overline{F}:=\{S\subset E\mid \exists S^{\prime }\in F,\overline{S^{\prime }}\subset S\}\text{ and}\mathrm{conv}(F):=\{S\subset E\mid \exists S^{\prime }\in F,\mathrm{conv}(S^{\prime })\subset S\}.$$

$$M_k^{\alpha }:=\{t\in \mathbb{A}\mid \alpha (t)=k\},D_k^{\alpha }:=\{t\in \mathbb{A}\mid \alpha (t)\le k\},$$

$$\forall (\lambda ,w)\in Y\times W^v,Z^{\lambda }\ast H_w=Z^{\lambda }{H}_w;$$

$$\forall i\in I,\forall w\in W^v,H_i\ast H_w=\{\begin{array}{ll}{H}_{r_{i}w}& \text{if }\ell (r_{i}w)=\ell (w)+1,\\ ({\sigma }_i-{\sigma }_i^{-1})H_w+H_{r_{i}w}& \text{if }\ell (r_{i}w)=\ell (w)-1;\end{array}$$

$$\forall (\lambda ,\mu )\in Y^2,Z^{\lambda }\ast Z^{\mu }=Z^{\lambda +\mu };$$

$$\forall \lambda \in Y,\forall i\in I,H_i\ast Z^{\lambda }-Z^{r_i(\lambda )}\ast H_i=b({\sigma }_i,{\sigma }_i^{\prime };Z^{-{\alpha }_i^{\vee }})(Z^{\lambda }-Z^{r_i(\lambda )}),$$

$$\text{with}b(t,u;z)=\frac{(t-t^{-1})+(u-u^{-1})z}{1-z^2}.$$

$$A=\left(\begin{array}{ccc}2& 0& -2\\ 0& 2& 0\\ -5& 0& 2\end{array}\right).$$

$$\{\begin{array}{l}\mathcal{R}[[Y^{+}]]:=\left\{a\in \mathcal{R}[[Y]]\mid \forall y\in Y\setminus Y^{+},{\pi }_{\lambda }(a)=0\right\}\\ \mathcal{R}[[Y{]]}^{W^v}:=\left\{a\in \mathcal{R}[[Y]]\mid \forall (\lambda ,w)\in Y\times W^v,{\pi }_{\lambda }(w\cdot a)={\pi }_{\lambda }(a)\right\}.\end{array}$$

$$F_{\nu }:=\{\lambda \in Y^{++}\mid {\pi }_{\nu }(a_{\lambda }E(\lambda ))\ne 0\}.$$

$$[1,w]:=\{u\in W^v\mid u\le w\}.$$

$$\mathrm{supp}(a):=\left\{(\lambda ,w)\in Y^{+}\times W^v\mid a_{\lambda ,w}\ne 0\right\}.$$

$${\mathrm{supp}}_Y(a):=\left\{\lambda \in Y^{+}\mid \exists w\in W^v,a_{\lambda ,w}\ne 0\right\}$$

$${\mathrm{supp}}_{W^v}(a):=\left\{w\in W^v\mid \exists \lambda \in Y^{+},a_{\lambda ,w}\ne 0\right\}.$$

$$\{w\in W^v\mid \exists \lambda \in Y^{+},(\lambda ,w)\in Z\}$$

$$\sum _{(\lambda ,w)\in Y^{+}\times W^v}{a}_{\lambda ,w}{Z}^{\lambda }{H}_w\text{and}\sum _{(\lambda ,w)\in Y^{+}\times W^v}{b}_{\lambda ,w}{Z}^{\lambda }{H}_w$$

$$\sum _{(\lambda ,w),({\lambda }^{\prime },w^{\prime })\in Y^{+}\times W^v}{\pi }_{\mu ,v}(a_{\lambda ,w}{b}_{{\lambda }^{\prime },w^{\prime }}{Z}^{\lambda }{H}_w\ast Z^{{\lambda }^{\prime }}{H}_{w^{\prime }})$$

$$\underset{(\nu ,t)\in R_i(\lambda )\times \{1,r_i\}}{⨁}\mathcal{R}\cdot Z^{\nu }{H}_t.$$

$$H_i\ast Z^{\lambda }=Z^{r_i(\lambda )}\ast H_i+({\sigma }_i-{\sigma }_i^{-1})Z^{\lambda }\frac{1-Z^{-{\alpha }_i(\lambda ){\alpha }_i^{\vee }}}{1-Z^{-{\alpha }_i^{\vee }}},$$

$$H_i\ast Z^{\lambda }=Z^{r_i(\lambda )}\ast H_i+({\sigma }_i-{\sigma }_i^{-1})\sum _{k=0}^{{\alpha }_i(\lambda )-1}{Z}^{\lambda -k{\alpha }_i^{\vee }}$$

$$H_i\ast Z^{\lambda }=Z^{r_i(\lambda )}\ast H_i+({\sigma }_i-{\sigma }_i^{-1})\sum _{k=1}^{-{\alpha }_i(\lambda )}{Z}^{\lambda +k{\alpha }_i^{\vee }}$$

$$H_i\ast Z^{\lambda }=Z^{r_i(\lambda )}\ast H_i+Z^{\lambda }\left(({\sigma }_i-{\sigma }_i^{-1})+({\sigma }_i^{\prime }-{\sigma }_i^{\prime -1})\right)Z^{-{\alpha }_i^{\vee }},$$

$$\underset{(\nu ,t)\in R_w(\lambda )\times [1,w]\cdot w^{\prime }}{⨁}\mathcal{R}\cdot Z^{\nu }{H}_t.$$

$$\underset{(\nu ,t)\in R_u(\lambda )\times [1,u]\cdot w}{⨁}\mathcal{R}{H}_i\ast Z^{\lambda }{H}_t\subset \underset{(\nu ,t)\in R_u(\lambda )\times [1,u]\cdot w}{⨁}\underset{({\nu }^{\prime },t^{\prime })\in R_i(\nu )\times \{1,r_i\}}{⨁}\mathcal{R}\cdot Z^{{\nu }^{\prime }}\ast H_{t^{\prime }}\ast H_t$$

$$\underset{n\to +\infty }{\lim }\ell (w_{{\mu }_n})=+\infty .$$

$$h({\mu }_n-\lambda )=h(w\cdot \lambda -\lambda )=\rho (w\cdot \lambda -\lambda )=(w^{-1}\rho -\rho )(\lambda ).$$

Full text

1 Introduction

Let $\mathbf{G}_{0}$ be a split reductive group over a non-Archimedean local field $\mathcal{K}$ and set $G_{0}=\mathbf{G}_{0}(\mathcal{K})$. An important tool to study complex representations of $G_{0}$ are Hecke algebras attached to each open compact subgroup of $G_{0}$: if $K$ is such a subgroup, the Hecke algebra $\mathcal{H}_{K}$ associated to $K$ is the convolution algebra of complex-valued $K$-bi-invariant functions on $G_{0}$ with compact support. Two choices of $K$ are of particular interest: the first one is when $K=K_{s}=\mathbf{G}(\mathcal{O})$ with $\mathcal{O}$ being the ring of integers of $\mathcal{K}$. In this case, $K_{s}$ is a maximal open compact subgroup of $G$ and $\mathcal{H}_{s}=\mathcal{H}_{K_{s}}$ is a commutative algebra called the spherical Hecke algebra of $G_{0}$. This algebra can be explicitly described through the Satake isomorphism: indeed, if $W^{v}$ denotes the Weyl group of $G_{0}$ and $Q^{\vee}$ is the coweight lattice of $G_{0}$, then $\mathcal{H}_{s}$ is isomorphic to the subalgebra $\mathbb{C}[Q^{\vee}]^{W^{v}}$ of $W^{v}-$invariant elements in the group algebra of $(Q^{\vee},+)$. A second interesting choice is when $K=K_{I}$ is an Iwahori subgroup of $G_{0}$: then $\mathcal{H}:=\mathcal{H}_{K_{I}}$ is called the Iwahori-Hecke algebra of $G_{0}$. This algebra comes with a basis (called the Bernstein-Lusztig basis) indexed by the affine Weyl group of $G_{0}$ and such that the product of two elements of this basis can be expressed via the Bernstein-Lusztig presentation [Lus89]. This presentation enables us to compute the center of $\mathcal{H}$ and to check that it is isomorphic to the spherical Hecke algebra of $G_{0}$. These results can be summarized as follows:

[TABLE]

where $S$ denotes the Satake isomorphism and $g$ comes from the Bernstein-Lusztig basis.

This article aims to study how far this theory can be extended to split Kac-Moody groups. Among the different definitions of Kac-Moody groups that are available in the literature, we choose to use the definition given by Tits in [Tit87] as it is more algebraic. Given $\mathbf{G}$ a split Kac-Moody group over $\mathcal{K}$, set $G=\mathbf{G}(\mathcal{K})$. To study $G$, Gaussent and Rousseau built in [GR08] an object $\mathcal{I}=\mathcal{I}(G)$ that they called a masure (also known as affine ordered hovel), and later extended by Rousseau [Rou16, Rou17]. This masure is a generalization of Bruhat-Tits buildings introduced in [BT72, BT84] as it gives back the Bruhat-Tits building of $G$ when $\mathbf{G}$ is reductive. As a set, $\mathcal{I}$ is a union of apartments that are all isomorphic to a standard one (denoted by $\mathbb{A}$ in the sequel) and $G$ acts on $\mathcal{I}$. We still have an arrangement of hyperplanes, called walls, but in general this arrangement is not locally finite anymore. This explains why faces in $\mathbb{A}$ are not sets anymore but filters. Another main difference with Bruhat-Tits buildings is that in general, two points of $\mathcal{I}$ are not necessarily contained in a common apartment.

Analogues of $K_{s}$ and $K_{I}$ (and more generally of parahoric subgroups) can be defined as fixers of some specific faces in $\mathcal{I}$. When $\mathbf{G}$ is an affine Kac-Moody group, Braverman, Kazhdan and Patnaik attached to $G$ a spherical Hecke algebra and an Iwahori-Hecke algebra [BK11, BKP16], and they obtained a Satake isomorphism as well as Bernstein-Lusztig relations. All these results were generalized to arbitrary Kac-Moody groups by Bardy-Panse, Gaussent and Rousseau [GR14, BPGR16]. In this framework, the Satake isomorphism appears as an isomorphism between $\mathcal{H}_{s}:=\mathcal{H}_{K_{s}}$ and $\mathbb{C}\llbracket Y\rrbracket^{W^{v}}$, where $Y$ denotes a lattice that can be, in first approximation, thought of as the coroot lattice (though it can be really different, even in the affine case) and $\mathbb{C}\llbracket Y\rrbracket$ is its Looijenga’s algebra (which is a completion of the group algebra $\mathbb{C}[Y]$, see Definition 4.6). So far, the analogy with the reductive case stops here. Indeed, let $\mathcal{H}$ be the Iwahori-Hecke algebra of $G$: following what happens in the reductive case [Par06], one would expect the center of $\mathcal{H}$ to be isomorphic to the spherical Hecke algebra $\mathcal{H}_{s}$. Unfortunately, this is not the case, as we prove that the center of $\mathcal{H}$ is actually more or less trivial (see Lemma 4.31). Moreover, in the non-reductive case, $\mathbb{C}\llbracket Y\rrbracket^{W^{v}}$ is a set of infinite formal series that cannot embed into $\mathcal{H}$, where all elements have finite support. All these reasons explain why we define a completion $\hat{\mathcal{H}}$ of $\mathcal{H}$ as follows: letting $(Z^{\lambda}H_{w})_{\lambda\in Y^{+},w\in W^{v}}$ be the Bernstein-Lusztig basis of $\mathcal{H}$, we define $\hat{\mathcal{H}}$ as the set of formal series $\hat{\mathcal{H}}$ whose support satisfies some conditions similar to what appears in the definition of $\mathbb{C}\llbracket Y\rrbracket$. One of our main results states that $\hat{\mathcal{H}}$ can be turned into an algebra when endowed with a well-defined convolution product compatible with the canonical inclusion of $\mathcal{H}$ into $\hat{\mathcal{H}}$ (see Theorem 4.21 and Corollary 4.23). We then determine the center of $\hat{\mathcal{H}}$ and show that it is isomorphic to $\mathbb{C}\llbracket Y\rrbracket^{W^{v}}$, as wanted (see Theorem 4.30). As before, these results can be summarized as follows:

[TABLE]

where $S$ denotes the Satake isomorphism and $g$ comes again from the Bernstein-Luzstig basis.

Another part of this paper is devoted to the construction of Hecke algebras attached to more general subgroups than $K_{s}$ and $K_{I}$. Recall that $K_{s}$ is the fixer of $\{0\}$ while $K_{I}$ is the fixer of the type [math] chamber $C^{+}_{0}$. When $G$ is reductive, any face $F$ between $\{0\}$ and $C_{0}^{+}$ corresponds to an open compact subgroup of $G$ (namely the parahoric subgroup associated with $F$) contained in its fixer $K_{F}$, hence one can use it to attach a Hecke algebra to $F$. This explains why it seems natural, in the non-reductive case, to wonder whether one can attach a Hecke algebra to $K_{F}$ for all faces $F$ between $\{0\}$ and $C_{0}^{+}$. We succeed in defining such an algebra when $F$ is spherical, which means that its fixer under the action of the Weyl group is finite. Our construction is very close to what is done for the Iwahori-Hecke algebra in [BPGR16]. We also prove that when $F$ is not spherical and different from $\{0\}$ (that cannot happen if $G$ is affine), this construction fails because the structural constants are infinite.

Finally, recall that up to now, there was no known topology on $G$ that generalizes the usual topology on $\mathbf{G}_{0}(\mathcal{K})$, for $\mathbf{G}_{0}$ a split reductive group over $\mathcal{K}$, in which $K_{s}$ and $K_{I}$ are open compact subgroups of $\mathbf{G}_{0}(\mathcal{K})$. We prove (see Theorem 3.1) that when $\mathbf{G}$ is not reductive, there is no way to turn $G$ into a topological group such that $K_{s}$ or $K_{I}$ (or, more generally, any given parahoric subgroup of $G$) is open compact. This result implies in particular that one cannot define smooth representations of $G$ in the same way as in the reductive case.

The paper is organized as follows: we first recall the definition of masures in Section 2. The reader only interested in Iwahori-Hecke algebras can read the two first sections and the last one, and skip the rest of this section. Section 3 is devoted to prove that $G$ cannot be turned into a topological group in which $K_{s}$ or $K_{I}$ is open compact. In Section 4, we define the completed Iwahori-Hecke algebra $\hat{\mathcal{H}}$ of $G$ and compute its center, as well as the center of $\mathcal{H}$. Finally, we use Section 5 to attach a Hecke algebra to any spherical face between $\{0\}$ and $C_{0}^{+}$ and to prove that this construction fails if $F$ is not spherical and different from $\{0\}$.

Remark 1.1**.**

As explained in Section 2, this paper is actually written in a more general framework, as we only need $\mathcal{I}$ to be an abstract masure and $G$ to be a strongly transitive group of (positive, type-preserving) automorphisms of $\mathcal{I}$. In particular, this applies to almost split (and not only split) Kac-Moody groups over local fields.

Acknowledgements

We warmly thank Stéphane Gaussent for suggesting our collaboration, for multiple discussions and for his useful comments on previous versions of this manuscript. We also thank Nicole Bardy-Panse and Guy Rousseau for discussions on this topic and for their comments on a previous version of this paper. Finally, we thank the referee for their valuable comments and suggestions, and for his/her interesting questions.

2 Masures: general framework

We recollect here some well-known facts. Further details are available in the first two sections of [Rou11].

2.1 Root generating system and Weyl groups

A Kac-Moody matrix (or generalized Cartan matrix) is a square matrix $A=(a_{i,j})_{i,j\in I}$ indexed by a finite set $I$, with integral coefficients, and such that:

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

$\forall(i,j)\in I^{2},(i\neq j)\Rightarrow(a_{i,j}\leq 0)$; 3. 3.

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

A root generating system 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 the finite set $I$, of two dual free $\mathbb{Z}$-modules $X$ and $Y$ of finite rank, and of a free family $(\alpha_{i})_{i\in I}$ (respectively $(\alpha_{i}^{\vee})_{i\in I}$) of elements in $X$ (resp. $Y$) called simple roots (resp. simple coroots) that satisfy $a_{i,j}=\alpha_{j}(\alpha_{i}^{\vee})$ for all $i,j$ in $I$. Elements of $X$ (respectively of $Y$) are called characters (resp. cocharacters).

Fix such a root generating system $\mathcal{S}=(A,X,Y,(\alpha_{i})_{i\in I},(\alpha_{i}^{\vee})_{i\in I})$ and set $\mathbb{A}:=Y\otimes\mathbb{R}$. Each element of $X$ induces a linear form on $\mathbb{A}$, hence $X$ can be seen as a subset of the dual $\mathbb{A}^{*}$. In particular, the $\alpha_{i}$’s (with $i\in I$) will be seen as linear forms on $\mathbb{A}$. This allows us to define, for any $i\in I$, an involution $r_{i}$ of $\mathbb{A}$ by setting $r_{i}(v):=v-\alpha_{i}(v)\alpha_{i}^{\vee}$ for any $v\in\mathbb{A}$. Note that the points fixed by $r_{i}$ are exactly the elements of $\ker\alpha_{i}$. We define the Weyl group of $\mathcal{S}$ as the subgroup $W^{v}$ of $\mathrm{GL}(\mathbb{A})$ generated by the finite set $\{r_{i},\ i\in I\}$. The pair $(W^{v},\{r_{i},\ i\in I\})$ is a Coxeter system, hence we can consider the length $\ell(w)$ with respect to $\{r_{i},\ i\in I\}$ of any element $w$ of $W^{v}$.

For any $x\in\mathbb{A}$, we set $\underline{\alpha}(x)=(\alpha_{i}(x))_{i\in I}\in\mathbb{R}^{I}$. Let $P^{\vee}:=\{v\in\mathbb{A}\mid\underline{\alpha}(v)\in\mathbb{Z}^{I}\}$ be the coweight-lattice, which is not a lattice when $\mathbb{A}_{\mathrm{in}}:=\bigcap_{i\in I}\ker\alpha_{i}$ is non-zero, $Q^{\vee}:=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}^{\vee}$ be the coroot-lattice and $Q^{\vee}_{\mathbb{R}}=\bigoplus_{i\in I}\mathbb{R}\alpha_{i}^{\vee}$. Furthermore setting $Q_{+}^{\vee}:=\bigoplus_{i\in I}\mathbb{N}\alpha_{i}^{\vee}$, we can define a pre-order $\leq_{Q^{\vee}}$ on $\mathbb{A}$ as follows: for any $x,y\in\mathbb{A}$, we say that $x\leq_{Q^{\vee}}y$ iff $y-x\in Q^{\vee}_{+}$. We also set $Q^{\vee}_{-}:=-Q^{\vee}_{+}$ for future reference.

There is an action of the Weyl group $W^{v}$ on $\mathbb{A}^{*}$ given by the following formula:

[TABLE]

Let $\Phi:=\{w\cdot\alpha_{i}\mid(w,i)\in W^{v}\times I\}$ be the set or real roots: then $\Phi$ is a subset of $Q:=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}$. We will also use the set $\Delta:=\Phi\cup\Delta_{\mathrm{im}}^{+}\cup\Delta_{\mathrm{im}}^{-}\subset Q$ of all roots, as defined in [Kac94]. Note that $\Delta$ is stable under the action of $W^{v}$. For any root $\alpha\in\Delta$, we set $\Lambda_{\alpha}:=\mathbb{Z}$ if $\alpha\in\Phi$, and $\Lambda_{\alpha}:=\mathbb{R}$ otherwise (i.e if $\alpha\in\Delta_{\mathrm{im}}:=\Delta_{\mathrm{im}}^{+}\cup\Delta_{\mathrm{im}}^{-}$). For any pair $(\alpha,k)\in\Delta\times\mathbb{Z}$, we set

[TABLE]

For any root $\alpha\in\Delta$, we also set $D(\alpha,+\infty):=\mathbb{A}$.

Finally, we let $W:=Q^{\vee}\rtimes W^{v}\subset\mathrm{GA}(\mathbb{A})$ be the affine Weyl group of $\mathcal{S}$, where $\mathrm{GA}(\mathbb{A})$ denotes the group of affine isomorphisms of $\mathbb{A}$. Note that $W\subset P^{\vee}\rtimes W^{v}$ and that $\alpha(P^{\vee})$ is contained in $\mathbb{Z}$ for any $\alpha\in Q$. Consequently, if $\tau$ is a translation of $\mathbb{A}$ of vector $p\in P^{\vee}$, then for any $\alpha\in Q$, $\tau$ acts by permutations on the set $\{D(\alpha,k),\ k\in\mathbb{Z}\}$. On the other hand, as $W^{v}$ stabilizes $\Delta$, any element of $W^{v}$ permutes the sets of the form $D(\alpha,k)$, where $\alpha$ runs over $\Delta$. Hence we have an action of $W$ on $\{D(\alpha,k),\ (\alpha,k)\in\Delta\times\mathbb{Z}\}$.

2.2 Vectorial faces and Tits cone

As in the reductive case, define the fundamental chamber as $C_{f}^{v}:=\{v\in\mathbb{A}\mid\forall i\in I,\,\alpha_{i}(v)>0\}$. For any subset $J$ in $I$, set

[TABLE]

then the closure $\overline{C_{f}^{v}}$ of $C_{f}^{v}$ is exactly the union of all $F^{v}(J)$’s for $J\subset I$. The positive vectorial faces (resp. negative vectorial faces) are defined as the sets $w\cdot F^{v}(J)$ (resp. $-w\cdot F^{v}(J)$) for $w\in W^{v}$ and $J\subset I$, and a vectorial face is either a positive vectorial face or a negative one. A positive chamber (resp. negative chamber) is a cone of the form $w\cdot C^{v}_{f}$ (resp. $-w\cdot C^{v}_{f}$) for some $w\in W^{v}$. Note that for any $x\in C^{v}_{f}$ and any $w\in W^{v}$, we have $(w\cdot x=1)\Rightarrow(w=1)$, which ensures that the action of $w\in W^{v}$ on the set of positive chambers is simply transitive.

Let $\mathcal{T}:=\bigcup_{w\in W^{v}}w\cdot\overline{C^{v}_{f}}$ be the Tits cone and $-\mathcal{T}$ be the negative cone. We can use it to define a $W^{v}$-invariant pre-order $\leq$ on $\mathbb{A}$ as follows:

[TABLE]

We also set $Y^{+}:=\mathcal{T}\cap Y$ and $Y^{++}:=Y\cap\overline{C^{v}_{f}}$. We can now recall the following simple but very useful result [GR14, Lem. 2.4 a)].

Lemma 2.1**.**

For any $\lambda\in Y^{++}$ and any $w\in W^{v}$, we have $w\cdot\lambda\leq_{Q^{\vee}}\lambda$.

2.3 Filters and masures

This section aims to recall what masures are. As stated in the introduction, the reader only interested in the completion of Iwahori-Hecke algebras can skip this section and go directly to Section 4.

Masures were first introduced for symmetrizable split Kac-Moody groups over a valued field whose residue field contains $\mathbb{C}$ by Gaussent and Rousseau [GR08]. Later, Rousseau axiomatized this construction in [Rou11], then generalized it with further developments to almost-split Kac-Moody groups over non-Archimedean local fields in [Rou16, Rou17]. For the reader familiar with this work, let us mention that we consider here semi-discrete masures which are thick of finite thickness.

2.3.1 Filters, sectors and rays

Definition 2.2**.**

A filter on a set $E$ is a non-empty set $F$ of non-empty subsets of $E$ that satisfies the following conditions:

• •

for any subsets $S_{1},S_{2}$ of $E$ that both belong to $F$, then $S_{1}\cap S_{2}$ belongs to $F$;

• •

for any subsets $S_{1},S_{2}$ of $E$ with $S_{1}$ in $F$ and $S_{1}\subset S_{2}$, then $S_{2}$ belongs to $F$.

Given a filter $F$ on a set $E$ and a subset $E^{\prime}$ of $E$, we say that $F$** contains $E^{\prime}$** if every element of $F$ contains $E^{\prime}$. If $E^{\prime}$ is non-empty, then the set $F_{E^{\prime}}$ of all subsets of $E$ containing $E^{\prime}$ is a filter on $E$ called the filter associated to $E^{\prime}$. By language abuse and to ease notations, we will sometimes write that $E^{\prime}$ is a filter, by identification of $E^{\prime}$ with $F_{E^{\prime}}$.

Now, let $F$ be a filter on a finite-dimensional real affine space $E$. We define its closure $\overline{F}$ as the filter of all subsets of $E$ that contain the closure of some arbitrary element of $F$, and its convex hull $\mathrm{conv}(F)$ as the filter of all subsets of $E$ that contain the convex hull of some arbitrary element of $F$. Said differently, we have:

[TABLE]

Given two filters $F_{1}$ and $F_{2}$ on the same set $E$, we say that $F_{1}$** is contained in $F_{2}$** iff any subset of $E$ contained $F_{2}$ is in $F_{1}$. Similarly, we say that the filter $F_{1}$ is contained in a subset $\Omega$ of $E$ iff any subset of $E$ contained in $\Omega$ is in $F_{1}$.

Let $\Omega$ be a subset of $\mathbb{A}$ containing an element $x$ in its closure $\overline{\Omega}$ of $\Omega$. The germ of $\Omega$ in $x$ is defined as the filter $\mathrm{germ}_{x}(\Omega)$ of all subsets of $\mathbb{A}$ containing some neighborhood of $x$ in $\Omega$. A sector in $\mathbb{A}$ is a translate $\mathfrak{s}:=x+C^{v}$ of a vectorial chamber $C^{v}=\pm w\cdot C_{f}^{v}$ (with $w\in W^{v}$) by an element $x\in\mathbb{A}$. The point $x$ is called the base point of the sector $\mathfrak{s}$ and the chamber $C^{v}$ is called its direction. One can easily check that the intersection of two sectors having the same direction is a sector with the same direction.

Given a sector $\mathfrak{s}:=x+C^{v}$ as above, the sector-germ of $\mathfrak{s}$ is the filter $\mathfrak{S}$ of all subsets of $\mathbb{A}$ containing an $\mathbb{A}$-translate of $\mathfrak{s}$. Note that it only depends on the direction $C^{v}$ of $\mathfrak{s}$. In particular, we denote by $\pm\infty$ the sector-germ of $\pm C^{v}_{f}$.

Finally, let $\delta$ be a ray with base point $x$ and let $y\not=x$ be another point on $\delta$ (which amounts to say that $\delta$ contains the interval $]x,y]=[x,y]\setminus\{x\}$, as well as the interval $[x,y]$). We say that $\delta$ is pre-ordered (resp. generic) if either $y\leq x$ or $x\leq y$ (resp. if $y-x\in\pm\mathring{\mathcal{T}}$, where $\pm\mathring{\mathcal{T}}$ denotes the interior of the cone $\pm\mathcal{T}$).

2.3.2 Enclosures and faces

We keep the notations introduced at the end of Section 2.1. Given a filter $F$ on $\mathbb{A}$, we define its enclosure $\mathrm{cl}_{\mathbb{A}}(F)$ as the filter of all subsets of $\mathbb{A}$ containing some element of $F$ of the form $\bigcap_{\alpha\in\Delta}D(\alpha,k_{\alpha})$, with $k_{\alpha}\in\mathbb{Z}\cup\{+\infty\}$ for any $\alpha\in\Delta$. Sets of the form $D(\alpha,k)$ with $\alpha\in\Phi$ and $k\in\mathbb{Z}$ are called half-apartments in $\mathbb{A}$. Sets of the form $M(\alpha,k):=\{t\in\mathbb{A}\mid\alpha(t)+k=0\}$ with $\alpha\in\Phi$ and $k\in\mathbb{Z}$ are called walls in $\mathbb{A}$.

A local face in $\mathbb{A}$ is a filter $F^{\ell}$ of the form $\mathrm{germ}_{x}(x+F^{v})$, where $x\in\mathbb{A}$ is called the vertex of $F^{\ell}$ and a vectorial face $F^{v}\subset\mathbb{A}$ is a vectorial face called the direction of $F^{\ell}$. To keep track of the elements $x$ and $F^{v}$, such a local face may be denoted $F^{\ell}(x,F^{v})$. A local face is said to be spherical when its direction is spherical: in this case, its pointwise stabilizer under the action of $W^{v}$ is a finite group.

A face in $\mathbb{A}$ is a filter $F=F(x,F^{v})$ associated with a point $x\in A$ and a vectorial face $F^{v}\subset A$ as follows: a subset $S$ of $\mathbb{A}$ belongs to $F$ iff it contains an intersection of half-spaces $D(\alpha,k_{\alpha})$ or open half-spaces $\mathring{D}(\alpha,k_{\alpha})$, with $\alpha\in\Delta$ and $k_{\alpha}\in\mathbb{Z}$, that also contains the local face $F^{\ell}(x,F^{v})$. Note that (local) faces can be ordered as follows: given two such faces $F,F^{\prime}$ in $\mathbb{A}$, we say that $F$** is a face of $F^{\prime}$** (or $F^{\prime}$** contains $F$, or $F^{\prime}$ dominates $F$**) when $F\subset\overline{F^{\prime}}$.

As explained at the end of Section 2.1, the action of $W$ on $\mathbb{A}$ permutes the sets of the form $D(\alpha,k)$, where $(\alpha,k)$ runs over $\Delta\times\mathbb{Z}$. In particular, this implies that $W$ permutes enclosures (resp. walls, faces) of $\mathbb{A}$.

The dimension of a face $F$ is the smallest dimension of an affine space generated by some element $S$ of $F$. Such an affine space is unique and is called the support of $F$. A local chamber (or local alcove) is a maximal local face, i.e a local face of the form $F^{\ell}(x,\pm w\cdot C^{v}_{f})$ for $x\in\mathbb{A}$ and $w\in W^{v}$. The fundamental local chamber is $C_{0}^{+}:=F^{\ell}(0,C^{v}_{f})$. A local panel is a spherical local face which is maximal among faces that are not chambers. Equivalently, a local panel is a spherical local face of dimension $\dim\mathbb{A}-1$. Analogue definitions of chambers and panels exist, see for instance [GR08, §1.4]. Finally, a local face is of type [math] (or: is a type [math] local face) if its vertex lies in $Y$. We denote by $F_{0}$ the local face $F^{\ell}(0,\mathbb{A}_{\mathrm{in}})$, where $\mathbb{A}_{\mathrm{in}}=F^{v}(I)=\bigcap_{i\in I}\ker(\alpha_{i})$. From now on, we will write type [math] face instead of type [math] local face to make it shorter.

Remark 2.3**.**

In [Rou11], Rousseau defines a notion of chimney that he uses in his axiomatization of masures. We do not define here what chimneys are: we only recall that each sector-germ is a splayed, solid chimney-germ, that each spherical face is contained in a solid chimney and that the action of $W$ on $\mathbb{A}$ permutes the chimneys and preserve their properties (being splayed or solid for instance). For more details about this, see [Rou11, §1.10].

2.3.3 Apartments and masure of type $\mathbb{A}$

An apartment of type $\mathbb{A}$ is a set $A$ together with a non-empty set $\mathrm{Isom}(\mathbb{A},A)$ of bijections (called Weyl-isomorphisms) such that, given $f_{0}\in\mathrm{Isom}(\mathbb{A},A)$, the elements of $\mathrm{Isom}(\mathbb{A},A)$ are exactly the bijections of the form $f_{0}\circ w$ with $w\in W$. All the isomorphisms considered in the sequel will be Weyl-isomorphisms, hence we will only write isomorphism instead of Weyl-isomorphism.

An isomorphism $\phi:A\to A^{\prime}$ between two apartments of type $\mathbb{A}$ is a bijection such that: $f\in\mathrm{Isom}(\mathbb{A},A)\iff\phi\circ f\in\mathrm{Isom}(\mathbb{A},A^{\prime})$. By construction, all the notions that are preserved under the action of $W$ can be extended to any apartment of type $A$. For instance, we can define sectors, enclosures, faces or chimneys in any apartment $A$ of type $\mathbb{A}$, as well as a pre-order $\leq_{A}$ on $A$.

We can now define the most important object of this section: the masures of type $\mathbb{A}$.

Definition 2.4**.**

A masure of type $\mathbb{A}$ is a set $\mathcal{I}$ endowed with a covering $\mathcal{A}$ of subsets (called apartments) such that the five following axioms hold.

(MA1) Any $A\in\mathcal{A}$ can actually be endowed with a structure of apartment of type $\mathbb{A}$.

(MA2) If $F$ is a point (resp. a germ of a preordered interval, a generic ray, a solid chimney) contained in an apartment $A$ and if $A^{\prime}$ is an other apartment containing $F$, then $A\cap A^{\prime}$ contains $\mathrm{cl}_{A}(F)$ and there exists an isomorphism from $A$ to $A^{\prime}$ that fixes $\mathrm{cl}_{A}(F)$.

(MA3) If $\mathfrak{R}$ is the 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 both $\mathfrak{R}$ and $F$.

(MA4) If two apartments $A$, $A^{\prime}$ contain both $\mathfrak{R}$ and $F$ as in (MA3), then there exists an isomorphism from $A$ to $A^{\prime}$ that fixes $\mathrm{cl}_{A}(\mathfrak{R}\cup F)$.

(MAO) If $x$ and $y$ are two points that are both contained in two apartments $A$ and $A^{\prime}$ and such that $x\leq_{A}y$, then the two segments $[x,y]_{A}$ and $[x,y]_{A^{\prime}}$ are equal.

Recall that saying that an apartment contains a germ of a filter means that it contains at least one element of this germ. Similarly, a map fixes a germ when it fixes at least one element of this germ.

From now on, $\mathcal{I}$ will denote a masure of type $\mathbb{A}$. We assume that $\mathcal{I}$ is thick of finite thickness, which means that the number of chambers ($={}$alcoves) containing a given panel is finite and greater or equal to three. We also assume that there exists a group $G$ of automorphisms of $\mathcal{I}$ that acts strongly transitively on $\mathcal{I}$, which implies that all the isomorphisms involved in the axioms above are induced by the action of elements of $G$. We fix an apartment in $\mathcal{I}$ that we identify with $\mathbb{A}$ and call the fundamental apartment of $\mathcal{I}$. As the action of $G$ on $\mathcal{I}$ is strongly transitive, the apartments of $\mathcal{I}$ are exactly the sets $g\cdot\mathbb{A}$ with $g\in G$. Let $N$ be the stabilizer of $\mathbb{A}$ in $G$: it defines a group of affine automorphisms of $\mathbb{A}$, and we assume that this group is $W^{v}\ltimes Y$. As we will see in Section 2.4, these assumptions are not very restrictive for our purpose as they are all satisfied by the masure attached to a split Kac-Moody group $G$ over a non-Archimedean local field (see also [GR08] and [Rou16]).

Remark 2.5**.**

In a recent work [Héb17a, §5], the second author gives a much simpler axiomatic for masures. To simplify the arguments, the reader can assume that $G$ is an affine Kac-Moody group, in which case the three axioms (MA2), (MA4) and (MAO) can be replaced by the following statement [Héb17a, Th. 5.38]: for any two apartments $A$ and $A^{\prime}$ in $\mathcal{I}$, we have $A\cap A^{\prime}=\mathrm{cl}(A\cap A^{\prime})$ and there exists an isomorphism from $A$ to $A^{\prime}$ that fixes $A\cap A^{\prime}$. This partially explains why the affine case is less technical, as it does not require to question the existence of isomorphisms that fix subsets of an intersection of apartments.

Remark 2.6**.**

Let $F$ be a local face of an apartment $A$ and $A^{\prime}$ be another apartment that contains $F$. Then $F$ is also a local face of $A^{\prime}$ and there exists an isomorphism from $A$ to $A^{\prime}$ that fixes $F$. Indeed, if $x$ is the vertex of $F$ and if $J$ is a germ of a pre-ordered segment based at $x$ and contained in $F$, then the enclosure of $J$ contains $F$ and the application of (MA2) to $J$ now proves the claim.

Remark 2.7**.**

Pick $w\in W$ and $\mathcal{F}$ a filter on $\mathbb{A}$ fixed by $w$: then $w$ fixes $\mathrm{cl}(\mathcal{F})$. Combined with the argument used in Remark 2.6, this proves here that for any vectorial face $F^{v}$ and any base point $x$, the fixer in $G$ of the face $F(x,F^{v})$ is exactly the fixer in $G$ of the corresponding local face $F^{\ell}(x,F^{v})$.

Remark 2.8**.**

As we noticed earlier, each apartment $A$ of $\mathcal{I}$ can be endowed with a pre-order $\leq_{A}$ induced by $\leq_{\mathbb{A}}$. Let $A$ be an apartment of $\mathcal{I}$ and $x,y$ be two points in $A$ such that $x\leq_{A}y$. By [Rou11, Prop. 5.4], we know that for any apartment $A^{\prime}$ of $\mathcal{I}$ that contains both $x$ and $y$, we also have $x\leq_{A^{\prime}}y$. We hence get a relation $\leq$ on $\mathcal{I}$ and [Rou11, Th. 5.9] ensures that this relation is a $G$-invariant pre-order on $\mathcal{I}$.

2.4 Masure attached to a split Kac-Moody group

As in [Tit87] or in [Rém02, Chap. 8], we consider the group functor $\mathbf{G}$ associated with the root generating system $\mathcal{S}$ fixed in Section 2.1. This functor goes from the category of rings to the category of groups and satisfies axioms (KMG1)–(KMG9) of [Tit87]. For any field $R$, the group $\mathbf{G}(R)$ is uniquely determined by these axioms [Tit87, Th. 1’]. Furthermore, this functor $\mathbf{G}$ contains a toric functor $\mathbf{T}$ (denoted by $\mathcal{T}$ in [Rém02]) that goes from the category of rings to the category of abelian groups, and two functors $\mathbf{U}^{\pm}$ going from the category of rings to the category of groups.

In particular, let $\mathcal{K}$ be a non-Archimedean local field. Denote by $\mathcal{O}$ its ring of integers, by $\varpi$ a fixed uniformizer of $\mathcal{O}$, by $q$ the cardinality of the residue class field $\mathcal{O}/\varpi\mathcal{O}$ and set $G:=\mathbf{G}(\mathcal{K})$ (as well as $U^{\pm}:=\mathbf{U}^{\pm}(\mathcal{K})$, $T:=\mathbf{T}(\mathcal{K})$, etc.). For any sign $\varepsilon\in\{+,-\}$ and any root $\alpha\in\Phi^{\varepsilon}$, there is an isomorphism $x_{\alpha}$ from $\mathcal{K}$ to a root group $U_{\alpha}$. For any integer $k\in\mathbb{Z}$, we get a subgroup $U_{\alpha,k}:=x_{\alpha}(\varpi^{k}\mathcal{O})$ of $U_{\alpha}$ (see [GR08, §3.1] for precise definitions). Let $\mathcal{I}$ denote the masure attached to $G$ in [Rou17]: then the following properties hold.

• •

The fixer of $\mathbb{A}$ in $G$ is $H:=\mathbf{T}(\mathcal{O})$ [GR08, Rem. 3.2].

• •

The fixer of $\{0\}$ in $G$ is $K_{s}:=\mathbf{G}(\mathcal{O})$ [GR08, Exam. 3.14]. Applying (MA2) to $\{0\}$ and using Remark 2.7, we get that $K_{s}$ is also the fixer in $G$ of the face $F_{0}$.

• •

For any pair $(\alpha,k)\in\Phi\times\mathbb{Z}$, the fixer of $D(\alpha,k)$ in $G$ is $H.U_{\alpha,k}$ [GR08, Exer. 4.2.7)].

• •

For any sign $\varepsilon\in\{+,-\}$, $U^{\varepsilon}$ is the fixer in $G$ of $\varepsilon\infty$ [GR08, Exam. 4.2.4)].

Moreover, each panel is contained in $q+1$ chambers, hence $\mathcal{I}$ is thick of finite thickness.

Remark 2.9**.**

The group $G$ is reductive iff $W^{v}$ is finite. In this case, $\mathcal{I}$ is the usual Bruhat-Tits building of $G$ and we have $\mathcal{T}=\mathbb{A}$ and $Y^{+}=Y$.

Acknowledgements

We warmly thank Stéphane Gaussent for suggesting our col- laboration, for multiple discussions and for his useful comments on previous versions of this manuscript. We also thank Nicole Bardy-Panse and Guy Rousseau for dis- cussions on this topic and for their comments on a previous version of this paper. Finally, we thank the referee for their valuable comments and suggestions, and for his/her interesting questions.

Funding

The first author was supported by the ANR grants ANR-14-CE25-0002-01 and ANR-16-CE40-0010- 01, by the GDR TLAG and by a CNRS grant PEPS-JCJC. The second author was supported by the ANR grant ANR-15-CE40-0012.

Contents

1. 1 Introduction

2. 2 Masures: general framework

3. 2.1 Root generating system and Weyl groups

4. 2.2 Vectorial faces and Tits cone

5. 2.3 Filters and masures

6. 2.4 Masure attached to a split Kac-Moody group

7. 3 A topological restriction on parahoric subgroups

8. 3.1 Statement of the result and idea of the proof

9. 3.2 Detailed proof of Theorem 3.1

10. 4 The completed Iwahori-Hecke algebra

11. 4.1 Definition of the usual Iwahori-Hecke algebra

12. 4.2 Almost finite sets in $Y$ and $Y^{+}$

13. 4.3 The Looijenga algebra $\mathcal{R}\llbracket Y\rrbracket$

14. 4.4 The completed Iwahori-Hecke algebra $\hat{\mathcal{H}}$

15. 4.5 Center of the Iwahori-Hecke algebras

16. 4.6 Some further remarks

17. 5 Hecke algebra associated with a parahoric subgroup

18. 5.1 Motivation from the reductive case

19. 5.2 Distance and spheres associated with a type [math] face

20. 5.3 Hecke algebra associated with a spherical type [math] face

21. 5.4 What about non-spherical type [math] faces?

3 A topological restriction on parahoric subgroups

3.1 Statement of the result and idea of the proof

In this section, we will prove that beside the reductive case, it is impossible to endow $G$ with a structure of topological group for which $K_{s}$ or $K_{I}$ are open compact subgroups, where $K_{I}$ denotes the (standard) Iwahori subgroup.111We recall that $K_{I}$ is the fixer in $G$ of the fundamental local chamber $C_{0}^{+}$. In fact, we will prove the following result, which is slightly more general.

Theorem 3.1**.**

Let $F$ be a type [math] face of $\mathbb{A}$ and let $K_{F}$ be its fixer in $G$. If $W^{v}$ is infinite, then there is no topology of topological group on $G$ for which $K_{F}$ is open and compact.

Let $F$ be a type [math] face of $\mathbb{A}$, i.e a local face whose vertex lies in $Y$. First note that, up to replacing $F$ by $h\cdot F$ for some well-chosen $h\in G$, which leads to consider the conjugate of $K_{F}$ under $h$ instead of $K_{F}$, we can assume that $F$ is contained in $C_{0}^{\pm}$. As the treatment of both cases is similar, we assume that $F$ is contained in $C_{0}^{+}$. To prove Theorem 3.1, it is enough to prove the existence of $g\in G$ such that $K_{F}/(K_{F}\cap g\cdot K_{F}.g^{-1})$ is infinite.

To explain the strategy of proof, we need to introduce some more notations. Let $\alpha\in\Phi^{+}$ and $(w,i)\in W^{v}\times I$ be such that $\alpha=w\cdot\alpha_{i}$. For any $k\in\mathbb{Z}$, set

[TABLE]

and let $K_{\alpha,k}$ be the fixer of $D^{\alpha}_{k}$ in $G$. Furthermore, pick a panel $P^{\alpha}_{k}$ in $M^{\alpha}_{k}$ and a chamber $C^{\alpha}_{k}$ contained in $\mathrm{conv}(M^{\alpha}_{k},M^{\alpha}_{k+1})$ that dominates $P^{\alpha}_{k}$.

For any $i\in I$, we let $q_{i}+1$ (resp. $q^{\prime}_{i}+1$) be the number of chambers containing $P^{\alpha_{i}}_{0}$ (resp. $P^{\alpha_{i}}_{1}$). By [Rou11, Prop. 2.9] and [Héb16, Lem. 3.2], $q_{i}$ and $q^{\prime}_{i}$ do not depend on the choices of the panels $P^{\alpha_{i}}_{0}$ and $P^{\alpha_{i}}_{1}$. (This fact will be explained in the proof of Lemma 3.4.) As $\alpha_{i}(\alpha_{i}^{\vee})=2$, and as there exists an element of $G$ that induces on $\mathbb{A}$ a translation of vector $\alpha^{\vee}_{i}$ (because we assumed that the stabilizer $N$ of $\mathbb{A}$ in $G$ induces $W^{v}\ltimes Y$ for group of affine automorphisms), the value of $1+q_{i}$ (resp. $1+q^{\prime}_{i}$) is also the number of chambers that contain $P_{2k}^{\alpha_{i}}$ (resp. $P_{2k+1}^{\alpha_{i}}$) for any integer $k$.

Let us now explain the basic idea of the proof. Pick $g\in G$ such that $g\cdot 0\in C^{v}_{f}$ and set $F^{\prime}:=g\cdot F$: then $K_{F}/(K_{F}\cap K_{F^{\prime}})$ is in bijection with $K_{F}\cdot F^{\prime}$. For $\alpha=w\cdot\alpha_{i}$, set $\tilde{K_{\alpha}}:=\bigcup_{k\in\mathbb{Z}}K_{\alpha,k}$: then $\mathbb{T}_{\alpha}:=\tilde{K_{\alpha}}\cdot\mathbb{A}$ is a semi-homogeneous extended tree with parameters $q_{i}$ and $q^{\prime}_{i}$. Using the thickness of $\mathcal{I}$, we can prove that the number $n_{\alpha}$ of walls between [math] and $g\cdot 0$ that are parallel to $\alpha^{-1}(\{0\})$ satisfies $|K_{\alpha,1}g\cdot 0|\geq 2^{n_{\alpha}}$, which implies that $|K_{F}g\cdot 0|\geq 2^{n_{\alpha}}$. As $n_{\alpha}$ can be made arbitrarily large (for a suitable choice of $\alpha$) when $W^{v}$ is infinite, this will end the proof.

3.2 Detailed proof of Theorem 3.1

Fix for now $\alpha=w\cdot\alpha_{i}$. Set $K_{\alpha}:=K_{\alpha,1}$ and pick a sector-germ $\mathfrak{q}$ contained in $D^{\alpha}_{0}$. By (MA3), we know that for any $x\in\mathcal{I}$, there exists an apartment $A_{x}$ that contains both $x$ and $\mathfrak{q}$. Axiom (MA4) implies the existence of an isomorphism $\phi_{x}:A_{x}\to\mathbb{A}$ that fixes $\mathfrak{q}$, and [Rou11, §2.6] ensures that $\phi_{x}(x)$ does not depend on the choice of the apartment $A_{x}$ nor on the isomorphism $\phi_{x}$, hence we can denote this element by $\rho_{\mathfrak{q}}(x)$. The map $\rho_{q}:\mathcal{I}\to\mathbb{A}$ is the retraction of $\mathcal{I}$ onto $\mathbb{A}$ centered at $\mathfrak{q}$, and its restriction to $\mathbb{T}_{\alpha}$ does not depend on the choice of $\mathfrak{q}\in D^{\alpha}_{0}$.

Remark 3.2**.**

Let $A$ be an apartment containing $\mathfrak{q}$ and $\phi=\rho_{q}|_{A}$ be the restriction to $A$ of the retraction map $\rho_{q}$. Then $\phi$ is the unique isomorphism of apartments that fixes $A\cap\mathbb{A}$. Indeed, (MA4) implies the existence of an isomorphism of apartments $\psi:A\to\mathbb{A}$ that fixes $\mathfrak{q}$. By definition, $\rho_{\mathfrak{q}}$ coincides with $\psi$ on $A$, hence $\phi=\psi$ is an isomorphism of apartments, and fixes $\mathbb{A}$, hence $\phi$ fixes $\mathbb{A}\cap A$. If $f:A\to\mathbb{A}$ is an isomorphism of apartments that fixes $A\cap\mathbb{A}$, then $f\circ\phi^{-1}:\mathbb{A}\to\mathbb{A}$ is an isomorphism of affine spaces that fixes $\mathfrak{q}$, hence it must be trivial, which proves that $f=\phi$ is unique.

Lemma 3.3**.**

Let $v\in\tilde{K}_{\alpha}$ and $x\in\mathbb{A}$ be such that $v\cdot x\in\mathbb{A}$. Then $v$ fixes $D^{\alpha}_{\lceil\alpha(x)\rceil}$.

Proof.

Set $A:=v\cdot\mathbb{A}$ and let $k\in\mathbb{Z}$ be such that $A\cap\mathbb{A}$ contains $D^{\alpha}_{k}$. By [Héb16, Lem. 3.2], $A\cap\mathbb{A}$ is a half-apartment,222Our definition of half-apartments is a bit different from the definition of [Héb16]: what we call half-apartments correspond to the true half-apartments of [Héb16]. hence there exists an integer $k_{1}\in\mathbb{Z}$ such that $A\cap A=D^{\alpha}_{k_{1}}$. Let $\phi:\mathbb{A}\to A$ be the isomorphism of apartments induced by $v$: Remark 3.2 ensures that $\phi$ fixes $A\cap\mathbb{A}$, which means that $v$ fixes $A\cap\mathbb{A}$. As $v\cdot x$ belongs to $A\cap\mathbb{A}$, we obtain that $v\cdot(v\cdot x)=v\cdot x$, hence $v\cdot x=x$ belongs to $D^{\alpha}_{k_{1}}$. This implies that $D^{\alpha}_{\lceil\alpha(x)\rceil}$ is contained in $D^{\alpha}_{k_{1}}=A\cap\mathbb{A}$, hence is fixed by $v$. ∎

Lemma 3.4**.**

Let $x\in\mathbb{A}$ and $M_{x}:=M^{\alpha}_{\lceil\alpha(x)\rceil}$. Then the map $f:K_{\alpha}\cdot x\to K_{\alpha}.M_{x}$ defined by $f(v\cdot x):=v\cdot M_{x}$ is well-defined and bijective.

Proof.

Let $v,v^{\prime}\in K_{\alpha}$ be such that $v\cdot x=v^{\prime}\cdot x$: by Lemma 3.3, we get that $v^{-1}\cdot v^{\prime}$ fixes $M_{x}$, hence $f$ is well-defined. Now assume that $v,v^{\prime}\in K_{\alpha}$ satisfy $v\cdot M_{x}=v^{\prime}.M_{x}$. Set $u:=v^{-1}v^{\prime}$ and $A:=u\cdot\mathbb{A}$: then $u\cdot M_{x}=M_{x}$ is contained in $A\cap\mathbb{A}$ hence $u$ fixes $D^{\alpha}_{\lceil\alpha(x)\rceil}$ by Lemma 3.3. In particular, we have $u\cdot x=x$, i.e $v\cdot x=v^{\prime}\cdot x$ and $f$ is injective. As $f$ is surjective by definition, the lemma is proved. ∎

Set $\alpha_{\mathcal{I}}:=\alpha\circ\rho_{\mathfrak{q}}$ and, for any integer $k\geq 0$, let $\mathcal{C}^{\alpha}_{k}$ be the set of all chambers $C$ that dominate some element of $K_{\alpha}.P_{k}$ and satisfy $\alpha_{\mathcal{I}}(C)>k$ (which means that there exists $X\in C$ such that $\alpha_{\mathcal{I}}(x)>k$ for all $x\in X$). Assume also that the chamber $C^{\alpha}_{k}$ chosen in Section 3.1 is not contained in $D^{\alpha}_{k}$.

Lemma 3.5**.**

For any integer $k\geq 0$, the map $g_{k}:K_{\alpha}M^{\alpha}_{k+1}\to\mathcal{C}^{\alpha}_{k}$ sending $v\cdot M^{\alpha}_{k+1}$ onto $v\cdot C^{\alpha}_{k}$ is well-defined and bijective.

Proof.

The proof of the first part of the assertion is as in Lemma 3.4: if $v,v^{\prime}\in K_{\alpha}$ are such that $v\cdot M^{\alpha}_{k+1}=v^{\prime}.M^{\alpha}_{k+1}$, then $u:=v^{-1}\cdot v^{\prime}$ satisfies $u\cdot M^{\alpha}_{k+1}\subset\mathbb{A}$ and Lemma 3.3 implies that $u\cdot C^{\alpha}_{k}=C^{\alpha}_{k}$, i.e $v\cdot C^{\alpha}_{k}=v^{\prime}\cdot C^{\alpha}_{k}$. As we moreover have $\alpha_{\mathcal{I}}(v\cdot C^{\alpha}_{k})=\alpha(C^{\alpha}_{k})>k$, $v\cdot C^{\alpha}_{k}=v^{\prime}\cdot C^{\alpha}_{k}$ belongs to $\mathcal{C}^{\alpha}_{k}$ and the map $g_{k}$ is well-defined.

Assume now that $v,v^{\prime}\in K_{\alpha}$ are such that $v\cdot C^{\alpha}_{k}=v^{\prime}\cdot C^{\alpha}_{k}$. Set $u:=v^{-1}v^{\prime}$ and let $X\in C^{\alpha}_{k}$ be an element fixed by $u$. Let $x\in X$ be such that $\alpha(x)>k$: by Lemma 3.3, $u$ fixes $M^{\alpha}_{k+1}\subset D^{\alpha}_{\lceil\alpha(x)\rceil}$, hence $g_{k}$ is injective.

It remains to check that $g_{k}$ is surjective, i.e that $\mathcal{C}^{\alpha}_{k}=K_{\alpha}\cdot C^{\alpha}_{k}$. If $C\in\mathcal{C}^{\alpha}_{k}$, then there exists $u\in K_{\alpha}$ such that $C$ dominates $u\cdot P^{\alpha}_{k}$. By [Rou11, Prop. 2.9.1)], there is an apartment $A$ that contains both $u\cdot D^{\alpha}_{k}$ and $C$, and Remark 3.2 now gives an explicit isomorphism $\phi:A\to\mathbb{A}$ that fixes $A\cap\mathbb{A}$. If $v\in K_{\alpha}$ induces $\phi$, then $\alpha_{\mathcal{I}}(C)=\alpha(v\cdot C)$, hence $\alpha(v\cdot C)>k$, and $v\cdot C\subset\mathbb{A}$ dominates $P^{\alpha}_{k}$, hence $v\cdot C=C^{\alpha}_{k}$, i.e $C=v^{-1}\cdot C^{\alpha}_{k}$, which ends the proof. ∎

Combining Lemmas 3.4 and 3.5, we get the following corollary.

Corollary 3.6**.**

For any $x\in\mathbb{A}$, if $k:=\max(1,\lceil\alpha(x)\rceil)$, then $|K_{\alpha}\cdot x|=q_{i}^{\prime}q_{i}q_{i}^{\prime}\cdots$ ($k-1$ factors).

Until the end of this section, we assume that $W^{v}$ is infinite.

Lemma 3.7**.**

Let $F$ be a type [math] face of $\mathbb{A}$. If $W^{v}$ is infinite, then there exists $g\in G$ such that $K_{F}/K_{F}\cap K_{g\cdot F}$ is infinite.

Proof.

Let $g\in G$ be such that $a:=g\cdot 0$ belongs to $C^{v}_{f}$ and set $F^{\prime}:=g\cdot F$. Let $(\alpha_{n})_{n\geq 0}$ be an injective sequence of positive real roots (i.e $\alpha_{n}\in\Phi^{+}$ for any non-negative integer $n$). As we have $K_{\alpha_{n}}\subset K_{F}$, hence $|K_{F}\cdot F^{\prime}|\geq|K_{\alpha_{n}}.a|$, for all $n\geq 0$, it is enough to check (by Corollary 3.6 and thickness of $\mathcal{I}$) that $\alpha_{n}(a)\to+\infty$ as $n\to+\infty$.

By definition, any $\alpha_{n}$ can be written as $\sum_{i\in I}\lambda_{n,i}\alpha_{i}$ with $\lambda_{n,i}\in\mathbb{Z}_{+}$ for all $(n,i)\in\mathbb{Z}_{+}\times I$. The injectivity of the sequence $(\alpha_{n})_{n\geq 0}$ implies that $\sum_{i\in I}\lambda_{n,i}\to+\infty$ as $n$ goes to $+\infty$, hence $\lim_{n\to+\infty}\alpha_{n}(a)=+\infty$ as required. ∎

Corollary 3.8**.**

Let $F$ be a type [math] face of $\mathcal{I}$. If $W^{v}$ is infinite, then there is no topology of topological group on $G$ for which $K_{F}$ is open and compact.

Proof.

If there was such a topology, then for any $g\in G$, $K_{F}$ and $K_{g\cdot F}=gK_{F}g^{-1}$ would be open and compact in $G$, hence $K_{F}\cap K_{g\cdot F}$ would have the same properties. This would imply the finiteness of the quotient $K_{F}/K_{F}\cap K_{g\cdot F}$ for any $g\in G$, which contradicts Lemma 3.7. ∎

Considering $F=F_{0}$ (resp. $F=C^{+}_{0}$), we obtain that $K_{s}$ (resp. $K_{I}$) cannot be open and compact in $G$ when $W^{v}$ is infinite, i.e when $G$ is not reductive. This shows how different reductive groups and (non-reductive) Kac-Moody groups are from this point of view.

4 The completed Iwahori-Hecke algebra

4.1 Definition of the usual Iwahori-Hecke algebra

Let us first recall briefly the construction of the Iwahori-Hecke algebra via its Bernstein-Lusztig presentation, as done in [BPGR16, §6.6]. Note that this definition requires some restrictions on the possible choices for the ring of scalars; nevertheless, choosing $\mathbb{C}$ or $\mathbb{Z}[\sqrt{q},\sqrt{q}^{-1}]$ is allowed when $G$ is a split Kac-Moody group over $\mathcal{K}$. Another definition of the Iwahori-Hecke algebra (as an algebra of functions on pairs of type [math] chambers in a masure) is given in [BPGR16, Def. 2.5] and allows more flexibility in the choice of scalars. This will be recalled in Section 5.

Let $\mathcal{R}_{1}:=\mathbb{Z}[(\sigma_{i},\sigma^{\prime}_{i})_{i\in I}]$, where $(\sigma_{i})_{i\in I}$ and $(\sigma^{\prime}_{i})_{i\in I}$ denote indeterminates that satisfy the following relations:

• •

if $\alpha_{i}(Y)=\mathbb{Z}$, then $\sigma_{i}=\sigma^{\prime}_{i}$;

• •

if $(i,j)\in I^{2}$ are such that $r_{i}$ and $r_{j}$ are conjugate (i.e such that $\alpha_{i}(\alpha^{\vee}_{j})=\alpha_{j}(\alpha^{\vee}_{i})=-1$), then $\sigma_{i}=\sigma_{j}=\sigma^{\prime}_{i}=\sigma^{\prime}_{j}$.

To define the Iwahori-Hecke algebra $\mathcal{H}$ associated with $\mathbb{A}$ and $(\sigma_{i},\sigma^{\prime}_{i})_{i\in I}$, we first introduce the Bernstein-Lusztig-Hecke algebra. Let ${{}^{\mathrm{BL}}\mathcal{H}}$ be the free $\mathcal{R}_{1}$-module with basis $(Z^{\lambda}H_{w})_{\lambda\in Y,w\in W^{v}}$. For short, set $H_{i}:=H_{r_{i}}$ for $i\in I$, as well as $H_{w}=Z^{0}H_{w}$ for $w\in W^{v}$ and $Z^{\lambda}=Z^{\lambda}H_{1}$ for $\lambda\in Y$. The Bernstein-Lusztig-Hecke algebra ${{}^{\mathrm{BL}}\mathcal{H}}$ is the module ${{}^{\mathrm{BL}}\mathcal{H}}$ equipped with the unique product $*$ that turns it into an associative algebra and satisfies the following relations (known as Bernstein-Lusztig relations):

[TABLE]

The existence and unicity of such a product $*$ comes from [BPGR16, Th. 6.2]. Following [BPGR16, §6.6], the Iwahori-Hecke algebra $\mathcal{H}_{\mathcal{R}_{1}}$ associated with $\mathbb{A}$ and $(\sigma_{i},\sigma^{\prime}_{i})_{i\in I}$ is now defined as the $\mathcal{R}_{1}$-submodule of ${{}^{\mathrm{BL}}\mathcal{H}}$ spanned by $(Z^{\lambda}H_{w})_{\lambda\in Y^{+},w\in W^{v}}$ (recall that $Y^{+}=Y\cap\mathcal{T}$ with $\mathcal{T}$ being the Tits cone). Note that for $G$ reductive, we recover the usual Iwahori-Hecke algebra of $G$.

Remark 4.1**.**

This construction is compatible with extension of scalars. Let indeed $\mathcal{R}$ be a ring that contains $\mathbb{Z}$ and $\phi:\mathcal{R}_{1}\to\mathcal{R}$ be a ring homomorphism such that $\phi(\sigma_{i})$ and $\phi(\sigma^{\prime}_{i})$ are invertible in $\mathcal{R}$ for all $i\in I$: then the Iwahori-Hecke algebra associated with $\mathbb{A}$ and $(\phi(\sigma_{i}),\phi(\sigma^{\prime}_{i}))_{i\in I}$ over $\mathcal{R}$ is $\mathcal{H}_{\mathcal{R}}=\mathcal{R}\otimes_{\mathcal{R}_{1}}\mathcal{H}_{\mathcal{R}_{1}}$.

Remark 4.2**.**

When $G$ is a split Kac-Moody group over $\mathcal{K}$, we can (and will) set $\sigma_{i}=\sigma^{\prime}_{i}=\sqrt{q}$ for all $i\in I$ and $\mathcal{R}=\mathbb{Z}[\sqrt{q}{}^{\pm 1}]$. The corresponding Iwahori-Hecke algebra $\mathcal{H}_{\mathcal{R}}$ will simply be denoted by $\mathcal{H}$.

4.2 Almost finite sets in $Y$ and $Y^{+}$

We fix a pair $(\mathcal{R},\phi)$ as in Remarks 4.1 and 4.2. In this section, we introduce a notion of almost finite sets in $Y$ and $Y^{+}$, that will be used to define the Looijenga algebra $\mathcal{R}\llbracket Y\rrbracket$ in the next section and the completed Iwahori-Hecke algebra $\hat{\mathcal{H}}=\hat{\mathcal{H}}_{\mathcal{R}}$ in Section 4.4.

4.2.1 Definition of almost finite sets

Definition 4.3**.**

A subset $E$ of $Y$ is almost finite (in $Y$) if there is a finite set $J\subset Y$ such that: $\forall\lambda\in E,\ \exists\nu\in j\mid\lambda\leq_{Q^{\vee}}\nu$.

Replacing $Y$ by $Y^{+}$ in the previous definition, we have the definition of almost finite sets in $Y^{+}$. Nevertheless, the following lemma (applied to $F=Y^{+}$) justifies why we do not set this other definition apart: it shows indeed that almost finiteness for $Y^{+}$ can already be seen in $Y$, which explains why we will just write almost finite sets with no more specification.

Lemma 4.4**.**

Let $E\subset Y$ be an almost finite set. For any subset $F$ of $Y$, there exists a finite set $J\subset F$ such that $F\cap E\subset\bigcup_{j\in J}(j-Q^{\vee}_{+})$.

Proof.

As $E$ is almost finite, we can assume that $E$ is contained in $y-Q^{\vee}_{+}$ for some well-chosen $y\in Y$. Let $J$ be the set of all elements in $F\cap E$ that are maximal in $F\cap E$ for the pre-order $\leq_{Q^{\vee}}$. As $E$ is almost finite, we already have: $\forall x\in E,\ \exists\nu\in\nobreak K\mid x\leq_{Q^{\vee}}\nu$. Let us prove that $J$ is finite, which will conclude the proof. To do this, we identify $Q^{\vee}$ with $\mathbb{Z}^{I}$ and set $J^{\prime}:=\{u\in Q^{\vee}\mid y-u\in J\}$. We define a comparison relation $\prec$ on $Q^{\vee}$ as follows: for all $x=(x_{i})_{i\in I}$ and $x^{\prime}=(x^{\prime}_{i})_{i\in I}$, we write $x\prec x^{\prime}$ when $x_{i}\leq x^{\prime}_{i}$ (in $\mathbb{Z}$) for all $i\in I$ and $x\not=x^{\prime}$. By definition of $J$, elements of $J^{\prime}$ are pairwise non comparable, hence [Héb17b, Lem. 2.2] implies that $J^{\prime}$ is finite, which requires that $J$ itself is finite and completes the proof. ∎

4.2.2 Examples of almost finite sets in $Y^{+}$

In the affine case

Suppose that $\mathbb{A}$ is associated with an affine Kac-Moody matrix $A$. By [Rou11, Rem. 5.10], we know that $\mathcal{T}=\delta^{-1}(\mathbb{R}^{*}_{+})\sqcup\mathbb{A}_{\mathrm{in}}$, where $\delta$ denotes the smallest positive imaginary root of $A$, and that $\delta$ is $W^{v}$-invariant, thus $\delta(\alpha^{\vee}_{i})>0$ for all $i\in I$. Therefore, an almost finite set of $Y^{+}$ is a set $E$ contained in $\bigcup_{i=1}^{k}(y_{i}-Q^{\vee}_{+})$ for some integer $k\geq 1$ and some $y_{1},\ldots,y_{k}\in Y^{+}$.

In the indefinite case

Unlike the finite or the affine case, when $\mathbb{A}$ is associated with an indefinite Kac-Moody matrix $A$, we have: $\forall y\in Y,\ y-Q^{\vee}_{+}\not\subseteq Y^{+}$. Indeed, due to the proof and the statement of [GR14, Lem. 2.9], there exists a linear form $\delta:\mathbb{A}\to\mathbb{R}$ such that $\delta(\mathcal{T})\geq 0$ and $\delta(\alpha^{\vee}_{i})<0$ for all $i\in I$. Consequently, if $y\in Y$ and $i\in I$, then $\delta(y-n\alpha^{\vee}_{i})<0$ for $n$ large enough, hence $y-Q^{\vee}_{+}$ is not contained in $Y^{+}$. However, $Y^{+}$ may be contained in $Q^{\vee}_{-}$, as stated by the following lemma.

Lemma 4.5**.**

We have $Y^{+}\subset Q^{\vee}_{-}$ iff we have $Y^{++}\subset Q^{\vee}_{-}$.

Proof.

As $Y^{++}$ is contained in $Y^{+}$, the direct implication is obvious. Assume conversely that $Y^{++}\subset Q^{\vee}_{-}$. By Lemma 2.1, we know that $\lambda\leq_{Q^{\vee}}\lambda^{++}$ for any $\lambda\in Y^{+}$, with $\lambda^{++}$ in $Y^{++}$, hence we must have $Y^{+}\subset Q^{\vee}_{-}$ and the proof is complete. ∎

We say that $\mathbb{A}$ is the essential realization of the Kac-Moody matrix $A$ when $\mathbb{A}_{\mathrm{in}}=\nobreak\{0\}$, or equivalently when $\dim_{\mathbb{R}}\mathbb{A}$ equals the size of the matrix $A$. If $\mathbb{A}$ is the essential realization of an indefinite matrix $A=\begin{pmatrix}2&a_{1,2}\\ a_{2,1}&2\end{pmatrix}$ of size $2$, with $a_{1,2}$ and $a_{2,1}$ negative integers, then $\mathbb{A}=\mathbb{R}\alpha^{\vee}_{1}\oplus\mathbb{R}\alpha^{\vee}_{2}$. For any integers $\lambda$ and $\mu$ of opposite sign (i.e such that $\lambda\mu<0$), we have $(2\lambda+a_{1,2}\mu)(a_{2,1}\lambda+2\mu)<0$, hence $Y^{++}$ is contained in $Q^{\vee}_{+}\cup Q^{\vee}_{-}$. By [Kac94, Th. 4.3], we get that $Q^{\vee}_{+}\cap Y^{++}=\{0\}$, hence $Y^{++}$ is contained in $Q^{\vee}_{-}$, and Lemma 4.5 implies that $Y^{+}$ is also contained in $Q^{\vee}_{-}$. Consequently, every subset of $Y^{+}$ is almost finite.

Note that this conclusion does not always hold when $A$ is of size $n\geq 3$. Indeed, assume for instance that $\mathbb{A}$ is the essential realization of the matrix

[TABLE]

Then $-2\alpha_{1}^{\vee}+\alpha_{2}^{\vee}-\alpha_{3}^{\vee}$ is in $Y^{++}$ but not in $Q^{\vee}_{-}$.

4.3 The Looijenga algebra $\mathcal{R}\llbracket Y\rrbracket$

We keep the previous notations. The next definition follows the definition of the algebra $A$ given in [Loo80, §4].

Definition 4.6**.**

Let $(e_{\lambda})_{\lambda\in Y}$ be a family of symbols that satisfy $e^{\lambda}e^{\mu}=e^{\lambda+\mu}$ for all $\lambda,\mu\in Y$. The Looijenga algebra $\mathcal{R}\llbracket Y\rrbracket$ of $Y$ over $\mathcal{R}$ is defined as the set of formal series $\sum_{\lambda\in Y}a_{\lambda}e^{\lambda}$ with $(a_{\lambda})_{\lambda\in Y}\in\mathcal{R}^{Y}$ having almost finite support.

For any element $\lambda\in Y$, let $\pi_{\lambda}:\mathcal{R}\llbracket Y\rrbracket\to\mathcal{R}$ be the “$\lambda$-the coordinate map” defined by \pi_{\lambda}\bigl{(}\sum_{\mu\in Y}a_{\mu}e^{\mu}\bigr{)}:=a_{\lambda}. Define $\mathcal{R}\llbracket Y^{+}\rrbracket$ and $\mathcal{R}\llbracket Y\rrbracket^{W^{v}}$ as follows:

[TABLE]

One can check that $\mathcal{R}\llbracket Y^{+}\rrbracket$ and $\mathcal{R}\llbracket Y\rrbracket^{W^{v}}$ are $\mathcal{R}$-subalgebras of $\mathcal{R}\llbracket Y\rrbracket$.

Definition 4.7**.**

A family $(a_{j})_{j\in J}\in(\mathcal{R}\llbracket Y\rrbracket)^{J}$ is summable if:

• •

for all $\lambda\in Y$, $\{j\in J\mid\pi_{\lambda}(a_{j})\not=0\}$ is finite;

• •

the set $\{\lambda\in Y\mid\exists j\in J,\,\pi_{\lambda}(a_{j})\not=0\}$ is almost finite.

Given a summable family $(a_{j})_{j\in J}\in(\mathcal{R}\llbracket Y\rrbracket)^{J}$, we set $\sum_{j\in J}a_{j}:=\sum_{\lambda\in Y}b_{\lambda}e^{\lambda}$, with $b_{\lambda}:=\sum_{j\in J}\pi_{\lambda}(a_{j})$ for any $\lambda\in Y$. For $\lambda\in Y^{++}$, set $E(\lambda):=\sum_{\mu\in W^{v}\cdot\lambda}e^{\mu}\in\mathcal{R}\llbracket Y\rrbracket$. (Note that this is well-defined by Lemma 2.1.) Finally, for any $\lambda\in\mathcal{T}$, let $\lambda^{++}$ be the unique element in $\overline{C^{v}_{f}}$ that has the same $W^{v}$-orbit as $\lambda$ (i.e such that $W^{v}\cdot\lambda=W^{v}\cdot\lambda^{++}$).

Lemma 4.8**.**

Let $y\in Y$. Then $W^{v}\cdot y$ is upper-bounded (for $\leq_{Q^{\vee}}$) iff $y$ belongs to $Y^{+}$.

Proof.

If $y$ belongs to $Y^{+}$, then Lemma 2.1 implies that $W^{v}\cdot y$ is upper-bounded by $y^{++}$. Assume conversely that $y\in Y$ is such that $W^{v}\cdot y$ is upper-bounded for $\leq_{Q^{\vee}}$ and let $x\in W^{v}\cdot y$ be a maximal element. For any $i\in I$, we have $r_{i}(x)\leq_{Q^{\vee}}x$, hence $\alpha_{i}(x)\geq 0$, which proves that $x$ belongs to $\overline{C^{v}_{f}}$ and implies that $y$ is in $Y^{+}$. ∎

Denote by $\mathrm{AF}_{\mathcal{R}}(Y^{++})$ the set of elements of $\mathcal{R}^{Y^{++}}$ having almost finite support.

Proposition 4.9**.**

The map $E:\mathrm{AF}_{\mathcal{R}}(Y^{++})\to\mathcal{R}\llbracket Y\rrbracket^{W^{v}}$ that sends $a\in\mathrm{AF}_{\mathcal{R}}(Y^{++})$ to $\sum_{\lambda\in Y^{++}}\pi_{\lambda}(a)E(\lambda)$ is well-defined and bijective. In particular, we have $\mathcal{R}\llbracket Y\rrbracket^{W^{v}}\subset\mathcal{R}\llbracket Y^{+}\rrbracket$.

Proof.

Let $a=(a_{\lambda})_{\lambda\in Y^{++}}$ be an element of $\mathrm{AF}_{\mathcal{R}}(Y^{++})$. As $a$ has almost finite support, there exists a finite set $J\subset Y$ such that: $\forall\lambda\in\mathrm{supp}(a),\ \exists\nu\in J\mid\lambda\leq_{Q^{\vee}}\nu$. We start by proving that $(a_{\lambda}E(\lambda))_{\lambda\in Y^{++}}$ is summable. Let $\nu\in Y$ and set

[TABLE]

For any $\lambda\in F_{\nu}$, $\nu$ belongs to $W^{v}\cdot\lambda$, hence Lemma 2.1 implies that $\nu\leq_{Q^{\vee}}\lambda$. As there exists moreover some $j\in J$ such that $\lambda\leq_{Q^{\vee}}j$, we get the finiteness of $F_{\nu}$. Now let $F:=\bigcup_{\nu\in Y}F_{\nu}$. We just saw that any element of $F$ is dominated (for $\leq_{Q^{\vee}}$) by some element of $J$, hence $F$ is by definition almost finite, and $(a_{\lambda}E(\lambda))_{\lambda\in Y^{++}}$ is summable.

By construction, $E(\lambda)$ is in $\mathcal{R}\llbracket Y\rrbracket^{W^{v}}$ for any $\lambda\in Y^{++}$, hence $\sum_{\lambda\in Y^{++}}a_{\lambda}E(\lambda)$ is in $\mathcal{R}\llbracket Y\rrbracket^{W^{v}}$ too and $E$ is well-defined. Now assume that $a\in\mathrm{AF}_{\mathcal{R}}(Y^{++})$ is non-zero and let $\nu$ be maximal (for $\leq_{Q^{\vee}}$) among the elements $\lambda$ of $Y^{++}$ such that $\pi_{\lambda}(a)\not=0$. Then $\pi_{\nu}(E(a))=\pi_{\nu}(a)\not=0$, hence $E(a)$ is non-zero and $E$ is injective. To prove $E$ is surjective, let $u=\sum_{\lambda\in Y}u_{\lambda}e^{\lambda}$ be any element of $\mathcal{R}\llbracket Y\rrbracket^{W^{v}}$ and let $\lambda\in\mathrm{supp}(u)$. As $\mathrm{supp}(u)$ is almost finite and $W^{v}$-invariant, $W^{v}\cdot\lambda$ is upper-bounded, hence Lemma 4.8 implies that $\lambda$ belongs to $Y^{+}$. This proves that $\mathrm{supp}(u)$ is contained in $Y^{+}$, and that $u=E((\pi_{\lambda}(u))_{\lambda\in\mathrm{supp}(u)^{++}})$ is in the image of $E$, which completes the proof. ∎

Remark 4.10**.**

Lemma 4.8 and Proposition 4.9 are not explicitly stated in [GR14], but their proof is basically contained in the proof of [GR14, Th. 5.4].

4.4 The completed Iwahori-Hecke algebra $\hat{\mathcal{H}}$

In this subsection, we define an $\mathcal{R}$-algebra $\hat{\mathcal{H}}$ as a “completion” of the usual Iwahori-Hecke algebra, what justifies the name of completed Iwahori-Hecke algebra given to $\hat{\mathcal{H}}$. In the next section, we will compute the centers of $\mathcal{H}$ and of $\hat{\mathcal{H}}$, and recover the reasons that motivated the introduction of $\hat{\mathcal{H}}$ in this context.

Endow $W^{v}$ with its Bruhat order $\leq$ and, for any $w\in W^{v}$, set

[TABLE]

This notation makes sense as $1\leq w$ for all $w\in W^{v}$. Let $\mathcal{B}:=\prod_{\lambda\in Y^{+},\ w\in W^{v}}\mathcal{R}$.

Definition 4.11**.**

For any $a=(a_{\lambda,w})_{(\lambda,w)\in Y^{+}\times W^{v}}$ in $\mathcal{B}$, the support of $a$ is the set

[TABLE]

The support of $a$ along $Y$ is the set

[TABLE]

and the support of $a$ along $W^{v}$ is the set

[TABLE]

Definition 4.12**.**

A subset $Z$ of $Y^{+}\times W^{v}$ is almost finite if

[TABLE]

is finite and if, for all $w\in W^{v}$, the set $\{\lambda\in Y^{+}\ |(\lambda,w)\in Z\}$ is almost finite (in the sense of Definition 4.3).

Let $\hat{\mathcal{H}}$ be the set of all elements in $\mathcal{B}$ with almost finite support. An element $(a_{\lambda,w})_{(\lambda,w)\in Y^{+}\times W^{v}}$ of $\hat{\mathcal{H}}$ will also be written as $\sum_{(\lambda,w)\in Y^{+}\times W^{v}}a_{\lambda,w}Z^{\lambda}H_{w}$. Any pair $(\lambda,w)\in Y^{+}\times W^{v}$ defines a projection map $\pi_{\lambda,w}:\hat{\mathcal{H}}\to\mathcal{R}$ defined by \pi_{\lambda,w}\bigl{(}\sum_{(\nu,u)\in Y^{+}\times W^{v}}a_{\nu,u}Z^{\nu}H_{u}\bigr{)}:=a_{\lambda,w}.

To extend the product $*$ to $\hat{\mathcal{H}}$, we start by proving that for any elements

[TABLE]

of $\hat{\mathcal{H}}$, and any pair $(\mu,v)\in Y^{+}\times W^{v}$, the sum

[TABLE]

is a finite sum, i.e that only finitely many terms $\pi_{\mu,v}(a_{\lambda,w}b_{\lambda^{\prime},w^{\prime}}Z^{\lambda}H_{w}*Z^{\lambda^{\prime}}H_{w^{\prime}})$ are non-zero. The key fact to prove this is that for any pair $(\lambda,w)\in Y\times W^{v}$, the support of $H_{w}*Z_{\lambda}$ along $Y^{+}$ is in the convex hull of $\{u\cdot\lambda,u\in[1,w]\}$. This fact comes from Lemma 4.15 below.

For any subset $E$ of $Y$ and any $i\in I$, set $R_{i}(E):=\mathrm{conv}(\{E,r_{i}(E)\})\subset E+Q^{\vee}$. When $E=\{\lambda\}$ is reduced to a single element, we write $R_{i}(\lambda)$ instead of $R_{i}(\{\lambda\})$. For any $w\in W^{v}$ and any $\lambda\in Y$, set $R_{w}(\lambda):=\bigcup R_{i_{1}}(R_{i_{2}}(\ldots(R_{i_{k}}(\lambda))\ldots))$, where the union is taken over all the reduced writings $r_{i_{1}}.r_{i_{2}}\ldots r_{i_{k}}$ of $w$.

Remark 4.13**.**

For any pair $(\lambda,w)\in Y\times W^{v}$, the set $R_{w}(\lambda)$ is actually finite. Indeed, given any finite set $E$ and any $i\in I$, the set $R_{i}(E)$ is bounded and contained in $E+Q^{\vee}$, hence must be finite. By induction, we get that for any integer $k\geq 0$ and any list $(i_{1},\ldots,i_{k})$ of elements of $I$, the set $R_{i_{1}}(R_{i_{2}}(\ldots(R_{i_{k}}(E))\ldots))$ is also finite. As $w$ has only finitely many reduced writings,333The number of reduced writings of $w$ is upper-bounded by $|I|^{\ell(w)}$. we obtain that $R_{w}(\lambda)$ is finite.

Lemma 4.14**.**

For all $i\in I$ and all $\lambda\in Y$, the product $H_{i}*Z^{\lambda}$ is in

[TABLE]

Proof.

Let $i\in I$ and $\lambda\in Y$. If $\sigma_{i}=\sigma^{\prime}_{i}$, then (BL4) implies that

[TABLE]

hence we have the following alternative.

• •

Either $\alpha_{i}(\lambda)=0$, and $H_{i}*Z^{\lambda}=Z^{\lambda}*H_{i}$, i.e $H_{i}$ and $Z^{\lambda}$ commute to each other.

• •

Or $\alpha_{i}(\lambda)>0$, in which case we have

[TABLE]

with $r_{i}(\lambda)$ and $\lambda-k\alpha^{\vee}_{i}$ in $R_{i}(\lambda)$ for all $k\in\llbracket 0,\alpha_{i}(\lambda)-1\rrbracket$.

• •

Or $\alpha_{i}(\lambda)<0$, in which case we have

[TABLE]

with $r_{i}(\lambda)$ and $\lambda+k\alpha^{\vee}_{i}$ in $R_{i}(\lambda)$ for all $k\in\llbracket 1,-\alpha_{i}(\lambda)\rrbracket$.

In any case, we have proven that $H_{i}*Z^{\lambda}$ is in $\bigoplus_{(\nu,t)\in R_{i}(\lambda)\times\{1,r_{i}\}}\mathcal{R}Z^{\nu}H_{t}$ when $\sigma_{i}=\sigma^{\prime}_{i}$.

If $\sigma_{i}\not=\sigma^{\prime}_{i}$, then $\alpha_{i}(Y)=2\mathbb{Z}$, so we have now

[TABLE]

and similar computations as those done in the $\sigma_{i}=\sigma^{\prime}_{i}$ case complete the proof. ∎

Lemma 4.15**.**

For all $w,w^{\prime}\in W^{v}$ and all $\lambda\in Y$, $H_{w^{\prime}}*Z^{\lambda}H_{w}$ is in

[TABLE]

Proof.

The proof goes by induction on $\ell(w^{\prime})\geq 0$. There is nothing to prove if $\ell(w^{\prime})=0$, and if $\ell(w^{\prime})=1$, this is exactly Lemma 4.14. Now, fix an integer $k\geq 1$ and a pair $(w,\lambda)\in W^{v}\times Y^{+}$, and assume that for any $u\in W^{v}$ satisfying $\ell(u)\leq k-1$, the product $H_{u}*Z^{\lambda}H_{w}$ belongs to $\bigoplus_{(\nu,t)\in R_{w}(\lambda)\times[1,w]\cdot u}\mathcal{R}\cdot Z^{\nu}H_{t}$. Let $w^{\prime}\in W^{v}$ be an element of length $k$ and write $w^{\prime}=r_{i}u$ for $i\in I$ and $u\in W^{v}$ of length $k-1$. Then $H_{w^{\prime}}*Z^{\lambda}H_{w}=H_{i}*H_{u}*Z^{\lambda}H_{w}$ belongs to $\bigoplus_{(\nu,t)\in R_{u}(\lambda)\times[1,u]\cdot w}\mathcal{R}H_{i}*Z^{\lambda}H_{t}$, with

[TABLE]

by Lemma 4.14. Using (BL2) for $t^{\prime}\in\{1,r_{i}\}$ and $t\in[1,u]\cdot w$, we get that $H_{t^{\prime}}*H_{t}$ belongs to $\mathcal{R}H_{r_{i}t}\oplus\mathcal{R}H_{t}$, hence to $\bigoplus_{v\in[1,w^{\prime}]\cdot w}\mathcal{R}H_{v}$, and the lemma follows. ∎

Lemma 4.16**.**

Let $(\lambda,w)\in Y\times W^{v}$. For any $\nu\in R_{w}(\lambda)$, there is a family $(a_{u})_{u\leq w}\in[0,1]^{[1,w]}$ such that $\sum_{u\in[1,w]}a_{u}=1$ and $\nu=\sum_{u\in[1,w]}a_{u}\cdot u\cdot\lambda$.

Proof.

This proof goes again by induction on $\ell(w)\geq 0$. There is nothing to do when $\ell(w)=0$. Let $k\in\mathbb{Z}$ be non-negative and assume that the statement of the lemma is true whenever $w\in W^{v}$ is of length $k$. Let $w^{\prime}\in W^{v}$ be of length $k+1$, fix $\lambda\in Y$ and pick $\nu\in R_{w^{\prime}}(\lambda)$. Then there exists some triple $(i,w,\nu^{\prime})\in I\times W^{v}\times W^{v}$, with $w$ of length $k$ and $\nu^{\prime}\in R_{w}(\lambda)$, such that $\nu\in R_{i}(\nu^{\prime})$. As we can write $\nu^{\prime}=\sum_{u\in[1,w]}a_{u}u\cdot\lambda$ for some family $(a_{u})_{u\in[1,w]}\in[0,1]^{[1,w]}$ and $\nu=s\nu^{\prime}+(1-s)r_{i}\nu^{\prime}$ for some $s\in[0,1]$, we finally get that $\nu=\sum_{u\in[1,w]}\left(sa_{u}u\cdot\lambda+(1-s)a_{u}r_{i}u\cdot\lambda\right)$ with $r_{i}u\in[1,w^{\prime}]$ for all $u\in[1,w]$, hence the lemma follows. ∎

Lemma 4.17**.**

For all $\lambda,\mu\in Y^{+}$, we have $(\lambda+\mu)^{++}\leq_{Q^{\vee}}\lambda^{++}+\mu^{++}$. 2. 2.

Let $(\lambda,w)\in Y^{+}\times W^{v}$. For all $\nu\in R_{w}(\lambda)$, we have $\nu^{++}\leq_{Q^{\vee}}\lambda^{++}$.

Proof.

Let $\lambda,\mu\in Y^{+}$ and $w\in W^{v}$ be such that $(\lambda+\mu)^{++}=w\cdot(\lambda+\mu)$. By Lemma 2.1, we have $w\cdot\lambda\leq_{Q^{\vee}}\lambda^{++}$ and $w\cdot\mu\leq_{Q^{\vee}}\mu^{++}$, hence we get the first statement. Together with Lemma 2.1, Remark 4.13 and Lemma 4.16, the first statement implies the second one. ∎

Define the height $h(x)$ of any $x=\sum_{i\in I}x_{i}\alpha^{\vee}_{i}\in Q^{\vee}$ by $h(x):=\sum_{i\in I}x_{i}$. For any $\lambda\in Y^{+}$, let $w_{\lambda}\in W^{v}$ be the element of minimal length such that $w^{-1}\cdot\lambda\in\overline{C^{v}_{f}}$: then we have $\lambda=w_{\lambda}\cdot\lambda^{++}$.

Lemma 4.18**.**

Let $\lambda\in Y^{++}$ and $(\mu_{n})_{n\geq 0}\in(W^{v}\cdot\lambda)^{\mathbb{Z}_{+}}$ be such that

[TABLE]

Then $\lim_{n\to+\infty}h(\mu_{n}-\lambda)=-\infty$.

Proof.

By Lemma 2.1, we know that $h(\mu_{n}-\lambda)$ is well-defined. For all $\alpha\in\Phi_{+}$, we have $\alpha(\lambda)\geq 0$. Assume that $\alpha(\lambda)=0$: then $\alpha_{i}(\lambda)=0$ for all $i\in I$, hence $r_{i}\lambda=\lambda$ for all $i\in I$ and $W^{v}\cdot\lambda$ is then reduced to $\{\lambda\}$, which contradicts the fact that $\lim_{n\to+\infty}\ell(w_{\mu_{n}})=+\infty$. This proves that $\alpha(\lambda)>0$ for all $\alpha\in\Phi_{+}$.

Now let $\rho:\mathbb{A}\to\mathbb{R}$ be a linear form satisfying $\rho(\alpha^{\vee}_{i})=1$ for all $i\in I$. Pick $n\in\mathbb{Z}_{+}$ and set $w:=w_{\mu_{n}}$. Then we have

[TABLE]

Thanks to [Kum02, Cor. (3)], we know that $w^{-1}\cdot\rho-\rho=-\sum_{\alpha\in\Phi_{w^{-1}}}\alpha$ with $\Phi_{w^{-1}}:=\Delta^{+}\cap w^{-1}\cdot\Delta^{-}$. By [Kum02, Lem. 3.14], we also know that $|\Phi_{w^{-1}}|=\ell(w^{-1})=\ell(w)$. Letting $n$ go to $+\infty$, we obtain that

[TABLE]

goes to $-\infty$, as required. ∎

Lemma 4.19**.**

For all $(\lambda,\mu,w)\in Y^{+}\times Y^{+}\times W^{v}$, the set $\{\nu\in W^{v}\cdot\lambda\mid\mu\in R_{w}(\nu)\}$ is finite.

Proof.

Let $N\geq 0$ be an integer that satisfies the following property:

[TABLE]

(Such an integer exists by Lemma 4.18.) Let $\nu\in W^{v}\cdot\lambda$ be such that $\mu$ belongs to $R_{w}(\nu)$ and set $u:=w_{\nu}$. Following Lemma 4.16, write $\mu=\sum_{x\in[1,w]}a_{x}x\cdot\nu$ with $(a_{x})_{x\in[1,w]}\in[0,1]^{[1,w]}$ such that $\sum_{x\in[1,w]}a_{x}=1$. For all $x\in[1,w]$, set $v(x):=w_{x\cdot\nu}$. Then there exists $x\in[1,w]$ such that $\ell(v(x))<N$. Indeed, suppose by contradiction that $\ell(v(x))\geq N$ for all $x\in[1,w]$. As

[TABLE]

we obtain from (2) that

[TABLE]

which is absurd. We can hence pick $\overline{u}\in[1,w]$ such that $\ell(v(\overline{u}))<N$. As $\overline{u}\cdot\nu=v(\overline{u})\cdot\nu^{++}$, we have $\ell(\overline{u}^{-1}v(\overline{u}))\geq\ell(u)$ by definition of $u$. It implies that $\ell(v(\overline{u}))+\ell(u)\geq\ell(v(\overline{u}))+\ell(\overline{u})\geq\ell(u)$, hence we have $\ell(u)\leq N+\ell(\overline{u})$, so $\ell(u)$ is upper-bounded and the lemma follows. ∎

To allow infinite sums in $\hat{\mathcal{H}}$, we need a suitable notion of summable families, as we have by Definition 4.7 for the Looijenga algebra $\mathbb{R}\llbracket Y\rrbracket$. This is the purpose of the next definition.

Definition 4.20**.**

A family $(a_{j})_{j\in J}\in\hat{\mathcal{H}}^{J}$ is summable when the following properties hold.

• •

For all $\lambda\in Y^{+}$, the set $\{j\in J\mid\exists w\in W^{v},\,\pi_{\lambda,w}(a_{j})\not=0\}$ is finite.

• •

$\bigcup_{j\in J}\mathrm{supp}(a_{j})$* is almost finite.*

If $(a_{j})_{j\in J}\in\hat{\mathcal{H}}^{J}$ is a summable family, we define $\sum_{j\in J}a_{j}\in\hat{\mathcal{H}}$ by

[TABLE]

The next result claims that the product of two summable families is well-defined. This is the crucial step in the process that turns $\hat{\mathcal{H}}$ into a convolution algebra for $*$. Recall that elements of $\mathcal{H}$ corresponds to elements of $\hat{\mathcal{H}}$ with finite support.

Theorem 4.21**.**

Let $(a_{j})_{j\in J}\in\mathcal{H}^{J}$ and $(b_{k})_{k\in J}\in\mathcal{H}^{K}$ be two summable families. Then $(a_{j}*b_{k})_{(j,k)\in J\times K}$ is summable and $\sum_{(j,k)\in J\times K}a_{j}*b_{k}$ only depends on the two elements $\sum_{j\in J}a_{j}$ and $\sum_{k\in K}b_{k}$ of $\hat{\mathcal{H}}$.

Proof.

For any $j\in J$ and $k\in K$, we can decompose $a_{j}$ and $b_{k}$ as follows:

[TABLE]

For any $\lambda\in Y^{+}$, we set

[TABLE]

For any triple $(u,v,\mu)\in W^{v}\times W^{v}\times Y^{+}$, the application of Lemma 4.15 to $H_{u}*\nobreak Z^{\mu}H_{v}$ gives a family $(z^{u,v,\mu}_{\nu,t})_{(\nu,t)\in R_{u}(\mu)\times[1,u]\cdot v}$ of scalars that satisfy

[TABLE]

Given $j\in J$ and $k\in K$, we then have

[TABLE]

This equality implies that $\mathrm{supp}_{W^{v}}(a_{j}*b_{k})$ is contained in $S^{a}_{j}.S^{b}_{k}$, where $S^{c}_{n}:=\bigcup_{w\in\mathrm{supp}_{W^{v}}(c_{n})}[1,w]$ for $c\in\{a,b\}$ and $n\in\{j,k\}$. This already gives the finiteness of

[TABLE]

If we set $S:=\bigcup_{j\in J}\mathrm{supp}(a_{j})\cup\bigcup_{k\in K}\mathrm{supp}(b_{k})$ and $S_{Y}\!:=\!\pi_{Y}(S)$, where $\pi_{Y}{:}\,Y\!\times\!W^{v}\!\to\!Y$ is the projection on the first coordinate, then $S$ and $S_{Y}$ are by construction both almost finite. We can hence choose an integer $N\geq 0$ and elements $\kappa_{1},\ldots,\kappa_{N}\in Y^{++}$ such that: for all $x\in S_{Y}$, there exists $i\in\llbracket 1,N\rrbracket$ such that $x^{++}\leq_{Q^{\vee}}\kappa_{i}$.

Now pick a pair $(\rho,s)\in Y^{+}\times W^{v}$. The image of $a_{j}*b_{k}$ by the projection $\pi_{\rho,s}$ is given by

[TABLE]

Set

[TABLE]

If $(\lambda,\nu)$ is an element of $F(\rho)$, choose some $(\mu,u)\!\in\!S_{Y}\!\times\!S_{W^{v}}$ such that $\nu\!\in\!R_{u}(\mu)$ and $\lambda+\nu=\rho$. Then Lemma 4.17 implies the existence of $n,m\in\llbracket 1,N\rrbracket$ such that $\lambda\leq_{Q^{\vee}}\lambda^{++}\leq_{Q^{\vee}}\kappa_{n}$ and $\nu\leq_{Q^{\vee}}\mu^{++}\leq_{Q^{\vee}}\kappa_{m}$, which proves that $F(\rho)$ is finite.

Set

[TABLE]

If $\mu$ is an element of $F^{\prime}(\rho)$ and if $(u,(\lambda,\nu))\in S_{W^{v}}\times F(\rho)$ is such that $\nu\in R_{u}(\mu)$, applying again Lemma 4.17 gives an integer $i\in\llbracket 1,N\rrbracket$ such that $\nu^{++}\leq_{Q^{\vee}}\mu^{++}\leq_{Q^{\vee}}\kappa_{i}$. This implies the finiteness of $F^{\prime}(\rho)^{++}$, which implies itself the finiteness of $F^{\prime}(\rho)$ by Lemma 4.19.

Set

[TABLE]

By construction, $L(\rho)$ is finite and for all $(j,k)\!\in\!J\!\times\!K$, the non-vanishing of $\pi_{\rho,s}(a_{j}*b_{k})$ implies that $(j,k)$ belongs to $L(\rho)$. Also, if $(\rho,s)$ is in

[TABLE]

then there exists $(\lambda,\mu)\in S_{Y}\times S_{Y}$, $u\in S_{W^{v}}$ and $\nu\in R_{u}(\mu)$ such that $\lambda+\nu=\rho$. Applying once more Lemma 4.17, we get integers $n,m\in\llbracket 1,N\rrbracket$ such that

[TABLE]

Summed up, all this shows that $\bigcup_{(j,k)\in J\times K}\mathrm{supp}(a_{j}*b_{k})$ is almost finite and that $(a_{j}*b_{k})_{(j,k)\in J\times K}$ is a summable family.

Moreover, we have

[TABLE]

where we set

[TABLE]

hence the lemma is proved. ∎

Definition 4.22**.**

For any summable families $(a_{j})_{j\in J}\in\mathcal{H}^{J}$ and $(b_{k})_{k\in J}\in\mathcal{H}^{K}$, we set

[TABLE]

Corollary 4.23**.**

The convolution product $*$ provides $\hat{\mathcal{H}}$ with a structure of associative $\mathcal{R}$-algebra.

Proof.

Theorem 4.21 ensures that $(\hat{\mathcal{H}},*)$ is an $\mathcal{R}$-algebra. The associativity of $*$ in $\hat{\mathcal{H}}$ comes from Theorem 4.21 and from the associativity of $*$ in $\mathcal{H}$. ∎

The algebra $\hat{\mathcal{H}}$ is called the completed Iwahori-Hecke algebra of $(\mathbb{A},(\sigma_{i},\sigma^{\prime}_{i})_{i\in I})$ over $\mathcal{R}$.

Example 4.24**.**

Let $\mathcal{I}$ be a thick masure of finite thickness on which a group $G$ acts strongly transitively. For any $i\in I$, pick a panel $P_{i}$ of $\{x\in\mathbb{A}\mid\alpha_{i}(x)=0\}$ and a panel $P^{\prime}_{i}$ of $\{x\in\mathbb{A}\mid\alpha_{i}(x)=1\}$. Let $1+q_{i}$ (resp. $1+q^{\prime}_{i}$) be the number of chambers in $\mathcal{I}$ that contain $P_{i}$ (resp. $P^{\prime}_{i}$) and set $\sigma_{i}=\sqrt{q_{i}},\ \sigma^{\prime}_{i}=\sqrt{q^{\prime}_{i}}$. Then $(\sigma_{i},\sigma^{\prime}_{i})_{i\in I}$ satisfy the relations stated at the beginning of Section 4 and the completed Iwahori-Hecke algebra of $(\mathbb{A},(\sigma_{i},\sigma^{\prime}_{i})_{i\in I})$ over $\mathcal{R}$ is called the completed Iwahori-Hecke algebra of $\mathcal{I}$ over $\mathcal{R}$.

4.5 Center of the Iwahori-Hecke algebras

The goal of this section is to compute the center of the Iwahori-Hecke algebra $\mathcal{H}$ and of its completed version $\hat{\mathcal{H}}$. Our proof is basically an adaptation to this context of the proof of [NR03, Th. 1.4]. In the sequel, we denote by $\mathcal{Z}(A)$ the center of any $\mathcal{R}$-algebra $A$.

4.5.1 The completed Bernstein-Lusztig bimodule $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$

To determine $\mathcal{Z}(\hat{\mathcal{H}})$, we want to compute elements of the form $Z^{\mu}*z*Z^{-\mu}$ for $z\in\mathcal{Z}(\hat{\mathcal{H}})$ and $\mu\in Y^{+}$. However, left and right multiplication by $\mathbb{Z}^{\lambda}$ are only defined in $\hat{\mathcal{H}}$ for $\lambda\in Y^{+}$. To extend multiplication by $Z^{\lambda}$ for arbitrary $\lambda\in Y$, we need to pass to a bigger space: indeed, if $\lambda\in Y$ is not in $Y^{+}$, multiplication by $Z^{\lambda}$ obviously does not stabilize $\hat{\mathcal{H}}$, as $Z^{\lambda}*1=Z^{\lambda}$ is not in $\hat{\mathcal{H}}$ in this case. The bigger space aforementioned is a “completion” $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$ of $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$ that contains $\hat{\mathcal{H}}$. Note that $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$ will not be equipped with a structure of algebra, but with a structure of $\mathcal{R}[Y]$-bimodule compatible with the convolution product $*$ on $\hat{\mathcal{H}}$.

Any $a=(a_{\lambda,w})\in\mathcal{R}^{Y\times W^{v}}$ will also be written as $a=\sum_{(\lambda,w)\in Y\times W^{v}}a_{\lambda,w}Z^{\lambda}H_{w}$. For such an $a$, we define the support of $a$ along $W^{v}$ as

[TABLE]

We set $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}:=\{a\in\mathcal{R}^{Y\times W^{v}}\mid\mathrm{supp}_{W^{v}}(a)\text{ is finite}\}$; note that ${{}^{\mathrm{BL}}\mathcal{H}}$ and $\hat{\mathcal{H}}$ can be seen as subspaces of $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$. For any pair $(\rho,s)\in Y\times W^{v}$, we have again a projection map $\pi_{\rho,s}:\overline{{{}^{\mathrm{BL}}\mathcal{H}}}\to\mathcal{R}$ defined by:

[TABLE]

Definition 4.25**.**

A family $(a_{j})_{j\in J}\in\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$ is summable if the following properties hold.

• •

For all pair $(\rho,s)\in Y\times W^{w}$, the set $\{j\in J\mid\pi_{s,\rho}(a_{j})\not=0\}$ is finite.

• •

$\bigcup_{j\in J}\mathrm{supp}_{W^{v}}(a_{j})$* is almost finite.*

If $(a_{j})_{j\in J}\in\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$ is a summable family, we define $\sum_{j\in J}a_{j}\in{\overline{{{}^{\mathrm{BL}}\mathcal{H}}}}$ as

[TABLE]

Lemma 4.26**.**

Let $(a_{j})_{j\in J}\in({{}^{\mathrm{BL}}\mathcal{H}})^{J}$ be a summable family in $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$ and $a:=\sum_{j\in J}a_{j}\in\ \overline{{{}^{\mathrm{BL}}\mathcal{H}}}$. For any $\mu\in Y$, $(a_{j}*Z^{\mu})_{j\in J}$ and $(Z^{\mu}*a_{j})_{j\in J}$ are summable families of $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$, and the elements $\sum_{j\in J}Z^{\mu}*a_{j}$ and $\sum_{j\in J}a_{j}*Z^{\mu}$ only depend on $a$ and $\mu$ (but not on the choice of the family $(a_{j})_{j\in J}$.

Moreover, setting $a\ \overline{*}\ Z^{\mu}:=\sum_{j\in J}a_{j}*Z^{\mu}$ and $Z^{\mu}\ \overline{*}\ a:=\sum_{j\in J}Z^{\mu}*a_{j}$, we define a convolution product that provides $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$ with a structure of $\mathcal{R}[Y]$-bimodule.

Proof.

Let $(a_{j})_{j\in J}\!\in\!({{}^{\mathrm{BL}}\mathcal{H}})^{J}$ be a summable family and set $S\!:=\bigcup_{j\in J}\mathrm{supp}_{W^{v}}(a_{j})$. For all pair $(\lambda,w)\in Y\times W^{v}$, set $J(\lambda,w):=\{j\in J\mid\pi_{\lambda,w}(a_{j})\neq 0\}$. For all $(\mu,\rho,s,j)\in Y\times Y\times W^{v}\times J$, we have $\pi_{\rho,s}\left(Z^{\mu}*a_{j}\right)=\pi_{\rho-\mu,s}(a_{j})$, hence the summability of $(Z^{\mu}*a_{j})_{j\in J}$ directly comes from the summability of $(a_{j})_{j\in J}$ and

[TABLE]

only depends on $a$ and $\mu$.

The corresponding statement for $(a_{j}*Z^{\mu})_{j\in J}$ is a little bit trickier to prove. Given $w\in W^{v}$, Lemma 4.15 gives a family $(z^{w}_{\nu,t})_{(\nu,t)\in R_{w}(\mu)\times[1,w]}$ of coefficients in $\mathcal{R}$ such that

[TABLE]

For $j\in J$, write $a_{j}=\sum_{(\lambda,w)\in Y\times W^{v}}a_{j,\lambda,w}Z^{\lambda}H_{w}$ with $a_{j,\lambda,w}\in\mathcal{R}$ for any pair $(\lambda,w)\in Y\times W^{v}$. Then we have, for all $(\mu,\rho,s,j)\in Y\times Y\times W^{v}\times J$:

[TABLE]

Fix $\mu\in Y$ and set $F(\rho,s):=\{j\in J\ |\pi_{\rho,s}(a_{j}*Z^{\mu})\not=0\}$ for all pair $(\rho,s)\in Y\times W^{v}$. By the previous computation, we have $F(\rho,s)\subset\bigcup_{(w,\nu)\in S\times R_{w}(\mu)}J(\rho-\nu,w)$, hence $F(\rho,s)$ is finite. Moreover, for any $j\in J$, $\mathrm{supp}_{W^{v}}(a_{j}*Z^{\mu})$ is contained in $\bigcup_{w\in S}[1,w]$, hence $\bigcup_{j\in J}\mathrm{supp}_{W^{v}}(a_{j}*Z^{\mu})$ is almost finite and $(a_{j}*Z^{\mu})_{j\in J}$ is summable. Also note that the calculation of $\pi_{\rho,s}\left(a_{j}*Z^{\mu}\right)$ we did above implies that for all $(\rho,s)\in Y\times W^{v}$, we have

[TABLE]

hence if $a=\sum_{(\lambda,w)\in Y\times W^{v}}a_{\lambda,w}Z^{\lambda}H_{w}$, then

[TABLE]

only depends on $a$ and $\mu$.

To conclude the proof, we are left to show that for any $(b,\mu,\mu^{\prime})\in Y^{3}$, we have

[TABLE]

To do this, write $b=\sum_{(\lambda,w)\in Y\times W^{v}}b_{\lambda,w}Z^{\lambda}H_{w}$ with $(b_{\lambda,w})\in\mathcal{R}^{Y\times W^{v}}$ and apply the first part of this lemma to $J=Y\times W^{v}$: by associativity of $*$ in ${{}^{\mathrm{BL}}\mathcal{H}}$, we get the required identities. ∎

Corollary 4.27**.**

For all $a\in\hat{\mathcal{H}}$ and $\mu\in Y^{+}$, we have $Z^{\mu}*a=Z^{\mu}\ \overline{*}\ a$ and $a*Z^{\mu}=a\ \overline{*}\ Z^{\mu}$.

This statement justifies that we will from now on denote $*$ instead of $\overline{*}$.

4.5.2 Computation of the centers

Lemma 4.28**.**

For all $a\in\mathcal{Z}(\hat{\mathcal{H}})$ and $\mu\in Y$, we have $a*Z^{\mu}=Z^{\mu}*a$.

Proof.

Write $\mu=\mu_{+}-\mu_{-}$ with $\mu_{+},\mu_{-}\in Y^{+}$. The associativity of $*$ proven in Lemma 4.26 implies that $Z^{\mu_{-}}*(Z^{-\mu_{-}}*a)=a$, hence $Z^{-\mu_{-}}*a=a*Z^{-\mu_{-}}$ and $Z^{\mu}*a=Z^{\mu_{+}}*a*Z^{-\mu_{-}}=a*Z^{\mu}$. ∎

For any $w\in W^{v}$, we introduce the following subsets of $\overline{{{}^{\mathrm{BL}}\mathcal{H}}}$:

[TABLE]

We let $\hat{\mathcal{H}}_{\not\geq w}:=\overline{{{}^{\mathrm{BL}}\mathcal{H}}}_{\not\geq w}\cap\hat{\mathcal{H}}$ and $\hat{\mathcal{H}}_{=w}:=\overline{{{}^{\mathrm{BL}}\mathcal{H}}}_{=w}\cap\hat{\mathcal{H}}$ be the corresponding subspaces in $\hat{\mathcal{H}}$.

Lemma 4.29**.**

Let $w\in W^{v}$ and $\lambda\in Y$.

We have

[TABLE] 2. 2.

There exists $S\in{\overline{{{}^{\mathrm{BL}}\mathcal{H}}}}_{\ngeq w}$ such that $H_{w}*Z^{\lambda}=Z^{w(\lambda)}H_{w}+S$.

Proof.

These statements are consequence of [BPGR16, Th. 6.2], of Lemma 4.15 and of Lemma 4.26. ∎

The following theorem is the heart of this section, as it describes the center of the completed Iwahori-Hecke algebra $\hat{\mathcal{H}}$. This generalizes a well-known theorem of Bernstein (see [Lus83, Th. 8.1], which seems to be the first published version of this result) and gives a recovery of the spherical Hecke algebra $\mathcal{H}_{s}$ as center of a natural Iwahori-Hecke algebra.

Theorem 4.30**.**

The center of the completed Iwahori-Hecke algebra $\hat{\mathcal{H}}$ is $\mathcal{Z}(\hat{\mathcal{H}})=R\llbracket Y\rrbracket^{W^{v}}$.

Proof.

Let $a=\sum_{\lambda\in Y^{+}}a_{\lambda}Z^{\lambda}$ be an element of $R\llbracket Y\rrbracket^{W^{v}}$ and $i\in I$. We can write $a=x+y$ with $x=\sum_{\lambda\in Y^{+}\cap\ker\alpha_{i}}a_{\lambda}Z^{\lambda}$ and $y=\sum_{\lambda\in Y^{+}\mid\alpha_{i}(\lambda)>0}a_{\lambda}(Z^{\lambda}+Z^{r_{i}(\lambda)})$. As $x$ and $y$ commute with $H_{i}$, we obtain that $a$ commutes with $H_{i}$ for all $i\in I$, hence we have $a\in\mathcal{Z}(\hat{\mathcal{H}})$ and $R\llbracket Y\rrbracket^{W^{v}}\subset\mathcal{Z}(\hat{\mathcal{H}})$.

Conversely, let $z$ be an element of $\mathcal{Z}(\hat{\mathcal{H}})\subset{\overline{{{}^{\mathrm{BL}}\mathcal{H}}}}$ and write

[TABLE]

First assume that the set

[TABLE]

is non empty and choose a pair $(\nu,m)\in F$ with $m$ maximal in $W^{v}$ (for the Bruhat order). Write $z=x+y$ with $x=\sum_{\lambda\in Y}x_{\lambda,m}Z^{\lambda}H_{m}\in\hat{\mathcal{H}}_{=m}$ and $y\in\hat{\mathcal{H}}_{\ngeq m}$. Lemmas 4.28 and 4.29 imply that for all $y\in Y$, we have

[TABLE]

with $y_{1}\in{\overline{{{}^{\mathrm{BL}}\mathcal{H}}}}_{\ngeq m}$. By projection on ${\overline{{{}^{\mathrm{BL}}\mathcal{H}}}}_{=m}$, we get that

[TABLE]

Now let $J\subset Y$ be a finite set that satisfies the following property:

[TABLE]

Pick $\gamma\in Y$ such that $c_{\gamma,m}\not=0$. Then for all $\mu\in Y$, we have $c_{\gamma+\mu-m(\mu),m}\not=0$, hence there exists $\nu(\mu)\in J$ such that $\gamma+\mu-m(\mu)\leq_{Q^{\vee}}\nu(\mu)$. In particular, pick $\mu\in Y\cap C^{v}_{f}$ and let $\nu\in J$ be such that for all integer $n\geq 0$, we have $\gamma+\sigma(n)(\mu-m(\mu))\leq_{Q^{\vee}}\nu$, where $\sigma:\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ is such that $\lim_{n\to+\infty}\sigma(n)=+\infty$. Then $\gamma+\sigma(1)(\mu-m(\mu))-\nu$ belongs to $Q^{\vee}$, and Lemma 2.1 implies that $\mu-m(\mu)$ is a non-zero element of $Q^{\vee}_{+}$. Hence for $n$ large enough, we have

[TABLE]

which contradicts the definition of $\nu$. Consequently, $F$ is empty and $z$ belongs to $R\llbracket Y\rrbracket$. We can hence simplify the above decomposition of $z$ and write $z=\sum_{\lambda\in Y}c_{\lambda}Z^{\lambda}$ with $c_{\lambda}=c_{\lambda,1}$. By Lemma 4.29, we know that for any $w\in W^{v}$, we have

[TABLE]

for some $y\in{\overline{{{}^{\mathrm{BL}}\mathcal{H}}}}_{\ngeq w}$. But $z$ commutes with $H_{w}$, so we also have

[TABLE]

By projection on $\hat{\mathcal{H}}_{=w}$, we get that

[TABLE]

hence $z$ is in $R\llbracket Y\rrbracket^{W^{v}}$, which ends the proof. ∎

As a consequence of Theorem 4.30, we get a description of the center of the usual Iwahori-Hecke algebra $\mathcal{H}$. Note that the proof relies on a characterization of finite $W^{v}$-orbits in $\mathbb{A}$ that will be proven independently at the end of Section 5 (see Corollary 5.23).

Before we state the result, let us recall some notations. If $A_{1},\ldots,A_{r}$ denote the indecomposable components of the Kac-Moody matrix $A$, we let $J^{f}$ be the set of all $j\in\llbracket 1,r\rrbracket$ such that $A_{j}$ is of finite type [Kac94, Th. 4.3] and $J^{\infty}$ be the complement of $J^{f}$ in $\llbracket 1,r\rrbracket$. Set $\mathbb{A}^{f}:=\bigoplus_{j\in J^{f}}\mathbb{A}_{j}$, let $\Phi_{j}$ be the root system of $\mathbb{A}_{j}$ and $\mathbb{A}_{j,in}:=\bigcap_{\phi\in\Phi_{j}}\ker\phi$ for all $j\in J^{f}$. Finally, set $Y^{f}:=Y\cap\mathbb{A}^{f}$, $\mathbb{A}^{\infty}_{\mathrm{in}}:=\bigoplus_{j\in J^{\infty}}\mathbb{A}_{j,in}$ and $Y^{\infty}_{\mathrm{in}}:=Y\cap A^{\infty}_{\mathrm{in}}$.

Lemma 4.31**.**

We have $\mathcal{Z}(\mathcal{H})=\mathcal{Z}(\hat{\mathcal{H}})\cap\mathcal{H}=R[Y^{f}\oplus Y^{\infty}_{\mathrm{in}}]$.

Proof.

Any $a\in\mathcal{Z}(\mathcal{H})$ is in $\mathcal{H}$ and satisfies $a*Z^{\lambda}H_{w}=Z^{\lambda}H_{w}*a$ for all $(\lambda,w)\in Y\times W^{v}$, hence belongs to $\mathcal{Z}(\hat{\mathcal{H}})$ by Theorem 4.21. As the other inclusion is clear, we already have that $\mathcal{Z}(\mathcal{H})=\mathcal{Z}(\hat{\mathcal{H}})\cap\mathcal{H}$. By Theorem 4.30, we get that $\mathcal{Z}(\mathcal{H})=\mathcal{H}\cap R\llbracket Y\rrbracket^{W^{v}}$, and Corollary 5.23 now implies that this intersection is reduced to $R[Y^{f}\oplus Y^{\infty}_{\mathrm{in}}]$, which ends the proof. ∎

Remark 4.32**.**

When $W^{v}$ is finite, it is well-known that $\mathcal{H}$ is a finitely generated $\mathcal{Z}(\mathcal{H})$-module, and it is natural to wonder whether the corresponding statement holds in the infinite case. Unfortunately, when $W^{v}$ is infinite, $\hat{\mathcal{H}}$ is not of finite type over $\mathcal{Z}(\hat{\mathcal{H}})$. Indeed, let $J$ be any finite set and pick any finite family $(h_{j})_{j\in J}\in\hat{\mathcal{H}}^{J}$. For all $(z_{j})_{j\in J}\in\mathcal{Z}(\hat{\mathcal{H}})^{J}$, we have

[TABLE]

hence $(h_{j})_{j\in J}$ cannot span $\hat{\mathcal{H}}$ over $\mathcal{Z}(\hat{\mathcal{H}})$.

4.6 Some further remarks

4.6.1 The special case of reductive groups

Assume in this paragraph that $G$ is reductive, in which case $\mathcal{T}=\mathcal{A}$ and $Y=Y^{+}$. Then almost finite sets as defined in [GR14] are finite sets: indeed, the Kac-Moody matrix $A$ is in this case a Cartan matrix, hence it satisfies condition (FIN) in [Kac94, Th. 4.3]. In particular, $Y^{++}$ is contained in $Q^{\vee}_{+}\oplus\mathbb{A}_{\mathrm{in}}$, so to be a subset of some \bigl{(}\bigcup_{i=1}^{k}(y_{i}-Q^{\vee}_{+})\bigr{)}\cap Y^{++} amounts to be finite. Though the algebra $\hat{\mathcal{H}}$ is still different from $\mathcal{H}$, as $\sum_{\mu\in Q^{\vee}_{+}}Z^{-\mu}$ is for instance an element of $\hat{\mathcal{H}}$ that is not in $\mathcal{H}$, they both have the same center. Indeed, we have the following result.

Proposition 4.33**.**

Let $R$ be any ring. Then $R\llbracket Y\rrbracket^{W^{v}}=R[Y]^{W^{v}}$ if and only if $W^{v}$ is finite.

Proof.

If $W^{v}$ is infinite, then for any $y\in Y\cap C^{v}_{f}$, the element $\sum_{w\in W^{v}}e^{w\cdot y}$ belongs to $R\llbracket Y\rrbracket^{W^{v}}$ but not to $R[Y]^{W^{v}}$, hence $R\llbracket Y\rrbracket^{W^{v}}\not=R[Y]^{W^{v}}$.

If $W^{v}$ is finite, let $w_{0}$ be the longest element of $W^{v}$. By [Hum92, §1.8], we know that $w_{0}.Q^{\vee}_{+}=Q^{\vee}_{-}$. If $E\subset Y$ is almost finite, there is some finite set $J$ and a family $(y_{j})_{j\in J}\in Y^{J}$ such that $E$ is contained in $\bigcup_{j\in J}(y_{j}-Q^{\vee}_{+})$. If E is furthermore $W^{v}$-invariant, then $E=w_{0}\cdot E$ is also contained in $\bigcup_{j\in J}(w_{0}\cdot y_{j}+Q^{\vee}_{+})$, hence any element $x\in E$ satisfies $w_{0}\cdot y_{j}\leq_{Q^{\vee}}x\leq_{Q^{\vee}}y_{j^{\prime}}$ for some $j,j^{\prime}\in J$. This implies that $E$ is finite and completes the proof. ∎

Using [Lus83, Th. 8.1], Theorem 4.30 and Lemma 4.31, we get from Proposition 4.33 that when $W^{v}$ is finite, we have:

[TABLE]

4.6.2 Iwahori-Hecke algebras and $K_{I}$ double cosets

In the completion process we used to define $\hat{\mathcal{H}}$, we used the Bernstein-Lusztig relations of $\mathcal{H}$. However, the Iwahori-Hecke algebra $\mathcal{H}$ is initially defined in a different way, namely as a convolution algebra of $K_{I}$-bi-invariant functions. In particular, a natural basis of $\mathcal{H}$ is given by characteristic functions of $K_{I}$ double cosets, and the Bernstein-Lusztig presentation comes afterwards. This leads naturally to address the following question: can we see the completed algebra at the level of $K_{I}$ double cosets, as it is the case for the spherical Hecke algebra?

Fix a ring $\mathcal{R}$ as before and let $\mathcal{C}_{0}$ be the set of positive type [math] chambers. Set $W^{+}:=W^{v}\ltimes Y^{+}$ and let $d^{W}$ be the distance defined in [BPGR16] (see also Section 5.2 below). Recall that $\mathcal{H}$ is isomorphic to $\bigoplus_{\mathbf{w}\in W^{+}}\mathcal{R}T_{\mathbf{w}}$ for the product defined by $T_{\mathbf{w}}*T_{\mathbf{v}}=\sum_{\mathbf{u}\in W^{+}}a^{\mathbf{u}}_{\mathbf{w},\mathbf{v}}$ for all elements $\mathbf{w},\mathbf{v}\in W^{+}$, provided that we set, for all $\mathbf{u}\in W^{+}$,

[TABLE]

For $x=(x_{\mathbf{w}})_{\mathbf{w}\in W^{+}}\in\mathcal{R}^{W^{+}}$, also write $x=\sum_{\mathbf{w}\in W^{+}}x_{\mathbf{w}}T_{\mathbf{w}}$. For now, we do not know whether it is possible to endow $\mathcal{R}^{W^{+}}$, or some subspace $\overline{\mathcal{H}}\subset\mathcal{R}^{W^{+}}$ containing $\mathcal{H}$, with a product that extends the convolution product of $\mathcal{H}$. At least in general, it seems difficult to embed $\widehat{\mathcal{H}}$ into $\mathcal{R}^{W^{+}}$. Indeed, assume for instance that $\mathcal{R}=\mathbb{C}$ and let $\pi:\mathbb{C}^{W^{+}}\to\mathbb{C}$ be the map defined by

[TABLE]

(Here, $1$ denotes the identity element in $W^{+}$.) When $G$ is reductive, we know for instance by [Opd03, Cor. 1.9] that for any $\lambda\in-Q^{\vee}$, $\pi(Z^{\lambda})$ is a positive real number, which makes it apparently hard to consider $\sum_{\lambda\in-Q^{\vee}}\frac{1}{\pi(Z^{\lambda})}Z^{\lambda}\in\widehat{\mathcal{H}}$ as an element of $\mathbb{C}^{W^{+}}$. In the non-reductive case, we do not know so far whether an analogue of [Opd03, Cor. 1.9] is true.

5 Hecke algebra associated with a parahoric subgroup

The goal of this section is to attach a Hecke algebra to other subgroups than $K_{s}$ or $K_{I}$, by generalizing previous constructions of [BKP16] and [BPGR16] for the Iwahori subgroup $K_{I}$. Our motivation comes from the reductive case, where Hecke algebras can be associated with any open compact subgroup (see Section 5.1 below). When $G$ is not reductive, we know from Theorem 3.1 that there is no reasonable topology on $G$, hence we cannot define “open compact” in our context. Nevertheless, there is still a notion of special parahoric subgroup, defined as the fixer of a type [math] face of the masure $\mathcal{I}$.

Given a special parahoric subgroup $K=K_{F}$ that fixes a spherical type [math] face $F$ satisfying $F_{0}\subset\overline{F}\subset\overline{C^{+}_{0}}$, we will generalize the construction done for $K_{I}$ by Bardy-Panse, Gaussent and Rousseau [GR14, BPGR16] to build a Hecke algebra associated with $K_{F}$. This requires some finiteness results that fails anytime $F\not=F_{0}$ is not spherical (see Section 5.4).

5.1 Motivation from the reductive case

To motivate our definition in the Kac-Moody case (see Definition 5.14), we start by recalling the classical setting for reductive groups. This section follows [Vig96, I.3.3], though the idea of considering Hecke algebras as spaces of bi-invariant functions goes back at least to Weil and Shimura [Shi59], and to Iwahori [Iwa64] and Iwahori-Matsumoto [IM65].

Assume that $G$ is reductive, in which case it is naturally endowed with a structure of topological group induced by the topology of $\mathcal{K}$. For any open compact subgroup $K$ of $G$, let $\mathbb{Z}_{c}(G/K)$ be the space of compactly supported functions $G\to\mathbb{Z}$ that are $K$-invariant under right multiplication. Define an action of $G$ on this space by setting $g\cdot f:=[x\mapsto f(g\cdot x)]$ for all pairs $(g,f)\in G\times\mathbb{Z}_{c}(G/K)$. The algebra $H(G,K):=\mathrm{End}_{G}(\mathbb{Z}_{c}(G/K))$ of $G$-equivariant endomorphisms of $\mathbb{Z}_{c}(G/K)$ is called the Hecke algebra of $G$ relative to $K$. If $\mathbb{Z}_{c}(G//K)$ is the ring of compactly supported functions $G\to\mathbb{Z}$ that are $K$-bi-invariant (for left and right multiplications), then we have a natural isomorphism of algebras $\Upsilon:H(G,K)\to\mathbb{Z}_{c}(G//K)$ given by $\Upsilon(\phi):=\phi(\mathds{1}_{K})$ for all $\phi\in H(G,K)$. This shows that $H(G,K)$ is a free $\mathbb{Z}$-algebra with canonical basis $\{e_{g}=\mathds{1}_{KgK},\ g\in K\backslash G/K\}$. The product of two elements $e_{g},e_{g^{\prime}}$ of this basis is given by

[TABLE]

(Note that the non-vanishing of $m(g,g^{\prime};g^{\prime\prime})$ implies that $Kg^{\prime\prime}K$ is contained in $KgKg^{\prime}K$.)

Extension of scalars works as follows: for any commutative ring $R$, the algebra $H_{R}(G,K):=H(G,K)\otimes_{\mathbb{Z}}R$ is called the Hecke algebra of $G$ over $R$ relative to $K$.

When $G$ is not reductive, we will replace open compact subgroups (that are not defined) by special parahoric subgroups. More precisely, let $K=K_{F}$ be the fixer in $G$ of a type [math] face $F$ that satisfies $F_{0}\subset\overline{F}\subset\overline{C}^{+}_{0}$. Following what is done in [BPGR16] in the Iwahori case, we will see $(KgK\cap g^{\prime\prime}Kg^{\prime-1}K)/K$ as intersection of “spheres” in $\mathcal{I}$ and prove that this intersection is finite when $F$ is spherical (see Lemma 5.11) but infinite when $F\not=F_{0}$ is not spherical (see Proposition 5.21). Hence for $F$ spherical, we will be able to define the Hecke algebra ${{}^{\mathrm{F}}}\mathcal{H}$ associated with $K_{F}$ as the free $\mathbb{Z}$-module with basis $\{e_{g}=\mathds{1}_{KgK},\ g\in K\backslash G^{+}/K\}$, where $G^{+}:=\{g\in G\mid g\cdot 0\geq 0\}$, equipped with the convolution product given by the analogue of formula (4) with $g^{\prime\prime}\in G^{+}$. To prove this, we use the fact that these results are already known when $F$ is a type [math] chamber (by [BPGR16]), and the finiteness of the number of type [math] chambers dominating $F$ as above. Note that the use of $G^{+}$ instead of $G$ in the definition of ${{}^{\mathrm{F}}}\mathcal{H}$ is related to the fact that two points of $\mathcal{I}$ do not always lie in a same apartment. This change of group already shows up in the spherical and the Iwahori cases (see [BPGR16, BK11, BKP16, GR14]).

From now on, we fix a type [math] face $F$ that satisfy $F_{0}\subset\overline{F}\subset\overline{C^{+}_{0}}$. We denote by $K=K_{F}$ its fixer in $G$ and by $W_{F}$ its pointwise fixer in $W^{v}$. Then $F$ is spherical when $W_{F}$ is finite. We also let $\Delta_{F}=G\cdot F$ be its orbit under the action of $G$ on $\mathcal{I}$. Note that we have a bijection $\Upsilon_{F}:G/K\to\Delta_{F}$ that maps $g\cdot K_{F}$ to $g\cdot F$.

5.2 Distance and spheres associated with a type [math] face

In this section, we define an “$F$-distance” (or “$W_{F}$-distance”) that generalizes the $W^{v}$-distance introduced in [GR14] and the $W$-distance defined in [BPGR16].

If $A$ (resp. $A^{\prime}$) is an apartment of $\mathcal{I}$ and if $E_{1},\ldots,E_{k}$ (resp. $E^{\prime}_{1},\ldots,E^{\prime}_{k}$) are subsets or filters of $A$ (resp. $A^{\prime}$), we denote by $\phi:(A,E_{1},\ldots,E_{k})\to(A^{\prime},E^{\prime}_{1},\ldots,E^{\prime}_{k})$ any isomorphism of apartment $\phi:A\to A^{\prime}$ induced by some element of $G$ and such that: $\forall i\in\llbracket 1,k\rrbracket$, $\phi(E_{i})=E^{\prime}_{i}$. When we do not want to precise which apartments $A$ and $A^{\prime}$ are chosen, we simply write $\phi:(E_{1},\ldots,E_{k})\to(E^{\prime}_{1},\ldots,E^{\prime}_{k})$.

We define a relation $\leq$ on $\Delta_{F}$ as follows: for $F_{1},F_{2}$ in $\Delta_{F}$, we write $F_{1}\leq F_{2}$ when $a_{1}\leq a_{2}$, where $a_{i}$ denotes the vertex of $F_{i}$ for $i\in\{1,2\}$. We then set

[TABLE]

For any $F^{\prime}\in\Delta_{F}\cap\mathbb{A}$, we set $[F^{\prime}]:=W_{F}\cdot F^{\prime}$.

Proposition 5.1**.**

For all $(F_{1},F_{2})\in\Delta_{F}\times_{\leq}\Delta_{F}$, there exists an apartment $A$ containing $F_{1}$ and $F_{2}$, and an isomorphism $\phi:(A,F_{1})\to(\mathbb{A},F)$. Moreover, $d^{F}(F_{1},F_{2}):=[\phi(F_{2})]$ only depends on the pair $(F_{1},F_{2})$.

Proof.

Given $(F_{1},F_{2})\in\Delta_{F}\times_{\leq}\Delta_{F}$, the existence of an apartment $A$ containing $F_{1}$ and $F_{2}$ comes from [Rou11, Prop. 5.1]. By construction, there is some $g\in G$ such that $F_{1}=g\cdot F$. Let $A^{\prime}:=g\cdot\mathbb{A}$: by (MA2), there exists an isomorphism $\psi:(A,F_{1})\to(A^{\prime},F_{1})$. Set $\psi^{\prime}:=g_{|\mathbb{A}}^{|A^{\prime}}$: then $\phi:=\psi^{\prime-1}\circ\psi$ has the required properties.

Let now $A_{1}$ be another apartment containing $F_{1}$ and $F_{2}$ and $\phi_{1}:(A_{1},F_{1})\to(\mathbb{A},F)$ be another suitable isomorphism. By [Héb17a, Th. 5.18], there exists an isomorphism $f:(A,F_{1},F_{2})\to(A_{1},F_{1},F_{2})$. We hence have the following commutative diagram:

[TABLE]

with the lower horizontal arrow that is induced by an element of $W_{F}$, hence $[\phi(F_{2})]=[\phi_{1}(F_{2})]$ does not depend on any choice and the proof is complete. ∎

Remark 5.2**.**

Proposition 5.1 does not require $F$ to be spherical, thought it is the most important case for us. When $F$ is spherical, one can use [BPGR16, Prop. 1.10c)] instead of [Héb17a, Th. 5.18]. Note that in the sequel, we will only use the $F$-distance attached to a non-spherical face for pairs of type [math] faces based at the same vertex. In this special case, [Héb17a, Th. 5.18] could be replace by [Rou11, Prop. 5.2].

Remark 5.3**.**

When $F=F_{0}$, we can identify $d^{F}$ with the “vectorial distance” $d^{v}$ of [GR14] through the usual bijections $\Delta_{F_{0}}\simeq G\cdot 0$ and $Y^{++}\simeq Y^{+}/W^{v}$.

When $F=C^{+}_{0}$, we have $W_{C^{+}_{0}}=\{1\}$, hence $[C]=\{C\}$ for all chamber $C\in\Delta_{C^{+}_{0}}$ and $d^{C^{+}_{0}}$ can be identified with the distance $d^{W}$ of [BPGR16], provided that each element $w$ of $W^{v}\ltimes Y^{+}$ is identified with the type [math] chamber $w\cdot C^{+}_{0}$.

Set

[TABLE]

Moreover, for any pair $(E,[R])\in\Delta_{F}\times[\Delta_{F}]$, set

[TABLE]

For any $E\in\Delta_{\geq F}^{\mathbb{A}}$, we choose some $g_{E}\in N$ such that $E=g_{E}\cdot F$. Such an element exists: indeed, let $g\in G$ be such that $E=g\cdot F$ and set $A:=g\cdot\mathbb{A}$. By (MA2) and [Rou11, 2.2.1)], we get an isomorphism $\phi:(A,g\cdot F)\to(\mathbb{A},g\cdot F)$. Letting $\psi=g_{|\mathbb{A}}^{|A}$, we have $\phi\circ\psi\in N$ such that $(\phi\circ\psi)(F)=\phi(E)=E$, hence $\phi\circ\psi$ comes from an element $g_{E}$ as required.

Lemma 5.4**.**

For all $[R]\in[\Delta_{F}]$, we have

[TABLE]

Proof.

For any $E\in\mathcal{S}^{F}(F,[R])$, there exists some $g\in K_{F}$ such that $g\cdot E=R=g_{R}\cdot F$, hence $\Upsilon_{F}^{-1}(E)$ belongs to $K_{F}g_{R}K_{F}/K_{F}$. Now let $x=k_{1}g_{R}k_{2}$ be an element of $K_{F}g_{R}K_{F}$: then we have $\Upsilon_{F}(x):=\Upsilon_{F}(k_{1}g_{R})=k_{1}g_{R}\cdot F=k_{1}\cdot R$. As $d^{F}$ is $G$-invariant, we obtain that

[TABLE]

hence $x$ is in $\Upsilon^{-1}_{F}(S^{F}(F,[R]))$ and the proof of the first equality is complete. The proof of the second equality is similar and left to the reader. ∎

5.3 Hecke algebra associated with a spherical type [math] face

Let $C$ and $C^{\prime}$ be two positive type [math] chambers base at some common vertex $x\in\mathcal{I}_{0}:=G\cdot 0$. We can (and will) identify $W$ with the set of type [math] chambers of $\mathbb{A}$ whose vertex lies in $Y^{+}$. Thus $d^{W}(C,C^{\prime})=d^{C^{+}_{0}}(C,C^{\prime})$ is in $W^{v}$ and we can set $d(C,C^{\prime}):=\ell(d^{W}(C,C^{\prime}))$.

Lemma 5.5**.**

Let $C$ be a positive type [math] chamber of $\mathcal{I}$ and let $x$ be its vertex. For any integer $n\geq 0$, the set $B_{n}(C)$ of all positive type [math] chambers $C^{\prime}$ of $\mathcal{I}$ based at $x$ and such that $d(C,C^{\prime})\leq n$ is a finite set.

Proof.

The argument goes by induction on $n\geq 0$, noticing that $B_{n}(C)$ contains $B_{m}(C)$ each time we have $n\geq m\geq 0$. As $\mathcal{I}$ is of finite thickness, the set $B_{1}(E)$ is finite for all $E\in G\cdot C^{+}_{0}$. Now let $n\geq 0$ be such that $B_{n}(E)$ is finite for all $E\in G\cdot C^{+}_{0}$ and take $C^{\prime}\in B_{n+1}(C)$. By [Rou11, Prop. 5.1], we can choose an apartment $A$ that contains $C$ and $C^{\prime}$. Let $\phi:(A,C)\to(\mathbb{A},C^{+}_{0})$ be an isomorphism of apartments: then we have $\phi(C^{\prime})=w\cdot C^{+}_{0}$ for some $w\in W^{v}$ of length at most $n+1$. We can assume that $\ell(w)=n+1$, otherwise $C^{\prime}$ is in $B_{n}(C)$ and there is nothing more to do. In this case, let $\tilde{w}\in W^{v}$ be such that $\ell(\tilde{w})=n$ and $d(\tilde{w}\cdot C_{0}^{+},\phi(C^{\prime}))=1$. Then $\tilde{C}:=\phi^{-1}(\tilde{w}\cdot C^{+}_{0})$ satisfies $d(C^{\prime},\tilde{C})=1$, hence $C^{\prime}$ belongs to $\bigcup_{C^{\prime\prime}\in B_{n}(C)}B_{1}(C^{\prime\prime})$, which is a finite set, and the proof is complete. ∎

5.3.1 Type of a type [math] face

Let $\mathcal{F}^{v}_{\mathbb{A}}$ be the set of all positive vectorial faces of $\mathbb{A}$ and $\mathcal{F}^{0}_{\mathbb{A}}$ be the set of all positive type [math] faces of $\mathbb{A}$ based at [math].

Lemma 5.6**.**

The map $f:\mathcal{F}^{v}_{\mathbb{A}}\to\mathcal{F}^{0}_{\mathbb{A}}$ that sends $F^{v}\in\mathcal{F}^{v}$ to $F^{\ell}(0,F^{v})\in\mathcal{F}^{0}_{\mathbb{A}}$ is bijective.

Proof.

The definition of local faces ensures that $f$ is well-defined and surjective. Now let $F^{v}_{1},F^{v}_{2}$ be two distinct elements of $F^{v}_{\mathbb{A}}$. As [math] is special, we have $F_{i}^{v}\in f(F_{i}^{v})$ for $i\in\{1,2\}$. But $F^{v}_{1}\cap F^{v}_{2}=\emptyset$ implies that $f(F^{v}_{1})\not=f(F^{v}_{2})$ (for otherwise, we would have $\emptyset\in f(F^{v}_{1})$, which does not make sense) and $f$ is thus injective, which ends the proof. ∎

For any positive type [math] face $F^{\prime}$ of $\mathcal{I}$, there is some type [math] face $F_{1}\leq C^{+}_{0}$ and some $g_{1}\in G$ such that $F^{\prime}=g_{1}\cdot F_{1}$. The set $J\subset I$ such that $F_{1}=F^{\ell}(0,F^{v}(J))$ is called the type of $F^{\prime}$ and denoted by $\tau(F^{\prime})$. This notion is well-defined: indeed, if we also have $F^{\prime}=g_{2}F^{\ell}(0,F^{v}(J^{\prime}))$ for some $g_{2}\in G$ and $J^{\prime}\subset I$, then $g:=g_{2}^{-1}g_{1}$ is such that $F^{\ell}(0,F^{v}(J))=g\cdot F^{\ell}(0,F^{v}(J^{\prime})$. By (MA2) and [Rou11, 2.2.1)], we can assume that $g$ lies in $N$, hence $g_{|\mathbb{A}}$ is in $W^{v}$ and Lemma 5.6 then implies that $F^{v}(J)=F^{v}(J^{\prime})$. By [Rou11, 1.3], this requires that $J=J^{\prime}$, as wanted.

Remark 5.7**.**

The type of a face is invariant under the action of $G$. Also note that for any type [math] chamber $C$ and any subset $J$ of $I$, there exists exactly one sub-face of $C$ with type $J$.

5.3.2 Finiteness results for spherical type [math] faces

From now on, we assume that the face $F$ is furthermore spherical.

Lemma 5.8**.**

For all $F^{\prime}\in\Delta_{F}$, the set $\mathcal{C}_{F^{\prime}}$ of all type [math] chambers of $\mathcal{I}$ containing $F^{\prime}$ is finite.

Proof.

Fix a chamber $C\in\mathcal{C}_{F^{\prime}}$, denote by $x$ its vertex and pick another chamber $C^{\prime}$ in $\mathcal{F^{\prime}}$. By [Rou11, Prop. 5.1], there exists an apartment $A$ that contains $C$ and $C^{\prime}$. We identify $A$ with $\mathbb{A}$ and fix the origin of $\mathbb{A}$ at $x$: then there exists $w\in W^{v}$ such that $C^{\prime}=w\cdot C$. If $J$ denotes the type of $F^{\prime}$, then $w\cdot F^{\prime}$ is also a sub-face of type $J$ in $C^{\prime}$, hence we have $w\cdot F^{\prime}=F^{\prime}$ by the unicity property stated in Remark 5.7. This means that $w$ belongs to $W_{F^{\prime}}$, which is a finite group as $F^{\prime}\in\Delta_{F}$ is spherical (because $F$ is). In particular, we must have $d(C,C^{\prime})\leq\max\{\ell(u),\ u\in W_{F^{\prime}}\}$, which ends the proof by Lemma 5.5. ∎

Lemma 5.9**.**

Let $(E_{1},E_{2})$ and $(E^{\prime}_{1},E^{\prime}_{2})$ be in $\Delta_{F}\times_{\leq}\Delta_{F}$. Then $d^{F}(E_{1},E_{2})=d^{F}(E^{\prime}_{1},E^{\prime}_{2})$ iff there exists an isomorphism $\phi:(E_{1},E_{2})\to(E^{\prime}_{1},E^{\prime}_{2})$.

Proof.

Assume that $d^{F}(E_{1},E_{2})=d^{F}(E^{\prime}_{1},E^{\prime}_{2})=:[R]$. For any choice of

[TABLE]

the map $\phi:=\psi^{\prime-1}\circ\psi$ satisfies the required property.

Conversely, suppose that there exists an isomorphism $\phi:(E_{1},E_{2})\mapsto(E^{\prime}_{1},E^{\prime}_{2})$. Pick $R\in d^{F}(E_{1},E_{2})$ and choose $\psi:(E_{1},E_{2})\to(F,R)$: then $\phi^{-1}\circ\psi$ is an isomorphism that maps $(E^{\prime}_{1},E^{\prime}_{2})$ to $(F,R)$. By Proposition 5.1, we thus have $d^{F}(E^{\prime}_{1},E^{\prime}_{2})=[R]=d^{F}(E_{1},E_{2})$, which ends the proof of the lemma. ∎

Lemma 5.10**.**

Let $(F_{1},F_{2})\in\Delta_{F}\times_{\leq}\Delta_{F}$ and $r:=d^{F}(F_{1},F_{2})$. Let $R\in\Delta_{\geq F}^{\mathbb{A}}$ be such that $r=[R]=W_{F}\cdot R$ and let $\mathcal{C}_{\mathbb{A}}(r)$ be the set of all chambers of $\mathbb{A}$ containing an element of $r$. For any type [math] chambers $C_{1}$ and $C_{2}$ respectively dominating $F_{1}$ and $F_{2}$, we have $d^{W}(C_{1},C_{2})\in\mathcal{C}_{\mathbb{A}}(r)$. Moreover, the set $\mathcal{C}_{\mathbb{A}}(r)$ is finite.

Proof.

Pick an apartment $A$ containing $C_{1}$ and $C_{2}$ and an isomorphism

[TABLE]

By Remark 5.7, $\phi(F_{1})$ is the unique sub-face of type $\tau(F)$ in $C^{+}_{0}$, hence we have $F=\phi(F_{1})$ and $\phi(F_{2})$ belongs to $W_{F}\cdot R=r$, which implies that $d^{W}(C_{1},C_{2})$ lies in $\mathcal{C}_{\mathbb{A}}(r)$.

Now note that the map sending a positive type [math] face $F^{\prime}$ on its type $\tau(F^{\prime})$ induces a bijection from the set of type [math] chambers of $\mathbb{A}$ containing $w\cdot R$ onto the fixer $W_{R}$ of $R$ in $W$. As $W_{R}$ is a conjugate of $W_{F}$, it is finite (as $F$ is spherical), hence so is $\mathcal{C}_{\mathbb{A}}(r)$. ∎

Lemma 5.11**.**

For any pair $(F_{1},F_{2})\in\Delta_{F}\times_{\leq}\Delta_{F}$ and any elements $r_{1},r_{2}\in[\Delta_{F}]$, the set $\mathcal{S}^{F}(F_{1},r_{1})\cap\mathcal{S}^{F}_{\mathrm{opp}}(F_{2},r_{2})$ is finite and its cardinality only depends on $r_{1},r_{2}$ and $r:=d^{F}(F_{1},F_{2})$.

Proof.

Denote by $\mathcal{S}$ the set of type [math] chambers containing an element of $\mathcal{S}^{F}(F_{1},r_{1})\cap\mathcal{S}^{F}_{\mathrm{opp}}(F_{2},r_{2})$ and let $C_{1}$ (resp. $C_{2}$) be a type [math] chamber that contains $F_{1}$ (resp. $F_{2}$). By Lemma 5.10, any chamber $C\in\mathcal{S}$ satisfies $d^{W}(C_{1},C)\in\mathcal{C}_{\mathbb{A}}(r_{1})$ and $d^{W}(C,C_{2})\in\mathcal{C}_{\mathbb{A}}(r_{2})$. This implies that $\mathcal{S}$ is contained in

[TABLE]

By [BPGR16, Prop. 2.3] and Lemma 5.10, this inclusion implies the finiteness of $\mathcal{S}$, which itself implies that $\mathcal{S}^{F}(F_{1},r_{1})\cap\mathcal{S}^{F}_{\mathrm{opp}}(F_{2},r_{2})$ is finite.

To prove the independence of the cardinality, assume that $(F^{\prime}_{1},F^{\prime}_{2})\in\Delta_{F}\times_{\leq}\Delta_{F}$ is such that $d^{F}(F^{\prime}_{1},F^{\prime}_{2})=r$. By Lemma 5.9, there exists an isomorphism

[TABLE]

Thus we have

[TABLE]

which ends the proof. ∎

Following Lemma 5.11, and with the same notations, we set

[TABLE]

Lemma 5.12**.**

For any elements $r_{1},r_{2}$ in $[\Delta_{F}]$, the set

[TABLE]

is finite.

Proof.

Denote by $\mathcal{E}$ the set of all triples $(C_{1},C^{\prime},C_{2})$ of type [math] chambers such that there exist sub-faces $F_{1}\subset C_{1}$, $F^{\prime}\subset C^{\prime}$ and $F_{2}\subset C_{2}$ of these chambers that satisfy $d^{F}(F_{1},F^{\prime})=r_{1}$ and $d^{F}(F^{\prime},F_{2})=r_{2}$. If $(C_{1},C^{\prime},C_{2})$ is in $\mathcal{E}$, then Lemma 5.10 implies that $d^{W}(C_{1},C^{\prime})\in\mathcal{C}_{\mathbb{A}}(r_{1})$ and $d^{W}(C^{\prime},C_{2})\in\mathcal{C}_{\mathbb{A}}(r_{2})$. This proves that $P:=\{d^{W}(C_{1},C_{2}),\ (C_{1},C^{\prime},C_{2})\in\mathcal{E}\}$ is contained in $\bigcup_{(\mathbf{w_{1}},\mathbf{w_{2}})\in\mathcal{C}_{\mathbb{A}}(r_{1})\times\mathcal{C}_{\mathbb{A}}(r_{2})}P_{\mathbf{w_{1}},\mathbf{w_{2}}}$, where the $P_{\mathbf{w_{1}},\mathbf{w_{2}}}$’s are the finite sets introduced in the proof of [BPGR16, Prop. 2.2]. By Lemma 5.10, this proves that $P$ is finite.

Let $(F_{1},F^{\prime},F_{2})\!\in\!\Delta_{F}\times_{\leq}\Delta_{F}\leq\Delta_{F}$ be such that $d^{F}(F_{1},F^{\prime})\!=\!r_{1}$ and $d^{F}(F^{\prime},F_{2})\!=\!r_{2}$. Then there is a triple $(C_{1},C^{\prime},C_{2})\in\mathcal{E}$ such that $F_{i}$ is a face of $C_{i}$ for $i\in\{1,2\}$. The distance $d^{F}(F_{1},F_{2})$ is of the form $W_{F}\cdot F^{\prime\prime}$ for some face $F^{\prime\prime}$ of $d^{W}(C_{1},C_{2})$, hence the lemma follows. ∎

5.3.3 Definition of the Hecke algebra ${{}^{\mathrm{F}}}\mathcal{H}$

Let $R$ be a commutative unitary ring.444Note that we do not require here any of the additional assumptions made on $\mathcal{R}$ in Section 4. Denote by ${{}^{\mathrm{F}}}\mathcal{H}={{}^{\mathrm{F}}}\mathcal{H}^{\mathcal{I}}_{R}$ the set of all functions $\varphi:G\backslash\Delta_{F}\times_{\leq}\Delta_{F}\to R$. For any $r\in[\Delta_{F}]$, let $T_{r}:\Delta_{F}\times_{\leq}\Delta_{F}\to R$ be defined as follows (where $\delta_{.,.}$ denotes the Kronecker symbol):

[TABLE]

One directly checks that ${{}^{\mathrm{F}}}\mathcal{H}$ is a free $R$-module with basis $\{T_{r},\ r\in[\Delta_{F}]\}$.

Theorem 5.13**.**

Define a product $*:{{{}^{\mathrm{F}}}\mathcal{H}}\times{{{}^{\mathrm{F}}}\mathcal{H}}\to{{{}^{\mathrm{F}}}\mathcal{H}}$ by the following formula:

[TABLE]

Then the product $*$ is well-defined and endow ${{}^{\mathrm{F}}}\mathcal{H}$ with a structure of associative algebra that has $T_{[F]}$ for identity element. Moreover, the product of any two elements of the basis $\{T_{r},\ r\in[\Delta_{F}]\}$ is given by the following formula:

[TABLE]

Proof.

Lemmas 5.11 and 5.12 imply that $*$ is well-defined and give the required formula for $T_{r_{1}}*T_{r_{2}}$ for any $(r_{1},r_{2})\in[\Delta_{F}]\times[\Delta_{F}]$. The associativity of $*$ directly comes from the definition, and a direct computation shows that $T_{[F]}$ is the identity element as we have $\mathcal{S}^{F}(F_{1},[F])=\{F_{1}\}$ for all $F_{1}\in\Delta_{F}$. ∎

Definition 5.14**.**

The algebra ${{}^{\mathrm{F}}}\mathcal{H}={{{}^{\mathrm{F}}}\mathcal{H}^{\mathcal{I}}_{R}}$ is called the Hecke algebra of $\mathcal{I}$ associated to $F$ (or: to $K_{F}$) over $R$.

Remark 5.15**.**

Given $g\in G^{+}$, there exists some element $R_{g}\in\Delta_{\geq F}^{\mathbb{A}}$ such that

[TABLE]

Let $(F_{1},F_{2})\in G\backslash\Delta_{F}\times_{\leq}\Delta_{F}$. We can always assume that $F_{1}=F$ and write $F_{2}=g\cdot F$ for some $g\in G$. One easily checks that $f(g):=K_{F}gK_{F}$ only depends on $(F_{1},F_{2})$, and that the corresponding map $f:G\backslash\Delta_{F}\times_{\leq}\Delta_{F}\to K_{F}\backslash G^{+}/K_{F}$ is bijective. Via $f$, we can identify ${{}^{\mathrm{F}}}\mathcal{H}$ with the set of all functions $K_{F}\backslash G^{+}/K_{F}\to R$. Under this identification, $T_{R_{g}}$ corresponds to $e_{g}:=\mathds{1}_{K_{F}gK_{F}}$ for all $g\in G^{+}$. Moreover, for any $g,g^{\prime}\in K_{F}\backslash G^{+}/K_{F}$, we have

[TABLE]

where $m(g,g^{\prime};g^{\prime\prime})=a^{[R_{g^{\prime\prime}}]}_{[R_{g}],[R_{g^{\prime}}]}$ for all $g^{\prime\prime}\in K_{F}\backslash G^{+}/K_{F}$. Using Lemmas 5.4 and 5.11, we get that $m(g,g^{\prime};g^{\prime\prime})=|(KgK\cap g^{\prime\prime}Kg^{\prime-1}K)/K|$, as in the reductive case (compare with (4)).

Remark 5.16**.**

For now, we do not know whether it is possible to define a completed Hecke algebra $\widehat{{{}^{\mathrm{F}}}\mathcal{H}}$ for any spherical face $F$ as above in the similar manner as what we did for the Iwahori-Hecke algebra. To generalize our completion process to this context, one would in particular need an analogue of Bernstein-Lusztig relations for arbitrary $F$.

5.4 What about non-spherical type [math] faces?

In [GR14], Gaussent and Rousseau defined the spherical Hecke algebra as a Hecke algebra associated with the non-spherical type [math] face $F_{0}$, and we noticed in Remark 5.3 that their distance $d^{v}$ matches with our $d^{F_{0}}$. Consequently, it seems natural to try to associate a Hecke algebra with any type [math] face $F$ between $F_{0}$ and $C^{+}_{0}$, i.e to see whether the extra assumption of being spherical can be suppressed.

In this section, we consider a non-spherical type [math] face $F$ such that $F_{0}\subsetneq F\subsetneq C_{0}^{+}$. Note that this implies that $A$ is an indefinite Kac-Moody matrix of size $\geq 3$: indeed, when $A$ is of finite type, then any type [math] face is spherical, and when $A$ is of affine type, the only non-spherical type [math] face of $C^{+}_{0}$ is $F_{0}$. In this last section, we will prove that the coefficients involved in the definition of the convolution product introduced earlier (see Theorem 5.13) are now infinite. The proof of this result requires the injectivity of the restriction map that sends $w\in W^{v}$ to $w_{|Q^{\vee}}$. This property is proved in [Kac94] for less general realizations of $A$ than the one we use, hence we will start by extending this property to our framework: this is the point of Lemma 5.18 below.

5.4.1 Realizations of a Kac-Moody matrix

Let $A=(a_{i,j})_{i,j\in\llbracket 1,n\rrbracket}$ be a Kac-Moody matrix. Following [Kac94, Chap. 1], we say that a realization of $A$ is a triple $(\mathcal{A},\Pi,\Pi^{\vee})$ where $\mathcal{A}$ denotes an $\mathbb{R}$-vector space,555Note that in [Kac94], complex vector spaces are used instead of real vector spaces. $\Pi=\{\alpha_{1},\ldots,\alpha_{n}\}$ a family of $n$ elements in $\mathcal{A}^{*}$ (the dual space of $\mathcal{A}$) and $\Pi^{\vee}=\{\alpha_{1}^{\vee},\ldots,\alpha_{n}^{\vee}\}$ a family of $n$ elements in $\mathcal{A}$, such that the following three properties hold.

1. (F)

The elements of $\Pi$ (resp. $\Pi^{\vee}$) are linearly independent in $\mathcal{A}^{*}$ (resp. in $\mathcal{A}$). 2. (C)

For all $i,j\in\llbracket 1,n\rrbracket$, $\alpha_{j}(\alpha_{i}^{\vee})=a_{i,j}$. 3. (D)

We have $n-\mathrm{rk}(A)=\dim_{\mathbb{R}}\mathcal{A}-n$.

A generalized free realization of $A$ is a triple $(\mathcal{A},\Pi,\Pi^{\vee})$ with $\mathcal{A},\Pi,\Pi^{\vee}$ defined as above but only satisfying properties (F) and (C). Two realizations $(\mathcal{A}_{1},\Pi_{1},\Pi^{\vee}_{1})$ and $(\mathcal{A}_{2},\Pi_{2},\Pi_{2}^{\vee})$ are said isomorphic if there exists an isomorphism of vector spaces $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ such that $\phi^{*}(\Pi_{1})=\Pi_{2}$ and $\phi(\Pi^{\vee}_{1})=\Pi_{2}^{\vee}$. We know by [Kac94, Prop. 1.1] that up to (non unique in general) isomorphism, $A$ admits a unique realization $(\mathcal{A}_{0},\Pi_{0},\Pi_{0}^{\vee})$.

Given a generalized free realization $(\mathcal{A},\Pi,\Pi^{\vee})$ of $A$, we let the inessential part of $\mathcal{A}$ be the subspace $\mathcal{A}_{\mathrm{in}}:=\bigcap_{i=1}^{n}\ker\alpha_{i}$. We also set

[TABLE]

The next lemma is easy to prove and thus left to the reader.

Lemma 5.17**.**

For any generalized free realization $\mathcal{A}$ of $A$, there exist subspaces $\mathcal{A}^{\prime}\subset\mathcal{A}$ and $B\subset\mathcal{A}_{\mathrm{in}}$ such that $\mathcal{A}^{\prime}$ is isomorphic to $\mathcal{A}_{0}$ (as realizations of $A$), $Q^{\vee}_{\mathcal{A}}\subset\mathcal{A}^{\prime}$ and $\mathcal{A}=\mathcal{A}^{\prime}\oplus B$.

Let $W^{v}_{\mathcal{A}}$ be the Weyl group of $\mathcal{A}$, i.e the subgroup of $\mathrm{GL}(\mathcal{A})$ generated by the $r_{i}$, $i\in I$ (where $r_{i}:\mathcal{A}\to\mathcal{A}$ sends any $x\in\mathcal{A}$ to $x-\alpha_{i}(x)\alpha_{i}^{\vee}$).

Lemma 5.18**.**

For any generalized free realization $\mathcal{A}$ of $A$, the map

[TABLE]

is injective.

Proof.

Write $\mathcal{A}=\mathcal{A}^{\prime}\oplus B$ with $\mathcal{A}^{\prime}$ and $B$ as in Lemma 5.17. For any $x\in\mathcal{A}$ and $w\in W^{v}_{\mathcal{A}}$, we have $w(x)-x\in Q^{\vee}_{\mathbb{R},\mathcal{A}}$, hence $\mathcal{A}^{\prime}$ is stable under the action of $W^{v}_{\mathbb{A}}$. Moreover, $W^{v}_{\mathcal{A}}$ fixes pointwise $\mathcal{A}_{\mathrm{in}}$, hence the restriction map $W^{v}_{\mathcal{A}}\to W^{v}_{\mathcal{A}^{\prime}}$ is a an isomorphism. As a consequence, we can assume that $\mathcal{A}=\mathcal{A}_{0}$. Now apply assertion (3.12.1) of the proof of [Kac94, Prop. 3.12] to $\Delta^{\vee}$ instead of $\Delta$: we get that the only $w\in W^{v}_{\mathcal{A}_{0}}$ satisfying $w_{|\Delta^{\vee}}=1$ is $w=1$. As $\Delta^{\vee}$ is contained in $Q^{\vee}_{\mathcal{A}}$, this ends the proof. ∎

5.4.2 Infinite intersections of spheres

From now on, we assume that $F$ is a non-spherical type [math] face of $\mathbb{A}$ that satisfies $F_{0}\subsetneq F\subsetneq C^{+}_{0}$. Recall that this implies that the fixer $W_{F}$ of $W$ is infinite. Indeed, we can assume that $F$ has [math] for vertex, which identifies $W_{F}$ with a subgroup of $W^{v}$. Let $F^{v}$ be the vectorial face such that $F=F^{\ell}(0,F^{v})$ and let us prove that $W_{F}$ is also the fixer $W_{F^{v}}$ of $F^{v}$ (which will prove the claim as $F^{v}$ is non spherical, hence $W^{F^{v}}$ is infinite by definition). If $w\in W_{F}$, let $X\in F$ be fixed by $w$: then $w$ fixes $\mathbb{R}^{*}_{+}.X\supset F^{v}$ hence $w$ is in $W_{F^{v}}$. Conversely, we have $W_{F^{v}}\subset W_{F}$ because $F^{v}\in F$ (as [math] is special), hence $W_{F}=W_{F^{v}}$ is infinite.

Remark 5.19**.**

By [Rou11, §1.3], the vectorial faces based at [math] form a partition of the Tits cone. Therefore, for any vectorial face $F^{v}$, if there exist some $u\in F^{v}$ and some $w\in W^{v}$ such that $w\cdot u\in F^{v}$, then $w\cdot F^{v}=F^{v}$. Consequently, for any $W^{\prime}\subset W^{v}$, $W^{\prime}\cdot F^{v}$ is infinite if and only if $W^{\prime}\cdot u$ is infinite for some $u\in F^{v}$, if and only if $W^{\prime}\cdot u$ is infinite for all $u\in F^{v}$.

The proof of the next proposition uses the graph of the matrix $A$, whose vertices are the elements $i\in I$ and whose arrows are the pairs $\{i,j\}$ such that $a_{i,j}\neq 0$.

Lemma 5.20**.**

Suppose that the matrix $A$ is indecomposable. For any non-spherical type [math] face $F$ of $\mathbb{A}$ that satisfies $F_{0}\subsetneq F\subsetneq C^{+}_{0}$, there exists $w\in W^{v}$ such that $W_{F}\cdot w\cdot F$ is infinite.

Proof.

Write $F=F^{\ell}(0,F^{v})$ with

[TABLE]

for some subset $J$ of $I$. Note that $J\neq I$ as $F_{0}$ is strictly contained in $F$. Let $k\in I$ be such that $W_{F}\cdot\alpha_{k}^{\vee}$ is infinite (such a $k$ exists by Lemma 5.18). As the graph of $A$ is connected [Kac94, 4.7], any element $j\in I\setminus J$ can be linked to $k$ via a finite sequence $j=j_{1},\ldots,j_{\ell}=k$ of elements of $I$ that satisfy $\prod_{m=1}^{\ell-1}a_{j_{m},j_{m+1}}\not=0$. We fix such a $j$ and such a sequence $j_{1},\ldots,j_{\ell}$.

Now pick $u\in F^{v}$ and let us show the existence of some $w\in W^{v}$ such that $\alpha_{k}(w\cdot u)\neq 0$. Given $x\in\mathbb{A}$ and $m\in\llbracket 1,\ell\rrbracket$, we say that $x$ satisfies $P_{m}$ when $\alpha_{j_{m}}(x)\not=0$ and $\alpha_{j_{m^{\prime}}}(x)=0$ for all $m^{\prime}\in\llbracket m+1,l\rrbracket$. If $x\in\mathbb{A}$ satisfies $P_{m}$ for some $m\in\llbracket 1,\ell-1\rrbracket$, then $x^{\prime}:=r_{j_{m}}(x)$ satisfies $\alpha_{j_{m+1}}(x^{\prime})=-\alpha_{j_{m}}(x)a_{j_{m},j_{m+1}}\not=0$ (recall that $x^{\prime}=x-\alpha_{j_{m}}(x)\alpha_{j_{m}}^{\vee}$), hence $x^{\prime}$ satisfies $P_{s}$ for some $s\in\llbracket m+1,\ell\rrbracket$. As $u$ is in $F^{v}$ and $j_{1}=j$ is in $I\setminus J$, we have $\alpha_{j_{1}}(u)>0$, hence $u$ satisfies $P_{m}$ for some $m\in\llbracket 1,\ell\rrbracket$. Replacing $u$ by $x$ in the previous argument and using successive iterations, we finally get some $w\in W^{v}$ such that $w\cdot u$ satisfies $P_{\ell}$, i.e such that $\alpha_{k}(w\cdot u)\neq 0$.

We conclude as follows: if $W_{F}\cdot w\cdot u$ is finite, then $W_{F}.r_{k}(w\cdot u)=W_{F}.(u-\alpha_{k}(w\cdot u)\alpha_{k}^{\vee})$ is infinite, hence at least one of the orbits $W_{F}\cdot w\cdot u$ or $W_{F}.r_{k}(w\cdot u)$ is infinite, which implies the required result by Remark 5.19. ∎

Let $A_{1},\ldots,A_{r}$ be the indecomposable components of the Kac-Moody matrix $A$. For any $i\in\llbracket 1,r\rrbracket$, pick a realization $\mathcal{A}_{i}$ of $A_{i}$: then $\mathbb{A}=\mathcal{A}_{1}\oplus\ldots\oplus\mathcal{A}_{r}$. Also note that $W^{v}$ decomposes as $W^{v}=W^{v}_{1}\times\ldots\times W^{v}_{r}$, where $W^{v}_{i}$ denotes the vectorial Weyl group of $\mathcal{A}_{i}$, and that we can decompose any face $F^{\prime}$ of $\mathbb{A}$ as $F^{\prime}=F^{\prime}_{1}\oplus\ldots\oplus F^{\prime}_{r}$ with $F^{\prime}_{i}\subset\mathcal{A}_{i}$.

Proposition 5.21**.**

Let $F^{\prime}=\bigoplus_{i=1}^{r}F^{\prime}_{i}$ be a type [math] face of $\mathbb{A}$. The following are equivalent.

1. i.

There exists $w\in W^{v}$ such that $W_{F^{\prime}}\cdot w\cdot F^{\prime}$ is infinite. 2. ii.

There exists $i\in\llbracket 1,r\rrbracket$ such that $F^{\prime}_{i}$ is non-spherical and different from $F_{i,0}:=F^{\ell}(0,\mathcal{A}_{i,in})$. (Recall that $F_{i,0}$ is the minimal type [math] face of $\mathcal{A}_{i}$ based at [math].)

Proof.

The decomposition of $F^{\prime}$ induces a decomposition of its fixer as $W_{F^{\prime}}=W_{F^{\prime}_{1}}\times\ldots\times W_{F^{\prime}_{r}}$. First assume the existence of some $w\in W^{v}$ such that $W_{F^{\prime}}\cdot w\cdot F^{\prime}$ is infinite and decompose $w$ as $w=(w_{1},\ldots,w_{r})$. Then

[TABLE]

hence there is (at least) an integer $i\in\llbracket 1,r\rrbracket$ such that $W_{F^{\prime}_{i}}\cdot w_{i}\cdot F^{\prime}_{i}$ is infinite. For such an $i$, $F^{\prime}_{i}$ must be non-spherical (otherwise $W_{F^{\prime}_{i}}$ would be finite) and different from $F_{i,0}$ (otherwise $W_{F^{\prime}_{i}}\cdot w_{i}\cdot F^{\prime}_{i}=F_{i,0}$). Hence (i) implies (ii). The reverse implication is a consequence of Lemma 5.20. ∎

The next proposition gives a counterexample to Lemma 5.11 for non-spherical faces, which explains why we needed this restriction in our construction.

Proposition 5.22**.**

Let $F^{\prime}$ be a face based at [math] for which there exists some $w\in W^{v}$ such that $W_{F^{\prime}}w\cdot F^{\prime}$ is infinite. Then $\mathcal{S}^{F^{\prime}}(F^{\prime},[w\cdot F^{\prime}])\cap\mathcal{S}_{\mathrm{opp}}^{F^{\prime}}(F^{\prime},[w^{-1}\cdot F^{\prime}])$ is infinite.

Proof.

It is enough to check that $W_{F^{\prime}}w\cdot F^{\prime}$ is contained in

[TABLE]

Let $E\in W_{F^{\prime}}\cdot w\cdot F^{\prime}$ and let $w_{E}\in W_{F^{\prime}}$ be such that $E=w_{E}\cdot w\cdot F^{\prime}$. As $w_{E}\cdot F^{\prime}=F^{\prime}$, we have $F^{\prime}\leq E\leq F^{\prime}$, hence $d^{F^{\prime}}(F^{\prime},E)=[E]=[w\cdot F^{\prime}]$ by definition of $d^{F^{\prime}}$. Now the isomorphism $(w_{E}\cdot w)^{-1}:\mathbb{A}\to\mathbb{A}$ maps $E$ to $F^{\prime}$ and $F^{\prime}$ to $w^{-1}\cdot w_{E}^{-1}\cdot F^{\prime}=w^{-1}\cdot F^{\prime}$, thus $d^{F^{\prime}}(E,F^{\prime})=[w^{-1}\cdot F^{\prime}]$. This shows that $E$ belongs to

[TABLE]

hence the proposition. ∎

Recall that the notations $Y^{f}$ and $Y^{\infty}_{\mathrm{in}}$ used in the next result were introduced in Section 4.5.2.

Corollary 5.23**.**

Let $\lambda\in Y^{+}$. Its $W^{v}$-orbit $W^{v}\cdot\lambda$ is finite iff $\lambda$ belongs to $Y^{f}\oplus Y^{\infty}_{\mathrm{in}}$.

Proof.

Given $\lambda\in Y^{+}$, write $\lambda=\sum_{j=1}^{r}\lambda_{j}$ with $\lambda_{j}\in\mathcal{A}_{j}$ for all $j\in\llbracket 1,r\rrbracket$. First assume that $\lambda$ is in $Y^{f}\oplus Y^{\infty}_{\mathrm{in}}$: then

[TABLE]

As $W^{v}_{j}$ is finite for any $j\in J^{f}$, the finiteness of $W^{v}\cdot\lambda$ follows from its decomposition above and the converse implication is proved.

Now assume that $\lambda$ is not in $Y^{f}\oplus Y^{\infty}_{\mathrm{in}}$. Let $j\in J^{\infty}$ be such that $\lambda_{j}\not\in\mathcal{A}_{j,\mathrm{in}}$ and let $F^{v}_{j}$ be the vectorial face of $\mathcal{A}_{j}$ that contains $\lambda_{j}$. By Remark 5.19, the map $W^{v}_{j}\cdot F^{v}_{j}\lambda_{j}\to W^{v}_{j}\lambda_{j}$ that sends $w\cdot F^{v}_{j}$ onto $w\cdot\lambda_{j}$ is well-defined and bijective. If $F^{v}_{j}$ is spherical, then its stabilizer is finite and $W^{v}_{j}\cdot F^{v}_{j}$ is thus infinite as $W^{v}_{j}$ is. If $F^{v}_{j}$ is non-spherical, then Lemma 5.20 produces an element $w_{j}\in W^{v}_{j}$ such that $W_{F_{j}}\cdot w_{j}\cdot F^{v}_{j}$ is infinite, where $W_{F_{j}}$ is also the fixer of $F^{v}_{j}$ in $W^{v}_{j}$. In any case, $W^{v}_{j}\cdot F^{v}_{j}$ is infinite, hence so is $W^{v}_{j}\cdot\lambda_{j}$, which ends the proof. ∎