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\datereceived{2020-09-03}
\dateaccepted{2022-07-14}
\dateepreuves{2022-08-18}

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\datepublished{2022-08-25}
\begin{document}
\frontmatter
\title[Erratum to ``Completed Iwahori-Hecke algebra...'']{Erratum to\\ ``Completed Iwahori-Hecke algebra and parahoric Hecke algebras for Kac-Moody~groups over local fields''}

\author[\initial{R.} \lastname{Abdellatif}]{\firstname{Ramla} \lastname{Abdellatif}}
\address{LAMFA -- UPJV, UMR CNRS 7352\\
80 039 Amiens Cedex 1, France}
\email{ramla.abdellatif@u-picardie.fr}
\urladdr{http://www.lamfa.u-picardie.fr/abdellatif/}

\author[\initial{A.} \lastname{Hébert}]{\firstname{Auguste} \lastname{Hébert}}
\address{Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502 CNRS\\
Boulevard des Aiguillettes, 54506 Vandœuvre-lès-Nancy, France}
\email{auguste.hebert@univ-lorraine.fr}
\urladdr{https://iecl.univ-lorraine.fr/membre-iecl/hebert-auguste/}

\begin{abstract}
We modify the definition of the completed Iwahori-Hecke algebra given in our previous article (J.\ Éc.\ Polytechnique {\bf 6}, 79--118) and explain why the construction we gave earlier is not correct as such.
\end{abstract}

\subjclass{20G44, 20C08}

\keywords{Kac-Moody groups, Hecke algebras, Iwahori-Hecke algebras, local fields}

\altkeywords{Groupes de Kac-Moody, algèbres de Hecke, algèbres d'Iwahori-Hecke, corps locaux}

\alttitle{Erratum à \guillemotleft Algèbres d'Iwahori-Hecke complétées et algèbres de Hecke parahoriques pour les groupes de Kac-Moody sur les corps locaux\guillemotright}

\begin{altabstract}
Nous modifions la définition de l'algèbre de Hecke complétée donnée dans notre précédent article (J.\ Éc.\ Polytechnique {\bf 6}, 79--118) et expliquons pourquoi la définition que nous avions donnée n'était pas correcte.
\end{altabstract}

\maketitle
\tableofcontents
\mainmatter

The goal of this erratum is to fix the construction of the \emph{completed Iwahori-Hecke algebra} given in \cite[\S 4.4]{abdellatif2019completed}, as the definition given there does not always provide an actual algebra (see Section~\ref{secCouter-examples} below). We define here an algebra that must be used instead. Consequently, the following modifications should be operated in \cite{abdellatif2019completed} : the paragraph before \cite[Prop.\,4.33]{abdellatif2019completed} is wrong and must be replaced by Section~\ref{SectionConstructionTildeHC} below; Theorem 4.21, Corollary 4.23 and Theorem 4.30 of \cite{abdellatif2019completed} are wrong as stated there and must respectively be replaced by Theorem~\ref{thmAH4.21}, Corollary~\ref{corAH4.23} and Theorem~\ref{thmAH4.30} below.

\section{Introduction}
Let $G$ be a split Kac-Moody group (as defined by Tits in \cite{tits1987uniqueness}) over a non-Archimedean local field $\mathcal{K}$. Given a ring $\RCC$ containing $\Z$ and satisfying light technical conditions (as in \cite[Rem.\,4.1]{abdellatif2019completed}), Braverman-Kazdhan-Patnaik \cite{braverman2011spherical, braverman2016iwahori} and Bardy--Panse-Gaussent-Rousseau \cite{gaussent2014spherical, bardy2016iwahori} associated to $G$ a spherical Hecke algebra $\HC_{s}$ and an Iwahori-Hecke algebra $\HC$, both defined over $\RCC$. Fixing a maximal split torus $T$ of $G$, and letting $Y$ (\resp $Y^{+}$) be the cocharacter lattice (\resp its intersection with the Tits cone) and $W^{v}$ be the Weyl group of $(G,T)$, then these authors moreover proved the existence of a Satake isomorphism from $\HC_{s}$ to $\mathscr{R}[\![Y]\!]^{W^v}$, where $\mathscr{R}[\![Y]\!]$ is the Looijenga algebra, which is a completion of the group algebra $\mathscr{R}[Y]$ of $Y$ over $\mathscr{R}$ (see \cite[Def.\,4.6]{abdellatif2019completed} for its definition). A striking difference with the classical case of reductive groups is that for $G$ non-reductive, this spherical Hecke algebra is not isomorphic to the center of the Iwahori-Hecke algebra~$\HC$.

In \cite{abdellatif2019completed}, we announced the definition of a completed Iwahori-Hecke algebra $\widehat{\HC}$ that contains $\HC$ and whose center is isomorphic to $\mathscr{R}[\![Y]\!]^{W^v}$, hence to $\HC_s$ \cite[Th.\,4.30]{abdellatif2019completed}. Nevertheless, the construction of $\widehat{\HC}$ we gave in \cite[\S 4.4, p.\,94--100]{abdellatif2019completed} is not correct as stated, since as such, $\widehat{\HC}$ is actually not stable in general under the convolution product. Indeed, the product of two elements of $\widehat{\HC}$ can lead to infinite coefficients, as will be seen below in Sections~\ref{exReductive_case} and \ref{ex_Counter_example}. This erratum corrects this mistake by defining a slightly different algebra $\widetilde{\HC}$, contained in the vector space $\widehat{\HC}$, for a suitable convolution product (see Corollary~\ref{corAH4.23}). The main point is to use the correct notion of almost-finiteness in the definition of the support of the elements of the completed algebra. We check here that $\widetilde{\HC}$ contains $\HC$ and that the center of $\widetilde{\HC}$ is isomorphic to $\RCC[\![Y]\!]^{W^{v}}$, hence to $\HC_{s}$ (see Theorem~\ref{thmAH4.30}), as aimed at first. Moreover note that this modified definition of the completed Iwahori-Hecke algebra suppresses the aforementioned gap between the reductive and non-reductive cases, as~for $G$ reductive, $\widetilde{\HC}$ is actually isomorphic to the classical Iwahori-Hecke algebra (see~Proposition~\ref{Prop44}).

This erratum is organized as follows. In Section~\ref{secCouter-examples}, we give two counter-examples to \cite[Th.\,4.21]{abdellatif2019completed}: one in the reductive case and one in the non-reductive case. Then we introduce the required modifications in the definition of the completed algebra to build $\widetilde{\HC}$ in Section~\ref{SectionConstructionTildeHC}. In particular, we explain in Section~\ref{secCenter} how to adapt the content of \cite[\S 4.4, p.\,94--100]{abdellatif2019completed} to prove that the center of $\widetilde{\HC}$ is isomorphic to $\mathscr{R}[\![Y]\!]^{W^v}$.

\subsubsection*{Acknowledgements} We thank the referees for their valuable comments and suggestions.

\section{Two counter-examples to \texorpdfstring{\cite[Th.\,4.21]{abdellatif2019completed}}{cite}}
\label{secCouter-examples}

We keep the notation of \cite[\S 2]{abdellatif2019completed}. Let us briefly recall that, as in \cite[\S 2.1]{abdellatif2019completed}, given a root generating system $\mathcal{S} = (A,X,Y,(\alpha)_{i \in I }, (\alpha_{i}^{\vee})_{i \in I})$, we set $\A := Y \otimes \R$, let~$W^{v}$ denote the Weyl group of $\mathcal{S}$, $Q^{\vee} := \bigoplus_{i \in I} \Z\alpha_{i}^{\vee}$ denote its coroot lattice, $Q^{\vee}_{\R,+}:=\bigoplus_{i\in I} \R_{+} \alpha_{i}^{\vee}$ and $Q^{\vee}_\R=\bigoplus_{i\in I} \R\alpha_{i}^{\vee}$.

We then have $\HC=\bigoplus_{\lambda\in Y^{+}, w\in W^{v}} \mathscr{R} Z^{\lambda} H_{w}$, where $Z^\lambda$ and $H_w$ are symbols that satisfy relations (BL1) to (BL4) of \cite[\S 4.1, page 91]{abdellatif2019completed}. The notion of support is defined in \cite[Def.\,4.11]{abdellatif2019completed}. In this section, we give two examples of elements $(a_{j})_{j\in J}, (b_{k})_{k\in K}$ in $\HC$ that are summable in $\widehat{\HC}$ in the sense of \cite[Def.\,4.20]{abdellatif2019completed}, but such that $(a_{j}*b_{k})_{j\in J,k\in K}$ is not summable: one is a reductive case (Section \ref{exReductive_case}), the other one is an affine Kac-Moody case (Section \ref{ex_Counter_example}). This prevents \cite[Th.\,4.21]{abdellatif2019completed} from being true as stated, and we explain in Section~\ref{positionpbsupport} why the problem stands in the notion of almost-finiteness initially used in the definition of $\widehat{\HC}$, and how to modify it to get a correct analogue of \cite[Th.\,4.21]{abdellatif2019completed}.

\subsection{A counter-example in the reductive case}
\label{exReductive_case}
Assume that the standard apartment $\A$ is associated with a Cartan matrix. Fix $\lambda\in Y=Y^{+}$ and $i\in I$. For $j\in \N$, set $a_{j}=Z^{\lambda-j\alpha_i^\vee}H_{i}$ and $b_{j}=Z^{\lambda-j\alpha_i^\vee}$. As $\lambda-\N\alpha_{i}^{\vee}$ is almost finite, $(a_j)_{j\in \N}$ and $(b_k)_{k\in \N}$ are summable in $\widehat{\HC}$. Now let $j,k\in \N$. By (BL4) (see \cite[\S 4.1 p.\,91]{abdellatif2019completed}), there exists $c_{j,k}\in \mathscr{R}[\![Y]\!]$ such that
\[
a_{j}*b_{k}=Z^{\lambda+r_i(\lambda)+(k-j)\alpha_{i}^{\vee}} H_{i}+c_{j,k}.
\]
This implies that $(\lambda+r_{i}(\lambda)+(k-j)\alpha_{i}^{\vee},r_{i})\in \supp(a_{j}*b_{k})$, hence $(a_{j}*b_{k})_{(j,k)\in \N^{2}}$ satisfies none of the conditions of \cite[Def.\,4.20]{abdellatif2019completed}, so it is not summable in $\widehat{\HC}$.

\subsection{A counter-example in the non-reductive case}
\label{ex_Counter_example}
Assume now that $\A$ is associated with an indecomposable affine Kac-Moody matrix $A$. Let $\delta:\A\to \R$ be the smallest positive imaginary root associated with $A$. Fix $\lambda\in Y^{+}$ such that $\delta(\lambda)>0$ and $i \in I$: then \cite[\S 4.2.2, p.\,92--93]{abdellatif2019completed} ensures that $\lambda-\N \alpha_{i}^{\vee}$ is an almost finite subset of $Y^{+}$. For $j\in \N$, set $a_{j}=Z^{\lambda-j\alpha_{i}^{\vee}}H_{i}$ and $b_{j}=Z^{\lambda-j\alpha_{i}^{\vee}}$: then the same process as in the reductive case (Section~\ref{exReductive_case} above) shows that $(a_j*b_k)_{(j,k)\in \N^2}$ satisfies none of the conditions of \cite[Def.\,4.20]{abdellatif2019completed}, hence is not summable in $\widehat{\HC}$, although both $(a_{j})$ and~$(b_{k})$ are summable in $\widehat{\HC}$.

\subsection{Position of the problem and modifications required}
\label{positionpbsupport}
The definition of $\widehat{\HC}$ given in \cite[p.\,95]{abdellatif2019completed} crucially relies on the notion of almost finiteness defined in \cite[Def.\,4.12]{abdellatif2019completed}. The problem is that almost finiteness is not preserved by the action of $W^v$ on $Y^+$: there can exist (depending on the Kac-Moody matrix $A$) an almost finite set $E$ such that $w\cdot E$ is not almost finite, for some $w\in W^v$. To fix this problem, we introduce in the next section a refined notion of almost-finiteness, namely the notion of \emph{$W^{v}$-almost finiteness}. Using this new notion, we define an algebra $\widetilde{\HC}$ through an analogous construction to the one done for $\widehat{\HC}$ in \cite[\S 4.4, pages 94-100]{abdellatif2019completed}. We then explain why the results and proofs stated for $\widehat{\HC}$ in \cite{abdellatif2019completed} are now valid for $\widetilde{\HC}$.

Before going further, let us list precisely what modifications are actually done in this erratum.
\begin{itemize}
\item
The notion of almost finiteness defined in \cite[Def.\,4.12]{abdellatif2019completed} must be replaced by the notion of \emph{$W^{v}$-almost finiteness} introduced in Definition \ref{DeffWvalmostfinite} below to define $\widetilde{\HC}$ as we defined $\widehat{H}$ but with the aforementioned replacement.
\item
The statement and proof of \cite[Th.\,4.21, Cor.\,4.23 \& Th.\,4.30]{abdellatif2019completed} must be respec\-tively replaced by the statement and proof of Theorem~\ref{thmAH4.21}, Corollary~\ref{corAH4.23} and Theorem~\ref{thmAH4.30} below.
\item
The content of the paragraph before \cite[Prop.\,4.33]{abdellatif2019completed}, which explains what happens in the reductive case, must be replaced by Section \ref{CPcasreductif} below.
\end{itemize}

\section{The completed Iwahori-Hecke algebra \texorpdfstring{$\widetilde{\HC}$}{HC}}
\label{SectionConstructionTildeHC}
The goal of this section is to build an algebra $\widetilde{\HC}$ that appears to be smaller than~$\widehat{\HC}$ (that is \emph{not} always an algebra) whose center is (still) isomorphic to $\RCC[\![Y]\!]^{W^v}$ and that (still) contains $\HC$ as the subalgebra of finitely supported elements. It actually boils down to defining the right notion of almost-finiteness and checking that what we did in \cite[\S 4]{abdellatif2019completed} transposes in this setting to define an actual algebra $\widetilde{\HC}$ with the required properties.
\subsection{$W^{v}$-almost finiteness and definition of $\widetilde{\HC}$}
\label{sectionalmostfiniteness}
The idea behind the use of the following refined notion of almost finiteness is that it is preserved by the action of~$W^v$.
\begin{Def}
\label{DeffWvalmostfinite}
Let $u\in W^v$.
\begin{itemize}
\item A subset $E$ of $Y^{+}$ is \textit{$u$-almost finite} if $u\cdot E$ is almost finite in the sense of \cite[Def.\,4.3]{abdellatif2019completed}.
\item A subset $E$ of $Y^{+}\times W^{v}$ is called \textit{$u$-almost finite} if its projection on $W^v$ is finite and if its projection on $Y^+$ is $u$-almost finite (as a subset of $Y^+$).
\item A subset of $Y^{+}$ or of $Y^{+} \times W^{v}$ is \emph{$W^{v}$-almost finite} if it is $u$-almost finite for any $u \in W^{v}$.
\end{itemize}
\end{Def}

As in \cite[p.\,95]{abdellatif2019completed}, we set $\mathscr{B}=\prod_{\lambda\in Y^{+},w\in W^{v}} \mathscr{R}$ and for $(\lambda, w) \in Y^{+} \times W^{v}$, we let $Z^{\lambda}H_{w}$ denote the element whose coefficients are all equal to $0$ apart from the coefficient indexed by $(\lambda, w)$, which is equal to $1$. This allows us to write $a=(a_{\lambda,w})_{(\lambda,w)\in Y^{+}\times W^{v}}\in \mathscr{B}$ as the formal linear combination
\[
a= \sum_{\substack{\lambda\in Y^{+}\\w\in W^{v}}} a_{\lambda,w}Z^{\lambda} H_{w}.
\]
Also recall that any $(\nu, u) \in Y^{+}\times W^{v}$ is associated to a projection map $\pi_{\nu,u}:\mathscr{B}\to \mathscr{R}$ defined by
\[
\pi_{\nu,u}\Biggl(\sum_{\substack{u'\in W^v\\\nu'\in Y^+}} c_{\nu',u'}Z^{\nu'}H_{u'}\Biggr) =:c_{\nu,u}
\]
for any $ \sum c_{\nu',u'}Z^{\nu'}H_{u'}\in \mathscr{B}$.

We can now define $\widetilde{\HC}$ as the set of elements of $\mathscr{B}$ with $W^{v}$-almost finite support. To prove that $\widetilde{\HC}$ can be endowed with a convolution product $*$ that turns it into an associative algebra containing $\HC$, we will basically follow the same steps as in \cite[\S 4.4]{abdellatif2019completed}, replacing the almost finiteness condition by the $W^{v}$-almost finiteness condition.

We let $\conv_{\R}(F)$ denote the convex hull of any part $F$ of $\A$, and we set $\conv(E) := \conv_{\R}(E) \cap Y$ for any subset $E$ of $Y$. Following \cite[p.\,95]{abdellatif2019completed}, recall that for any part~$E$ of~$Y$ and any $i\in I$, we let $R_{i}(E)=\conv(E\cup r_i(E)) \subset E + Q^{\vee}$ and that, for~any pair $(\lambda, w) \in Y^{+} \times W^{v}$, we 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 $w$. The next two results replace \cite[Rem.\,4.13]{abdellatif2019completed} and act as preparation for the proof of Lemma \ref{lemAH_Lemma_4.15} below, which replaces \cite[Lem.\,4.15]{abdellatif2019completed}.

\begin{lemma}\label{lemInclusion_R_convex_hull}
For any $(\lambda, w) \in Y^{+} \times W^{v}$, we have
\[R_w(\lambda)\subset \conv(\{u\cdot \lambda \mid u\in [1,w]\}),\]
where $[1,w] := \{ u \in W^{v} \mid u \leq w \}$ is defined as in \cite[bottom of p.\,94]{abdellatif2019completed}.
\end{lemma}

\begin{proof}
We prove this result by induction on $\ell(w)$. If $\ell(w)=0$, there is nothing to prove, so let $w \in W^{v}$ be an element of length $\ell(w) \geq 1$ and assume by induction that the lemma holds for any element $w' \in W^{v}$ such that $\ell(w') < \ell(w)$. Let $\mu \in R_{w}(\lambda)$, then there exists $i \in I$ such that $w' := r_{i}w$ satisfies $w' < w$ and $\mu \in R_{i}\left(R_{w'}(\lambda)\right) = \conv(R_{w'}(\lambda), r_{i}R_{w'}(\lambda))$. As $\ell(w') < \ell(w)$, we have $R_{w'}(\lambda)\subset \conv(\{u\cdot \lambda \mid u\in [1,w']\})$ by induction hypothesis. Since \cite[Cor.\,1.3.19]{kumar2002kac} ensures that $\{1,r_{i}\}\cdot[1,w']\subset [1,w]$, we obtain that
\[
\conv(\{u\cdot \lambda \mid u\in [1,w']\})\cup r_{i}\cdot \conv(\{u\cdot \lambda \mid u\in [1,w']\})\subset \conv(\{u\cdot \lambda \mid u\in [1,w]\}).
\]
Consequently, we get that
\[
\begin{aligned}
\mu\in \conv&(R_{w'}(E),r_{i}\cdot R_{w'}(E))\\
& \subset \conv(\conv(\{u\cdot \lambda \mid u\in [1,w']\}) \cup r_{i}\cdot \conv(\{u\cdot \lambda \mid u\in [1,w']\}))\\
&\subset \conv(\conv(\{u\cdot \lambda \mid u\in [1,w]\})) = \conv(\{u\cdot \lambda \mid u\in [1,w]\}).
\end{aligned}
\]
This proves that $R_{w}(\lambda)$ is contained in $\conv(\{u\cdot \lambda \mid u\in [1,w]\})$, hence the lemma.
\end{proof}

\begin{lemma}\label{lemFiniteness_mu_such_Ru(mu)ninu}
Let $E$ be a $W^{v}$-almost finite subset of $Y^{+}$.
Then, for any pair $(\nu, w) \in Y^{+} \times W^{v}$, the set $\{\mu \in E \mid \nu \in R_{w}(\mu)\}$ is finite.
\end{lemma}

\begin{proof}
Let $E \subset Y^{+}$ and $(\nu, w) \in Y^{+} \times W^{v}$ be as in the statement. Applying the definition of almost finiteness \cite[Def.\,4.3]{abdellatif2019completed} to $u\cdot E$ for any $u \in [1,w]$ provides a finite set $F\subset Y^{+}$ such that :
\[
\forall u \in [1,w], \ \forall \mu \in u\cdot E, \ \exists \lambda \in F \mid \mu \leq_{Q^{\vee}} \lambda.
\]
Set $\mathcal{X}:=\{\mu \in E \mid \nu \in R_{w}(\mu)\}$ and pick some $\mu \in \mathcal{X}$. As $\nu$ belongs to $R_{w}(\mu)$, Lemma~\ref{lemInclusion_R_convex_hull} implies the existence of $(t_{u})_{u \in [1,w]}\in [0,1]^{[1,u]}$ such that
\[
\sum_{u \in [1,w]}t_{u}=1 \ \text{ and } \ \nu=\sum_{u\in [1,w]} t_{u}u\cdot \mu.
\]
For any $u \in [1,w]$, choose $\lambda(u)\in F$ such that $u\cdot \mu\leq_{Q^{\vee}} \lambda(u)$ and write $\lambda(u)-u\cdot \mu$ as $ \sum_{i\in I} n_{i}(u) \alpha_{i}^{\vee}$ with $n_i(u)\in\N$ for all $i\in I$. Then we have:
\[
\nu = \sum_{u\in [1,w]}t_{u}u\cdot \mu = \sum_{u \in [1,w]}t_{u}\lambda(u)-\sum_{\substack{u\in [1,w]\\ i\in I}} t_{u}n_{i}(u)\alpha_{i}^{\vee}.
\]
Set $a(\mu) := \sum_{u \in [1,w]}t_{u}\lambda(u)\in \conv_\R(F)$ and $q(\mu) := \sum_{u \in [1,w], \ i\in I} t_{u}n_{i}(u)\alpha_{i}^{\vee}\in Q^{\vee}_{\R,+}$. Since $F$ is finite, $\conv_\R(F)$ is bounded. As $q(\mu) = a(\mu) - \nu$ lies in $\conv_{\R}(F) - \nu$, the set $\{q(\mu')\mid \mu'\in \mathcal{X}\}$ is bounded too. Moreover, as $\sum_{u\in [1,w]} t_u=1$, there exists $u'\in [1,w]$ such that $t_{u'}\geq \sfrac{1}{\left\vert[1,w]\right\vert}$. Letting $f_{j}(x)$ denote the $j$-th coordinate of $x \in Q^{\vee}_{\R}$ in the basis $(\alpha_{i}^{\vee})_{i\in I}$ for all $j \in J$, we have:
\[
\forall i \in I, \quad f_{i}\left(q(\mu)\right)=\sum_{u\in [1,w]}t_{u}n_{i}(u) \geq t_{u'}n_{i}(u') \geq 0.
\]
We hence obtain that:
\[
\forall i \in I, \quad 0 \leq n_{i}(u') \leq \frac{ \sup_{\mu' \in \mathcal{X}} f_{i}\left(q(\mu')\right)}{t_{u'}}\leq \frac{ \sup_{\mu'\in \mathcal{X}} f_{i}\left(q(\mu')\right)}{\left\vert[1,u]\right\vert}.
\]
Consequently, if we set $ N :=\max_{i\in I} \Psfrac{\sup_{\mu'\in \mathcal{X}} f_{i}\left(q(\mu')\right)}{\left\vert[1,u]\right\vert}$, then we have:
\[
u'\cdot\mu \in \lambda(u') - \sum_{i\in I} \llbracket 0,N\rrbracket\alpha_{i}^{\vee} \subset F-\sum_{i\in I}\llbracket 0,N\rrbracket\alpha_{i}^{\vee}.
\]
This proves that $\mu$ lies in $ \bigcup_{u\in [1,w]} u^{-1}\cdot(F- \sum_{i\in I}\llbracket 0,N\rrbracket\alpha_{i}^{\vee})$, hence that $\mathcal{X}$ is contained in the finite set $ \bigcup_{u \in [1,w]} u^{-1}\cdot(F- \sum_{i\in I}\llbracket 0,N\rrbracket\alpha_{i}^{\vee})$, which proves that $\mathcal{X}$ is finite too, as claimed.
\end{proof}

The next lemma replaces \cite[Lem.\,4.15]{abdellatif2019completed}: the only modification consists in replacing $[1,w]w'$ by $[1,w']w$ in the aforementioned statement. In particular, the proof follows the exact same lines as \cite[p.\,96]{abdellatif2019completed}, hence we do not rewrite it here.

\begin{lemma}\label{lemAH_Lemma_4.15}
For all $w,w' \in W^{v}$ and all $\lambda \in Y$, $H_{w'}*Z^{\lambda}H_{w}$ is in
\[
\bigoplus_{(\nu, t) \in R_{w'}(\lambda) \times [1,w']\cdot w} \mathscr{R}\cdot Z^{\nu}H_{t}.
\]
\end{lemma}

Using the definitions of $\supp_{W^{v}}$ and $\supp_{Y}$ given by \cite[Def.\,4.11]{abdellatif2019completed}, one can straightforward deduce from Lemma \ref{lemAH_Lemma_4.15} the following inclusions.
\begin{lemma}\label{corSupport_convolution}
For all $a,b\in \HC$, we have:
\begin{enumerate}
\item[\tt $(1)$] $ \supp_{Y}(a*b)\subset \supp_{Y}(a)+\bigcup_{\substack{w \in \supp_{W^{v}}(a)\\\lambda \in \supp_Y(b)}}R_{w}(\lambda)$;
\item[\tt $(2)$] $ \supp_{W^{v}}(a*b)\subset \bigcup_{\substack{v \in \supp_{W^{v}}(a)\\w \in \supp_{W^{v}}(b)}} [1,v]\cdot w$.
\end{enumerate}
\end{lemma}

Before we give the definition of summable families in $\widetilde{\HC}$, we prove two more statements related to $W^{v}$-almost finiteness in $Y^{+}$.
\begin{lemma}\label{lemConvex_hull_almost_finite_set}
For any almost finite set $E\subset Y^{+}$, $\conv(E)$ is also almost finite.
\end{lemma}

\begin{proof}
Let $E \subset Y^{+}$ be an almost finite set and let $F$ be a finite set such that:
\[
\forall \lambda \in E, \ \exists \mu \in F \mid \lambda \leq_{Q^{\vee}} \mu.
\]
Given $\lambda\in \conv(E)$, there exist $n\in \N$, $t_{1},\ldots, t_{n} \in [0,1]$ and $\lambda_{1},\ldots,\lambda_{n} \in E$ such that
\[
\sum_{i=1}^{n} t_{i} = 1 \ \text{ and } \ \sum_{i=1}^{n} t_{i}\lambda_{i}=\lambda.
\]
For each index $i \in \llbracket 1,n \rrbracket$, choose $\kappa_{i} \in F$ such that $\lambda_{i} \leq_{Q^\vee} \kappa_{i}$: then
\[
\sum_{i=1}^{n} t_{i} \kappa_{i} - \sum_{i=1}^{n} t_{i} \lambda_{i} \in \bigoplus_{i\in I} \R_{+} \alpha_{i}^{\vee}.
\]
We can hence write $ \sum_{i=1}^{n} t_{i} \kappa_{i}-\sum_{i=1}^{n} t_{i}\lambda_{i} =\sum_{i\in I} x_{i}\alpha_{i}^{\vee}$ for some nonnegative real numbers $(x_{i})_{i \in I}$. Now let $(x'_{i})\in [0,1[^{I}$ be such that $x_{i} + x'_{i}$ lies in $\N$ for all $i\in I$ and set $\nu := \sum_{i\in I} (x_{i}+x'_{i})\alpha_{i}^{\vee}+\lambda$. Then we have
\[
\nu\geq_{Q^\vee} \lambda\quad\text{and}\quad \textstyle\nu\in (\conv_\R(F)+\bigoplus_{i\in I} [0,1]\alpha_i^\vee)\cap Y.
\]
Since $F$ is finite, $ (\conv_\R(F)+\bigoplus_{i\in I} [0,1]\alpha_{i}^{\vee})\cap Y$ is a finite set that can be taken as $J$ in \cite[Def.\,4.3]{abdellatif2019completed} for $\conv(E)$, and the lemma is proved.
\end{proof}

\begin{lemma}\label{lemU_almost_finiteness_R_w}
Let $E$ be a $W^{v}$-almost finite subset of $Y^{+}$. Then, for any $w\in W^{v}$, the set $ \bigcup_{\lambda\in E} R_{w}(\lambda)$ is $W^{v}$-almost finite.
\end{lemma}

\begin{proof}
Let $w \in W^{v}$. By Lemma~\ref{lemInclusion_R_convex_hull}, we have
\[
\bigcup_{\lambda \in E} R_{w}(\lambda) \subset \bigcup_{\lambda \in E} \conv(\{u\cdot \lambda, \ u\in [1,w]\}) \subset \conv\biggl(\bigcup_{u\in [1,w]} u\cdot E\biggr).
\]
Let $v\in W^{v}$. Since $[1,w]$ is finite, the set $ v\cdot \bigcup_{u\in [1,w]} u\cdot E$ is almost finite, hence Lemma~\ref{lemConvex_hull_almost_finite_set} implies that $ v\cdot \conv(\bigcup_{u\in [1,w]} u\cdot E)=\conv(v\cdot \bigcup_{u\in [1,w]} u\cdot E)$ is almost finite. This proves that $ \conv(\bigcup_{u\in [1,w]} u\cdot E)$ is $v$-almost finite for any $v \in W^{v}$, and the lemma is proved.
\end{proof}

\subsection{$\widetilde{\HC}$ is an associative algebra}
This subsection contains the main modification of the paper, as it aims to prove that $\widetilde{\HC}$ is actually an associative algebra. To do this, we first need to introduce the correct definition of summable families, which is the counterpart of \cite[Def.\,4.20]{abdellatif2019completed} for $W^{v}$-almost finite sets.

\begin{Def}\label{defSummable_family}
A family $ (a_j)_{j\in J}\in(\widetilde{\HC})^{J}$ is \emph{summable in $\widetilde{\HC}$} when the two following properties hold:
\begin{enumerate}\renewcommand{\theenumi}{\roman{enumi}}
\item\label{defSummable_familyi}
for any $\lambda \in Y^{+}$, the set $\{j \in J \mid \exists w \in W^{v}, \ \pi_{\lambda, w}(a_{j}) \not= 0\}$ is finite;
\item\label{defSummable_familyii}
the set $ \bigcup_{j \in J} \supp(a_{j}):=\bigcup_{j\in J}\{(\lambda,w)\in Y^+\times W^v\mid \pi_{\lambda,w}(a_j)\neq 0\}$ is $W^{v}$\nobreakdash-al\-most finite.
\end{enumerate}
Given a summable family $(a_{j})_{j \in J} \in(\widetilde{\HC})^{J}$, we define $ \sum_{j \in J} a_{j} \in \widetilde{\HC}$ by the following formula:
\[
\sum_{j \in J} a_{j} := \hspace*{-2mm}\sum_{(\lambda, w) \in Y^{+} \times W^{v}}\hspace*{-2mm} a_{\lambda, w}Z^{\lambda}H_{w}, \text{ with } a_{\lambda,w} := \sum_{j \in J} \pi_{\lambda,w}(a_{j}) \text{ for all } (\lambda, w) \in Y^{+} \times W^{v}.
\]
\end{Def}

\begin{lemma}\label{lemFiniteness_possibilities_sum}
For any almost finite subsets $E,E'$ of $Y^{+}$ and any $\rho\in Y^{+}$, the set
\[
E'':= \{\lambda\in E \mid \exists \nu\in E', \ \lambda+\nu=\rho\}
\] is finite.
\end{lemma}

\begin{proof}
By definition, there exists a finite set $F\subset Y^{+}$ such that:
\[
\forall \mu\in E\cup E', \ \exists \kappa \in F \mid \mu\leq_{Q^{\vee}} \kappa.
\]
Now let $\lambda\in E''$ and $\nu\in E'$ be such that $\lambda+\nu=\rho$. Then we have:
\[
\exists (\kappa, \kappa') \in F^{2} \mid \rho-\kappa'\leq_{Q^{\vee}} \lambda \leq_{Q^{\vee}} \kappa,
\]
and the lemma follows.
\end{proof}

The next lemma is the cornerstone that ensures that the convolution product on~$\widetilde{\HC}$ is well-defined.
\begin{lemma}\label{lem_Key_lemma}
Let $(a_{\lambda}),(b_{\mu})\in \RCC^{Y^{+}}$ be such that $\{\lambda\in Y^{+}\mid a_{\lambda} \neq 0\text{ or }b_{\lambda}\neq 0\}$ is $W^{v}$-almost finite. Then, for any $w \in W^{v}$, $(a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z_{\mu})_{(\lambda,\mu)\in (Y^{+})^{2}}$ is summable in $\widetilde{\HC}$. Said differently, if $ \sum_{\lambda\in Y^{+}} a_{\lambda} Z^{\lambda} H_{w},\sum_{\mu\in Y^{+}} b_{\mu} Z^{\mu} \in \widetilde{\HC}$, then
\[
\biggl(\sum_{\lambda \in Y^{+}} a_{\lambda} Z^{\lambda} H_{w}\biggr)*\biggl( \sum_{\mu\in Y^{+}} b_{\mu} Z^{\mu} \biggr):=\sum_{\lambda,\mu\in Y^{+}} a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z^{\mu}\] is a well-defined element of $\widetilde{\HC}$.
\end{lemma}

\begin{proof}
Set
\[
S_{a} :=\{\lambda\in Y^{+} \mid a_{\lambda} \neq 0\},\quad S_{b}=\{\mu\in Y^{+} \mid b_{\mu}\neq 0 \}\quad\text{and}\quad E= \bigcup_{\mu\in S_{b}} R_{w}(\mu).
\]
Note that $E$ is almost finite by Lemma \ref{lemU_almost_finiteness_R_w}.
Given $\mu\in Y^{+}$, Lemma~\ref{lemAH_Lemma_4.15} ensures the existence of $(z_{\mu}^{v,\nu})_{\nu\in R_{w}(\mu),v\in [1,w]}\in \RCC^{R_{w}(\mu)\times [1,w]}$ such that
\[
H_{w}*Z^{\mu} =\sum_{\nu\in R_{w}(\mu),v\in [1,w]} z_{\mu}^{v,\nu} Z^{\nu} H_{v}.
\]
Let us fix $v\in [1,w]$. Given any $\lambda,\rho\in Y^{+}$, we have
\begin{equation}
\label{eqConvolution}
\pi_{\rho,v}(a_{\lambda} Z^{\lambda} H_{w}* b_{\mu} Z^{\mu}) =\sum_{\nu\in R_{w}(\mu) \mid \lambda+\nu=\rho} a_{\lambda} b_{\mu} z_{\mu}^{v,\nu}.
\end{equation}
Set $ F_{1} : = \left\{\lambda\in Y^{+} \mid \exists \mu\in Y^{+}, \pi_{\rho,v}(a_{\lambda}Z^{\lambda} H_{w}*b_{\mu}Z^{\mu})\neq 0 \right\} \subset S_{a}$ and let $\lambda \in F_{1}$. By~equality \eqref{eqConvolution}, there exists $\nu\in E$ such that $\lambda+\nu=\rho$. Since $\lambda$ lies in $S_{a}$, applying Lemma~\ref{lemFiniteness_possibilities_sum} to $E$ and $S_{a}$ implies that $F_{1}$ is finite.

Fix now $\lambda\in F_{1}$ and set $ F_{2}(\lambda)=\{\mu\in Y^{+} \mid \pi_{\rho,v}(a_{\lambda} Z^{\lambda} H_{w}* b_{\mu} Z^{\mu})\neq 0\} \subset S_{b}$.
Given $\mu \in F_2(\lambda)$, we know from equality \eqref{eqConvolution} that $\rho-\lambda\in R_{w}(\mu)$. As $S_{b}$ is $W^{v}$-almost finite, Lemma~\ref{lemFiniteness_mu_such_Ru(mu)ninu} yields the finiteness of $F_2(\lambda)$, hence the finiteness of \[
F_{v} := \{(\lambda,\mu)\in (Y^{+})^2 \mid \pi_{\rho,v}(a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z^{\mu})\neq 0\}=\bigcup_{\lambda\in F_{1}} F_{2}(\lambda).
\]
Finally, we obtain that
\[
\{(\lambda,\mu)\in (Y^{+})^2 \mid \exists v\in W^{v}, \pi_{\rho,v}(a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z^{\mu})\neq 0\}=\bigcup_{v\in [1,w]} F_{v}
\]
is finite, which proves that $(a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z^{\mu})_{(\lambda,\mu)\in (Y^{+})^{2}}$ satisfies condition \eqref{defSummable_familyi} of Definition~\ref{defSummable_family}.

Now let $\lambda,\mu\in Y^{+}$. Then equality \eqref{eqConvolution} ensures that:
\[
\supp(a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z^{\mu})\subset \left(\lambda+R_{w}(\mu) \right)\times [1,w]\subset \left(\lambda +E\right)\times [1,w].
\]
In particular, we have
\[
\bigcup_{(\lambda,\mu)\in (Y^{+})^2} \supp(a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z^{\mu})\subset (S_{a}+E)\times [1,w].
\]
As $E$ is $W^{v}$-almost finite, and as the sum of two $W^{v}$-almost finite sets is $W^{v}$-almost finite, we obtain that $(a_{\lambda} b_{\mu} Z^{\lambda} H_{w}*Z^{\mu})_{(\lambda,\mu)\in (Y^{+})^2}$ satisfies condition \eqref{defSummable_familyii} of Definition~\ref{defSummable_family}, which ends the proof.
\end{proof}

\begin{lemma}\label{lem_right_mulitiplication_summable}
For any summable family $(a_{j})_{j\in J}\in (\widetilde{\HC})^{J}$ in $\widetilde{\HC}$ and any $i\in I$, the family $(a_{j}*H_{i})_{j\in J}$ is summable in $\widetilde{\HC}$.
\end{lemma}
\begin{proof}
By definition, there exist a $W^{v}$-almost finite subset $E$ of $Y^{+}$ and a finite subset $F$ of $W^{v}$ such that: $\forall j \in J, \ \supp(a_{j})\subset E\times F$. By (BL2) of \cite[p.\,91]{abdellatif2019completed}, we~get that
\[
\forall j \in J, \quad \supp(a_{j}*H_{i})\subset E\times (F\cup F\cdot r_{i}),\]
which proves that $ \bigcup_{j\in J} \supp(a_{j}*H_{i})$ is $W^{v}$-almost finite. Now let $\lambda\in Y^{+}$, $w\in W^{v}$ and $j\in J$. By (BL2) again, we know that $\pi_{\lambda,w}(a_{j}*H_{i})\neq 0$ implies $\pi_{\lambda,w}(a_{j})\neq 0$ or $\pi_{\lambda,wr_{i}}(a_{j})\neq 0$. Since there are finitely many such $j \in J$, this completes the proof of the lemma.
\end{proof}

The next statement replaces \cite[Th.\,4.21]{abdellatif2019completed} and its proof basically follows the same lines as the proof of \cite[Th.\,4.21]{abdellatif2019completed}, replacing almost finiteness by $W^{v}$-almost finiteness. Recall that the elements of $\HC$ correspond to the elements of $\widetilde{\HC}$ with finite support.

\begin{theorem}\label{thmAH4.21}
Let $(a_{j})_{j \in J} \in \left(\HC\right)^{J}$ and $(b_{k})_{k \in J} \in \left(\HC\right)^{K}$ be two families that are both summable in $\widetilde{\HC}$. Then $(a_{j} *b_{k})_{(j,k) \in J \times K}$ is summable in $\widetilde{\HC}$ 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 $\widetilde{\HC}$.
\end{theorem}

\begin{proof}
For $j\in J$ and $k\in K$, write
\[
a_{j} := \sum_{v\in W^{v}} a_{v,j}* H_{v}\quad\text{and}\quad b_{k} := \sum_{w\in W^{v}} b_{w,k} *H_{w},
\]
with $(a_{v,j})_{j\in J}\in \RCC[\![Y]\!]^J$ and $(b_{w,k})_{k\in K}\in \RCC[\!Y]\!]^K$ for any $v,w\in W^{v}$. Given $v,w\in W^{v}$, Lemma~\ref{lem_Key_lemma} ensures that $(a_{v,j}*H_{v}*b_{w,k})_{(j,k)\in J\times K}$ is summable in $\widetilde{\HC}$. By~induction on~$\ell(w)$ and using Lemma~\ref{lem_right_mulitiplication_summable}, we get that $(a_{v,j}*H_{v}*b_{w,k}*H_{w})_{(j,k)\in J\times K}$ is summable in $\widetilde{\HC}$. Moreover, as $(a_{j})$ and $(b_{k})$ are summable in~$\widetilde{\HC}$, there are at most finitely many $v,w\in W^{v}$ satisfying $(a_{v,j})_{j\in J}\neq 0$ and $(b_{w,k})_{k\in K}\neq 0$. Consequently, the family $(a_{j}*b_{k})_{(j,k) \in J \times K}$ is summable in $\widetilde{\HC}$.

Now, given any triple $(u,v,\mu)\!\in\!W^{v}\!\times\!W^{v}\!\times\!Y^{+}$, applying Lemma~\ref{lemAH_Lemma_4.15} to~\hbox{$H_{u}\!*\!Z^{\mu}H_{v}$} gives a family $(z^{u,v,\mu}_{\nu, t})_{(\nu, t)\in R_{u}(\mu) \times [1,u].v}$ of scalars that satisfy
\[
H_{u} * Z^{\mu}H_{v} = \sum_{(\nu, t) \in R_{u}(\mu) \times [1,u]\cdot v} z^{u,v,\mu}_{\nu,t}Z^{\nu}H_{t}.
\]

For any pair $(\rho, s) \in Y^{+} \times W^{v}$, we have
\[
\begin{aligned}
\pi_{\rho,s}\biggl(\sum_{(j,k) \in J \times K}\hspace*{-1mm} a_{j}*b_{k} \biggr) &= \sum_{(\lambda,u), (\mu,v) \in Y^{+} \times W^{v}}\ \sum_{\nu \in R_{u}(\mu) \mid \lambda + \nu = \rho}\ \sum_{(j,k) \in J\times K}\hspace*{-1mm} a_{j,\lambda, u} b_{k,\mu,v} z^{u,v,\mu}_{\nu,s}\\
&= \sum_{(\lambda,u), (\mu,v) \in Y^{+} \times W^{v}}\ \sum_{\nu \in R_{u}(\mu) \mid \lambda + \nu = \rho}\hspace*{-2mm} a_{\lambda,u}b_{\mu,v}z^{u,v,\mu}_{v,s},
\end{aligned}
\]
where we set
\[
\sum_{j \in J} a_{j} = \sum_{(\lambda, u) \in Y^{+} \times W^{v}} a_{\lambda,u}Z^{\lambda}H_{u}\quad\text{and}\quad \sum_{k \in K} b_{k} = \sum_{(\mu, v) \in Y^{+} \times W^{v}} b_{\mu,v} Z^{\mu}H_{v},
\]
hence the theorem is proved.
\end{proof}
The mistake done in the former proof of \cite[Th.\,4.20]{abdellatif2019completed} is to implicitly assume that $ S_{Y}=\bigcup_{j\in J} \supp_{Y}(a_{j})\cup \bigcup_{k\in K}\supp_{Y} (b_{k})$ is such that $\{\lambda^{++}\mid \lambda\in S_{Y}\}$ is almost finite, which is not true in general, as shown by the counter-examples given in Section~\ref{secCouter-examples}. As spotted by the referee, the same kind of subtlety underlies a mistake made by Looijenga in his seminal 1980 work \cite[(4.1), end of the first paragraph]{looijenga1980invariant}.

For $a=\sum_{(\lambda,v)\in Y^+\times W^v} a_{\lambda,v} Z^\lambda H_v\in \widetilde{\HC}$ and $b=\sum_{(\mu,w)\in Y^+\times W^v}b_{\mu,w} Z^\mu H_w\in \widetilde{\HC}$, we set \[a*b=\sum_{(v,\lambda),(w,\mu)\in Y^+\times W^v} a_{\lambda,v}b_{\mu,w} Z^\lambda H_v* Z^\mu H_w,\] which is well-defined by Theorem~\ref{thmAH4.21}. We can now formulate the statement that replaces \cite[Cor.\,4.23]{abdellatif2019completed}, providing the required structure on $\widetilde{\HC}$. Its proof is the same as \cite[Cor.\,4.23]{abdellatif2019completed}, replacing \cite[Th.\,4.21]{abdellatif2019completed} by Theorem \ref{thmAH4.21} above.

\begin{corollary}\label{corAH4.23}
The convolution product $*$ equips $\widetilde{\HC}$ with the structure of an associative algebra over $\mathscr{R}$ that contains $\HC$ as subspace of finitely supported elements.
\end{corollary}

\subsection{The center of $\widetilde{\HC}$ is isomorphic to $\HC_{s}$}
\label{secCenter}
Recall that the definition of the Looijenga algebra $\mathscr{R}[\![Y]\!]$ and its variants $\mathscr{R}[\![Y]\!]^{W^{v}}$ and $\mathscr{R}[\![Y^{+}]\!]$ is given by \cite[Def.\,4.6]{abdellatif2019completed}. Also, we proved in \cite[Prop.\,4.9]{abdellatif2019completed} that $\mathscr{R}[\![Y]\!]^{W^{v}}$ is a subspace of $\mathscr{R}[\![Y^{+}]\!]$. Now note that the latter can be seen as a subspace of $\mathscr{B}$, so it makes sense to compare these algebras with the algebra $\widetilde{\HC}$ we built earlier.

Given $a \in \mathscr{R}[\![Y]\!]^{W^v}$, we have $\supp(a)=\supp_{Y}(a)\times \{1\}$, with $\supp_Y(a)$ being $W^{v}$-invariant and almost finite, hence $\supp(a)$ is $W^{v}$-almost finite. In particular, this implies that $\mathscr{R}[\![Y]\!]^{W^{v}}$ is contained in $\widetilde{\HC}$. However, note that in general, $\mathscr{R}[\![Y]\!]$ may not be entirely contained in $\widetilde{\HC}$, as can be seen for instance in Example~\ref{ex_Counter_example}.

Replacing $\widehat{\HC}$ by $\widetilde{\HC}$ in the proof of \cite[Th.\,4.30]{abdellatif2019completed} provides the following theorem, which replaces \cite[Th.\,4.30]{abdellatif2019completed}.
\begin{theorem}\label{thmAH4.30}
The center of the algebra $\widetilde{\HC}$ is $\mathscr{Z}(\widetilde{\HC})=\mathscr{R}[\![Y]\!]^{W^{v}}$, hence is isomorphic to $\HC_{s}$ via the Satake isomorphism.
\end{theorem}

\begin{Rem}
By \cite[Lem.\,4.5]{abdellatif2019completed}, we know that if $\A$ is associated with an indefinite size $2$ Kac-Moody matrix, then any subset of $Y^{+}$ is almost finite, hence any subset of $Y^{+}$ is $W^{v}$-almost finite.
\end{Rem}

\Subsection{The reductive case}
\label{CPcasreductif}
\begin{lemma}
\label{LemReductif}
Assume that $\A$ is associated with a Cartan matrix $A$. Then a subset of $Y^{+}=Y$ is $W^{v}$-almost finite if, and only if, it is finite.
\end{lemma}
\begin{proof}
Thanks to \cite[Lem.\,5.17]{abdellatif2019completed}, we may assume that $ \bigcap_{i\in I} \ker \alpha_i=\{0\}$. Let $A_{1},\ldots,A_{r}$ denote the indecomposable components of $A$: then $\A=\bigoplus_{i=1}^{r}\A_{i}$, where~$\A_{i}$ is a realization of $A_{i}$ (as defined in \cite[5.4.1]{abdellatif2019completed}) for all $i\in \llbracket 1,r\rrbracket$. For $i\in \llbracket 1,r\rrbracket$, we~deno\-te by $W^{v}_{i}$ the Weyl group of $\A_{i}$, by $Q^{\vee}_{i}$ its coroot lattice and by $Y_{i}$ its cocharacter lattice, so that we have $W^{v}=W^{v}_{1}\times \ldots \times W^{v}_{r}$, $Q^{\vee} = \bigoplus_{i=1}^{r} Q^{\vee}_{i}$ and $Y= \bigoplus_{i=1}^{r} Y_{i}$.

Now let $E$ be a $W^{v}$-almost finite subset of $Y$. For $w\in W^{v}$, set
\[
E_{w}:=E\cap w\cdot\overline{C^{v}_{f}}.
\]
Since $E$ is $w^{-1}$-almost finite, there exists a finite set $F\subset Y$ such that:
\[
\forall \lambda \in E,\ \exists \mu \in F \mid \lambda \leq_{Q^{\vee}} \mu.
\]
For $i\in \llbracket 1,r\rrbracket$, set $Y_{i}^{++}:=Y_{i} \cap \overline{C^{v}_{f,i}}$, where $C^{v}_{f,i}$ denotes the fundamental chamber of $\A_{i}$: then we know from \cite[Th.\,4.3]{kac1994infinite} that $Y_{i}^{++}\subset Q^{\vee}_{i,+}$. Given $\lambda = (\lambda_{1}, \ldots, \lambda_{r}) \in E_{w}$, let $\mu = (\mu_{1}, \ldots, \mu_{r}) \in F$ be such that $\lambda \leq_{Q^{\vee}} \mu$. Then we have $0\leq_{Q^{\vee}_{i}} \lambda_{i} \leq_{Q^{\vee}_{i}}\mu_{i}$ for all $i\in \llbracket 1, r\rrbracket$, hence $E_{w}$ must be finite. As $W^{v}$ is finite, we get that $E$ is finite too, and the lemma is proved as the converse statement is straightforward.
\end{proof}
Since $\HC$ corresponds to the subspace of elements of $\widetilde{\HC}$ with finite support, we directly obtain the following result from Lemma \ref{LemReductif}, which states that $\widetilde{\HC}$ is just the usual Iwahori-Hecke algebra in the reductive case. This replaces the first paragraph of \cite[\S 4.6.1, p.\,105]{abdellatif2019completed}.

\begin{proposition}
\label{Prop44}
If $\A$ is associated with a Cartan matrix, then $\widetilde{\HC} = \HC$.
\end{proposition}

\backmatter
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\end{document}