}\langle\nu(m),\alpha\rangle\leq 0(\Leftarrow m\in M^{\pm}\text{ and
}\alpha\in\Phi^{\mp}\cap\Phi_{red})\\\ \supset U_{\alpha+r}&\text{if
}\langle\nu(m),\alpha\rangle\geq 0(\Leftarrow m\in M^{\pm}\text{ and
}\alpha\in\Phi^{\pm}\cap\Phi_{red})\end{cases}$

By Proposition 2.9, we have

$m^{-1}U_{\Upomega}^{-}m=\prod_{\alpha\in\Phi^{-}\cap\Phi_{red}}^{\prec}U_{\alpha+f_{\Upomega}(\alpha)+\langle\nu(m),\alpha\rangle}.$

for a fixed ordering ${\prec}$ of the set $\Psi$. Therefore, if $m$ is
antidominant:

$\displaystyle m(U_{\Psi}\cap K_{\Upomega})m^{-1}\subset U_{\Psi}\cap
K_{\Upomega},\quad m^{-1}U_{\Upomega}^{-}m\subset U_{\Upomega}^{-}\quad$
$\displaystyle\text{and}\quad m\,U_{\Upomega}^{+}m^{-1}\subset
U_{\Upomega}^{+}.\qed$

###  3.7. Double cosets

Recall that $(W_{\textit{\emph{\text{aff}}}},\mathcal{S}(\mathfrak{a}))$ and
$(W_{\textit{\emph{\text{aff}}}}^{\flat},\mathcal{S}^{\flat})$ are isomorphic
Coxeter systems. Denote by $\ell^{\flat}\colon
W_{\textit{\emph{\text{aff}}}}^{\flat}\to\mathds{N}$ the length function of
the Coxeter system. Inflate the map $\ell^{\flat}$ to a length function
$\widetilde{W}^{\flat}\simeq
W_{\textit{\emph{\text{aff}}}}^{\flat}\rtimes\widetilde{\Omega}^{\flat}\to\mathds{N}$,
for which $\widetilde{\Omega}^{\flat}$ is exactly the subset of elements of
length equals to $0$888Equivalently, the length of an element in
$w\in\widetilde{W}$ is the number of walls between the fixed alcove
$\mathfrak{a}$ and the alcove $w(\mathfrak{a})$.:

$\ell^{\flat}(w)=\ell^{\flat}(w_{\textit{\emph{\text{aff}}}})\text{ if
}w=w_{\textit{\emph{\text{aff}}}}u\text{ with
}w_{\textit{\emph{\text{aff}}}}\in
W_{\textit{\emph{\text{aff}}}}^{\flat},u\in\widetilde{\Omega}^{\flat}.$

Furthermore, we consider the Chevalley–Bruhat (partial) order
$\overset{\flat}{\leq}$ 999Let $(W,S)$ be a Coxeter system and define a
partial order on $W$ as follows: Fix a reduced word $w=s_{1}s_{2}\dots s_{k}$.
We say $v\leq w$ if and only if there is a reduced subword
$s_{i_{1}}s_{i_{2}}\dots s_{i_{j}}=v$ such that $1\leq
i_{1}<i_{2}<\dots<i_{j}\leq k$. on $W_{\textit{\emph{\text{aff}}}}^{\flat}$.
Now extend this order to $\widetilde{W}^{\flat}$ as follows: we say
$(w_{1},u_{1})\overset{\flat}{\leq}(w_{2},u_{2})\in
W_{\textit{\emph{\text{aff}}}}^{\flat}\rtimes\widetilde{\Omega}^{\flat}$ if
and only if $w_{1}\overset{\flat}{\leq}w_{2}$ and $\lambda_{1}=\lambda_{2}$.

######  Remark 3.1.

Thanks to Remark 3.4, the projection
$\square^{\flat}\colon\widetilde{W}\twoheadrightarrow\widetilde{W}^{\flat}$
preserves the length and the Chevalley–Bruhat order and we may and will write
$\ell^{\flat}$ (resp. $\overset{\flat}{\leq}$) simply as $\ell$ (resp.
${\leq}$).

######  Remark 3.2.

The Chevalley–Bruhat ordering of
$(W_{\textit{\emph{\text{aff}}}}^{\flat},\mathcal{S}^{\flat})$ can be defined
in several ways, their equivalence is established in [Deo77]. One interesting
equivalent definition is: For $w,w^{\prime}\in
W_{\textit{\emph{\text{aff}}}}$, $w\leq w^{\prime}$ if there exists
$\alpha_{1}\cdots\alpha_{k}\in\Phi^{+}$ such that (i)
$w^{\prime}=ws_{\alpha_{1}}\cdots s_{\alpha_{k}}$ and (ii)
$ws_{\alpha_{1}}\cdots s_{\alpha_{j-1}}(\alpha_{j})\in\Phi^{+}$ for all $1\leq
j\leq k$.

######  Lemma 3.3.

For any $w\in\widetilde{W}^{\flat}$ and $s\in\mathcal{S}^{\flat}$ we have

$I^{\flat}sI^{\flat}wI^{\flat}=\begin{cases}I^{\flat}swI^{\flat},&\text{ if
}w<sw\quad(\ell(sw)=\ell(w)+1),\\\ I^{\flat}wI^{\flat}\sqcup
I^{\flat}swI^{\flat},&\text{ if }sw<w\quad(\ell(sw)=\ell(w)-1).\end{cases}$

###### Proof.

Since the quadruplet
$(G_{1}^{\flat},I^{\flat},N_{1}^{\flat},\mathcal{S}^{\flat})$ is a Tits
system, it verifies the axioms listed below Corollary 3.2. Therefore, for any
$w\in W_{\textit{\emph{\text{aff}}}}^{\flat}$ and $s\in\mathcal{S}^{\flat}$

$I^{\flat}sI^{\flat}wI^{\flat}=\begin{cases}I^{\flat}swI^{\flat},&\text{ if
}w<sw,\\\ I^{\flat}wI^{\flat}\sqcup I^{\flat}swI^{\flat},&\text{ if
}sw<w.\end{cases}$

To finish the proof, recall that elements of $\widetilde{\Upomega}^{\flat}$
are of zero length and their lifts in $N_{\mathfrak{a}}^{\dagger}$ normalizes
$I^{\flat}$, therefore for any triplet
$(s,w_{\textit{\emph{\text{aff}}}},u)\in\mathcal{S}^{\flat}\times
W_{\textit{\emph{\text{aff}}}}^{\flat}\times\widetilde{\Upomega}^{\flat}$, we
have
$I^{\flat}sI^{\flat}wI^{\flat}=I^{\flat}sI^{\flat}w_{\textit{\emph{\text{aff}}}}I^{\flat}u$,
which completes the proof. ∎

######  Remark 3.4.

The first equality in the lemma above can be easily generalized: for any
$w,w^{\prime}\in\widetilde{W}^{\flat}$

$\displaystyle I^{\flat}wI^{\flat}w^{\prime}I^{\flat}$
$\displaystyle=I^{\flat}ww^{\prime}I^{\flat},\quad\text{ if and only if
}\quad\ell(ww^{\prime})=\ell(w)+\ell(w^{\prime}).\qed$

Let us end this section with the following results, they will be needed later.
3.6.

######  Corollary 3.5.

Let $x,y\in\widetilde{W}^{\flat}$ then

$I^{\flat}xI^{\flat}yI^{\flat}\subset\bigsqcup_{z\leq
y}I^{\flat}xzI^{\flat}\text{ and
}I^{\flat}xI^{\flat}yI^{\flat}\subset\bigsqcup_{z\leq x}I^{\flat}zyI^{\flat}.$

###### Proof.

One argues by induction on the size of a minimal word for $y$. Let
$y=s_{1}\in\mathcal{S}^{\flat}$, we have

$I^{\flat}xI^{\flat}s_{1}I^{\flat}=\begin{cases}I^{\flat}xs_{1}I^{\flat},&\text{
if }x<xs_{1},\\\ I^{\flat}xI^{\flat}\sqcup I^{\flat}xs_{1}I^{\flat},&\text{ if
}xs_{1}<x.\end{cases}$

So it suffices to take $z\in\\{y\\}$ if $x<xs_{1}$ and $z\in\\{s_{1}y,y\\}$ if
$xs_{1}<x$. Let $y=s_{1}\cdots s_{r}$ a reduced word for $y$. Put
$y^{\prime}=\prod_{i=2}^{r}s_{{I^{\flat}}}$, we then have

$\displaystyle
I^{\flat}xI^{\flat}yI^{\flat}=I^{\flat}xI^{\flat}s_{1}y^{\prime}I^{\flat}$
$\displaystyle{=}I^{\flat}xI^{\flat}s_{1}I^{\flat}y^{\prime}I^{\flat}$
$\displaystyle\subset\begin{cases}\bigsqcup_{z\leq
y^{\prime}}I^{\flat}xs_{1}zI^{\flat},&\text{ if }x<xs_{1},\\\ \bigsqcup_{z\leq
y^{\prime}}I^{\flat}xzI^{\flat}\cup\bigsqcup_{z\leq
y^{\prime}}I^{\flat}xs_{1}zI^{\flat},&\text{ if }xs_{1}<x\end{cases}.$

where the last inclusion is just the recursion hypothesis. Now if $z\leq
y^{\prime}$ then clearly $z<y$ and $s_{1}z\leq y$, hence in both cases we have
$I^{\flat}xI^{\flat}yI^{\flat}\subset\bigsqcup_{z\leq y}I^{\flat}xzI^{\flat}$
for any $x\in\widetilde{W}^{\flat}$ and $y\in
W_{\textit{\emph{\text{aff}}}}^{\flat}$.

Now to extend this fact to any $y\in\widetilde{W}^{\flat}$, write
$y=y_{\textit{\emph{\text{aff}}}}u\in
W_{\textit{\emph{\text{aff}}}}^{\flat}\rtimes\widetilde{\Omega}^{\flat}$.
Because any lift of $u$ in $N_{\mathfrak{a}}^{\dagger}$ normalizes $I^{\flat}$
one has

$I^{\flat}xI^{\flat}y_{\textit{\emph{\text{aff}}}}uI^{\flat}=I^{\flat}xI^{\flat}y_{\textit{\emph{\text{aff}}}}I^{\flat}u\subset\bigsqcup_{z\leq
y_{\textit{\emph{\text{aff}}}}}I^{\flat}xzI^{\flat}u=\bigsqcup_{z\leq
y_{\textit{\emph{\text{aff}}}}}I^{\flat}xzuI^{\flat}\subset\bigsqcup_{z^{\prime}\leq
y}I^{\flat}xz^{\prime}I^{\flat}.$

This conclude the proof of the first inclusion, from which the second
inclusion follows readily by upon applying the inverse map $w\mapsto w^{-1}$.
∎

######  Corollary 3.6.

Let $x,y\in\widetilde{W}^{\flat}$. If $U^{+}x\cap
I^{\flat}yI^{\flat}\neq\emptyset$ then $x\leq y$ in the Chevalley–Bruhat
order.

###### Proof.

The proof resembles the proof of the claim in [HKP10, Lemma 1.6.1]. Let
$x,y\in\widetilde{W}^{\flat}$ such that $U^{+}x\cap
I^{\flat}yI^{\flat}\neq\emptyset$. There exist then $u\in U^{+}$ such that
$ux\in I^{\flat}yI^{\flat}$. Choose any $m\in M^{-}$ such that $mum^{-1}\in
U^{+}\cap I^{\flat}$. Therefore,

$I^{\flat}mxI^{\flat}=I^{\flat}muxI^{\flat}=I^{\flat}miyi^{\prime}I^{\flat}\subset
I^{\flat}mI^{\flat}yI^{\flat}.$

Given Corollary 3.5, this shows $I^{\flat}mxI^{\flat}\subset\bigsqcup_{z\leq
y}I^{\flat}mzI^{\flat}$, and consequently $x\leq y$. ∎

###  3.8. Volumes and $q$-powers

######  Lemma 3.1.

Let $\mathcal{F}\subset\mathcal{A}$ be any facet. For any $n\in N$, we have
$q_{\mathcal{F},n}:=[{K}_{\mathcal{F}}n{K}_{\mathcal{F}}:{K}_{\mathcal{F}}]=[{K}_{\mathcal{F}}^{\flat}n{K}_{\mathcal{F}}^{\flat}:{K}_{\mathcal{F}}^{\flat}]$.

###### Proof.

This is a generalization of [Ros15, Proposition 4.2.3], in which a similar
statement is proved when $n\in W$, $M^{\flat}=M^{1}$ and
$\mathcal{F}=\mathfrak{a}$. Here, we present a different argument:

Using Lemma 3.1, note that
${K}_{\mathcal{F}}^{\flat}n{K}_{\mathcal{F}}^{\flat}=K_{\mathcal{F}}n{K}_{\mathcal{F}}^{\flat}$,
the map
$K_{\mathcal{F}}\to{K}_{\mathcal{F}}^{\flat}n{K}_{\mathcal{F}}^{\flat}/{K}_{\mathcal{F}}^{\flat}$,
given by $x\mapsto xn{K}_{\mathcal{F}}^{\flat}$ is then clearly surjective.
Two elements $x,y\in{K}_{\mathcal{F}}$ give the same image if and only if
$xn\in yn{K}_{\mathcal{F}}^{\flat}=\sqcup_{m\in
M^{\flat}/M_{1}}ynm{K}_{\mathcal{F}}$, if and only if there exists $m\in
M^{\flat}$ such that $n^{-1}y^{-1}xn\in m{K}_{\mathcal{F}}$.

Given that $K_{\mathcal{F}}\subset\ker\kappa_{G}$, we see that
$\kappa_{G}(m)=\kappa_{G}(n^{-1}y^{-1}xn)=1$, i.e. $m\in G_{1}\cap
M^{1}=M_{1}$ (Lemma 2.11). This shows that
${K}_{\mathcal{F}}n{K}_{\mathcal{F}}/{K}_{\mathcal{F}}\cong{K}_{\mathcal{F}}/n{K}_{\mathcal{F}}n^{-1}\cap{K}_{\mathcal{F}}\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}{K}_{\mathcal{F}}^{\flat}n{K}_{\mathcal{F}}^{\flat}/{K}_{\mathcal{F}}^{\flat}.$
∎

######  Lemma 3.2.

Let $w=s_{1}\cdots s_{r}\in\widetilde{W}^{\flat}$ be a reduced word for $w$.
Then, $q_{w}:=q_{\mathfrak{a},w}=\prod_{i=1}^{r}q_{\mathfrak{a},s_{i}}$.
Moreover, $q_{w}$ is a power of $q$ for any $w\in\widetilde{W}^{\flat}$.

###### Proof.

Let $s\in\mathcal{S}^{\flat}$ and $w\in\widetilde{W}^{\flat}$ such that
$\ell(sw)=\ell(w)+1$. By Lemma 3.3, we know that
$I^{\flat}swI^{\flat}=I^{\flat}sI^{\flat}\cdot I^{\flat}wI^{\flat}$. By
assumption, $s\neq w$, hence $I^{\flat}sI^{\flat}\cap
I^{\flat}wI^{\flat}\neq\emptyset$ and accordingly $q_{sw}=q_{s}q_{w}$.
Therefore, the equality
$q_{w}:=q_{\mathfrak{a},w}=\prod_{i=1}^{r}q_{\mathfrak{a},s_{i}}$ follows by
induction on the length $r$ of $w$. Consequently, to show that $q_{w}$ is a
power of $q$ it suffices to show the claim for all elements of $\mathcal{S}$.
For any $s\in\mathcal{S}$ (corresponding to the affine root
$\alpha_{s}\in\Sigma_{\textit{\emph{\text{aff}}}}$) we have, as in the proof
of [Mac71, Lemma 2.7.4]

$\displaystyle IsI/I$ $\displaystyle\cong U_{\alpha_{s}}sI/I$ [Mac71, Lemma
2.6.7 (1)] $\displaystyle\cong U_{\alpha_{s}}/U_{\alpha_{s}}\cap
I_{\nu(s)\cdot\mathfrak{a}}$ $\displaystyle\cong
U_{\alpha_{s}}/U_{\alpha_{s}+1}$ [Mac71, Lemma 2.6.5] $\displaystyle.\qed$

This completes the proof of the lemma since the size of the last quotient is a
power of $q$.

######  Lemma 3.3.

For any $n\in N$, we have a bijection

$U^{+}I\cap
nIn^{-1}I/I\cong\prod_{\Phi_{red}\cap\Phi^{+}}^{\prec_{+}}U_{\alpha+f_{\nu(n)\cdot\mathfrak{a}}(\alpha)}/U_{\alpha+\max(0,f_{\nu(n)\cdot\mathfrak{a}}(\alpha))}$

for any fixed ordering $\prec_{\pm}$ of $\Phi^{\pm}$. In particular,
$|U^{+}I\cap nIn^{-1}I|_{I}$ is a power of $q$.

###### Proof.

Write $nIn^{-1}=I_{\nu(n)\cdot\mathfrak{a}}$. Using Corollary 3.4, we have

$U^{+}I\cap I_{\nu(n)\cdot\mathfrak{a}}I=(U^{+}M_{1}I^{-}\cap
I_{\nu_{N}(n)\cdot\mathfrak{a}})I=I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}I.$

This equality yields the bijection
$I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}I/I\cong
I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}/I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}\cap
I^{+}$.

By Corollary 3.4 again together with d and e of Proposition 2.5, we have

$I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}=\prod_{\Phi_{red}\cap\Phi^{+}}^{\prec_{+}}U_{\alpha+f_{\nu(n)\cdot\mathfrak{a}}(\alpha)}\text{
and }I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}\cap
I^{+}=\prod_{\Phi_{red}\cap\Phi^{+}}^{\prec_{+}}U_{\alpha+\max(0,f_{\nu(n)\cdot\mathfrak{a}}(\alpha))}$

for any fixed ordering $\prec_{\pm}$ of $\Phi^{\pm}$. Therefore, we get a
bijection

$I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}I/I\cong\prod_{\Phi_{red}\cap\Phi^{+}}^{\prec_{+}}U_{\alpha+f_{\nu(n)\cdot\mathfrak{a}}(\alpha)}/U_{\alpha+\max(0,f_{\nu(n)\cdot\mathfrak{a}}(\alpha))}.$

This completes the proof of the lemma since the size of each quotient in the
right product is a power of $q$. ∎

######  Corollary 3.4.

Let $n=m\cdot w$ for any $(m,w)\in\Lambda_{M}^{-}\times W_{a_{\circ}}$, we
have $U^{+}I\cap nIn^{-1}I=I$.

###### Proof.

This corollary follows readily using Lemma 3.1 and Lemma 3.2. We can also
deduce it from Lemma 3.3: Let $n=m\cdot w$ for any $m\in\Lambda_{M}^{-}$ and
any $w\in W_{a_{\circ}}$. By Remark 2.6, we have
$f_{\nu_{N}(n)\cdot\mathfrak{a}}(\alpha)=-\langle\nu(m),\alpha\rangle+f_{\mathfrak{a}}(w^{-1}(\alpha))$.
In particular, if $n\in\Lambda_{M}^{-}\cup W_{a_{\circ}}$, then
$\langle\nu(m),\alpha\rangle\geq 0$ for $\alpha\in\Phi_{red}\cap\Phi^{+}$ and
$n_{\alpha}^{-1}\leq f_{\mathfrak{a}}(w^{-1}(\alpha))\leq 0$ then
$I_{\nu_{N}(n)\cdot\mathfrak{a}}^{+}\subset I$ and so $U^{+}I\cap
I_{\nu(n)\cdot\mathfrak{a}}I=I$ by Lemma 3.3. ∎

###  3.9. Double cosets decomposition

######  Lemma 3.1.

Let $\mathcal{F}$ and $\mathcal{F}^{\prime}$ be two facets of $\mathfrak{a}$.
For any $w\in\widetilde{W}^{\flat}$ we have

$K_{\mathcal{F}}^{\flat}wK_{\mathcal{F}^{\prime}}^{\flat}=I^{\flat}W_{\mathcal{F}}wW_{\mathcal{F}^{\prime}}I^{\flat}.$

We will be interested later in the particular case $w=m\in\Lambda_{M}^{-}$ and
$\mathcal{F}=\mathcal{F}^{\prime}$, for which we have
$K_{\mathcal{F}}mK_{\mathcal{F}}=\sqcup_{(w_{1},w_{2})\in
J_{m,\mathcal{F}}}Iw_{1}mw_{2}I$, where $J_{m,\mathcal{F}}\simeq
W_{\mathcal{F}}^{m}\backslash W_{\mathcal{F}}\times W_{\mathcal{F}}$. Here,
$w\in W_{\mathcal{F}}^{m}$ (the isotropy subgroup of $m$ in $W_{\mathcal{F}}$)
acts on $(w_{1},w_{2})\in W_{\mathcal{F}}\times W_{\mathcal{F}}$ as follow
$(w_{1}w,w^{-1}w_{2})$.

###### Proof.

For $w=m\in\Lambda_{M}^{-,\flat}$, the equality can easily be obtained by
combining Proposition 3.1 and Lemma 3.2. However, here is an argument for a
general $w\in\widetilde{W}^{\flat}$. For any
$s_{1},\cdots,s_{x}\in\mathcal{T}_{\mathcal{F}}^{\flat}$ and
$t_{1},\cdots,t_{y}\in\mathcal{T}_{\mathcal{F}^{\prime}}^{\flat}$, Lemma 3.5
shows

$I^{\flat}s_{1}\cdots s_{x}I^{\flat}wI^{\flat}t_{1}\cdots
t_{y}I^{\flat}\subset
I^{\flat}W_{\mathcal{F}}wW_{\mathcal{F}^{\prime}}I^{\flat}.$

But $W_{\mathcal{F}}^{\flat}$ (resp. $W_{\mathcal{F}^{\prime}}^{\flat}$) is
generated by $\mathcal{T}_{\mathcal{F}}^{\flat}$ (resp.
$\mathcal{T}_{\mathcal{F}^{\prime}}^{\flat}$) and hence Lemma 3.1 yields
$K_{\mathcal{F}}^{\flat}wK_{\mathcal{F}^{\prime}}^{\flat}\subset
I^{\flat}W_{\mathcal{F}}wW_{\mathcal{F}^{\prime}}I^{\flat}$. The opposite
inclusion is obvious. ∎

###  3.10. The modular function

Let $\delta_{B}$ be the modular function on the fixed minimal parabolic $B$
containing $M$ given by the normalized absolute value of the determinant of
the adjoint action on $\text{Lie}\,U^{+}$:

$\delta_{B}(m):=|\det(\text{Ad}(m)_{\text{Lie}U^{+}})|_{F},\quad\forall m\in
M$

where $|\cdot|_{F}$ is the fixed normalized absolute value of $F$.

######  Lemma 3.1.

Let $\Upomega\subset\mathcal{A}_{ext}$ be any bounded subset containing any
alcove, then for any $m\in M^{-}$

$[K_{\Upomega}mK_{\Upomega}:K_{\Upomega}]=\delta_{B}(m)^{-1}.$

In particular, if $\Upomega$ is an alcove $\mathfrak{b}$, then
$q_{m}=[I_{\mathfrak{b}}mI_{\mathfrak{b}}:I_{\mathfrak{b}}]=\delta_{B}(m)^{-1}$
and so $q_{w(m)}=\delta_{B}(m)^{-1}$ for all $m\in M^{-}$.

###### Proof.

Consider the map $i_{1}\mapsto i_{1}mK_{\Upomega}$ from the set $K_{\Upomega}$
to $K_{\Upomega}mK_{\Upomega}/K_{\Upomega}$. This yields a bijection

$K_{\Upomega}mK_{\Upomega}/K_{\Upomega}\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}K_{\Upomega}/K_{\Upomega}\cap
mK_{\Upomega}m^{-1}.$

Here, $K_{\Upomega}\cap mK_{\Upomega}m^{-1}=K_{\Upomega}\cap
K_{\nu(m)(\Upomega)}$ is the
$K_{\Upomega_{m}^{\text{conv}}}={P}(F^{un})_{\Upomega_{m}^{\text{conv}}}^{\circ}\cap
G_{1}$ the $G_{1}$-fixator of the convex hull of
$\Upomega_{m}^{\text{conv}}:=\Upomega\cup\nu(m)(\Upomega)$. Since $m\in
M^{-}$, then $K_{\Upomega}\cap
mK_{\Upomega}m^{-1}=U_{\Upomega}^{-}M_{1}mU_{\Upomega}^{+}m^{-1}$ and so

$\displaystyle[K_{\Upomega}:K_{\Upomega}\cap mK_{\Upomega}m^{-1}]$
$\displaystyle=[U_{\Upomega}^{+}:mU_{\Upomega}^{+}m^{-1}]=\delta_{B}^{-1}(m).$

For the last statement, it suffices to observe that
$q_{w(m)}=q_{\mathcal{F},w^{-1}(\mathfrak{a})}=\delta_{B}(m)^{-1}$ for any
$m\in M^{-}.$∎

######  Remark 3.2.

A similar proof of [Mac71, Proposition 3.2.15] shows that for any
$m\in\Lambda_{M}^{-}$ we have

$q_{a_{\circ},m}=[KmK:K]=\delta_{B}(m)^{-1}\frac{\sum_{w\in
W_{a_{\circ}}}q_{w}^{-1}}{\sum_{w\in W_{a_{\circ}}^{m}}q_{w}^{-1}}$

where, $W_{a_{\circ}}^{m}$ is the isotropy group of $m$ in $W_{a_{\circ}}$.

###  3.11. Relative Hecke algebras

For any ring $R$, set $\mathcal{C}_{c}(G,R)$ for the $R$-module of locally
constant and compactly supported functions $f\colon G\to R$. Let $H$ be any
open compact subgroup of $G$. The group $G$ has a unique left invariant
measure $\mu_{H}$ normalized by $H$ on $\mathds{Q}$ [Vig96, §2.4]:

$\mathcal{C}_{c}(G,\mathds{Q})\longrightarrow\mathds{Q},\quad
f\longmapsto\int_{G}f(g)d\mu_{H}(g)\text{ such that }\int_{G}{\bf
1}_{H}(g)d\mu_{H}(g)=1.$

The vector space $\mathcal{C}_{c}(G,\mathds{Q})$ acquire the structure of a
$\mathds{Q}$-algebra without a unit, when endowed with the convolution product
with respect to $\mu_{H}$:

$f*_{H}f^{\prime}\colon
x\mapsto\int_{G}f(g)f^{\prime}(g^{-1}x)d\mu_{H}(g)\quad(f,f^{\prime}\in\mathcal{C}_{c}(G,\mathds{Q})).$

######  Remark 3.1.

The above expression of the convolution can be rewritten

$\displaystyle f*_{H}f^{\prime}(x)$
$\displaystyle=\int_{G}f(xg)f^{\prime}(g^{-1})d\mu_{H}(g).$

This shows that for $f,f^{\prime}\in\mathcal{C}_{c}(G,\mathds{Q})$, if $f$ is
$X$-invariant on the left and $f^{\prime}$ is $Y$-invariant on the right, then
$f*_{H}f^{\prime}$ is $X$-invariant on the left and $Y$-invariant on the
right.

The group $G$ acts on the underlying $\mathds{Q}$-vector space of this algebra
by translation, on the left and on the right as follows:

${(G\times
G)\times\mathcal{C}_{c}(G,\mathds{Q})}$${\mathcal{C}_{c}(G,\mathds{Q})}$${((g,g^{\prime}),f)}$${((g,g^{\prime})\cdot
f\colon x\mapsto f(g^{-1}xg^{\prime})).}$

######  Lemma 3.2.

Let $X,Y$ be two open compact subgroups of $G$. For any $g,h\in G$, one has

$\displaystyle(i)$ $\displaystyle\mathbf{1}_{gY}*_{H}\mathbf{1}_{hX}$
$\displaystyle=|Y\cap hXh^{-1}|_{H}\mathbf{1}_{gYhX}$ $\displaystyle(ii)$
$\displaystyle\mathbf{1}_{gY}*_{H}\mathbf{1}_{XhX}$
$\displaystyle=\frac{|Y\cap X|_{H}}{|X\cap
hXh^{-1}|_{H}}\mathbf{1}_{gYX}*_{H}\mathbf{1}_{hX},$

where, the notation $|\square|_{H}$ denotes the volume of $\square$ with
respect to the measure $\mu_{H}$.

###### Proof.

(i) Observe that the function

$\mathbf{1}_{gY}*_{H}\mathbf{1}_{hX}(a)=\int_{G}\mathbf{1}_{gY}(b)\mathbf{1}_{hX}(b^{-1}a)d\mu_{H}(b),$

can only be nonzero on the set $gYhX$. Let $a\in gYhX$ and write it as
$a=gyhx$, thus

$\displaystyle\mathbf{1}_{gY}*_{H}\mathbf{1}_{hX}(a)=|gY\cap
gyhxXh^{-1}|_{H}=|Y\cap hXh^{-1}|_{H}$

where, the third equality holds by left invariance of $\mu_{H}$. For (ii), we
have

$\displaystyle\mathbf{1}_{gY}*_{H}\mathbf{1}_{XhX}$
$\displaystyle=\frac{1}{|X\cap
hXh^{-1}|_{H}}(\mathbf{1}_{gY}*_{H}\mathbf{1}_{X})*_{H}\mathbf{1}_{hX}=\frac{|Y\cap
X|_{H}}{|X\cap hXh^{-1}|_{H}}\mathbf{1}_{gYX}*_{H}\mathbf{1}_{hX}.\qed$

######  Example 3.3.

For $g,g^{\prime}\in G$, write $Hg^{\prime}H$ as disjoint union
$\sqcup_{h^{\prime}}h^{\prime}g^{\prime}H$, then by lemma above:

$\mathbf{1}_{gH}*_{H}\mathbf{1}_{Hg^{\prime}H}=\mathbf{1}_{gHg^{\prime}H}=\mathbf{1}_{\sqcup_{h^{\prime}}gh^{\prime}g^{\prime}H}=\sum_{h^{\prime}}\mathbf{1}_{gh^{\prime}g^{\prime}H}.$

If also $HgH=\sqcup_{h}hgH$, then
$\mathbf{1}_{HgH}*_{H}\mathbf{1}_{Hg^{\prime}H}=(\sum_{h}\mathbf{1}_{hgH})*_{H}\mathbf{1}_{Hg^{\prime}H}=\sum_{h,h^{\prime}}\mathbf{1}_{hgh^{\prime}g^{\prime}H}$.

Now, we associate to the pair $(G,H)$ different $\mathds{Z}$-modules.

######  Definition 3.4.

For any commutative ring $A$, we define $\mathcal{C}_{c}(G/H,A)$ to be the
$A$-module of compactly supported functions $f\colon G\to A$ that are
$H$-invariant on the right. It has the following canonical basis $\\{{\bf
1}_{gH}\colon g\in G/H\\}$. In addition, the group left action of $G$ on
$\mathcal{C}_{c}(G,\mathds{Q})$ as defined above, restricts to a left action
on $\mathcal{C}_{c}(G/H,\mathds{Z})$. We also define $\mathcal{H}_{c}(G\sslash
H,A)\subset\mathcal{C}_{c}(G/H,A)$ to be the $A$-algebra101010By Remark 3.1
and Example 3.3, we see that $\mathcal{H}_{H}(A):=\mathcal{C}_{c}(G\sslash
H,\mathds{Z})\subset\mathcal{C}_{c}(G\sslash H,\mathds{Q})$ is stable under
$*_{H}$. of functions $f\colon G\to A$, that are also $H$-invariant on the
left. We call $\mathcal{H}_{c}(G\sslash H,A)$ the Hecke algebra relative to
$H$ with values in $A$ and denote $\mathcal{H}_{H}(A)$. We have
$\mathcal{H}_{H}(A)=\mathcal{H}_{H}(\mathds{Z})\otimes_{\mathds{Z}}A$.

The Hecke algebra $\mathcal{H}_{H}(\mathds{Z})$ is a free $\mathds{Z}$-module,
it has the canonical basis $\\{{\bf 1}_{HgH}\colon g\in HgH\\}$.

######  Example 3.5.

According to Proposition 3.1, for any facet $\mathcal{F}\subset\mathcal{A}$,
the Hecke algebra $\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ has the
following canonical $\mathds{Z}$-basis

$\\{{\bf 1}_{K_{\mathcal{F}}^{\flat}wK_{\mathcal{F}}^{\flat}}\text{ for }w\in
W_{\mathcal{F}}^{\flat}\backslash\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}\\}.$

When $\mathcal{F}=\\{a_{\circ}\\}$, $\mathcal{H}_{K^{\flat}}(\mathds{Z})$ will
be called the ${\flat}$-special–Hecke algebra. By Lemma 3.3, one has
$\Lambda_{M}^{-,{\flat}}\cong
W_{a_{\circ}}^{\flat}\backslash\widetilde{W}^{\flat}/W_{a_{\circ}}^{\flat}$,
so one can exhibit a natural $\mathds{Z}$-basis for
$\mathcal{H}_{K^{\flat}}(\mathds{Z})$ as follows $\\{h_{m}^{\flat}:={\bf
1}_{K^{\flat}mK^{\flat}}\text{ for }m\in\Lambda_{M}^{-,{\flat}}\\}$. When
$\mathcal{F}=\mathfrak{a}$, $\mathcal{H}_{I^{\flat}}(\mathds{Z})$ is called
the ${\flat}$-Iwahori–Hecke algebra and the following set forms a
$\mathds{Z}$-basis $\\{i_{w}^{\flat}:={\bf 1}_{I^{\flat}wI^{\flat}}\text{ for
}w\in\widetilde{W}^{\flat}\\}.$

When we are in the case $M^{\flat}=M_{1}$, we systematically omit the prefix
${\flat}$ in all of the above nomenclatures and notations.

Let $End_{\mathds{Z}[G]}\mathcal{C}_{c}(G/H,\mathds{Z})$ denote the ring of
$G$-equivariant endomorphisms of $\mathcal{C}_{c}(G/H,\mathds{Z})$.

######  Proposition 3.6.

The following map is an isomorphism of rings111111the superscript opp
indicates the opposite ring:

${\mathcal{H}_{H}(\mathds{Z})}$${End_{\mathds{Z}[G]}\mathcal{C}_{c}(G/H,\mathds{Z})^{\text{\emph{opp}}}}$${h}$${e_{h}\colon
f\mapsto f*_{H}h,}$$\scriptstyle{\simeq}$

###### Proof.

We first prove that the above map is an isomorphism of $\mathds{Z}$-modules.
The injectivity being clear, we prove surjectivity. Any $G$-equivariant
endomorphism $e$ of $\mathcal{C}_{c}(G/H,\mathds{Z})$ is uniquely defined by
$e({\bf 1}_{H})$. Since ${\bf 1}_{H}$ is $H$-invariant on the left, $e({\bf
1}_{H})$ must also be $H$-invariant on the left. This shows that $e$ is the
image of the Hecke element $e({\bf 1}_{H})$.

Finally, for any $h_{1},h_{2}\in\mathcal{H}_{H}(\mathds{Z})$, we clearly have

$\displaystyle e_{h_{1}*_{H}h_{2}}({\bf 1}_{H})$ $\displaystyle=e_{h_{2}}\circ
e_{h_{1}}({\bf 1}_{H})$

As we previously said, elements of
$End_{\mathds{Z}[G]}\mathcal{C}_{c}(G/H,\mathds{Z})$ are uniquely determined
by the image they give to ${\bf 1}_{H}$, this shows that
$e_{h_{1}*_{H}h_{2}}=e_{h_{2}}\circ e_{h_{1}}$ and ends the proof of the
lemma. ∎

######  Remark 3.7.

The map

${\mathcal{H}_{H}(\mathds{Z})}$${\mathcal{H}_{H}(\mathds{Z})}$${h}$${h^{\vee}\colon
g\mapsto h(g^{-1}),}$$\scriptstyle{\simeq}$

is an involution and for any $h_{1},h_{2}\in\mathcal{H}_{H}(\mathds{Z})$ we
have $(h_{1}*_{H}h_{2})^{\vee}=h_{2}^{\vee}*_{H}h_{1}^{\vee}$.

The following lemma is meant to clarify the multiplicative structure of the
algebra $\mathcal{H}_{H}(\mathds{Z})$.

######  Lemma 3.8.

For $g,g^{\prime}\in G$, we have

${\bf 1}_{HgH}*_{H}{\bf 1}_{Hg^{\prime}H}=\sum_{g^{\prime\prime}\in
C_{g,g^{\prime}}}c^{H}(g,g^{\prime},g^{\prime\prime}){\bf
1}_{Hg^{\prime\prime}H},\quad
c^{H}(g,g^{\prime},g^{\prime\prime}):=\big{|}HgH\cap
g^{\prime\prime}Hg^{\prime-1}H\big{|}_{H}$

where, $C_{g,g^{\prime}}$ denotes a set of representatives for $H\backslash
HgHg^{\prime}H/H$.

###### Proof.

First, we know that the function ${\bf 1}_{HgH}*_{H}{\bf 1}_{Hg^{\prime}H}$ is
$H$-biinvariante (Remark 3.1), therefore, using the canonical basis of
$\mathcal{H}_{H}(\mathds{Z})$, it can be written as follows

${\bf 1}_{HgH}*_{H}{\bf 1}_{Hg^{\prime}H}=\sum_{g^{\prime\prime}\in
C_{g,g^{\prime}}}c^{H}(g,g^{\prime},g^{\prime\prime}){\bf
1}_{Hg^{\prime\prime}H},\quad
c^{H}(g,g^{\prime},g^{\prime\prime})\in\mathds{Z}_{\geq 0}$

for some finite set $C_{g,g^{\prime}}$ and integral coefficients
$c^{H}(g,g^{\prime},g^{\prime\prime})\neq 0$, for each $g^{\prime\prime}\in
C_{g,g^{\prime}}$.

Secondly, the integral defining ${\bf 1}_{HgH}*_{H}{\bf 1}_{Hg^{\prime}H}(a)$
is nonzero only if $a\in HgHg^{\prime}H$. Therefore, if
$Hg^{\prime\prime}H\subset HgHg^{\prime}H$, we have
$c^{H}(g,g^{\prime},g^{\prime\prime})={\bf 1}_{HgH}*_{H}{\bf
1}_{Hg^{\prime}H}(g^{\prime\prime})=\big{|}HgH\cap
g^{\prime\prime}Hg^{\prime-1}H\big{|}_{H}\in\mathds{Z}_{\geq 0}$. For each
double coset $Hg^{\prime\prime}H\subset HgHg^{\prime}H$, write
$g^{\prime\prime}=gh^{\prime\prime}g^{\prime}$ for some $h^{\prime\prime}\in
H$, thus

$c^{H}(g,g^{\prime},g^{\prime\prime})=\big{|}HgH\cap
gh^{\prime\prime}g^{\prime}Hg^{\prime-1}H\big{|}_{H}\geq\big{|}gH\big{|}_{H}=1.$

This shows that $C_{g,g^{\prime}}$ is indeed a set of representatives for
$H\backslash HgHg^{\prime}H/H$. ∎

######  Remark 3.9.

For any $g,g^{\prime}\in G$, we have

$\displaystyle\int_{G}{\bf 1}_{HgH}*_{H}{\bf 1}_{Hg^{\prime}H}(a)d\mu_{H}(a)$
$\displaystyle=\int_{G}\left(\int_{G}{\bf
1}_{Hg^{\prime}H}(b^{-1}a)d\mu_{H}(a)\right){\bf 1}_{HgH}(b)d\mu_{H}(b)$
$\displaystyle=\left|Hg^{\prime}H\right|_{H}\int_{G}{\bf
1}_{HgH}(b)d\mu_{H}(b)=\left|HgH\right|_{H}\left|Hg^{\prime}H\right|_{H}.$

This shows that the linear functional
${d_{H}\colon\mathcal{H}_{H}(\mathds{Z})}$${\mathds{Z},}$ defined on the
canonical basis elements by ${\bf 1}_{HgH}\longmapsto\left|HgH\right|_{H}$, is
an homomorphism of rings. Consequently,

${\bf 1}_{HgH}\in\mathcal{H}_{H}(\mathds{Z})^{\times}\Longleftrightarrow
d_{H}\left({\bf 1}_{HgH}\right)=1\Longleftrightarrow g\in N_{G}(H).$

######  Remark 3.10.

Let $\Upomega\subset\mathcal{A}_{ext}$ be any bounded subset containing the
alcove $\mathfrak{a}$. Proposition 2.9 shows that the product map yields an
isomorphism $U^{-}_{\Upomega}\times M_{1}\times
U^{+}_{\Upomega}\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}K_{\Upomega}$.

For any $m_{1},m_{2}\in M^{-}$, applying Lemma 3.2 shows that
$K_{\Upomega}m_{1}K_{\Upomega}m_{2}K_{\Upomega}=K_{\Upomega}m_{1}m_{2}K_{\Upomega}$
and $K_{\Upomega}m_{1}K_{\Upomega}\cap
m_{1}m_{2}K_{\Upomega}m_{2}^{-1}K_{\Upomega}=m_{1}K_{\Upomega}$. Accordingly,
by Lemma 3.8, we have

${\bf 1}_{K_{\Upomega}m_{1}K_{\Upomega}}*_{K_{\Upomega}}{\bf
1}_{K_{\Upomega}m_{2}K_{\Upomega}}={\bf
1}_{K_{\Upomega}m_{1}m_{2}K_{\Upomega}}.$

Therefore, we get an homomorphism of rings
${\mathds{Z}[\Lambda_{M}^{-}]}$${\mathcal{H}_{K_{\Upomega}}(\mathds{Z})}$ with
commutative image.

######  Lemma 3.11.

Let $H^{\prime}\subset H$ be two open compact subgroups of $G$ (e.g.
${H^{\prime}}=I$ and ${H}=K$). For any ring $R$ in which $[H:H^{\prime}]$ is
invertible, we have a natural isomorphism of algebras

${(\mathcal{H}_{H}(R),+,*_{H})}$${(e_{H}*_{H^{\prime}}\mathcal{H}_{H^{\prime}}(R)*_{H^{\prime}}e_{H},+,*_{H^{\prime}}),\quad
h}$${{|{H}|_{H^{\prime}}}^{-1}h.}$$\scriptstyle{\simeq}$

where $e_{H}:={|{H}|_{H^{\prime}}}^{-1}{\bf 1}_{H}$ is an idempotent of the
relative Hecke algebra $\mathcal{H}_{H^{\prime}}(R)$.

###### Proof.

This is clear. ∎

###  3.12. ”Averaging” maps are homomorphisms

######  Lemma 3.1.

Let $\mathcal{F}\subset\mathcal{A}$ be any facet. For any $w\in N$, we have

${\bf
1}_{{K}_{\mathcal{F}}^{\flat}w{K}_{\mathcal{F}}^{\flat}}=\mathbf{1}_{K_{\mathcal{F}}wK_{\mathcal{F}}}*_{K_{\mathcal{F}}}\mathbf{1}_{{K}_{\mathcal{F}}^{\flat}}={\bf
1}_{{K}_{\mathcal{F}}^{\flat}}*_{K_{\mathcal{F}}}{\bf
1}_{K_{\mathcal{F}}wK_{\mathcal{F}}}=\mathbf{1}_{K_{\mathcal{F}}w{K}_{\mathcal{F}}^{\flat}}=\sum_{m\in
M^{\flat}/M_{1}}{\bf 1}_{K_{\mathcal{F}}wmK_{\mathcal{F}}}.$

In particular, $\mathbf{1}_{{K}_{\mathcal{F}}^{\flat}}\in
Z(\mathcal{H}_{K_{\mathcal{F}}}(\mathds{Z}))$.

###### Proof.

Using Lemma 3.2, one shows that for any $w\in N$

$|{K}_{\mathcal{F}}\cap
w{K}_{\mathcal{F}}^{\flat}w^{-1}|_{{K}_{\mathcal{F}}}\mathbf{1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}^{\flat}}=|{K}_{\mathcal{F}}\cap
w{K}_{\mathcal{F}}w^{-1}|_{{K}_{\mathcal{F}}}\mathbf{1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}}*_{{K}_{\mathcal{F}}}\mathbf{1}_{{K}_{\mathcal{F}}^{\flat}}$

Note that ${K}_{\mathcal{F}}\cap w{K}_{\mathcal{F}}^{\flat}w^{-1}\subset
G_{1}$, hence it equals ${K}_{\mathcal{F}}\cap w({K}_{\mathcal{F}}^{\flat}\cap
G_{1})w^{-1}={K}_{\mathcal{F}}\cap w{K}_{\mathcal{F}}w^{-1}$. So
$\mathbf{1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}}*_{{K}_{\mathcal{F}}}\mathbf{1}_{{K}_{\mathcal{F}}^{\flat}}=\mathbf{1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}^{\flat}}.$
By Lemma 3.1, we have
${K}_{\mathcal{F}}w{K}_{\mathcal{F}}^{\flat}={K}_{\mathcal{F}}^{\flat}w{K}_{\mathcal{F}}^{\flat}$,
hence

$\displaystyle{\bf 1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}^{\flat}}$
$\displaystyle={\bf
1}_{{K}_{\mathcal{F}}^{\flat}w{K}_{\mathcal{F}}^{\flat}}={\bf
1}_{{K}_{\mathcal{F}}^{\flat}}*_{{K}_{\mathcal{F}}}{\bf
1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}}.$

Finally, since $M^{\flat}$ normalizes $K_{\mathcal{F}}$ and
${K}_{\mathcal{F}}^{\flat}=\sqcup_{m\in M^{\flat}/M_{1}}m{K}_{\mathcal{F}}$,
one has $\sum_{m\in M^{\flat}/M_{1}}{\bf
1}_{{K}_{\mathcal{F}}wm{K}_{\mathcal{F}}}={\bf
1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}^{\flat}}$. ∎

######  Corollary 3.2.

The ”averaging” map
$\iota^{\flat}\colon\mathcal{H}_{K_{\mathcal{F}}}(\mathds{Z})\to\mathcal{H}_{{K}_{\mathcal{F}}^{\flat}}(\mathds{Z})$
defined on basis elements by

${\bf
1}_{K_{\mathcal{F}}wK_{\mathcal{F}}}\mapsto\mathbf{1}_{{K}_{\mathcal{F}}^{\flat}w{K}_{\mathcal{F}}^{\flat}}$

is a surjective homomorphism of algebras.

###### Proof.

Clearly it is a surjective morphism of $\mathds{Z}$-modules. It suffices then
to verify that $\iota$ respects the algebra structure. For any two elements
$w,w^{\prime}\in N$, Lemma 3.1 shows

$\displaystyle\iota({\bf
1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}}*_{{K}_{\mathcal{F}}}{\bf
1}_{{K}_{\mathcal{F}}w^{\prime}{K}_{\mathcal{F}}})$ $\displaystyle={\bf
1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}}*_{{K}_{\mathcal{F}}}{\bf
1}_{{K}_{\mathcal{F}}w^{\prime}{K}_{\mathcal{F}}}*_{{K}_{\mathcal{F}}}{\bf
1}_{{K}_{\mathcal{F}}^{\flat}}$ $\displaystyle={|M^{\flat}/M_{1}|}^{-1}{\bf
1}_{{K}_{\mathcal{F}}w{K}_{\mathcal{F}}}*_{{K}_{\mathcal{F}}}{\bf
1}_{{K}_{\mathcal{F}}^{\flat}}*_{{K}_{\mathcal{F}}}{\bf
1}_{{K}_{\mathcal{F}}w^{\prime}{K}_{\mathcal{F}}}*_{{K}_{\mathcal{F}}}{\bf
1}_{{K}_{\mathcal{F}}^{\flat}}$ $\displaystyle={\bf
1}_{{K}_{\mathcal{F}}^{\flat}w{K}_{\mathcal{F}}^{\flat}}*_{{K}_{\mathcal{F}}^{\flat}}{\bf
1}_{{K}_{\mathcal{F}}^{\flat}w^{\prime}{K}_{\mathcal{F}}^{\flat}}\qed$

###  3.13. Iwahori–Matsumoto presentation and other properties

Recall that $\mathcal{S}\subset N_{1}/M_{1}$ corresponds to the set of
orthogonal reflections with respect to the walls of the fixed alcove
$\mathfrak{a}$ (Using the identification $N_{1}/M_{1}\simeq
W_{\textit{\emph{\text{aff}}}}$).

######  Theorem 3.1 (The Iwahori–Matsumoto Presentation).

The Iwahori–Hecke ring $\mathcal{H}_{I^{\flat}}(\mathds{Z})$ is the free
$\mathds{Z}$-module with basis $(i_{w}^{\flat})_{w\in\widetilde{W}^{\flat}}$
endowed with the unique ring structure satisfying

  * •

The braid relations:
$i_{w}^{\flat}*_{I^{\flat}}i_{w^{\prime}}^{\flat}=i_{ww^{\prime}}^{\flat}\text{
if }w,w^{\prime}\in\widetilde{W}^{\flat}\text{ such that
}\ell(w)+\ell(w^{\prime})=\ell(ww^{\prime})$.

  * •

The quadratic relations:
$(i_{s}^{\flat})^{2}=q_{s}i_{1}^{\flat}+(q_{s}-1)i_{s}^{\flat}\text{ if
}s\in\mathcal{S}^{\flat}$.

This is a very known result, at least for Iwahori–Hecke attached to extended
affine Weyl groups. See for example [Gar97, §6.3], [GP00], [CR62],[Car93]. For
the general case, it was showed in [Vig16, Theorem 2.1] for
$\mathds{Z}$-coefficients or [Ros15, Proposition 4.1.1] for
$\mathds{C}$-coefficients. However, since we have enough ingredients, we prove
here using a slightly different argument:

###### Proof.

  1. (1)

The Braid relation : Let $w,w^{\prime}\in\widetilde{W}^{\flat}$ such that
$\ell(w)+\ell(w^{\prime})=\ell(ww^{\prime})$, then by Remark 3.4 together with
Remark 3.8 we obtain that
$i_{w}^{\flat}*_{I^{\flat}}i_{w^{\prime}}^{\flat}=c^{I^{\flat}}(w,w^{\prime},ww^{\prime})i_{ww^{\prime}}^{\flat}$
where $c^{I^{\flat}}(w,w^{\prime},ww^{\prime})=|I^{\flat}wI^{\flat}\cap
ww^{\prime}I^{\flat}w^{\prime-1}I|_{{I^{\flat}}}$. Combining Remark 3.9 and
Remark 3.4 again shows that $c^{I^{\flat}}(w,w^{\prime},ww^{\prime})$ must be
$1$.

  2. (2)

For the quadratic relations one has by Lemma 3.3
$I^{\flat}sI^{\flat}sI^{\flat}=I^{\flat}\cup I^{\flat}sI^{\flat}$ for any
$s\in\mathcal{S}$. Thus,
$(i_{s}^{\flat})^{2}=c^{I^{\flat}}(s,s,1)i_{1}^{\flat}+c^{I^{\flat}}(s,s,s)i_{s}^{\flat}$
where $c^{I^{\flat}}(s,s,1)=|I^{\flat}sI^{\flat}|_{I^{\flat}}=q_{s}$ and
$c^{I^{\flat}}(s,s,s)=|I^{\flat}sI^{\flat}\cap
sI^{\flat}sI^{\flat}|_{I^{\flat}}$ by Remark 3.8.

Clearly $sI^{\flat}sI^{\flat}\subset
I^{\flat}sI^{\flat}sI^{\flat}=I^{\flat}\sqcup I^{\flat}sI^{\flat}$, thus
$sI^{\flat}sI^{\flat}\setminus I^{\flat}\subset I^{\flat}sI^{\flat}$ but since
$I^{\flat}\cap I^{\flat}sI^{\flat}=\emptyset$ we must have
$I^{\flat}sI^{\flat}\cap sI^{\flat}sI^{\flat}=sI^{\flat}sI^{\flat}\setminus
I^{\flat}$, hence $|I^{\flat}sI^{\flat}\cap
sI^{\flat}sI^{\flat}|_{I^{\flat}}=|sI^{\flat}sI^{\flat}|_{I^{\flat}}-1=|I^{\flat}sI^{\flat}|_{I^{\flat}}-1=q_{s}-1$.∎

######  Lemma 3.2.

For any $w,w^{\prime}\in\widetilde{W}^{\flat}$, we have

$i_{w}^{\flat}*_{I^{\flat}}i_{w^{\prime}}^{\flat}=i_{ww^{\prime}}^{\flat}\Leftrightarrow
I^{\flat}ww^{\prime}I^{\flat}=I^{\flat}wI^{\flat}w^{\prime}I^{\flat}\Leftrightarrow\ell(w)+\ell(w^{\prime})=\ell(ww^{\prime})\Leftrightarrow
q_{w}q_{w^{\prime}}=q_{ww^{\prime}}.$

###### Proof.

The first two equivalences can be easily deduced using Remark 3.4 and Lemma
3.8. Let us prove the last one. If
$i_{w}^{\flat}*_{I^{\flat}}i_{w^{\prime}}^{\flat}=i_{ww^{\prime}}^{\flat}$,
Remark 3.9 implies that $q_{w}q_{w^{\prime}}=q_{ww^{\prime}}$. Now, assume
that $q_{w}q_{w^{\prime}}=q_{ww^{\prime}}$, then by Remark 3.9 again and Lemma
3.8

$(c^{I^{\flat}}(w,w^{\prime},ww^{\prime})-1)\int_{G}i_{ww^{\prime}}^{\flat}(a)d\mu_{I^{\flat}}(a)+\sum_{w^{\prime\prime}\in
C_{w,w^{\prime}}\setminus\\{ww^{\prime}\\}}c^{I^{\flat}}(w,w^{\prime},w^{\prime\prime})\int_{G}i_{w^{\prime\prime}}^{\flat}(a)d\mu_{I^{\flat}}(a)=0.$

As we saw in the proof of Theorem 3.1, we have
$c^{I^{\flat}}(w,w^{\prime},ww^{\prime})=1$. Accordingly,

$\sum_{w^{\prime\prime}\in
C_{w,w^{\prime}}\setminus\\{ww^{\prime}\\}}c^{I^{\flat}}(w,w^{\prime},w^{\prime\prime})\int_{G}i_{w^{\prime\prime}}^{\flat}(a)d\mu_{I^{\flat}}(a)=0.$

But if $\\{ww^{\prime}\\}\subsetneq C_{w,w^{\prime}}$, then all
$c^{I^{\flat}}(w,w^{\prime},w^{\prime\prime})=\big{|}I^{\flat}wI^{\flat}\cap
w^{\prime\prime}I^{\flat}w^{\prime-1}I^{\flat}\big{|}_{I^{\flat}}\geq 1$. This
can not be possible, unless $C_{w,w^{\prime}}=\\{ww^{\prime}\\}$, i.e.
$I^{\flat}ww^{\prime}I^{\flat}=I^{\flat}wI^{\flat}w^{\prime}I^{\flat}$, i.e.
$i_{w}^{\flat}*_{I^{\flat}}i_{w^{\prime}}^{\flat}=i_{ww^{\prime}}^{\flat}$.
This concludes the proof of the lemma. ∎

The above Iwahori–Matsumoto presentation yields the following consequences:

######  Corollary 3.3.

  1. (i)

For any $w,w^{\prime}\in\widetilde{W}^{\flat}$, if
$\ell(w)+\ell(w^{\prime})=\ell(ww^{\prime})$ then
$I_{w^{-1}(\mathfrak{a})}I\cap
I_{w^{\prime}(\mathfrak{a})}I=(I_{w^{-1}(\mathfrak{a})}\cap
I_{w^{\prime}(\mathfrak{a})})I=I$.

  2. (ii)

The $\mathds{Z}$-linear map $z\mapsto i_{z}^{\flat}$ embeds the group algebra
$\mathds{Z}[\widetilde{\Omega}^{\flat}]$ into
$\mathcal{H}_{I^{\flat}}(\mathds{Z})$.

  3. (iii)

For all $w\in W_{\textit{\emph{\text{aff}}}}^{\flat}$ and
$z\in\widetilde{\Omega}^{\flat}$, one has
$i_{w}^{\flat}i_{z}^{\flat}=i_{z}^{\flat}i_{z^{-1}wz}^{\flat}$.

  4. (iv)

Let $w\in W_{\textit{\emph{\text{aff}}}}^{\flat}$ and $w=s_{1}s_{2}\cdots
s_{\ell(w)}$ a reduced expression. Using Lemma 3.3, one has

$q_{w}=q_{s_{1}}q_{s_{2}}\cdots q_{s_{\ell(w)}}.$ (1)

  5. (v)

Let $w=w_{\textit{\emph{\text{aff}}}}\cdot z\in\widetilde{W}^{\flat}$ with
$w_{\textit{\emph{\text{aff}}}}=s_{1}\cdots s_{\ell(w)}$ a reduced word in
$W_{\textit{\emph{\text{aff}}}}^{\flat}$ and $z\in\widetilde{\Omega}^{\flat}$,
we have

$\displaystyle(i_{w}^{\flat})^{-1}=i_{z^{-1}}^{\flat}(i_{w_{\textit{\emph{\text{aff}}}}}^{\flat})^{-1}=q_{w}^{-1}{i_{z^{-1}}^{\flat}(i_{s_{\ell(w)}}^{\flat}-q_{s_{\ell(w)}}+1)\cdots(i_{s_{1}}^{\flat}-q_{s_{1}}+1)}.$

which lives in $\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$ given Lemma 3.2.

  6. (vi)

If we do not include $q^{-1}$ in the coefficients ring, we still have

$i_{w}^{\flat}(i_{w}^{\flat})^{*}=(i_{w}^{\flat})^{*}i_{w}^{\flat}=q_{w},\quad\text{in
}\mathcal{H}_{I^{\flat}}(\mathds{Z}),$

where,
$(i_{w}^{\flat})^{*}:=(i_{s_{\ell(w)}}^{\flat}-q_{s_{\ell(w)}}+1)\cdots(i_{s_{1}}^{\flat}-q_{s_{1}}+1)$.

  7. (vii)

For any $s\in\mathcal{S}^{\flat}$, set
$\mathscr{P}_{s}^{l}:=\\{w\in\widetilde{W}^{\flat}\colon
w<sw,\,i.e.\,\ell(sw)=\ell(w)+1\\}$ and
$\mathscr{P}_{s}^{r}:=\\{w\in\widetilde{W}^{\flat}\colon
w>ws,\,i.e.\,\ell(ws)=\ell(w)-1\\}$ then
$(\mathscr{P}_{s}^{l})^{-1}=\mathscr{P}_{s}^{r}$ and
$\widetilde{W}=\mathscr{P}_{s}^{l}\sqcup
s\mathscr{P}_{s}^{l}=\mathscr{P}_{s}^{r}\sqcup\mathscr{P}_{s}^{r}s$ [Bou68,
Ch. IV, §1.7]. For any $s\in\mathcal{S}^{\flat}$ and any
$w\in\widetilde{W}^{\flat}$, we have

$i_{s}^{\flat}*_{{I^{\flat}}}i_{w}^{\flat}=\begin{cases}i_{sw}^{\flat},&\text{
if }w\in\mathscr{P}_{s}^{l},\\\
q_{s}i_{w}^{\flat}+(q_{s}-1)i_{sw}^{\flat},&\text{ if }w\in
s\mathscr{P}_{s}^{l}.\end{cases}\text{ and
}i_{w}^{\flat}*_{{I^{\flat}}}i_{s}^{\flat}=\begin{cases}i_{ws}^{\flat},&\text{
if }w\in\mathscr{P}_{s}^{r},\\\
q_{s}i_{w}^{\flat}+(q_{s}-1)i_{ws}^{\flat},&\text{ if
}w\in\mathscr{P}_{s}^{r}s.\end{cases}$

###### Proof.

These properties follow readily from Theorem 3.1.∎

######  Proposition 3.4.

Set
$\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z}):=\mathcal{H}(G_{1}^{\flat}\sslash
I^{\flat},\mathds{Z})$ and let
$\mathcal{H}_{I^{\flat}}^{\Upomega}(\mathds{Z})$ be the subalgebra of
functions supported on cosets of the form $\sigma I^{\flat}$ for
$\sigma\in\widetilde{\Upomega}^{\flat}$. We have an isomorphism of
$\mathds{Z}$-algebras

$\mathcal{H}_{I^{\flat}}(\mathds{Z})\simeq\mathds{Z}[\widetilde{\Upomega}^{\flat}]\otimes^{\prime}\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z})$

where, $\otimes^{\prime}$ indicates that the tensor modules is endowed with a
”twisted” multiplication: for any
$\sigma,\sigma^{\prime}\in\widetilde{\Upomega}^{\flat}$ and any
$w,w^{\prime}\in\widetilde{W}^{\flat}$

$(\sigma\otimes i_{w})(\sigma^{\prime}\otimes
i_{w^{\prime}})=\sigma\sigma^{\prime}\otimes
i_{\sigma^{\prime-1}w\sigma^{\prime}}*_{I^{\flat}}i_{w^{\prime}}.$

###### Proof.

This an immediate consequence of Proposition 3.1 and Lemma 3.1 together with
ii and iii of Corollary 3.3. ∎

######  Remark 3.5.

The restriction of the surjective morphism (Corollary 3.2) of
$\iota^{\flat}\colon\mathcal{H}_{I}({\mathds{Z}})\twoheadrightarrow\mathcal{H}_{I^{\flat}}({\mathds{Z}})$
to $\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}(\mathds{Z})$ is injective
(Remark 3.4) and yields an isomorphism of algebras
$\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}(\mathds{Z})\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z})$
since $G_{1}^{\flat}\sslash I^{\flat}\simeq G_{1}\sslash I$. Thanks to Remark
3.4, the following diagram is commutative

${\mathcal{H}_{I}(\mathds{Z})}$${\mathds{Z}[\widetilde{\Upomega}]\otimes^{\prime}\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}(\mathds{Z})}$${\mathcal{H}_{I^{\flat}}(\mathds{Z})}$${\mathds{Z}[\widetilde{\Upomega}^{\flat}]\otimes^{\prime}\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z})}$$\scriptstyle{\simeq}$$\scriptstyle{\iota^{\flat}}$$\scriptstyle{\square^{\flat}\otimes\iota^{\flat}}$$\scriptstyle{\simeq}$

###  3.14. Root data

Recall that we have defined in §2.5 the root system $\Sigma$ and that we have
$\Lambda_{\textit{\emph{\text{aff}}}}^{\vee}=\mathds{Z}[\Sigma]$. Set
$\Sigma^{\flat}\subset\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}$ for the
subset corresponding to $\Sigma$ under the isomorphism
$\Lambda_{\textit{\emph{\text{aff}}}}\simeq\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}$
(§3.1). We consider the root datum
$\mathcal{D}_{\textit{\emph{\text{aff}}}}^{\flat}:=({\Lambda}_{\textit{\emph{\text{aff}}}}^{\flat},\Sigma^{\flat,\vee},{\Lambda}_{\textit{\emph{\text{aff}}}}^{\flat,\vee},\Sigma^{\flat})\simeq\mathcal{D}_{\textit{\emph{\text{aff}}}}$.

Define the free abelian group $\Lambda_{M}^{1,\vee}\colon=\\{x\in
X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}\colon\langle
x,\nu_{M}(\Lambda_{M}^{1})\rangle_{\mathds{R}}\subset\mathds{Z}\\}$. We obtain
in this way a perfect pairing
$\langle\cdot,\cdot\rangle\colon\Lambda_{M}^{1}\times\Lambda_{M}^{1,\vee}\to\mathds{Z}$
(§2.2). Therefore, we have in our hands another root datum

$\mathcal{D}:=({\Lambda}_{M}^{1},\Sigma^{1,\vee},{\Lambda}_{M}^{1,\vee},\Sigma^{1},\Delta^{1,\vee}).$

In [Lus89, §3.2], Lusztig attaches to any quadruplet
$(\mathcal{D}^{\prime},\mathbf{W},v=?,L)$ (where, $\mathcal{D}^{\prime}$ is a
root datum, $\mathbf{W}$ the extended affine group associated with
$\mathcal{D}^{\prime}$, $v$ an indeterminate §3.2 in loc. cit. and
$L\colon\mathbf{W}\to\mathds{N}$ a function §3.1 in loc. cit. verifying an
additivity condition) a Hecke algebra
$\mathcal{H}(\mathcal{D}^{\prime},\mathbf{W},v=?,L)$ generated by elements
$\\{T_{w}\colon w\in\mathbf{W}\\}$, denoted by $H$ in loc. cit..

######  Corollary 3.1.

The map $i_{w}\mapsto T_{w}$ yields two isomorphisms of Hecke algebras

$\mathcal{H}_{{I}^{1}}(\mathds{Z}[q^{-1}])\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}\mathcal{H}(\mathcal{D},\widetilde{W}_{\textit{\emph{\text{aff}}}},v=q,L)\text{
and
}\mathcal{H}_{{I}}^{\textit{\emph{\text{aff}}}}(\mathds{Z}[q^{-1}])\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}\mathcal{H}(\mathcal{D}_{\textit{\emph{\text{aff}}}},{W}_{\textit{\emph{\text{aff}}}},v=q,L)$

where, $L\colon W\to\mathbb{N},w\mapsto-\frac{1}{2}\omega(q_{w})$.

###### Proof.

The corollary follows readily from the Iwahori–Matsumoto presentation in
Theorem 3.1 and Remark 3.5, because
$\mathcal{H}(\mathcal{D}^{\prime},\mathbf{W},v=?,L)$ is characterized
precisely by the same Braid relation [Lus89, §2.1] and quadratic relations
[Lus89, §3.2]. ∎

###  3.15. Properties of dominance $\Lambda_{M}^{\flat}$

######  Lemma 3.1.

We have the following properties regarding $\Lambda_{M}^{\flat}$:

  1. 1 -

If $m_{1},m_{2},\dots,m_{k}\in\Lambda_{M}^{\flat}$, then there exists
$m_{\circ}\in\Lambda_{M}^{-\flat}$ such that
$m_{\circ}+m_{{I^{\flat}}}\in\Lambda_{M}^{-,\flat}$121212We will denote the
operation on $\Lambda_{M}$ additively. for all $1\leq i\leq k$.

  2. 2 -

If $m_{1},m_{2}\in\Lambda_{M}^{-,\flat}$, then
$\ell(m_{1}+m_{2})=\ell(m_{1})+\ell(m_{2})$.

For any $w\in{W}$, we have:

  3. 3 -

$\ell(mw)=\ell(m)+\ell(w)$ if $m\in\Lambda_{M}^{-,\flat}$ and
$\ell(wm)=\ell(w)+\ell(m)$ if $m\in\Lambda_{M}^{+,\flat}$.

  4. 4 -

$\ell(w(m))=\ell(wmw^{-1})=\ell(m)$ for all $m\in\Lambda_{M}^{\flat}$.

###### Proof.

These are well-known facts for extended affine Weyl groups, so in particular
for $\mathcal{H}_{{I}^{1}}(\mathds{Z}[q^{-1}])$. Accordingly, the lemma
follows immediately because
$\square^{\beta}\colon\Lambda_{M}^{\flat}\to\Lambda_{M}^{1}$ is
$W$-equivariant and preserves dominance and the length $\ell$. See also
[Ros15, Lemma 5.2.1]. However, for the reader’s convenience, let us show them
using the ingredients we prepared so far:

1\. Once we show it for $k=2$ (which is obvious), take $m_{\circ}=\sum
m_{\circ,i}$ such that $m_{\circ,i}+m_{i}\in\Lambda_{M}^{-}$ and
$m_{\circ,i}\in\Lambda_{M}^{-}$.

2\. If $m,m^{\prime}$ are both antidominant then by Lemma 3.2
$I^{\flat}mm^{\prime}I^{\flat}=I^{\flat}mI^{\flat}m^{\prime}I^{\flat}$, hence
$\ell(mm^{\prime})=\ell(m)+\ell(m^{\prime})$ by Lemma 3.2.

3\. If $w\in W$, using this time both Lemma 3.2 and Lemma 3.1, we get
$I^{\flat}mwI^{\flat}=I^{\flat}mI^{\flat}wI^{\flat}$, hence
$\ell(mw)=\ell(m)+\ell(w)$ by Lemma 3.2.

4\. Thanks to Remark 3.4, this follows from the extended affine case. ∎

###  3.16. The Weyl dot-action

Set
$\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}:=\mathds{Z}[\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}]\subset\mathcal{R}^{\flat}:=\mathds{Z}[\Lambda_{M}^{\flat}]$.

The map $mM^{\flat}\mapsto{\mathbf{1}}_{mM^{\flat}}$ yields a natural
identification between $\mathcal{R}^{\flat}$ (resp.
$\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}$) and the Hecke algebra
$\mathcal{H}(M\sslash M^{\flat},\mathds{Z})$ (resp.
$\mathcal{H}(M^{\textit{\emph{\text{aff}}},\flat}\sslash
M^{\flat},\mathds{Z})$). The action of the Weyl group on $\Lambda_{M}^{\flat}$
extends linearly to an action on $\mathcal{R}^{\flat}$. For integrality
reasons that will become clearer in the coming sections, we are interested in
a variant of this action:

######  Definition 3.1 (Dot-action).

We define a twisted action of $W$ on $\mathcal{H}(M\sslash
M^{\flat},\mathds{Z}[q^{-1}])$, by

$\dot{w}(r):=m\mapsto\delta_{B}(m)^{1/2}\delta_{B}(w^{-1}(m))^{-1/2}r(w^{-1}(m)),$

for any $w\in W$ and any $r\in\mathcal{H}(M\sslash
M^{\flat},\mathds{Z}[q^{-1}])$. In particular, for any $m\in M$,

$\dot{w}({\bf 1}_{mM^{\flat}})=\delta_{B}(w(m))^{1/2}\delta_{B}(m)^{-1/2}{\bf
1}_{w(m)M^{\flat}}.$

Thus, upon identifying $\mathcal{H}(M\sslash M^{\flat},\mathds{Z}[q^{-1}])$
with ${\mathds{Z}[q^{-1}]\otimes_{\mathds{Z}}\mathcal{R}^{\flat}}$, the above
dot-action is given on basis elements by
$\dot{w}(mM^{\flat})=\left(\frac{\delta_{B}(w(m))}{\delta_{B}(m)}\right)^{1/2}w(m)M^{\flat}\in\Lambda_{M}^{\flat}$,
and extended $\mathds{Z}[q^{-1}]$-linearly to
$\mathds{Z}[q^{-1}]\otimes_{\mathds{Z}}\mathcal{R}^{\flat}$.

######  Remark 3.2.

(i) From now on, we write $\dot{W}$ to specify that we are dealing with the
dot-action of $W$ and not the standard one. (ii) The natural projection map
${\square}^{\flat}\colon\Lambda_{M}\to\Lambda_{M}^{\flat}$ is
$\dot{W}$-equivariant, i.e. $(\dot{w}(m))^{\flat}=\dot{w}(m^{\flat})$ for any
$w\in W$ and any $m\in\Lambda_{M}$.

Define $c(m,n):=\delta_{B}(nmn^{-1})^{1/2}\delta_{B}(m)^{-1/2}$ for any $m\in
M$ and any $n\in N$. Note that since the $\delta_{B}$ is trivial on the
compact $M^{1}$ it factors through $\Lambda_{M}^{1}$, and so $c(m,n)$ factors
then through the image of $(m,n)$ in $\Lambda_{M}^{1}\times W$. The following
lemma will be used in the sequel.

######  Lemma 3.3.

For any $m\in M$ and $n\in N$ with image $w$ in $W$, we have

  1. (i)

$c(m,n)=\prod_{\alpha\in\Phi_{red}^{-}\cap
w^{-1}(\Phi_{red}^{+})}\delta_{\alpha}(m)\in q^{\mathds{Z}}$.

  2. (ii)

If $m\in M^{-}$ then $c(m,n)$ is positive power of $q$.

We could have extracted (ii) from a similar result proved in [Mac71, Lemma
3.2.8 & Proposition 3.2.4] for simply connected groups.

###### Proof.

(i) Recall that the modulus character $\delta_{B}\colon B=MU^{+}\to
q^{\mathds{Z}},mu\mapsto|\det\text{Ad}_{\text{Lie}(U^{+})}(m)|$ (it is trivial
on $U^{+}$)131313Being an absolute value of a determinant on an $F$-vector
space, the modulus function has clearly its values in $q^{\mathds{Z}}$.. Given
that
$\text{Lie}(\mathbf{U}^{+})=\oplus_{\alpha\in\Phi_{red}^{+}}\text{Lie}(\mathbf{U}_{\alpha})$,
we have

$\delta_{B}(m)=\prod_{\alpha\in\Phi_{red}^{+}}\delta_{\alpha}(m),\text{ where
}\delta_{\alpha}(m):=|\det\text{Ad}_{\text{Lie}(\mathbf{U}_{\alpha})}(m)|.$

Since $n\mathbf{U}_{\alpha}n^{-1}=\mathbf{U}_{w(\alpha)}$, we have
$n{\text{Lie}(\mathbf{U}_{\alpha})}n^{-1}={\text{Lie}(\mathbf{U}_{w(\alpha)})}$
and so

$\displaystyle\delta_{\alpha}(w(m))=|\det\text{Ad}_{\text{Lie}(\mathbf{U}_{\alpha})}(w(m))|=|\det\text{Ad}_{\text{Lie}(\mathbf{U}_{w^{-1}(\alpha)})}(m)|=\delta_{w^{-1}(\alpha)}(m).$

Accordingly,

$\displaystyle
c(m,n)=\frac{\prod_{\alpha\in\Phi_{red}^{+}}\delta_{w^{-1}(\alpha)}(m)^{1/2}}{\prod_{\alpha\in\Phi_{red}^{+}}\delta_{\alpha}(m)^{1/2}}$
$\displaystyle=\frac{\prod_{\alpha\in
w^{-1}(\Phi_{red}^{+})\cap\Phi_{red}^{-}}\delta_{\alpha}(m)^{1/2}}{\prod_{\alpha\in\Phi_{red}^{+}\cap
w^{-1}(\Phi_{red}^{-})}\delta_{\alpha}(m)^{1/2}}$
$\displaystyle={\prod_{\alpha\in\Phi_{red}^{-}\cap
w^{-1}(\Phi_{red}^{+})}\delta_{\alpha}(m)}\quad\quad\quad\quad(\delta_{-\alpha}=\delta_{\alpha}^{-1}).$

(ii) For any $m\in M^{-}$ and any $\alpha\in\Phi_{red}^{-}$, one has
$\omega(\delta_{\alpha}(m))\leq 0$. Hence, $c(m,n)\in q^{\mathds{N}}$ for any
$n\in N$. ∎

##  4\. Twisted integral Bernstein morphism

###  4.1. Freeness of the $\mathcal{R}^{\flat}$-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$

Let $\mathcal{F}\subset\mathbb{A}(\mathbf{G},\mathbf{S})_{red}$ be any fixed
facet. Following the ideas of Bernstein [BD84], as exposed by the approach of
[HKP10], we define the universal unramified principal series right
$\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})=\mathcal{C}_{c}(M^{\flat}U^{+}\backslash
G/K_{\mathcal{F}},\mathds{Z})$, this is the set of $\mathds{Z}$-valued
functions supported on finitely many double cosets. The
$\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$-module structure of
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ comes from the natural
right action by convolution with respect to the normalized measure
$\mu_{K_{\mathcal{F}}^{\flat}}$ giving $K_{\mathcal{F}}^{\flat}$ volume $1$.

The natural identification between
$\mathcal{R}^{\flat}\simeq\mathcal{H}(M\sslash M^{\flat},\mathds{Z})$ allows
us to endow the $\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ with a left
$\mathcal{R}^{\flat}$-action as follows: define for any
$\psi\in\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ and
$r\in\mathcal{R}^{\flat}$:

$r\cdot\psi(g):=\int_{M}r(a)\psi(a^{-1}g)d\mu_{M^{\flat}}(a)\quad(\forall g\in
G)$

here, $d\mu_{M^{\flat}}(a)$ is the Haar measure on $M$ giving $M^{\flat}$
volume $1$. We will see in Lemma 4.3 a more concrete description of this
action, it will confirm that the resulting $r\cdot\psi$ is indeed
$\mathds{Z}$-valued. It is clear that the actions of
$\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ and $\mathcal{R}$ on
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ commute:
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ is an
$(\mathcal{R}^{\flat},\mathcal{H}_{{I^{\flat}}}{K_{\mathcal{F}}^{\flat}}(\mathds{Z}))$-bimodule.

######  Remark 4.1.

Here, we have defined an twisted action without using the modulus character.
The standard action is defined as follows

$\displaystyle r\cdot_{\text{twist}}\psi(g)$
$\displaystyle:=\int_{M}\delta_{B}^{1/2}(a)r(a)\psi(a^{-1}g)d\mu_{M_{1}}(a).\qed$

In Theorem 4.3, we will generalize [HKP10, Lemma 1.6.1]. For this purpose, we
begin by the following two lemmas. For any $n\in N$ set
$v_{n,\mathcal{F}}^{\flat}:={\bf 1}_{U^{+}nK_{\mathcal{F}}^{\flat}}$.

######  Lemma 4.2.

The family

$\\{v_{w,\mathcal{F}}^{\flat}\colon
w\in\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}\\},$

forms a $\mathds{Z}$-basis for the $\mathds{Z}$-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$.

###### Proof.

This is an immediate consequence of $U^{\pm}\backslash
G/K_{\mathcal{F}}^{\flat}\cong\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}$
(Proposition 3.1). ∎

######  Lemma 4.3.

For any $w\in N$ and $m\in\Lambda_{M}^{\flat}$ we have

$m\cdot v_{n,\mathcal{F}}^{\flat}=v_{mn,\mathcal{F}}^{\flat}.$

So the action of $\mathcal{R}^{\flat}$ on
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ is induced by the left
action of $\Lambda_{M}^{\flat}$ on $M^{\flat}U^{+}\backslash
G/{K_{\mathcal{F}}^{\flat}}$.

###### Proof.

Let $n\in N$ and $r=mM^{\flat}\in\mathcal{R}$ for some $m\in M$. Recall that
$\mathcal{R}^{\flat}$ is identified with the Iwahori–Hecke algebra for $M$;
$r=mM^{\flat}\leftrightarrow r={\bf 1}_{mM^{\flat}}$. The integral defining
$r\cdot v_{n,\mathcal{F}}^{\flat}(b)$ is (by definition) non-zero only if
$b\in
mM^{\flat}U^{+}n{K_{\mathcal{F}}^{\flat}}=M^{\flat}U^{+}mn{K_{\mathcal{F}}^{\flat}}$,
hence $r\cdot v_{n,\mathcal{F}}^{\flat}=\text{s}\,v_{mn,\mathcal{F}}^{\flat}$
where $s$ is the scalar $r\cdot v_{n,\mathcal{F}}^{\flat}(mw)=|mM^{\flat}|=1$.
∎

The following proposition gives a precise description of the
$\mathcal{R}^{\flat}$-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$.

######  Proposition 4.4.

The $\mathcal{R}^{\flat}$-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ is free of rank
$r_{\mathcal{F}}:=|W/\jmath_{W}(W_{\mathcal{F}})|$, with basis
$\\{v_{w,\mathcal{F}}^{\flat},\,w\in D_{\mathcal{F}}^{\flat}\\}$ where
$D_{\mathcal{F}}^{\flat}\subset W_{a_{\circ}}$ is any fixed set of
representatives for $W/\jmath_{W}(W_{\mathcal{F}})$. In particular,
$\\{v_{w,\mathfrak{a}}^{\flat},\,w\in W_{a_{\circ}}\\}$ is a canonical basis
for the $\mathcal{R}^{\flat}$-module $\mathcal{M}_{I^{\flat}}(\mathds{Z})$.

###### Proof.

Given that we have a bijection of $\Lambda_{M}^{\flat}$-sets
$\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}\cong\Lambda_{M}^{\flat}\times(W/\jmath_{W}(W_{\mathcal{F}}))$
(Lemma 3.2), the proposition follows readily from Lemma 4.2 and Lemma 4.3. ∎

######  Remark 4.5.

Proposition 4.4 yields an alternative proof of the main theorem of [Lan02]
giving the dimension of the space of $K_{\mathcal{F}}$-fixed vectors of an
unramified principal series representation of $G$.

######  Remark 4.6.

By Lemma 3.1 and Corollary 3.2, the map
$\psi^{\flat}\colon\mathcal{M}_{K_{\mathcal{F}}}(\mathds{Z})\to\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$,
$v\mapsto v*_{K_{\mathcal{F}}}\mathbf{1}_{K_{\mathcal{F}}^{\flat}}$ is an
homomorphism of
bi-$(\Lambda_{M}-\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z}))$-modules.
Here, the left action $\Lambda_{M}$ on
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})=\mathcal{M}_{K_{\mathcal{F}}}(\mathds{Z})*_{K_{\mathcal{F}}}\mathbf{1}_{K_{\mathcal{F}}^{\flat}}$
is well defined and factors through $\Lambda_{M}^{\flat}$.

###  4.2. Properties of the $\mathcal{H}_{I^{\flat}}(\mathds{Z})$-module
$\mathcal{M}_{I^{\flat}}(\mathds{Z})$

In this subsection we will focus on the case $\mathcal{F}=\mathfrak{a}$, for
this write $v_{w,\mathfrak{a}}^{\flat},w\in\widetilde{W}^{\flat}$ simply as
$v_{w}^{\flat}$. We would like to describe the structure of
$\mathcal{M}_{I^{\flat}}(\mathds{Z})$, this time as a
$\mathcal{H}_{I^{\flat}}(\mathds{Z})$-module. However, a satisfactory result
can only be obtained after enlarging the coefficients ring $\mathds{Z}$ to
$R:=\mathds{Z}[q^{\pm 1}]$.

######  Proposition 4.1.

The action of $\mathcal{H}_{I^{\flat}}(\mathds{Z})$ on
$\mathcal{M}_{I^{\flat}}(\mathds{Z})$ has the following rules: for any $w\in
W_{a_{\circ}}^{\flat}$ and $m\in\Lambda_{M}^{\flat}$, we have

  1. 1 -

$v_{1}^{\flat}*_{I^{\flat}}i_{w}^{\flat}=v_{w}^{\flat}$,

  2. 2 -

$v_{m}^{\flat}*_{I^{\flat}}i_{w}^{\flat}=v_{mw}^{\flat}$,

  3. 3 -

$v_{1}^{\flat}*_{I^{\flat}}i_{m}^{\flat}=v_{m}^{\flat}$ if
$m\in\Lambda_{M}^{-,\flat}$.

  4. 4 -

$v_{1}^{\flat}*_{I^{\flat}}(i_{m}^{\flat})^{-1}=v_{-m}^{\flat}\in\mathcal{M}_{I^{\flat}}(\mathds{Z}[q^{-1}])$
if $m\in\Lambda_{M}^{-,\flat}$.

###### Proof.

1 - & 3 - Let $n\in N$. If the quantity
$v_{1}^{\flat}*_{I^{\flat}}i_{n}^{\flat}(b)=\int_{U^{+}{I}^{\flat}}i_{n}^{\flat}(a^{-1}b)d\mu_{I^{\flat}}(a)$
is non-zero for some $b$, then $b\in
U^{+}I^{\flat}nI^{\flat}=U^{+}I^{-,\flat}nI^{\flat}$ (the equality follows
using the Iwahori factorization of $I^{\flat}$ and $M^{\flat}\triangleleft
N$).

If $n\in N_{1,a_{\circ}}\cup M^{-}$, then we must have $b\in U^{+}nI^{\flat}$
by Corollary 3.4. Accordingly,
$v_{1}^{\flat}*_{I^{\flat}}i_{w}^{\flat}=\text{s}\,v_{w}^{\flat}$, with
$s=v_{1}^{\flat}*_{I^{\flat}}i_{n}^{\flat}(n)=U^{+}I^{\flat}\cap
nI^{\flat}n^{-1}I^{\flat}\in\mathds{Z}$, which is $1$ by Corollary 3.4.

2 - Using the previous rule and Lemma 4.3:
$v_{m}^{\flat}*_{I^{\flat}}i_{w}^{\flat}=m\cdot
v_{1}^{\flat}*_{I^{\flat}}i_{w}^{\flat}=m\cdot v_{w}^{\flat}=v_{mw}^{\flat}$.

4 - Using fact 3, we have $v_{-m}^{\flat}={-m}\cdot v_{1}^{\flat}={-m}\cdot
v_{m}^{\flat}*_{I^{\flat}}(i_{m}^{\flat})^{-1}=v_{1}^{\flat}*_{I^{\flat}}(i_{m}^{\flat})^{-1}$.
∎

######  Corollary 4.2.

For any $m\in\Lambda_{M}^{\flat}$, any $w\in W$ and any
$m_{\circ}\in\Lambda_{M}^{-,\flat}$ such that
$m+m_{\circ}\in\Lambda_{M}^{-,\flat}$ (which always exists by Proposition
3.1), we have in $\mathcal{M}_{I^{\flat}}(\mathds{Z}[q^{-1}])$

$v_{mw}^{\flat}=v_{1}^{\flat}*_{I^{\flat}}i_{m+m_{\circ}}^{\flat}*_{I^{\flat}}(i_{m_{\circ}}^{\flat})^{-1}*_{I^{\flat}}i_{w}^{\flat}=v_{1}^{\flat}*_{I^{\flat}}(i_{m_{\circ}}^{\flat})^{-1}*_{I^{\flat}}i_{m+m_{\circ}}^{\flat}*_{I^{\flat}}i_{w}^{\flat}.$

###### Proof.

This follows readily from Proposition 4.1. ∎

Accordingly, any $v\in\mathcal{M}_{I^{\flat}}(\mathds{Z}[q^{-1}])$ can be
written as $v_{1}^{\flat}*_{I^{\flat}}h_{v}$ for some
$h_{v}\in\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$, i.e. the following
homomorphism of right $\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$-modules

$\displaystyle\hbar_{I^{\flat}}\colon\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})\longrightarrow\mathcal{M}_{I^{\flat}}({\mathds{Z}[q^{-1}]}),\quad
h\longmapsto v_{1}^{\flat}*_{I^{\flat}}h.$

is surjective. Actually, more is true:

######  Theorem 4.3.

The homomorphism $\hbar_{I^{\flat}}$ is an isomorphism.

###### Proof.

Given Corollary 4.2, we just need to show the injectivity of
$\hbar_{{I^{\flat}}}$. In doing so, we will actually show that this map is
”upper triangular with respect to the Chevalley–Bruhat order, with invertible
diagonals”.

Recall that $\mathcal{H}_{I^{\flat}}(\mathds{Z})$ (resp.
$\mathcal{M}_{I^{\flat}}(\mathds{Z})$) admits the $\mathds{Z}$-basis
$i_{y}^{\flat}$ (resp. $v_{x}^{\flat}$) for $x,y\in\widetilde{W}^{\flat}$. For
any $y\in\widetilde{W}^{\flat}$, write
$v_{1}^{\flat}*_{I^{\flat}}i_{y}^{\flat}=\sum_{x\in\widetilde{W}^{\flat}}c_{x,y}v_{x}^{\flat}$
with $c_{x,y}=v_{1}^{\flat}*_{I^{\flat}}i_{y}^{\flat}(x)\in\mathds{Z}$. The
function $v_{1}^{\flat}*_{I^{\flat}}i_{y}^{\flat}$ can be nonzero only on the
set $U^{+}I^{\flat}yI^{\flat}$, so by Corollary 3.6 one gets

$c_{x,y}\neq 0\Leftrightarrow U^{+}x\cap
I^{\flat}yI^{\flat}\neq\emptyset\Rightarrow x\leq y.$

Therefore,

$v_{1}^{\flat}*_{I^{\flat}}i_{y}^{\flat}=\sum_{x\in\widetilde{W}^{\flat}\colon
x\leq y}c_{x,y}v_{x}^{\flat}.$

Now, write any $h\in\ker\hbar_{I^{\flat}}$ as
$\sum_{y\in\widetilde{W}^{\flat}}r_{y}i_{y}^{\flat}$, where the coefficients
$\\{r_{y}\\}$ are nonzero precisely on some finite set
$I_{h}\subset\widetilde{W}^{\flat}$. Assume that $h\neq 0$ and let $z\in
I_{h}$ be any element maximal with respect to the Chevalley–Bruhat order in
$I_{h}$. The coefficient $c_{z,z}=|U^{+}I^{\flat}\cap
zI^{\flat}z^{-1}I^{\flat}|_{I^{\flat}}\geq 1$ and it is actually a power of
$q$ by Lemma 3.3. By maximality of $z$ in $I_{h}$, the term $v_{z}^{\flat}$
can not be killed in the sum $v_{1}^{\flat}*_{I^{\flat}}h=\sum_{y\in
I_{h}}\sum_{x\in\widetilde{W}^{\flat}\colon x\leq
y}c_{x,y}r_{y}v_{x}^{\flat}$, hence $v_{1}^{\flat}*_{I^{\flat}}h$ can not
equal zero. Therefore, $\ker\hbar_{I^{\flat}}$ is trivial and consequently
$\hbar_{I^{\flat}}$ is an isomorphism. ∎

Theorem 4.3, implies immediately the following

######  Corollary 4.4.

The right $\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})$-module
$\mathcal{M}_{I^{\flat}}({\mathds{Z}[q^{-1}]})$ is free and of rank $1$, with
canonical generator $v_{1}^{\flat}$. This yields a canonical isomorphism

${\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})}$${\text{End}_{\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})}(\mathcal{M}_{I^{\flat}}({\mathds{Z}[q^{-1}]})),\quad
h}$${(v_{1}^{\flat}*_{I^{\flat}}h^{\prime}\mapsto
v_{1}*_{{I^{\flat}}}h*_{{I^{\flat}}}h^{\prime}).}$$\scriptstyle{\simeq}$

######  Corollary 4.5.

Let $e\in\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$ be an idempotent and set

$v_{e}:=v_{1}^{\flat}*_{I^{\flat}}e,\mathcal{H}_{e}({\mathds{Z}[q^{-1}]}):=e*_{I^{\flat}}\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})*_{I^{\flat}}e\text{
and
}\mathcal{M}_{e}({\mathds{Z}[q^{-1}]}):=\mathcal{M}_{I^{\flat}}({\mathds{Z}[q^{-1}]})*_{I^{\flat}}e$

The following homomorphism of right
$\mathcal{H}_{e}({\mathds{Z}[q^{-1}]})$-modules is injective

$\displaystyle\hbar_{e}\colon\mathcal{H}_{e}({\mathds{Z}[q^{-1}]})\longrightarrow\mathcal{M}_{e}({\mathds{Z}[q^{-1}]}),\quad
h\longmapsto v_{e}*_{I^{\flat}}h.$

###### Proof.

It is clear that $\hbar_{e}$ is a well defined homomorphism of right
$\mathcal{H}_{e}({\mathds{Z}[q^{-1}]})$-modules. Let $h\in\ker\hbar_{e}$, then
$0=v_{e}*_{I^{\flat}}h=v_{1}^{\flat}*_{I^{\flat}}h$. Accordingly,
$\ker\hbar_{e}\subset\ker\hbar_{I^{\flat}}=\\{0\\}$ by Theorem 4.3. ∎

######  Remark 4.6.

Let $R$ be a ring in which ${|{K^{\flat}}:I^{\flat}|}$ is invertible. We can
naturally identify $\mathcal{H}_{K^{\flat}}(R)$ with the two-sided ideal
$e_{K^{\flat}}*_{{I^{\flat}}}\mathcal{H}_{{I^{\flat}}}(R)*_{{I^{\flat}}}e_{K^{\flat}}\subset\mathcal{H}_{{I^{\flat}}}(R)$,
where $e_{K^{\flat}}$ is the idempotent ${|{K^{\flat}}:I^{\flat}|}^{-1}{\bf
1}_{K^{\flat}}$ (see Lemma 3.11). In a similar way,
$\mathcal{M}_{K^{\flat}}(R)$ can also be naturally identified with
$\mathcal{M}_{{I^{\flat}}}(R)*_{{I^{\flat}}}e_{K^{\flat}}\subset\mathcal{M}_{{I^{\flat}}}(R)$
and the right action of $\mathcal{H}_{K^{\flat}}(R)$ will correspond to the
right action of
$e_{K^{\flat}}*_{{I^{\flat}}}\mathcal{H}_{{I^{\flat}}}(R)*_{{I^{\flat}}}e_{K^{\flat}}$.

###  4.3. Decomposition of $\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$
and the twisted Bernstein map

We begin this section by considering two sub-algebras of the Iwahori–Hecke
algebra $\mathcal{H}_{{I^{\flat}}}(\mathds{Z})$:

  * •

The finite Hecke algebra:

$\mathcal{H}_{{I^{\flat}}}(\mathds{Z})^{0}:=\mathcal{C}_{c}(K^{\flat}\sslash
I^{\flat},\mathds{Z}).$

A $\mathds{Z}$-basis for $\mathcal{H}_{{I^{\flat}}}(\mathds{Z})^{0}$ is given
by $\\{i_{w}^{\flat}\\}_{w\in W_{a_{\circ}}^{\flat}}$ (Proposition 3.1).

  * •

The $(\mathcal{R}^{\flat},\mathcal{H}_{{I^{\flat}}}(\mathds{Z}))$-bimodule
structure on $\mathcal{M}_{{I^{\flat}}}(\mathds{Z})$ induces a homomorphism of
algebras

$\mathcal{R}^{\flat}\hookrightarrow\text{End}_{\mathcal{H}_{{I^{\flat}}}(\mathds{Z})}(\mathcal{M}_{{I^{\flat}}}(\mathds{Z})),$

which is actually an embedding since $\mathcal{M}_{{I^{\flat}}}(\mathds{Z})$
is free over $\mathcal{R}^{\flat}$. Accordingly, thanks to Theorem 4.3, one
gets another embedding of algebras:

$\displaystyle\dot{\Theta}_{\text{\text{Bern}}}^{\flat}\colon\mathcal{R}^{\flat}\hookrightarrow\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]}),\quad
m\mapsto\dot{\Theta}_{m}^{\flat},$

characterized by the property: $m\cdot
v_{1}^{\flat}=v_{1}^{\flat}*_{{I^{\flat}}}\dot{\Theta}_{m}^{\flat}$, for any
$m\in\mathcal{R}^{\flat}$.

######  Lemma 4.1.

For any $m\in\Lambda_{M}^{\flat}$ and any $m_{\circ}\in\Lambda_{M}^{-,\flat}$
such that $m+m_{\circ}\in\Lambda_{M}^{-,\flat}$, we have

$\dot{\Theta}_{m}^{\flat}=i_{m+m_{\circ}}^{\flat}*_{{I^{\flat}}}(i_{m_{\circ}}^{\flat})^{-1}=(i_{m_{\circ}}^{\flat})^{-1}*_{{I^{\flat}}}i_{m+m_{\circ}}^{\flat}.$

In particular, $\dot{\Theta}_{m}^{\flat}=i_{m}^{\flat}$ if
$m\in\Lambda_{M}^{-,\flat}$ and
$\dot{\Theta}_{m}^{\flat}=(i_{-m}^{\flat})^{-1}$ if
$m\in\Lambda_{M}^{+,\flat}$.

###### Proof.

These equalities follow readily from Corollary 4.2. ∎

######  Remark 4.2.

(i) By Remark 4.6, since $\mathbf{1}_{I^{\flat}}$ is central in
$\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$, the following diagram is
commutative

${\mathcal{R}}$${\mathcal{H}_{I}({\mathds{Z}[q^{-1}]})}$${\mathcal{R}^{\flat}}$${\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})}$$\scriptstyle{\dot{\Theta}_{\text{\text{Bern}}}}$$\scriptstyle{\square^{\flat}}$$\scriptstyle{\iota^{\flat}}$$\scriptstyle{\dot{\Theta}_{\text{\text{Bern}}}^{\flat}}$

In other words,
$\iota^{\flat}(\dot{\Theta}_{m})=\dot{\Theta}_{{m}^{\flat}}^{\flat},m\in\mathcal{R}$.

(ii) The Iwahori–Hecke operators $\dot{\Theta}_{m}^{\flat}$ generalizes the
element denoted by $\widetilde{T}$ in Lusztig’s [Lus83, §7] (up to the factor
$\delta_{B}(m)^{1/2}$), which does not appear here. The untwisted version of
the map $\dot{\Theta}_{\text{Bern}}$ was attributed by Lusztig in loc. cit.
(for split reductive groups) to Bernstein, this is the reason we use the
subscript Bern.

######  Corollary 4.3.

For any $w\in\widetilde{W}^{\flat}$, we have

$i_{w}^{\flat}=\sum_{x=m_{x}w_{x}\leq
w}\dot{\Theta}_{m_{x}}^{\flat}*_{I^{\flat}}i_{w_{x}}^{\flat}.$

###### Proof.

The lemma follows readily from the proof of Theorem 4.3 and Corollary 4.2. ∎

######  Lemma 4.4.

For any $m\in\Lambda_{M}^{\flat}$, on has

$(\dot{\Theta}_{m}^{\flat})^{\vee}=(i_{w_{\circ}}^{\flat})^{-1}*_{I^{\flat}}\dot{\Theta}_{-w_{\circ}(m)}^{\flat}*_{I^{\flat}}i_{w_{\circ}}^{\flat}$

where $w_{\circ}\in W$ is the longest element.

###### Proof.

It suffices to show it for elements in $\Lambda_{M}^{-,\flat}$, so let
$m\in\Lambda_{M}^{-,\flat}$ we also have
$-w_{\circ}(m)\in\Lambda_{M}^{-,\flat}$ and accordingly by Lemma 3.1
$\ell(w_{\circ}(-m)\cdot
w_{\circ})=\ell(w_{\circ}m^{-1})=\ell(w_{\circ})+\ell(-m)=\ell(w_{\circ})+\ell(w_{\circ}(-m))$
hence

$\displaystyle i_{-w_{\circ}(m)}^{\flat}*_{I^{\flat}}i_{w_{\circ}}^{\flat}$
$\displaystyle=i_{w_{\circ}(-m)\cdot
w_{\circ}}^{\flat}=i_{w_{\circ}m^{-1}}^{\flat}=i_{w_{\circ}}^{\flat}*_{I^{\flat}}i_{-m}^{\flat}.\qed$

In our setting, it is easy to obtain a statement similar to [HKP10, Lemma
1.7.1]:

######  Lemma 4.5.

The homomorphism of
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-modules

$(\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]})\otimes_{\mathds{Z}}\mathcal{H}_{{I^{\flat}}}^{0}(\mathds{Z})\longrightarrow\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]}),\quad
m\otimes h^{0}\longmapsto\dot{\Theta}_{m}^{\flat}*_{{I^{\flat}}}h^{0},$

is an isomorphism. Composing this homomorphism with $h\mapsto
v_{1}^{\flat}*_{{I^{\flat}}}h$ yields the isomorphism of
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-modules
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}\mathcal{H}_{{I^{\flat}}}^{0}({\mathds{Z}[q^{-1}]})\longrightarrow\mathcal{M}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$,
given by $m\otimes i_{w}^{\flat}\longmapsto v_{mw}^{\flat}$, for $w\in
W_{a_{\circ}}^{\flat}$ and $m\in\Lambda_{M}^{\flat}$.

###### Proof.

Given Theorem 4.3, this lemma follows immediately from Corollary 4.2 which
shows that the composition

$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}\mathcal{H}_{{I^{\flat}}}^{0}({\mathds{Z}[q^{-1}]})\longrightarrow\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})\longrightarrow\mathcal{M}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]}),$

yields an isomorphism of
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-modules. ∎

A direct application of Lemma 4.5 gives an explicit
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-basis for the
Iwahori–Hecke algebra:

######  Corollary 4.6.

The algebra $\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$ is a free left
(and right)
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-module, with
canonical basis $\\{i_{w}^{\flat}\colon w\in W_{a_{\circ}}^{\flat}\\}$. The
sets $\\{\dot{\Theta}_{m}^{\flat}*_{{I^{\flat}}}i_{w}^{\flat}\colon
m\in\Lambda_{M}^{\flat},w\in W_{a_{\circ}}^{\flat}\\}$ and
$\\{i_{w}^{\flat}*_{{I^{\flat}}}\dot{\Theta}_{m}^{\flat}\colon
m\in\Lambda_{M}^{\flat},w\in W_{a_{\circ}}^{\flat}\\}$ are both
${\mathds{Z}[q^{-1}]}$-basis for
$\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$.

###### Proof.

The fact that $\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$ is a free left
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-module, with
canonical basis $\\{i_{w}^{\flat}\colon w\in W_{a_{\circ}}^{\flat}\\}$ and the
set $\\{\dot{\Theta}_{m}^{\flat}*_{{I^{\flat}}}i_{w}^{\flat}\colon
m\in\Lambda_{M}^{\flat},w\in W_{a_{\circ}}^{\flat}\\}$ is
${\mathds{Z}[q^{-1}]}$-basis for
$\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$ is clear from Corollary 4.5.
Let us show the remaining statements. We may argue in two different ways:

(i) Consider
$\mathcal{M}^{-}_{{I^{\flat}}}(\mathds{Z})=\mathds{Z}[I^{\flat}\backslash
G/U^{-}]$ instead, this is a
$(\mathcal{H}_{I^{\flat}}(\mathds{Z}),\mathcal{R}^{\flat})$-bi-module.
Rewriting the whole story of §4 in this context shows that
$\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$ is a free right
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-module and so
$\\{i_{w}^{\flat}*_{{I^{\flat}}}\dot{\Theta}_{m}^{\flat}\colon
m\in\Lambda_{M}^{\flat},w\in W_{a_{\circ}}^{\flat}\\}$ is
${\mathds{Z}[q^{-1}]}$-basis.

(ii) The map
$\mathcal{H}_{{I^{\flat}}}^{0}(\mathds{Z})\to\mathcal{H}_{{I^{\flat}}}^{0}(\mathds{Z}),i_{w}^{\flat}\to
i_{w_{\circ}}^{\flat}*_{{I^{\flat}}}(i_{w}^{\flat})^{\vee}$ is an isomorphism
of $\mathds{Z}$-modules, the map
$\Lambda_{M}^{\flat}\to\Lambda_{M}^{\flat},m\mapsto-w_{\circ}(m)$ is an
automorphism of rings and
$\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])\to\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}]),h\mapsto((i_{w_{\circ}}^{\flat})^{-1}*_{I^{\flat}}h)^{\vee}$
is an automorphism of $\mathds{Z}[q^{-1}]$-modules. Therefore, the
homomorphism of ${\mathds{Z}[q^{-1}]}$-modules

$(\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]})\otimes_{\mathds{Z}}\mathcal{H}_{{I^{\flat}}}^{0}(\mathds{Z})\longrightarrow\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]}),\quad
m\otimes
i_{w}^{\flat}\longmapsto\left((i_{w_{\circ}}^{\flat})^{-1}*_{{I^{\flat}}}\dot{\Theta}_{-w_{\circ}(m)}^{\flat}*_{{I^{\flat}}}i_{w_{\circ}}^{\flat}*_{{I^{\flat}}}(i_{w}^{\flat})^{\vee}\right)^{\vee},$

is an isomorphism. Given Lemma 4.4, this concludes the proof of the corollary.
∎

######  Remark 4.7.

If we replace $G$ by $G_{1}^{\flat}$ then
$\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}$ plays exactly the role of
$\Lambda_{M}^{\flat}$. For example, as in Corollary 4.5, we have two
decomposition:

$\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z}[q^{-1}])=\bigoplus_{w\in
W}i_{w}^{\flat}*_{I^{\flat}}\Theta_{\text{Bern}}^{\flat}(\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}])=\bigoplus_{w\in
W}\Theta_{\text{Bern}}^{\flat}(\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}])*_{I^{\flat}}i_{w}^{\flat}.$

###  4.4. Further properties of $\dot{\Theta}^{\flat}$-elements

By Corollary 3.1, various results in [Lus89] applies to
$\mathcal{H}_{I^{1}}({\mathds{Z}[q^{-1}]})$ (i.e. $M^{\flat}=M^{1}$) and
$\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})$. In this
subsection, we will extend some results proved by Lusztig for
$\mathcal{H}_{I^{1}}(\mathds{Z}[q^{-1}])$ to
$\mathcal{H}_{{I}}(\mathds{Z}[q^{-1}])$.

From now on, whenever needed, we identify $\mathcal{R}^{\flat}$ with its image
$\dot{\Theta}_{\text{\text{Bern}}}^{\flat}(\mathcal{R}^{\flat})$ and
$\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}$ with
$\dot{\Theta}_{\text{\text{Bern}}}^{\flat}(\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat})$.
Let $\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}$ be the fraction field
of $\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}$.

######  Lemma 4.1.

We have a commutative diagram

${\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})}$${\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})}$${\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})}$${\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{I}({\mathds{Z}[q^{-1}]})}$${\mathcal{L}_{\textit{\emph{\text{aff}}}}^{1}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{1}}\mathcal{H}_{I^{1}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})}$${\mathcal{L}_{\textit{\emph{\text{aff}}}}^{1}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{1}}\mathcal{H}_{I^{1}}({\mathds{Z}[q^{-1}]})}$$\scriptstyle{\simeq}$$\scriptstyle{\iota^{\flat}}$

###### Proof.

By Remark 3.5, we have a commutative diagram

${\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})}$${\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})}$${\mathcal{H}_{{I}^{1}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})}$${\mathcal{H}_{{I}^{1}}({\mathds{Z}[q^{-1}]})}$$\scriptstyle{\simeq}$$\scriptstyle{\iota^{1}}$

Therefore, It remains to show why the horizontal ones and the upper vertical
ones are injective. For this, we only need to show that no non-zero element in
$\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}$ is a zero left/right
divizor in $\mathcal{H}_{I}({\mathds{Z}[q^{-1}]})$: Let
$a\in\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}$ and
$b\in\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})$ such that
$a*_{I^{\flat}}b=0$. By the decomposition of Remark 3.4,
$b=\sum_{\sigma\in\widetilde{\Upomega}^{\flat}}\sigma\otimes b_{\sigma}$ for
some
$b_{\sigma}\in\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})$.
Hence $\sigma(a)*_{I^{\flat}}b_{\sigma}=0$ for all
$\sigma\in\widetilde{\Upomega}^{\flat}$. Thanks to the decomposition
$\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})=\oplus_{w\in
W_{a_{\circ}}^{\flat}}(\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}])*_{I^{\flat}}i_{w}^{\flat}$
(Remark 4.7) we see that $a$ must be a zero-divisor in
$\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}$ which forces it to be zero
since this latter is a domain. If we start with a right zero, i.e.
$b*_{I^{\flat}}a=0$, then using decomposition
$\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})=\oplus_{w\in
W_{a_{\circ}}^{\flat}}i_{w}^{\flat}*_{I^{\flat}}(\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}])$
one shows similarly that $a$ must be zero. ∎

######  Lemma 4.2.

For any $m\in\Lambda_{M}^{\flat},w\in W_{a_{\circ}}^{\flat}$ and
$s\in\mathcal{T}_{a_{\circ}}^{\flat}=\mathcal{S}^{\flat}\cap
W_{a_{\circ}}^{\flat}$, we have
$\dot{\Theta}_{-m}^{\flat}*_{I^{\flat}}i_{w}^{\flat}*_{I^{\flat}}\dot{\Theta}_{s(m)}^{\flat}\in\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})$.

###### Proof.

For any $m\in\Lambda_{M}^{\flat}$ and $s\in\mathcal{T}_{a_{\circ}}^{\flat}$,
by 1 Proposition 3.1, one can find an element
$m_{\circ}\in\Lambda_{M}^{-,\flat}$ such that $m_{\circ}-m$ and
$m_{\circ}+s(m)$ both lie in $\Lambda_{M}^{-,\flat}$. Hence, for any $w\in
W_{a_{\circ}}^{\flat}$, we have

$\dot{\Theta}_{-m}^{\flat}*_{I^{\flat}}i_{w}^{\flat}*_{I^{\flat}}\dot{\Theta}_{s(m)}^{\flat}=(i_{m_{\circ}}^{\flat})^{-1}*_{I^{\flat}}i_{(m_{\circ}-m)w}^{\flat}*_{I^{\flat}}i_{m_{\circ}+s(m)}^{\flat}*_{I^{\flat}}(i_{m_{\circ}}^{\flat})^{-1}.$

Let $z_{m_{\circ}}$ be the projection of $m_{\circ}$ in
$\widetilde{\Upomega}^{\flat}$. By v Corollary 3.3, we have

$i_{m_{\circ}}^{\flat}\in\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})^{\times}*_{I^{\flat}}i_{z_{m_{\circ}}}^{\flat}=i_{z_{m_{\circ}}}^{\flat}*_{I^{\flat}}\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})^{\times}.$

Accordingly, the support

$\text{Supp}(\dot{\Theta}_{-m}^{\flat}*_{I^{\flat}}i_{w}^{\flat}*_{I^{\flat}}\dot{\Theta}_{s(m)}^{\flat})\subset
I^{\flat}z_{m_{\circ}}^{-1}m_{\circ}m^{-1}wI^{\flat}m_{\circ}s(m)z_{m_{\circ}}^{-1}I^{\flat}\subset
G_{1}^{\flat}.$

Therefore,
$\dot{\Theta}_{-m}^{\flat}*_{I^{\flat}}i_{w}^{\flat}*_{I^{\flat}}\dot{\Theta}_{s(m)}\in\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})$.
∎

######  Proposition 4.3.

For any $s\in\mathcal{T}_{a_{\circ}}^{\flat}$ and any
$m\in\Lambda_{M}^{\flat}$, we have

$\dot{\Theta}_{m}^{\flat}*_{I^{\flat}}(i_{s}^{\flat}+1)-(i_{s}^{\flat}+1)*_{I^{\flat}}\dot{\Theta}_{\dot{s}(m)}^{\flat}=(\dot{\Theta}_{m}^{\flat}-\dot{\Theta}_{\dot{s}(m)}^{\flat})\mathcal{G}(\alpha)\in\mathcal{R}^{\flat}$

where,
$\mathcal{G}(\alpha)\in\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}$ is
the element defined in [Lus89, §3.8 & §3.13] for the algebra
$\mathcal{H}(\mathcal{D}_{\textit{\emph{\text{aff}}}},{W}_{\textit{\emph{\text{aff}}}},v=q,L)$.

###### Proof.

Let $m\in M$. We want to apply [Lus89, Proposition 3.9]. Since in loc. cit.
the notation ${\theta}_{m}$ does not equal our $\dot{\Theta}_{m}$, we first
compare the two to make things clear. Let $m_{1},m_{2}\in M^{-}$ verifying
$m=m_{1}-m_{2}$, hence by Corollary 3.1 and [Lus89, Definition §3.3 (a)]

${\theta}_{m}=\frac{q_{m_{2}}^{\frac{1}{2}}}{q_{m_{1}}^{\frac{1}{2}}}i_{m_{1}}*_{I}i_{m_{2}}^{-1}\overset{\text{Lemma
\ref{qmdeltam}}}{=}\delta_{B}(m)^{\frac{1}{2}}\dot{\Theta}_{m}.$

Set
$A_{s}(m):=(i_{s}+1)-\theta_{-m}(i_{s}+1)\theta_{s(m)}-(1-\theta_{s(m)-m})\mathcal{G}(\alpha)$.
Clearly, the term
$(1-\theta_{s(m)-m})\mathcal{G}(\alpha)\in\mathcal{L}_{\textit{\emph{\text{aff}}}}$,
but its image by $\iota^{1}$ lies in
$\mathcal{R}_{\textit{\emph{\text{aff}}}}^{1}$ [Lus89, §2.5 (b)], hence
$(1-\theta_{s(m)-m})\mathcal{G}(\alpha)\in\mathcal{R}_{\textit{\emph{\text{aff}}}}$.
Accordingly, by Lemma 4.2, we have
$A_{s}(m)\in\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})$.

Thanks to [Lus89, Proposition 3.9], we know that $\iota^{1}(A_{s}(m))=0$ in
$\mathcal{H}_{I^{1}}({\mathds{Z}[q^{-1}]})$ since by Remark 4.2 we have
$\iota^{1}(\dot{\Theta}_{m})=\dot{\Theta}_{{m}^{1}}^{1}$ (recall that $m^{1}$
denotes the image of $m$ in $\Lambda_{M}^{1}$) . Therefore $A_{s}(m)=0$, given
that
$\iota^{1}\colon\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})\to\mathcal{H}_{I^{1}}({\mathds{Z}[q^{-1}]})$
is injective.

In conclusion, we proved that for any reflexion $s\in\mathcal{T}_{a_{\circ}}$
and any $m\in M$

$\dot{\Theta}_{m}(i_{s}+1)-c(m,s)(i_{s}+1)\dot{\Theta}_{s(m)}=(\dot{\Theta}_{m}-c(m,s)\dot{\Theta}_{s(m)})\mathcal{G}(\alpha)$

which is proves the lemma (see Remark 4.2). ∎

###  4.5. $\dot{W}$-invariant elements are central

For $m\in\Lambda_{M}^{-,\flat}$, set

$r_{m}^{\flat}:=\sum_{w\in W/W_{m}}\dot{w}(m)=\sum_{w\in W/W_{m}}c(m,w)w(m)$

where, $W_{m}$ denotes the isotropy subgroup of $m$ in $W$ (for the standard
Weyl action).

Let ${\mathcal{R}^{\flat}}^{\dot{W}}\subset{\mathcal{R}^{\flat}}$ denotes the
$\mathds{Z}$-subalgebra of elements invariant under the dot-action, which
again will be identified with its
$\dot{\Theta}_{\text{\text{Bern}}}^{\flat}({\mathcal{R}^{\flat}}^{\dot{W}})$.
So $r_{m}^{\flat}$ also denotes its image $\sum_{w\in
W/W_{m}}\dot{\Theta}_{\dot{w}(m)}^{\flat}=\sum_{w\in
W/W_{m}}c(m,w)\dot{\Theta}_{{w}(m)}^{\flat}$.

By Lemma 3.3, all $r_{m}\in{\mathcal{R}^{\flat}}$ for all
$m\in\Lambda_{M}^{-,\flat}$. In consequence, Lemma 3.3 implies that
$\\{r_{m}^{\flat},m\in\Lambda_{M}^{-,\flat}\\}$ is a $\mathds{Z}$-basis for
${\mathcal{R}^{\flat}}^{\dot{W}}$.

######  Corollary 4.1.

For any $m\in\Lambda_{M}^{\flat}$ and any $s\in\mathcal{T}_{a_{\circ}}$,
$\dot{\Theta}_{m}^{\flat}+\dot{\Theta}_{\dot{s}(m)}^{\flat}$ commutes with
$i_{s}^{\flat}$. In consequence,

${\mathcal{R}^{\flat}}^{\dot{W}}\subset
Z(\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})).$

###### Proof.

The first statement is an immediate consequence of Proposition 4.3. For the
second statement, we observe that for any $s\in\mathcal{T}_{a_{\circ}}$
(Definition 2.2) and any $m\in\Lambda_{M}^{\flat}$, we have

$|W/W^{m}|r_{m}^{\flat}=\sum_{w\in\dot{W}}\dot{\Theta}_{\dot{w}(m)}^{\flat}=\sum_{w\in\mathcal{P}_{s}^{l}\cap{W}}\dot{\Theta}_{\dot{w}(m)}^{\flat}+\dot{\Theta}_{\dot{s}\dot{w}(m)}^{\flat}.$

So $r_{m}^{\flat}$ commutes with $i_{s}^{\flat}$ for all
$s\in\mathcal{S}^{\flat}$. This completes the proof since $W$ is generated by
$\mathcal{S}^{\flat}\cap W_{a_{\circ}}^{\flat}$. ∎

##  5\. Twisted integral Satake morphisms

###  5.1. Twisted Satake homomorphism: first basic properties

In this section, we focus on the level $K^{\flat}$. By Proposition 4.4, the
$\mathcal{R}^{\flat}$-module $\mathcal{M}_{K^{\flat}}(\mathds{Z})$ is free of
rank $1$ (with canonical basis element the spherical vector
$v_{1,K}^{\flat}:=v_{1,a_{\circ}}^{\flat}={\bf 1}_{U^{+}K^{\flat}}$).
Accordingly, we have a $\mathds{Z}$-algebra homomorphism
$\mathcal{H}_{K^{\flat}}(\mathds{Z})\to\mathcal{R}^{\flat}$, which we denote
by $\dot{\mathcal{S}}_{M}^{G^{\flat}}$. It is called the Satake transform and
is characterized by

$v_{1,K}^{\flat}*_{K^{\flat}}h=\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)\cdot
v_{1,K}^{\flat},\text{ for all }h\in\mathcal{H}_{K^{\flat}}(\mathds{Z}).$

This is actually an embedding of $\mathds{Z}$-algebras (using the fact that
$v_{1,K}^{\flat}=v_{1}^{\flat}*_{I^{\flat}}{\bf 1}_{K^{\flat}}$ and Corollary
4.5). In particular, the algebra $\mathcal{H}_{K^{\flat}}(\mathds{Z})$ is a
commutative.

######  Proposition 5.1.

The image of $\dot{\mathcal{S}}_{M}^{G}$ is $\dot{W}$-invariant, i.e.
${\dot{\mathcal{S}}_{M}^{G^{\flat}}\colon\mathcal{H}_{K^{\flat}}(\mathds{Z})}$${{\mathcal{R}^{\flat}}^{\dot{W}}.}$

###### Proof.

We give two proofs in the following two subsubsections; the first is
”algebraic” while the second is of ”analytical” taste. The first is short
while the second is explicit:

First, note that by Corollary 3.2 and Remark 3.2, it is sufficient to
show${\dot{\mathcal{S}}_{M}^{G}\colon\mathcal{H}_{K^{\flat}}(\mathds{Z})}$${{\mathcal{R}}^{\dot{W}}.}$

####  5.1.1. Algebraic argument

###### Proof.

Let
$h\in\mathcal{H}_{K}(\mathds{Z})\subset\mathcal{H}_{I}(\mathds{Z}[q^{-1}])$.
We have

$v_{1}*_{I}h=v_{1,K}*_{K}h=\dot{\mathcal{S}}_{M}^{G}(h)\cdot
v_{1,K}=\sum_{w\in W_{a_{\circ}}}\dot{\mathcal{S}}_{M}^{G}(h)\cdot v_{w}.$

Hence, $h=\dot{\mathcal{S}}_{M}^{G}(h)*_{I}\sum_{w\in
W_{a_{\circ}}}i_{w}=\dot{\mathcal{S}}_{M}^{G}(h)*_{I}{\bf 1}_{K}$. Now,
because $h$ is $K$-bi-equivariant, we have for any
$s\in\mathcal{T}_{a_{\circ}}$

$i_{s}*_{I}h=i_{s}*_{I}{\bf 1}_{K}*_{K}h=h=h*_{I}i_{s}.$

Therefore, for any $s\in\mathcal{S}\cap W_{a_{\circ}}$:

$0=i_{s}*_{I}\dot{\mathcal{S}}_{M}^{G}(h)*_{I}{\bf
1}_{K}-\dot{\mathcal{S}}_{M}^{G}(h)*_{I}i_{s}*_{I}{\bf
1}_{K}=\left(i_{s}*_{I}\dot{\mathcal{S}}_{M}^{G}(h)-\dot{\mathcal{S}}_{M}^{G}(h)*_{I}i_{s}\right)*_{I}\sum_{w\in
W_{a_{\circ}}}i_{w}.$

Therefore, given that
$i_{s}*_{I}\dot{\mathcal{S}}_{M}^{G}(h)-\dot{\mathcal{S}}_{M}^{G}(h)*_{I}i_{s}\in\mathcal{R}$
by Proposition 4.3, by Corollary 4.6 we must have

$i_{s}*_{I}\dot{\mathcal{S}}_{M}^{G}(h)-\dot{\mathcal{S}}_{M}^{G}(h)*_{I}i_{s}=0,\forall
s\in\mathcal{T}_{a_{\circ}}.$

This concludes the first proof of the proposition, since $W_{a_{\circ}}$ is
generated by $\mathcal{T}_{a_{\circ}}$. ∎

####  5.1.2. Analytical argument

First, we need an explicit integral formula for the Satake transform:

######  Lemma 5.2.

Let $h$ be any function in $\mathcal{H}_{K}(\mathds{Z})$, then its Satake
twisted transform is explicitly given by

$\dot{\mathcal{S}}_{M}^{G}(h)\colon m\mapsto\int_{U^{+}}h(um)d\mu_{U^{+}}(u),$

where $d\mu_{U^{+}}$ is the Haar measure on $U^{+}$ giving $1$ on $U^{+}\cap
K$.

###### Proof.

We just need to evaluate $v_{1,K}*_{K}h=\dot{\mathcal{S}}_{M}^{G}(h)\cdot
v_{1,K}$ on both sides: On the one hand, we have for $m\in M$

$\displaystyle(\dot{\mathcal{S}}_{M}^{G}(h)\cdot
v_{1,K})(m)=\dot{\mathcal{S}}_{M}^{G}(h)(m).$

On the second hand we have141414We refer to [Car79, §4.1] for all integration
formulas used in the sequel thanks to the Iwasawa decomposition $G=BK$
(Corollary 3.2)

$\displaystyle(v_{1,K}*_{K}h)(m)$
$\displaystyle=\int_{B}\int_{K}v_{1,K}(bk)h(k^{-1}b^{-1}m)d\mu_{B}(b)\,dk$
$\displaystyle=\int_{B}v_{1,K}(b)h(b^{-1}m)d\mu_{B}(b)\quad\quad\quad(K\text{
invariance})$
$\displaystyle=\int_{M}\int_{U^{+}}v_{1,K}(au)h(u^{-1}a^{-1}m)d\mu_{M}(a)d\mu_{U^{+}}(u)$
$\displaystyle=\int_{U^{+}}h(u^{-1}m)d\mu_{U^{+}}(u)\quad\quad\quad\quad(v_{1,K}(au)\neq
0\Rightarrow a\in M_{1})$
$\displaystyle=\int_{U^{+}}h(um)d\mu_{U^{+}}(u)\quad\quad\quad\quad\quad(U^{+}\text{
is unimodular})$

Where, $d\mu_{B}$ (respectively $d\mu_{U^{+}}$ and $dk$) is the Haar left
invariant measure on $B$ (respectively $U^{+}$ and $K$) giving volume $1$ to
$B\cap K$ (respectively $U^{+}\cap K$ and $K$). ∎

######  Remark 5.3.

The function
$m\mapsto\det(\text{Ad}_{\text{Lie}(U^{+})}(m)-\text{Id}_{\text{Lie}(U^{+})})$
is polynomial and nonzero. The set of regular elements is, by definition, the
dense set of elements of $M$ which do not annihilate the nonzero polynomial
function
$m\mapsto\det(\text{Ad}_{\text{Lie}(U^{+})}(m)-\text{Id}_{\text{Lie}(U^{+})})$.

######  Lemma 5.4.

Let $h$ be any function in $\mathcal{H}_{K}(\mathds{Z})$ and $m\in M$ any
regular elements (Definition 5.3). Then

$\int_{U^{+}}h(um)d\mu_{U^{+}}(u)=\Delta(m)\int_{G/S}h(gmg^{-1})\frac{d\mu_{K}(g)}{ds}$

where, the Haar measure $ds$ on $S=\mathbf{S}(F)$ is normalized by
$\int_{M/S}\frac{d\mu_{M_{1}}}{ds}=1$.

###### Proof.

The same reasoning of [Car79, Lemma 4.1] proves the lemma. ∎

######  Remark 5.5.

As opposed to [HR10, Remark 10.1.1], the twisted Satake transform defined
here, is dependent on the choice of the minimal parabolic $\mathbf{B}$ which
contains $\mathbf{M}$ as a Levi factor.

Given Lemma 5.2, we need to prove that

$\dot{\mathcal{S}}_{M}^{G}(h)(m)=\dot{w}\left(\dot{\mathcal{S}}_{M}^{G}(h)(m)\right)$

for all functions $h\in\mathcal{H}_{K}(\mathds{Z})$, $m\in M$ and $w\in W$. By
continuity, it suffices to show this equality for all $m$ in the dense set of
elements of regular (Remark 5.3) which are semi-simple as elements in $G$. For
these elements, we know by Lemma 5.4 that

$\dot{\mathcal{S}}_{M}^{G}(h)(m)=\Delta(m)\int_{G/S}h(gmg^{-1})\frac{d\mu_{K}(g)}{ds}$

On the one hand, by [Car79, (23)], one easily shows that
$\Delta(\dot{w}(m))=\Delta(m)$. On the other hand, since $h$ is $K$-bi-
invariant, one has $h(n_{w}gn_{w}^{-1})=h(g)$ for all $g\in G$. The compact
subgroup $N\cap K$ acts by inner automorphisms on $G$ and on $S$ and hence
leaves invariant the $G$-measure on $G/S$. Accordingly,

$\dot{w}\left(\dot{\mathcal{S}}_{M}^{G}(h)(m)\right)=\Delta(w\bullet
m)\int_{G/S}h(g(n_{w}mn_{w}^{-1})g^{-1})\frac{d\mu_{K}(g)}{ds}=\Delta(m)\int_{G/S}h(gmg^{-1})\frac{d\mu_{K}(g)}{ds}=\dot{\mathcal{S}}_{M}^{G}(h)(m).$

This ends the ”analytical” argument for Proposition 5.1. ∎

######  Remark 5.6.

The following two diagrams are commutative

${\mathcal{H}_{K}(\mathds{Z})}$${\mathcal{R}^{\dot{W}}}$${\mathcal{H}_{K}(\mathds{Z})}$${\mathcal{R}^{\dot{W}}}$${\mathcal{H}_{K^{\flat}}(\mathds{Z})}$${{\mathcal{R}^{\flat}}^{\dot{W}}}$${\mathcal{H}_{K^{\flat}}(\mathds{Z})}$${{\mathcal{R}^{\flat}}^{\dot{W}}}$$\scriptstyle{\dot{\mathcal{S}}_{M}^{G}}$$\scriptstyle{\iota^{\flat}=-*_{K_{\mathcal{F}}}\mathbf{1}_{K_{\mathcal{F}}^{\flat}}}$$\scriptstyle{\square^{\flat}}$$\scriptstyle{\dot{\mathcal{S}}_{M}^{G}}$$\scriptstyle{\dot{\mathcal{S}}_{M}^{G^{\flat}}}$$\scriptstyle{\dot{\mathcal{S}}_{M}^{G^{\flat}}}$

Here, the last right vertical map is the natural inclusion
$\Lambda_{M}^{\flat}\to\Lambda_{M}$ given by
$mM^{\flat}\mapsto\sum_{m^{\prime}\in M^{\flat}/M_{1}}mm^{\prime}M_{1}$.

###  5.2. Twisted Satake isomorphism

Motivated by arithmetic problems for Shimura varieties, Haines and Rostami
already established in [HR10],
$\mathcal{S}_{M}^{G}\colon\mathcal{H}_{K}(\mathds{C})\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}\mathcal{R}^{W}\otimes_{\mathds{Z}}\mathds{C}$.
In this section, we extend their result to $\mathds{Z}$-coefficients provided
by the twisted Satake homomorphism $\dot{\mathcal{S}}_{M}^{G}$.

######  Theorem 5.1.

The twisted Satake homomorphism
$\dot{\mathcal{S}}_{M}^{G^{\flat}}\colon\mathcal{H}_{K^{\flat}}(\mathds{Z})\to{\mathcal{R}^{\flat}}^{\dot{W}}$
is a canonical isomorphism of ${\mathds{Z}}$-algebras.

As for Proposition 5.1, we give in the following two subsubsections two
proofs; the first is ”algebraic” while the second is of ”combinatorial” taste.
We will see in §6, yet a third method which will give simultaneously this type
of isomorphisms for all parahoric subgroup $K_{\mathcal{F}}$ with
$\mathcal{F}\subset\mathfrak{a}$.

####  5.2.1. Algebraic argument

###### Proof.

######  Lemma 5.2 (Integral compatibility).

The Satake and Bernstein twisted $\mathds{Z}$-homorphisms are compatible,
i.e., the following diagram (of $\mathds{Z}$-modules) is commutative:

${\mathcal{H}_{K^{\flat}}(\mathds{Z})\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]}$${{\mathcal{R}^{\flat}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]}$${Z(\mathcal{H}_{{I^{\flat}}}(\mathds{Z}[q^{-1}]))}$$\scriptstyle{\dot{\mathcal{S}}_{M}^{G^{\flat}}\otimes\text{Id}}$$\scriptstyle{\dot{\Theta}_{Bern}^{\flat}}$$\scriptstyle{-*_{I^{\flat}}{\bf
1}_{K^{\flat}}}$

The injectivity of the horizontal and right diagonal maps is Corollary 4.1 and
Proposition 5.1. Let $h\in\mathcal{H}_{K^{\flat}}({\mathds{Z}})$, we have

$\displaystyle v_{1}*_{I^{\flat}}h$
$\displaystyle=v_{1}*_{I^{\flat}}e_{K^{\flat}}*_{{I^{\flat}}}h$
$\displaystyle\quad(e_{K^{\flat}}*_{I^{\flat}}h=h)$
$\displaystyle={[K^{\flat}:I^{\flat}]}(v_{1}^{\flat}*_{I^{\flat}}e_{K^{\flat}})*_{K^{\flat}}h$
$\displaystyle\quad(\text{passing from $*_{I^{\flat}}$ to $*_{K^{\flat}}$})$
$\displaystyle={[K^{\flat}:I^{\flat}]}\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)\cdot(v_{1}^{\flat}*_{I^{\flat}}e_{K^{\flat}})$
$\displaystyle(\text{Definition of $\dot{\mathcal{S}}_{M}^{G^{\flat}}$})$
$\displaystyle={[K^{\flat}:I^{\flat}]}(\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)\cdot
v_{1}^{\flat})*_{I^{\flat}}e_{K^{\flat}}$
$\displaystyle={[K^{\flat}:I^{\flat}]}(v_{1}^{\flat}*_{I^{\flat}}\dot{\Theta}_{\text{\emph{Bern}}}^{\flat}({\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)}))*_{I^{\flat}}e_{K^{\flat}}$
$\displaystyle(\text{Definition of
$\dot{\Theta}_{\text{\text{Bern}}}^{\flat}$})$
$\displaystyle=v_{1}^{\flat}*_{I^{\flat}}\dot{\Theta}_{\text{\emph{Bern}}}\circ{\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)}*_{I^{\flat}}\mathbf{1}_{K^{\flat}}$

Hence,
$h=\dot{\Theta}_{\text{\emph{Bern}}}^{\flat}\circ{\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)}*_{I^{\flat}}\mathbf{1}_{K^{\flat}}=\mathbf{1}_{K^{\flat}}*_{I^{\flat}}\dot{\Theta}_{\text{\emph{Bern}}}^{\flat}\circ{\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)}$.∎

Accordingly, Theorem 5.1, follows readily from:

######  Lemma 5.3.

For any $h\in\mathcal{H}_{K^{\flat}}(\mathds{Z})$, and any
$f\in{\mathcal{R}^{\flat}}^{\dot{W}}$ we have

$h=\dot{\Theta}_{\text{Bern}}^{\flat}(\dot{\mathcal{S}}_{M}^{G^{\flat}}(h))*_{I^{\flat}}{\bf
1}_{K^{\flat}}\text{ and
}\dot{\mathcal{S}}_{M}^{G^{\flat}}(\dot{\Theta}_{\text{Bern}}^{\flat}(f)*_{I^{\flat}}{\bf
1}_{K^{\flat}})=f.$

In other words,
$(-*_{I^{\flat}}{}_{I^{\flat}})\circ\dot{\Theta}_{\text{Bern}}^{\flat}$ is the
inverse of $\dot{\mathcal{S}}_{M}^{G^{\flat}}$.

Let $f\in{\mathcal{R}^{\flat}}^{\dot{W}}$ so
$\dot{\Theta}_{\text{Bern}}^{\flat}(f)\in
Z(\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}]))$. Then, clearly
$h_{f}=\dot{\Theta}_{\text{Bern}}^{\flat}(f)*_{I^{\flat}}{\bf
1}_{K^{\flat}}={\bf
1}_{K^{\flat}}*_{I^{\flat}}\dot{\Theta}_{\text{Bern}}^{\flat}(f)\in\mathcal{H}_{K^{\flat}}(\mathds{Z})$.
Now, given that
$v_{1,K}^{\flat}*_{K^{\flat}}h_{f}=v_{1}^{\flat}*_{I^{\flat}}h_{f}=f\cdot
v_{1}^{\flat}*_{I^{\flat}}{\bf 1}_{K^{\flat}}$, one concludes that
$\dot{\mathcal{S}}_{M}^{G^{\flat}}(h_{f})=f.$

If we start with any element $h\in\mathcal{H}_{K^{\flat}}(\mathds{Z})$, the
same reasoning shows that

$\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)=\dot{\mathcal{S}}_{M}^{G^{\flat}}(\dot{\Theta}_{\text{Bern}}^{\flat}\circ\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)*_{I^{\flat}}{\bf
1}_{K^{\flat}})$

and Proposition 5.1 then implies
$h=\dot{\Theta}_{\text{Bern}}^{\flat}\circ\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)*_{I^{\flat}}{\bf
1}_{K^{\flat}}.$ ∎

####  5.2.2. Combinatorial argument

Thanks to Remark 5.6, it is sufficient to show that
$\dot{\mathcal{S}}_{M}^{G}\colon\mathcal{H}_{K}(\mathds{Z})\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}{\mathcal{R}}^{\dot{W}}$
($M^{\flat}=M_{1}$) to prove Theorem 5.1. So, we focus on this case in this
subsection.

######  Definition 5.4.

Let $\precsim^{\flat}$ be the partial order on $\Lambda_{M}^{\flat}$ defined
as follows

$m\succsim^{\flat}m^{\prime}\text{ if and only if
}m-m^{\prime}\in\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}\text{ and
}\nu(m-m^{\prime})=\sum_{\Delta^{\vee}}n_{\alpha}\alpha^{\vee},\text{ with
}n_{\alpha}\in\mathds{N}.$

######  Remark 5.5.

(i) When $M^{\flat}=M^{1}$, this is the usual order on the lattice
$\Lambda_{M}^{1}$. (ii) Note that if $m,m^{\prime}\in\Lambda_{M}$ then
$m^{\prime}\precsim m$ if and only if
$m^{\prime\flat}\precsim^{\flat}m^{\flat}$, because (i) $\nu$ factors through
$M/\ker\nu$ and a fortiori through $\Lambda_{M}^{1}$, and (ii) because
$\square^{\flat}\colon\Lambda_{M}\twoheadrightarrow\Lambda_{M}^{\flat}$
restricts to an isomorphism
$\Lambda_{\textit{\emph{\text{aff}}}}\to\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}$.

!

######  Lemma 5.6.

If $m\in\Lambda_{M}^{-}$, then $U^{+}m^{-1}K\cap Km^{-1}K=m^{-1}K$.

###### Proof.

We intend to use [BT72, 4.3.6 Corollaire] for $G_{1}$ by taking (with loc.
cit. notations) $\Upomega=\\{a_{\circ}\\}$ and
$\Upomega^{\prime}=\\{a_{\circ}+\nu(m)\\}$ and $n=1$. Since
$\Upomega\subset\Lambda_{M}^{-}$, we have
$a_{\circ}\subset\Upomega^{\prime}+\overline{\mathcal{C}}^{+}$, so the
condition of [BT72, 4.3.6 Corollaire] is fulfilled and we have $U^{+}K\cap
K_{a_{\circ}+\nu(m)}K=K$. This shows the lemma. ∎

######  Lemma 5.7.

Let $m,m^{\prime}\in\Lambda_{M}$, if $U^{+}m^{\prime}\cap KmK\neq\emptyset$
then $m^{\prime}\precsim m$.

###### Proof.

If the above intersection is nonempty, i.e. $u\in
U^{+}\cap{m^{\prime}}^{-1}KmK$. There exists then $m_{0}\in M$ such that
$m_{0}um_{0}^{-1}\in I$, hence, the alcove
$\mathfrak{a}^{\prime}=\nu(m_{0})(\mathfrak{a})$ verifies

$I_{\mathfrak{a}^{\prime}}m^{\prime}m^{-1}\cap
KK_{a_{\circ}+\nu(m)}\neq\emptyset$

So in particular, $m-m^{\prime}\in\Lambda_{\textit{\emph{\text{aff}}}}$. We
may then apply [BT72, 4.3.16 Corollaire] for $G_{1}$, by taking (with loc.
cit. notations) $y=a_{\circ}+\nu(m)$,
$x=a_{\circ},n=m^{\prime}-m\in\Lambda_{\textit{\emph{\text{aff}}}}$,
$C=\mathfrak{a}^{\prime}$ and $D=\mathcal{C}^{-}$ our fixed negative chamber.
Then, the assumption $y\in x+\mathcal{C}^{-}$ is fulfilled because
$m\in\Lambda_{M}^{-}$. Therefore, we have

$y+\nu(m^{\prime}-m)=a_{\circ}+\nu(m^{\prime})\leq_{D}y=a_{\circ}+\nu(m).$

The order $\leq_{D}$ in loc. cit.151515We refer to [Bou68, $n^{\circ}$1 §6 p.
155] for more details on the order used in [BT72, 4.3.16 Corollaire].
translates here to
$\nu(m-m^{\prime})=\sum_{\alpha\in\Delta}n_{\alpha}\alpha^{\vee},n_{\alpha}\geq
0$. ∎

###### Second proof of Theorem 5.1.

This resembles the proof in [HR10, §10.2] for the untwisted Satake transform
with coefficients in $\mathds{C}$, which remains morally valid in our twisted
context, although we deal with the ”lexicographic” order on $\Lambda_{M}^{-}$
in a different manner. We need to show that the following morphism of
$\mathds{Z}$-modules

${\mathcal{H}_{K}({\mathds{Z}})}$${\mathcal{R}^{\dot{W}}\cdot
v_{1,K},h}$${\dot{\mathcal{S}}_{M}^{G}(h)*_{K}v_{1,K}}$

is ”upper triangular with respect to some total order, with invertible
diagonals”. The proof is similar to the proof of Theorem 4.3.

Recall that $\\{h_{x}\colon x\in\Lambda_{M}^{-}\\}$ forms a $\mathds{Z}$-basis
for $\mathcal{H}_{K}(\mathds{Z})$, we also have a $\mathds{Z}$-basis for
$\mathcal{R}^{\dot{W}}\cdot v_{1,K}$ given by $\\{r_{m}\cdot v_{1,K}\colon
m\in\Lambda_{M}^{-}\\}$. Fix an element $x\in\Lambda_{M}^{-}$, we have

$v_{1,K}*_{K}h_{x}=\dot{\mathcal{S}}_{M}^{G}(h_{x})\cdot
v_{1,K}=\sum_{m\in\Lambda_{M}^{-}}s_{x,m}r_{m},\text{ where
}s_{x,m}=|U^{+}m^{-1}K\cap Kx^{-1}K|_{K}\geq 0.$

For $m\in\Lambda_{M}^{-}$, $r_{m}$ appears in the above summation, i.e.
$s_{x,m}\neq 0$ if and only if $U^{+}m\cap KxK\neq\emptyset$. Accordingly, by
Lemma 5.7, we obtain

$v_{1,K}*_{K}h_{x}=\sum_{m\precsim x}s_{x,m}r_{m}.$

Therefore, since the monoid $\Lambda_{M}^{-}$ is countable and any element
$x\in\Lambda_{M}^{-}$ has only finitely many predecessors with respect to the
partial order $\precsim$, there exists a lexicographic (total) ordering
$x_{1},x_{2},\cdots$ for the elements of $\Lambda_{M}^{-}$. The above
discussion shows then, that the matrix of the transformation
$h\mapsto\dot{\mathcal{S}}_{M}^{G}(h)$ with respect to the bases
$\\{h_{x_{i}}\\}_{1}^{\infty}$ and $\\{{x_{i}}\\}_{1}^{\infty}$ is upper
triangular.

Finally, note that the diagonals $s_{x,x}=|U^{+}x^{-1}K\cap Kx^{-1}K|_{K}$ are
equal to $1$ by Lemma 5.6. This observation shows that the triangular matrix
$\mathcal{S}=\left(s_{x,m}\right)_{x,m\in\Lambda_{M}^{-}}$ is indeed
invertible, which concludes the combinatorial proof of Theorem 5.1. ∎

######  Remark 5.8.

By the previous proof, for any $m\in\Lambda_{M}^{-}$, we have
$\dot{\mathcal{S}}_{M}^{G}(h_{m})=\sum_{m^{\prime}\precsim
m}s_{m,m^{\prime}}r_{m^{\prime}}$. In particular, if $m$ is minuscule, hence
minimal with respect to the partial order $\precsim$, then
$\dot{\mathcal{S}}_{M}^{G}(h_{m})=r_{m}$. This yields an alternative proof
(and generalizes) for Kottwitz’s lemmas [Kot84, Lemma 2.3.3 & Lemma 2.3.7
(b)].

######  Remark 5.9 (Comparison of the untwisted and twisted Satake
isomorphisms).

Consider the $\mathds{C}$-linear map

${\eta_{B}\colon\mathcal{R}\otimes_{\mathds{Z}}\mathds{C}}$${\mathcal{R}\otimes_{\mathds{Z}}\mathds{C};m\otimes
c}$${\delta_{B}(m)^{1/2}m\otimes c.}$

Observe that for any $m\in\Lambda_{M}$, we have $\eta_{B}(\sum_{w\in
W/W_{m}}w(m)={\delta(m)^{1/2}}r_{m}$. Hence,
$\eta_{B}\colon\mathcal{R}\otimes_{\mathds{Z}}\mathds{C}\to\mathcal{R}\otimes_{\mathds{Z}}\mathds{C}$
is actually an isomorphism of $\mathds{C}$-algebras with $W$-action where the
target is endowed with the dot-action (Definition 3.1) and the source with the
usual action. Let
$\mathcal{S}_{M}^{G}\colon\mathcal{H}_{K}(\mathds{C})\to\mathcal{R}\otimes_{\mathds{Z}}\mathds{C}$
be the ”standard” Satake homomorphism [HR10, §9.2], then we have

$\eta_{B}\circ\mathcal{S}_{M}^{G}=\dot{\mathcal{S}}_{M}^{G}\otimes\text{id}_{\mathds{C}}.$

###  5.3. Positivity properties of the Satake isomorphism

In this subsection we show that the natural generalization of Rapoport’s
positivity result for unramified groups [Rap00, Theorem 1.1] and Haines listed
properties [Hai00, Theorem 1.2] remains true in general.

######  Theorem 5.1.

For any $x\in\Lambda_{M}^{-}$, write
$\dot{\mathcal{S}}_{M}^{G}(h_{x})=\sum_{m\in\Lambda_{M}^{-}}s_{x,m}r_{m}$. The
following statements are equivalent

  1. (1)

$s_{x,m}>0$,

  2. (2)

$U^{+}m\cap KxK\neq\emptyset$,

  3. (3)

$U^{+}m\cap Iwxw^{\prime}I\neq\emptyset$, for some $w,w^{\prime}\in
W_{a_{\circ}}$,

  4. (4)

${x}\succsim{m}$.

###### Proof.

The equivalence between (1) and (2) was already established while proving
Theorem 5.1, in which it is also proved (1) $\Rightarrow$ (4). The equivalence
between (2) and (3) is clear from Lemma 3.1.

Now, we show (4) $\Rightarrow$ (1): C. Schwer proved in [Sch06] such an
implication for the case of extended affine Weyl groups attached to root data.
We shall apply his result to
$\mathcal{D}:=({\Lambda}_{M}^{1},\Sigma^{1,\vee},{\Lambda}_{M}^{1,\vee},\Sigma^{1},\Delta^{1,\vee})$.
As we saw in Remark 5.6, the following diagram is commutative
$\left(\dot{\mathcal{S}}_{M}^{G}(h_{x})\right)^{1}=\dot{\mathcal{S}}_{M}^{G^{1}}\iota^{1}(h_{x})$,
hence

$\sum_{m\precsim
x}s_{x,m}r_{m^{1}}=\sum_{y\precsim^{1}x^{1}}s_{x^{1},y}r_{y}\in{\mathcal{R}^{1}}^{\dot{W}}$

By Remark 5.5, $m\precsim x$ if and only if ${m}^{1}\precsim{x}^{1}$.
Accordingly, $\sum_{m\precsim x}s_{x,m}r_{m^{1}}=\sum_{m\precsim
x}s_{x^{1},m^{1}}r_{m^{1}}\in{\mathcal{R}^{1}}^{\dot{W}}$ and so
$s_{x,m}=s_{x^{1},m^{1}}$, for all $m\precsim x$. We can actually see it
directly; because $M^{1}$ normalizes $K$ and
$x-m\in\Lambda_{\textit{\emph{\text{aff}}}}$, we get

$s_{x^{1},m^{1}}=|(U^{+}K\cap mKx^{-1}K^{1})K^{1}|_{K^{1}}=|(U^{+}K\cap
mKx^{-1}K)K^{1}|_{K^{1}}=s_{x,m}.$

Thanks to [Sch06, Corollary 4.8], we have the implication $x^{1}\precsim
m^{1}\Rightarrow s_{m^{1},x^{1}}>0$, which ends the proof of the theorem. ∎

##  6\. A compressible algebra and centers of parahoric-Hecke algebras

###  6.1. Normalized Intertwiners

The dot-action of $W$ on ${\mathcal{R}^{\flat}}$ stabilizes
${\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}$, hence it extends to a
dot-action of $W$ on
$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{R}^{\flat}$.
Now, since $W$ action’s is locally finite, we have by [Bou64, §1 n∘ 9,
Proposition 23] two canonical isomorphisms:

$\mathcal{L}^{\flat}:=\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{R}^{\flat}\simeq{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}\mathcal{R}^{\flat}\text{
and
}{\mathcal{L}^{\flat}}^{\dot{W}}\simeq{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}{\mathcal{R}^{\flat}}^{\dot{W}}$

Tensoring the decomposition provided by Corollary 4.6 with
${\mathcal{L}^{\flat}}^{\dot{W}}$ over
${\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}$, yields

${\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})=\bigoplus_{w\in
W_{a_{\circ}}^{\flat}}i_{w}^{\flat}*_{I^{\flat}}\mathcal{L}^{\flat}=\bigoplus_{w\in
W}\mathcal{L}^{\flat}*_{I^{\flat}}i_{w}^{\flat}.$ (2)

######  Lemma 6.1.

For any $f\in\mathcal{L}^{\flat}$, and $s\in\mathcal{S}^{\flat}$ we have

$f*_{I^{\flat}}(i_{s}^{\flat}+1)-(i_{s}^{\flat}+1)*_{I^{\flat}}\dot{s}(f)=(f-\dot{s}(f))*_{I^{\flat}}\mathcal{G}(\alpha).$

###### Proof.

It suffices to show this statement for $f\in\mathcal{R}^{\flat}$, which
follows readily from Proposition 4.3. ∎

For any $s\in\mathcal{T}_{a_{\circ}}$, set

$\mathscr{k}_{s}^{\flat}:=-1+({i}_{s}^{\flat}+1)\mathcal{G}(\alpha)^{-1}\in\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}.$

######  Proposition 6.2.

  1. (a)

There exists a unique group homomorphism

$\mathcal{K}^{\flat}\colon\dot{W}_{a_{\circ}}\hookrightarrow\left(\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})\right)^{\times}$

such that $\mathcal{K}^{\flat}(s)=\mathscr{k}_{s}^{\flat}$ for all
$s\in\mathcal{T}_{a_{\circ}}$.

  2. (b)

For any $f\in\mathcal{L}^{\flat}$, we have
$f*_{{I}^{\flat}}\mathscr{k}_{w}^{\flat}=\mathscr{k}_{w}^{\flat}*_{{I}^{\flat}}\dot{w}^{-1}(f)$.

  3. (c)

$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{{I}^{\flat}}({\mathds{Z}[q^{-1}]})=\bigoplus_{w\in
W}\mathcal{L}^{\flat}*_{I^{\flat}}\mathscr{k}_{w}^{\flat}=\bigoplus_{w\in
W}\mathscr{k}_{w}^{\flat}*_{I^{\flat}}\mathcal{L}^{\flat}$.

###### Proof.

(a) The condition forces the image of $\mathcal{K}$ to be in
$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\Lambda_{\textit{\emph{\text{aff}}}}}\mathcal{H}_{{}^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})=\bigoplus_{w\in
W}\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}*_{I^{\flat}}i_{w}^{\flat}$.
The existence and unicity follows then from [Lus89, Proposition 5.2 (a)]
applied to
$\mathcal{H}_{{I}^{\flat}}^{\textit{\emph{\text{aff}}}}({\mathds{Z}[q^{-1}]})\simeq\mathcal{H}({W}_{\textit{\emph{\text{aff}}}},\mathcal{D}_{\textit{\emph{\text{aff}}}},v=q,L)$.

(b) Because $\mathcal{K}$ is a morphism of groups, it suffices to check the
statement (b) for $w=s\in\mathcal{T}_{a_{\circ}}^{\flat}$, which follows from
Lemma 6.1 and the definition of $\mathscr{k}_{s}^{\flat}$.

(c) We proceed as in the proof of [Lus89, Proposition 5.5 (a)]. Let
$X=\sum_{w\in W}\mathcal{L}^{\flat}*_{I^{\flat}}\mathscr{k}_{w}$. Since
$W_{a_{\circ}}$ is generated by $\mathcal{T}_{a_{\circ}}$, it suffices to show
that $i_{s}^{\flat}\in X$, for all $s\in\mathcal{T}_{a_{\circ}}$. But, by
definition, we have
${i}_{s}^{\flat}\in\mathcal{L}^{\flat}(1+\mathscr{k}_{s}^{\flat})$. Therefore,
$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\Lambda_{\textit{\emph{\text{aff}}}}}\mathcal{H}_{I^{\flat}}^{\flat}({\mathds{Z}[q^{-1}]})=X$.
We saw in (2) p. 2, that
$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})$
is free over $\mathcal{L}$ of rank $|W|$. Therefore,
$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})=\oplus_{w\in
W}\mathcal{L}^{\flat}*_{I^{\flat}}\mathscr{k}_{w}^{\flat}$. The other equality
is proved in a similar way. ∎

###  6.2. A skew group ring

######  Proposition 6.1.

Let $\mathcal{L}^{\flat}*\dot{W}$ be the skew group ring of $\dot{W}$ over
$\mathcal{L}^{\flat}$ (See Definition .1). We have an isomorphism of
$\mathcal{L}^{\flat}$-algebras:

$\mathcal{L}^{\flat}*\dot{W}\to\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{{I}^{\flat}}({\mathds{Z}[q^{-1}]}),r\cdot
w\mapsto r*_{I^{\flat}}\mathscr{k}_{w}^{\flat}.$

###### Proof.

This follows readily from (b) and (c) of Proposition 6.2. ∎

###  6.3. An Azumaya algebra

######  Lemma 6.1.

  1. (i)

The skew group ring $\mathcal{L}_{\textit{\emph{\text{aff}}}}*\dot{W}$ is the
matrix algebra
$\mathbb{M}_{|W|}(\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\dot{W}})$.

  2. (ii)

The two ring extensions
$\mathcal{L}_{\textit{\emph{\text{aff}}}}/\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\dot{W}}$
and $\mathcal{L}^{\flat}/{\mathcal{L}^{\flat}}^{\dot{W}}$ are
${\dot{W}}$-Galois.

###### Proof.

(i) There is an obvious map of algebras
$\mathcal{L}_{\textit{\emph{\text{aff}}}}*{\dot{W}}\to
End_{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\dot{W}}}(\mathcal{L}_{\textit{\emph{\text{aff}}}})$
where, an element $a\in\mathcal{L}_{\textit{\emph{\text{aff}}}}$ is taken to
multiplication by $a$ on $\mathcal{L}_{\textit{\emph{\text{aff}}}}$, and
$w\in{\dot{W}}$ is taken to the obvious map
$w:\mathcal{L}_{\textit{\emph{\text{aff}}}}\to\mathcal{L}_{\textit{\emph{\text{aff}}}}$.
By an argument similar to the proof of linear independence of characters, one
can check that this map is injective. Now since each side has dimension
$|W|^{2}$ as vector space over
$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\dot{W}}$, this map is an
isomorphism.

(ii) We deduce from (i) that
$\mathcal{L}_{\textit{\emph{\text{aff}}}}*{\dot{W}}$ is Azumaya over
$\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\dot{W}}$. Accordingly, Theorem .4
shows that
$\mathcal{L}_{\textit{\emph{\text{aff}}}}/\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\dot{W}}$
is ${\dot{W}}$-Galois, and so is
$\mathcal{L}^{\flat}/{\mathcal{L}^{\flat}}^{\dot{W}}$ using Corollary .1. ∎

######  Proposition 6.2.

The ${\mathds{Z}[q^{-1}]}$-algebra
$A=\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{{I}^{\flat}}({\mathds{Z}[q^{-1}]})$
is compressible, that is

$Z(eAe)=eZ(A)=e\mathcal{L}^{\dot{W}}$

for any idempotent $e$ of $A$.

###### Proof.

By Theorem .4 and Lemma 6.1 (ii), the algebra $\mathcal{L}^{\flat}*\dot{W}$ is
Azumaya over $\mathcal{L}^{{\flat}}$, so the proposition is a consequence of
Theorem .1. ∎

###  6.4. The center of parahoric–Hecke algebras

######  Theorem 6.1.

For any facet $\mathcal{F}\subset\overline{\mathfrak{a}}$, we have

$Z\left(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Z}[q^{-1}]})\right)=\dot{\Theta}_{\text{Bern}}^{\flat}({\mathcal{R}^{\flat}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}])*_{I^{\flat}}{\bf
1}_{K_{\mathcal{F}}^{\flat}}.$

In particular, the set $\\{r_{m}^{\flat}*_{I^{\flat}}{\bf
1}_{K_{\mathcal{F}}^{\flat}}:m\in\Lambda_{M}^{-,\flat}\\}$ forms a basis for
the $\mathds{Z}[q^{-1}]$-module
$Z\left(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Z}[q^{-1}]})\right)$.

######  Lemma 6.2.

For any idempotent $e\in\mathcal{H}_{{I}^{\flat}}({\mathds{Q}})$, we have

$Z(e({\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}\mathcal{H}_{{I}^{\flat}}({\mathds{Q}}))e)=e{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{e{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}Z(e(\mathcal{H}_{{I}^{\flat}}({\mathds{Q}}))e).$

###### Proof.

Given Corollary 4.1, we have an inclusion
$Z(e({\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}\mathcal{H}_{{I}^{\flat}}({\mathds{Q}}))e)\supset
e{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{e{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}Z(e(\mathcal{H}_{{I}^{\flat}}({\mathds{Q}}))e).$
The opposite one is true because:

(i) The morphism $e\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})e\to
e{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{e{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}e\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})e$,
$h\mapsto e\otimes h$ is injective. This follows readily from the proof of
Lemma 4.1, in which we proved that
$\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$ is
${\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}$-torsion free,
so in particular $e\mathcal{H}_{I^{\flat}}({\mathds{Q}})e$ is also
$e{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}$-torsion free
and ${\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\simeq
e{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}$.

(ii) Any element in the left hand side can be written as $\ell\otimes h$, for
some $\ell\in e{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}$
and $h\in e\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})e$, hence we must have
$h\in Z(e\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})e)$. This shows the
lemma. ∎

###### Proof of Theorem 6.1.

Let $\mathcal{F}\subset\overline{\mathfrak{a}}$ be any facet and
$e_{\mathcal{F}}$ the idempotent $[K_{\mathcal{F}}^{\flat}:I^{\flat}]{\bf
1}_{K_{\mathcal{F}}^{\flat}}$. By Proposition 6.2 and Lemma 6.2, we have

$e_{\mathcal{F}}^{\flat}\mathcal{L}^{\dot{W}}=e_{\mathcal{F}}^{\flat}{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{e_{\mathcal{F}}{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}e_{\mathcal{F}}{\mathcal{R}^{\flat}}^{\dot{W}}=e_{\mathcal{F}}{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{e_{\mathcal{F}}{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}Z(e_{\mathcal{F}}\mathcal{H}_{I^{\flat}}({\mathds{Q}})e_{\mathcal{F}}).$

For any $h\in
Z(e_{\mathcal{F}}\mathcal{H}_{I^{\flat}}({\mathds{Q}})e_{\mathcal{F}})$, there
are then $s,r\in{\mathcal{R}^{\flat}}^{\dot{W}}$ such that $sh-re=0$. We have
then an equality
$sv_{1,\mathcal{F}}^{\flat}*_{K_{\mathcal{F}}^{\flat}}h=rv_{1,\mathcal{F}}^{\flat}$.
By Propoisition 4.4, we deduce that
$v_{1,\mathcal{F}}^{\flat}*_{K_{\mathcal{F}}^{\flat}}h=tv_{1,\mathcal{F}}^{\flat}$
for some $t\in{\mathcal{R}^{\flat}}^{\dot{W}}\otimes\mathds{Q}$, this shows
$v_{1}^{\flat}*_{I^{\flat}}h=v_{1}^{\flat}t*_{I^{\flat}}e_{\mathcal{F}}$ and
accordingly $h=t*_{I^{\flat}}e_{\mathcal{F}}$ by Corollary 4.4. In conclusion,

${\bf
1}_{K_{\mathcal{F}}^{\flat}}*_{I}({\mathcal{R}^{\flat}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Q})=Z(e_{\mathcal{F}}\mathcal{H}_{I^{\flat}}({\mathds{Q}})e_{\mathcal{F}})=Z(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Q}})).$

Let $h\in Z(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Z}[q^{-1}]}))$, by
the previous discussion, there exists $a/b\in\mathds{Q}$ and
$r\in{\mathcal{R}^{\flat}}^{\dot{W}}$ such that $h=\frac{a}{b}{\bf
1}_{K_{\mathcal{F}}^{\flat}}*_{I^{\flat}}\dot{\Theta}_{Bern}^{\flat}(r)$, but
by definition the term ${\bf
1}_{K_{\mathcal{F}}^{\flat}}*_{I^{\flat}}\dot{\Theta}_{Bern}^{\flat}(r)\in\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$,
hence one must have $\frac{a}{b}\in\mathds{Z}[q^{-1}]$. Consequently, we have

$\displaystyle Z(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Z}[q^{-1}]}))$
$\displaystyle={\bf
1}_{K_{\mathcal{F}}^{\flat}}*_{I^{\flat}}\dot{\Theta}_{Bern}^{\flat}({\mathcal{R}^{\flat}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]).\qed$

The linear independence of the elements $r_{m}^{\flat}*_{I^{\flat}}{\bf
1}_{K_{\mathcal{F}}^{\flat}}(m\in\Lambda_{M}^{-,\flat})$ is clear from the
injectivity of $-*_{I^{\flat}}{\bf 1}_{K_{\mathcal{F}}^{\flat}}$ and Corollary
4.6.

###  6.5. Compatibility of the twisted Satake and twisted Bernstein
isomorphisms

From the above, we see that
$Z(\mathcal{H}_{K_{\mathcal{F}}}({\mathds{Z}[q^{-1}]}))$ is a free
$\mathcal{R}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]$-module of rank
one, with basis the canonical vector ${\bf 1}_{K_{\mathcal{F}}}$. Define
$\dot{\mathcal{S}}_{\mathcal{F}}\colon
Z(\mathcal{H}_{K_{\mathcal{F}}}({\mathds{Z}[q^{-1}]}))\to\mathcal{R}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]$,
characterized by $\dot{\mathcal{S}}_{\mathcal{F}}(h)*_{I}{\bf
1}_{K_{\mathcal{F}}}=h$. By Lemma 5.3, we have
$\dot{\mathcal{S}}_{\mathcal{F}}=\dot{\mathcal{S}}_{M}^{G}$ if
$\mathcal{F}=\\{a_{\circ}\\}$. The Satake and Bernstein twisted isomorphisms
are compatible in the following sense:

######  Theorem 6.1.

Let
$\mathcal{F}\subset\overline{\mathcal{F}}^{\prime}\subset\overline{\mathfrak{a}}$,
be two facets, i.e. $K_{\mathcal{F}}\supset K_{\mathcal{F}^{\prime}}\supset
I$. The following diagram of $\mathds{Z}[q^{-1}]$-algebras is commutative:

${Z(\mathcal{H}_{K_{\mathcal{F}}}(\mathds{Z}[q^{-1}]))}$${{\mathcal{R}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]}$${Z(\mathcal{H}_{K_{\mathcal{F}^{\prime}}}(\mathds{Z}[q^{-1}]))}$${Z(\mathcal{H}_{I}(\mathds{Z}[q^{-1}]))}$$\scriptstyle{\dot{\mathcal{S}}_{\mathcal{F}}}$$\scriptstyle{\simeq}$$\scriptstyle{\dot{\Theta}_{Bern}}$$\scriptstyle{\simeq}$$\scriptstyle{-*_{K_{\mathcal{F}^{\prime}}}{\bf
1}_{K_{\mathcal{F}}}}$$\scriptstyle{\simeq}$$\scriptstyle{-*_{I}{\bf
1}_{K_{\mathcal{F}^{\prime}}}}$$\scriptstyle{\simeq}$

###### Proof.

Similar to the proof of Lemma 5.2. ∎

######  Remark 6.2.

Given Theorem 5.1, we have, in particular, an inclusion
$Z(\mathcal{H}_{{I^{\flat}}}(\mathds{Z}))\subset{\dot{\Theta}_{\text{\emph{Bern}}}}({\mathcal{R}^{\dot{W}}})$.

## Appendix. Skew group rings

This appendix is devoted to some properties of $G$-Galois extensions of
commutative rings.

###  .6. $G$-Galois extensions of commutative rings

######  Definition .1.

Let $S$ be a commutative ring and $G$ a finite group of ring automorphisms of
$S$. Write $S*G$ for the free $S$-module with basis $\\{g\in G\\}$. We make
$S*G$ into a ring as follows: for $s,s^{\prime}\in S$ and $g,g^{\prime}\in G$,
define $(s\cdot g)(s^{\prime}\cdot g^{\prime})=sg(s^{\prime})\cdot
gg^{\prime}$. The ring $S*G$ is usually called a skew group ring or a trivial
crossed product.

######  Definition .2.

Let $R$ be a commutative ring. Two $R$-algebras homomorphisms $f,g\colon A\to
B$ are said to be strongly distinct if for every nonzero idempotent $e\in B$
there exists $x\in A$ such that $f(x)e\neq g(x)e$.

######  Remark .3.

Strong distinctness is equivalent to Dedekind’s lemma in the context of
separable algebras. The elements of $G$ are all pairwise strongly distinct if
and only if $G$ is free over $S$ in $\text{End}_{R}(S)$ [FP97, Proposition
2.1].

######  Theorem .4.

Let $S$ be a commutative ring extension and $G$ a finite group of ring
automorphisms of $S$. Set $R:=S^{G}$. The ring $S$ is called a Galois
extension of $R$ with Galois group $G$ or $S/R$ is a $G$-Galois ring
extension, if one of the following equivalent statements is satisfied:

  1. (1)

$S$ is separable over $R$ and the elements of $G$ are all pairwise strongly
distinct.

  2. (2)

There exist elements $x_{i},y_{i}$ in $S$, $1\leq i\leq n$ for some integer
$n$, such that $\sum_{i}x_{i}g(y_{i})=\delta_{1,g}\mathbf{1}_{S}$, for all
$g\in G$. The set $\\{x_{i},y_{i}\colon 1\leq i\leq n\\}$ is called a
$G$-Galois system for $S$.

  3. (3)

$S$ is a finitely generated projective $R$-module and the map
$\varphi:S*G\to\text{End}_{R}(S)$, given by $\varphi(xg)(y)=xg(y)$, is an
isomorphism of $S$-modules and $R$-algebras.

  4. (4)

For any left $S*G$-module $M$ the map $\mu\colon S\otimes_{R}M^{G}\to
M,(s\otimes m)\to sm$ is an isomorphism of $S$-modules.

  5. (5)

The map $S\otimes_{R}S\to\prod_{g\in G}S,(x,y)\mapsto(xg(y))_{g\in G}$ is an
isomorphism of $S$-algebras.

  6. (6)

For each maximal ideal $\mathfrak{m}$ of $S$ and each $1\neq g\in G$ there
exists $y\in S$ such that $g(y)-y\not\in\mathfrak{m}$.

  7. (7)

$S\pi S=S*G$ where $\pi=\sum_{g\in G}g$.

  8. (8)

$S*G$ is Azumaya over $R$.

  9. (9)

$S*G$ is H-separable over $S$.

When this is the case, $S$ is a maximal commutative $R$-subalgebra of $S*G$
with $(S*G)\otimes_{R}S\simeq\mathbb{M}_{|G|}(S)$ and
$S\otimes_{R}(S*G)\simeq\mathbb{M}_{|G|}(S)$.

###### Proof.

The equivalence of the first seven statement is [CHR65, Theorem 1.3]. For (8),
first note that $S$ can be seen as an $(S;R)$-subbimodule of $S*G$ via $x\to
x\pi$, hence the notation $S\pi S$ makes sense. Because $\pi g=\pi$ we see
that $S\pi S$ is a two sided ideal of $S*G$, hence $S\pi S=S*G$ if and only if
$\mathbf{1}_{S}\in S\pi S$, which is equivalent to the existence of
$x_{i},y_{i}$ in $S$, $1\leq i\leq n$ for some integer $n$, such that
$\mathbf{1}_{S}=\sum_{i}x_{i}\pi y_{i}=\sum_{g\in G}(\sum_{i}x_{i}g(y_{i}))g$,
i.e. $\sum_{i}x_{i}g(y_{i})=\delta_{1,g}\mathbf{1}_{S}$, for all $g\in G$. For
8 and 9, this is [Ike81, Theorem 2] or [AS95, Corollary 2]. ∎

######  Remark .5.

If $S/R$ is $G$-Galois, then $\text{Rank}_{R}S=|G|$ [FP97, Lemma 2.3].

###  .7. Properties

######  Proposition .1.

For any $G$-Galois extension $S/R$:

(i) $S$ is faithfully flat over $R$.

(ii) $\text{Tr}\colon S\to R$ is surjective.

(iii) The $R$-submodule $R$ of $S$ is a direct summand of $S$.

(iv) For any $R$-algebra $T$, $T\otimes_{R}S/S$ is a $G$-Galois extension iff
$S/R$ is $G$-Galois.

(v) Let $S/R$ and $S^{\prime}/R$ be $G$-Galois. Then every $G$-equivariant
$R$-algebra homomorphism $f:S\to S^{\prime}$ of $G$-Galois extensions is an
isomorphism.

###### Proof.

See [Gre92, Chapitre 0, §1]. ∎

###  .8. An obvious corollary

######  Corollary .1.

Let $T$ be a commutative ring with a unity $\mathbf{1}_{T}$ and $G$ be a
finite group $G$ that acts faithfully on it by ring automorphisms. Let
$S\subset T$ be a subring such that $\mathbf{1}_{T}=\mathbf{1}_{S}$. If
$S/S^{G}$ is $G$-Galois then so is $T/T^{G}$.

###### Proof.

Thanks to Theorem .4, there exist elements $x_{i},y_{i}$ in $S\subset T$,
$1\leq i\leq n$, such that $\sum_{i}x_{i}g(y_{i})=\delta_{1,g}\mathbf{1}_{S}$,
for all $g\in G$. The corollary follows readily since
$\mathbf{1}_{T}=\mathbf{1}_{S}$. ∎

###  .9. Compressibility

######  Theorem .1.

If $A$ is an Azumaya algebra over a commutative ring $R$ then it is also
compressible that is if $Z(eAe)=eZ(A)=eR$ for each idempotent $e$ of $A$.

###### Proof.

This is [AP84, Theorem 2]. ∎

######  Example .2.

If $S/R$ is $G$-Galois then $S*G$ is compressible.

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