$A\subset\Pi_{\mathbf{v}}$.

  1. (1)

We have the implication $p_{B,A}\neq 0\Rightarrow B\in S_{A}$.

  2. (2)

We have

$S_{A}=\\{wC~{}|~{}w\in W_{\mathbf{v}},~{}C\preceq
A~{}\text{and}~{}C\subset\mathscr{C}^{+}_{\mathbf{v}}\\}.$

  3. (3)

Assume that there exists some facet
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ such that $A\in
A^{+}_{\mathbf{v}}W_{\mathbf{g}}$, and that $\mathbf{v}\subset
x\mathscr{C}^{+}_{0}$ for some $x\in W_{0}\backslash\\{\mathrm{id}\\}$ (this
simply means that $\mathbf{v}$ is inside another quarter with vertex $\\{0\\}$
than $\mathscr{C}^{+}_{0}$). Then we have

$D\in S_{A}\Rightarrow D\not\subset\rho^{\vee}+\mathscr{C}^{+}_{0}.$

###### Proof.

  1. (1)

This is [Soe97, Proposition 4.22].

  2. (2)

Let us write

$S^{\prime}_{A}=\\{wC~{}|~{}w\in W_{\mathbf{v}},~{}C\preceq
A~{}\text{and}~{}C\subset\mathscr{C}^{+}_{\mathbf{v}}\\}.$

The inclusion $S_{A}\subset S^{\prime}_{A}$ just follows from the fact that
any alcove $C\in\mathcal{A}$ has a $W_{\mathbf{v}}$ conjugate $wC$ inside of
$\mathscr{C}^{+}_{\mathbf{v}}$. Conversely, if $C$ is an alcove such that
$C\subset\mathscr{C}^{+}_{\mathbf{v}}$ and $C\preceq A$, then we have
$wC\preceq C$ for every $w\in W_{\mathbf{v}}$ thanks to Lemma 5.22, so
$wC\preceq A$ for every $w\in W_{\mathbf{v}}$. This proves the inclusion
$S^{\prime}_{A}\subset S_{A}$, and concludes the proof of the first point.

  3. (3)

Let $C\subset\mathscr{C}^{+}_{\mathbf{v}}$ be an alcove such that $C\preceq
A$.

     * •

We claim that $C$ belongs to $A^{+}_{\mathbf{v}}W_{\mathbf{g}}$. Since
$\preceq$ is invariant under translations, we can translate everything by
$-\mu$, where $\mathbf{v}=\\{\mu\\}$, so that we are reduced to the case where
$\mathbf{v}=\\{0\\}$. More precisely, if we denote by $w_{\mu}\in W$ the
element such that $w_{\mu}\mathbf{a}_{1}=A^{+}_{\mathbf{v}}$, by $w\in
W_{\mathbf{g}}$ the element such that $A=A^{+}_{\mathbf{v}}w$ and set
$\omega_{\mu}:=t_{-\mu}w_{\mu}$, then we get that
$\omega_{\mu}\mathbf{a}_{1}=\mathbf{a}_{1}$ and

$C\preceq A\Longleftrightarrow t_{-\mu}C\preceq
t_{-\mu}A=t_{-\mu}(w_{\mu}\mathbf{a}_{1}w)=t_{-\mu}(w_{\mu}w\mathbf{a}_{1})=\omega_{\mu}w\mathbf{a}_{1}\\\
=(\omega_{\mu}w\omega_{\mu}^{-1})\omega_{\mu}\mathbf{a}_{1}=(\omega_{\mu}w\omega_{\mu}^{-1})\mathbf{a}_{1}.$

So now, if we set $\mathbf{g}^{\prime}:=\omega_{\mu}\mathbf{g}$, we get the
inequality $t_{-\mu}C\preceq w^{\prime}\mathbf{a}_{1}$, with
$w^{\prime}=\omega_{\mu}w\omega_{\mu}^{-1}\in W_{\mathbf{g}^{\prime}}$. Since
the alcoves $t_{-\mu}C$ and $w^{\prime}\mathbf{a}_{1}$ belong to
$\mathcal{A}^{+}$ (because they are translates by $-\mu$ of alcoves in
$\mathscr{C}^{+}_{\mathbf{v}}$), we get that $t_{-\mu}C\leq
w^{\prime}\mathbf{a}_{1}$, so that $t_{-\mu}C=w^{\prime\prime}\mathbf{a}_{1}$
for some $w^{\prime\prime}\leq w^{\prime}$, from which we easily deduce that
$w^{\prime\prime}\in W_{\mathbf{g}^{\prime}}$; this means that
$w^{\prime\prime}=\omega_{\mu}u\omega_{\mu}^{-1}$ for some $u\in
W_{\mathbf{g}}$, so

$C=t_{\mu}(\omega_{\mu}u\omega_{\mu}^{-1})\mathbf{a}_{1}=t_{\mu}(\omega_{\mu}u\omega_{\mu}^{-1})t_{-\mu}t_{\mu}\mathbf{a}_{1}=(w_{\mu}u)(w_{\mu}^{-1}t_{\mu})\mathbf{a}_{1}=(w_{\mu}u)\omega_{\mu}^{-1}\mathbf{a}_{1}\\\
=w_{\mu}u\mathbf{a}_{1}=A^{+}_{\mathbf{v}}u.$

     * •

Denote by $\mathbf{h}$ the $W$-conjugate of $\mathbf{g}$ contained in
$\overline{A^{+}_{\mathbf{v}}}$. In particular, we get that $\mathbf{h}$
belongs to both $\overline{A^{+}_{\mathbf{v}}}$ and $\overline{C}$ (because
$C\in A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ thanks to the previous paragraph), so
that $\overline{C}\cap\overline{A^{+}_{\mathbf{v}}}\neq\emptyset$. From this
and the description of $S_{A}$ given in (2), we deduce that for any alcove
$D\in S_{A}$, there exists some $w\in W_{\mathbf{v}}$ such that

$\overline{D}\cap w\overline{A^{+}_{\mathbf{v}}}\neq\emptyset$

(just write $D=wC$ for some alcove $C$ as above). But
$wA^{+}_{\mathbf{v}}\subset x\mathscr{C}^{+}_{0}$ for any $w\in
W_{\mathbf{v}}$ (because $x^{-1}wA^{+}_{\mathbf{v}}$ is an alcove containing
$x^{-1}\mathbf{v}\subset\mathscr{C}^{+}_{0}$ in its closure, so that
$x^{-1}wA^{+}_{\mathbf{v}}\subset\mathscr{C}^{+}_{0}$ thanks to Proposition
5.5), so we get ultimately that if an alcove $D\in S_{A}$ is contained in
$\mathscr{C}^{+}_{0}$, then $\overline{D}$ must have a non-empty intersection
with a hyperplane separating $\mathscr{C}^{+}_{0}$ from
$x\mathscr{C}^{+}_{0}$, so that $D$ is not contained in
$\rho^{\vee}+\mathscr{C}^{+}_{0}$.

∎

######  Lemma 5.26.

Assume that the root system $\mathfrak{R}^{\vee}$ is irreducible. Let
$\lambda$ be a nonzero weight contained in
$\overline{\mathscr{C}^{+}_{0}}\cap\mathbb{X}^{\vee}$, $\mathbf{v}$ be a
special facet, $C$ be an alcove contained in $\Pi_{\mathbf{v}}$ and put
$A:=\lambda+C$. If
$\overline{A}\cap\overline{A^{+}_{\mathbf{v}}}\neq\emptyset$, we must have
$C=A^{+}_{\mathbf{v}}$.

###### Proof.

In order to simplify the notations, we may and will reduce ourselves to the
case where $\mathbf{v}=\\{0\\}$. Let us write

$C=\\{\mu\in
E~{}|~{}n_{\alpha}-1<\langle\mu,\alpha\rangle<n_{\alpha}~{}\forall\alpha\in\mathfrak{R}_{+}\\}$

for some integers $n_{\alpha}$. First notice that if
$\alpha^{\prime},\alpha\in\mathfrak{R}_{+}$ are such that
$\alpha\leq\alpha^{\prime}$ (i.e. $\alpha^{\prime}-\alpha$ is a sum of
positive roots), then we must have $n_{\alpha}\leq n_{\alpha^{\prime}}$.
Indeed, since $C\subset\mathscr{C}^{+}_{0}$, we know that
$\langle\mu,\alpha\rangle\leq\langle\mu,\alpha^{\prime}\rangle$ for every
$\mu\in C$, so in particular we get that

$n_{\alpha}-1<\langle\mu,\alpha\rangle\leq\langle\mu,\alpha^{\prime}\rangle<n_{\alpha^{\prime}},$

proving the claim.

Since $C\subset\Pi$, we know that $n_{\alpha}=1$ for every $\alpha\in S_{0}$,
and thus $n_{\alpha}\geq 1$ for every $\alpha\in\mathfrak{R}_{+}$. The
assumption that $\overline{A}\cap\overline{\mathbf{a}_{1}}\neq\emptyset$
implies that we have

$n_{\alpha}+\langle\lambda,{\alpha}\rangle\in\\{1,2\\}\quad\text{for
every}~{}\alpha\in\mathfrak{R}_{+}.$

Therefore we must have $\langle\lambda,{\alpha}\rangle\in\\{0,1\\}$ for every
$\alpha\in\mathfrak{R}_{+}$. Since $\lambda\neq 0$, there exists some
$\alpha_{0}\in S_{0}$ such that $\langle\lambda,\alpha_{0}\rangle>0$. So let
$\beta=\sum_{S_{0}}c_{\alpha}\alpha\in\mathfrak{R}_{+}$ be the longest root
(cf. [Bou, Chap. 6, Proposition 25], this is where we need the assumption that
$\mathfrak{R}$ is irreducible). In particular we have $c_{\alpha}\geq 1$ for
every $\alpha\in S_{0}$, so that $\langle\lambda,\beta\rangle>0$, and thus
$n_{\beta}=1$. But for every $\alpha\in\mathfrak{R}_{+}$, we have
$\alpha\leq\beta$, so we get that $n_{\alpha}\leq n_{\beta}$, and finally
$n_{\alpha}=1$. This means that $C=\mathbf{a}_{1}$. ∎

The following corollary can be visualized on Figure 3 below.

######  Corollary 5.27.

Let $\mathbf{v}$ be a special facet,
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ be a non-special facet and $A\in
A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ be the alcove which is maximal in
$A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ for $\preceq$. Then $A$ belongs to
$\Pi_{\mathbf{v}}$.

###### Proof.

Let us first explain why $A\subset\mathscr{C}^{+}_{\mathbf{v}}$. Denote by
$\mathbf{h}^{\prime}$ the element of $\mathcal{A}_{\mathbf{g}}$ which belongs
to the closure of $A^{+}_{\mathbf{v}}$. We have the inclusion
$A^{+}_{\mathbf{v}}\subset\mathscr{C}^{+}_{\mathbf{v}}$, so we know that
$\mathbf{h}^{\prime}$ is included in
$\overline{\mathscr{C}^{+}_{\mathbf{v}}}$. If $C\in
A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ is an alcove which is not included in
$\mathscr{C}^{+}_{\mathbf{v}}$, then there exists a hyperplane $H$ of the
boundary of $\mathscr{C}^{+}_{\mathbf{v}}$ which contains
$\mathbf{h}^{\prime}$ and separates $C$ from $\mathscr{C}^{+}_{\mathbf{v}}$,
so we clearly have the inclusion $C\subset E^{-}_{H}$, and therefore $C\prec
s_{H}C$. Since $s_{H}\in W_{\mathbf{h}^{\prime}}$, the alcove $s_{H}C$ must
belong111111Indeed, write $C=w\mathbf{a}_{1}$ for some $w\in W$, then
$\mathbf{h}^{\prime}=w\mathbf{g}$ and $w^{-1}s_{H}w\in W_{\mathbf{g}}$, so
that $s_{H}C=C(w^{-1}s_{H}w)\in
CW_{\mathbf{g}}=A^{+}_{\mathbf{v}}W_{\mathbf{g}}$. to
$A^{+}_{\mathbf{v}}W_{\mathbf{g}}$. This shows that the maximal element in
$A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ for $\preceq$ must be contained in
$\mathscr{C}^{+}_{\mathbf{v}}$, whence $A\subset\mathscr{C}^{+}_{\mathbf{v}}$.

Now let us show that $A\subset\Pi_{\mathbf{v}}$. For this we may and will
assume until the end of this proof that $\mathfrak{R}$ is irreducible. Assume
that this inclusion does not hold, then because
$A\subset\mathscr{C}^{+}_{\mathbf{v}}$, we can write $A=\lambda+C$ for some
nonzero weight $\lambda\in\overline{\mathscr{C}^{+}_{0}}\cap\mathbb{X}^{\vee}$
and some alcove $C\subset\Pi_{\mathbf{v}}$. But we also have
$\overline{A}\cap\overline{A^{+}_{\mathbf{v}}}\neq\emptyset$ (both of those
sets contain $\mathbf{h}^{\prime}$), therefore we can apply Lemma 5.26 (here
we use the assumption that $\mathfrak{R}$ is irreducible) to see that
$C=A^{+}_{\mathbf{v}}$. So we are in the following situation:

(5.8) $\lambda+A^{+}_{\mathbf{v}}\in A^{+}_{\mathbf{v}}W_{\mathbf{g}}.$

Let $\mu\in\mathbb{X}^{\vee}$ be such that $\\{\mu\\}=\mathbf{v}$, and denote
by $w_{\mu}\in W$ the element such that
$w_{\mu}\mathbf{a}_{1}=A^{+}_{\mathbf{v}}$. Putting
$\omega_{\mu}:=t_{-\mu}w_{\mu}$ and applying $t_{-\mu}$ to (5.8), we get that

$\lambda+\omega_{\mu}\mathbf{a}_{1}\in\omega_{\mu}\mathbf{a}_{1}W_{\mathbf{g}}=(\omega_{\mu}W_{\mathbf{g}}\omega_{\mu}^{-1})\omega_{\mu}\mathbf{a}_{1},$

and, using the fact that $\omega_{\mu}$ fixes the fundamental alcove
$\mathbf{a}_{1}$, we obtain

(5.9) $\lambda+\mathbf{a}_{1}\in
W_{\mathbf{g}^{\prime}}\mathbf{a}_{1},~{}\text{with}~{}\mathbf{g}^{\prime}:=\omega_{\mu}\mathbf{g}.$

Since $\omega_{\mu}\in\Omega$ (because $\omega_{\mu}$ fixes $\mathbf{a}_{1}$)
and $\mathbf{g}$ is a non-special facet, $\mathbf{g}^{\prime}$ is also a non-
special facet121212Indeed, the action of $W$ (and hence of $\Omega$) on $E$
obviously permutes the set of hyperplanes $\mathcal{H}$, so it also permutes
the set of special facets by construction. included in the closure of the
fundamental alcove. From (5.9) we deduce that
$\mathbf{g}^{\prime}\subset\lambda+\overline{\mathbf{a}_{1}}$. But
$\lambda\in\overline{\mathscr{C}^{+}_{0}}\cap\mathbb{X}^{\vee}$ is nonzero, so
$\langle\lambda,\beta\rangle\geq 1$, where $\beta\in\mathfrak{R}_{+}$ is the
longest root. Since $\mathbf{g}^{\prime}\subset\overline{\mathbf{a}_{1}}$, we
have that $\langle\mu,\beta\rangle\leq 0$ for every
$\mu\in-\lambda+\mathbf{g}^{\prime}$ (because
$\langle\mu^{\prime},\beta\rangle\leq 1$ for all
$\mu^{\prime}\in\overline{\mathbf{a}_{1}}$); therefore the inclusion
$-\lambda+\mathbf{g}^{\prime}\subset\overline{\mathbf{a}_{1}}$ is possible
only when $\langle\mu,\beta\rangle=0$ for all
$\mu\in-\lambda+\mathbf{g}^{\prime}$. Since $\alpha\leq\beta$ for all
$\alpha\in\mathfrak{R}_{+}$ we get that

$0\leq\langle\mu,\alpha\rangle\leq\langle\mu,\beta\rangle=0~{}\forall\alpha\in\mathfrak{R}_{+},\mu\in-\lambda+\mathbf{g}^{\prime}$

which means that $\mu=0$ and so $\mathbf{g}^{\prime}=\\{\lambda\\}$,
contradicting our assumption that $\mathbf{g}^{\prime}$ is not a special
facet. This concludes the proof of the lemma. ∎

We are now ready to prove our central result.

######  Proposition 5.28.

Let $\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ be a non-special facet,
$\mathbf{h}\in\mathcal{A}_{\mathbf{g}}^{+}$ be such that
$\mathbf{h}\subset\rho^{\vee}+\overline{\mathscr{C}^{+}_{0}}$ and
$B\subset\rho^{\vee}+\mathscr{C}^{+}_{0}$ be an alcove containing $\mathbf{h}$
in its closure. Let also $\mathbf{v}$ be a special facet such that
$\mathbf{v}\subset\overline{B}$, and $A$ be the alcove which is maximal in
$A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ for $\leq$. Then we have
$A\subset\Pi_{\mathbf{v}}$ and

$n_{Br,A}(1)=1\qquad\forall r\in W_{\mathbf{g}}.$

###### Proof.

Let $\mathbf{h}^{\prime}$ be the element of $\mathcal{A}_{\mathbf{g}}$ which
is included in $\overline{A^{+}_{\mathbf{v}}}$. By construction (i.e. because
$\mathbf{v}\subset\rho^{\vee}+\overline{\mathscr{C}_{0}^{+}}$), we have that
$A^{+}_{\mathbf{v}}\subset\rho^{\vee}+\mathscr{C}^{+}_{0}$, so that in
particular $\mathbf{h}^{\prime}\subset\mathscr{C}^{+}_{0}$. Thus, applying
Proposition 5.5, we know that all the alcoves of
$A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ are contained in $\mathcal{A}^{+}$ (because
they contain $\mathbf{h}^{\prime}$ in their closure), so that the orders
$\preceq$ and $\leq$ coincide in $A^{+}_{\mathbf{v}}W_{\mathbf{g}}$. Therefore
$A$ is maximal in $A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ for $\preceq$, so is
included in $\Pi_{\mathbf{v}}$ thanks to Corollary 5.27.

By Lemma 5.24 and because $A\subset\rho^{\vee}+\mathscr{C}^{+}_{0}$, we have
that

(5.10) $\sum_{C}n_{C,A}N_{C}=\sum_{x\in W_{0}}(-1)^{l(x)}\sum_{C}p_{C,x\ast
A}~{}\mathrm{res}~{}C.$

Our next goal is to use this formula to show that

$n_{C,A}=p_{C,A}\qquad\text{for
all}~{}C\subset\rho^{\vee}+\mathscr{C}^{+}_{0}.$

For that we will prove that when the alcove
$C\subset\rho^{\vee}+\mathscr{C}^{+}_{0}$ is fixed and $x\in
W_{0}\backslash\\{\mathrm{id}\\}$, we have $p_{C,x\ast A}=0$; this will follow
from a careful study of the periodic polynomials.

Fix $x\in W_{0}\backslash\\{\mathrm{id}\\}$, and notice that $x\ast
A\subset\Pi_{x\mathbf{v}}$ since $A\subset\Pi_{\mathbf{v}}$. We can then apply
the first point of Lemma 5.25:

$p_{D,x\ast A}\neq 0\Longrightarrow D\in S_{x\ast A}.$

But we have that $x\ast A\in A^{+}_{x\mathbf{v}}W_{\mathbf{g}}$ (because
$x\ast A$ and $A^{+}_{x\mathbf{v}}$ are just translates of $A$ and
$A^{+}_{\mathbf{v}}$ respectively, by the same translation of
$\mathbb{Z}\mathfrak{R}^{\vee}$) and $x\mathbf{v}\subset
x\mathscr{C}^{+}_{0}$, so the third point of Lemma 5.25 tells us that

$D\in S_{x\ast A}\Rightarrow D\not\subset\rho^{\vee}+\mathscr{C}^{+}_{0}.$

This fact, combined with (5.10), implies that we have

(5.11) $n_{C,A}=p_{C,A}\qquad\text{for
all}~{}C\subset\rho^{\vee}+\mathscr{C}^{+}_{0}.$

On the other hand, recall the result of Lemma 5.23:

(5.12) $p_{wC,A}(1)=p_{C,A}(1)~{}\forall w\in W_{\mathbf{v}}.$

Thus, for any alcove $C$ included in $\rho^{\vee}+\mathscr{C}^{+}_{0}$, (5.11)
and (5.12) yield

(5.13) $n_{wC,A}(1)=n_{C,A}(1)\qquad\forall w\in W_{\mathbf{v}}~{}\text{such
that}~{}wC\subset\rho^{\vee}+\mathscr{C}^{+}_{0}.$

By construction, there exist elements $w\in W_{\mathbf{v}},~{}r_{0}\in
W_{\mathbf{g}}$ such that $B=wA^{+}_{\mathbf{v}}$ and
$A^{+}_{\mathbf{v}}=Ar_{0}$, so that $B=wAr_{0}$. Since
$B\subset\rho^{\vee}+\mathscr{C}^{+}_{0}$, we can apply (5.13) to get that

$n_{B,A}(1)=n_{Ar_{0},A}(1).$

Finally, since $A$ is maximal in $AW_{\mathbf{g}}$ for $\leq$, we can apply
successively the first point of Proposition 5.4 (where the facet $\mathbf{q}$
from this proposition is our current $\mathbf{g}$, and where the elements
$w\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$, $w^{\prime}\in{{}_{\mathrm{f}}W}$ are
such that $w\mathbf{a}_{1}=A$ and $w^{\prime}\mathbf{a}_{1}=B$) to obtain

$n_{Br,A}(1)=n_{B,A}(1)=n_{Ar_{0},A}(1)=n_{A,A}(1)=1~{}\forall r\in
W_{\mathbf{g}}.$

∎

See Figure 3 for an illustration of the setting of the previous Proposition.

$B$$A^{+}_{\mathbf{v}}$$A$ Figure 3. Proposition 5.28 in type $C_{2}$. The
black dot is $\mathbf{v}$ and the thick black lines represent the affine
hyperplanes fixed by the reflections in $W_{\mathbf{v}}$. The red dots are the
$W_{\mathbf{v}}$-conjugates of $\mathbf{h}$, and the set of gray alcoves
represents $\\{Br,~{}r\in W_{\mathbf{g}}\\}$.

######  Remark 5.29.

Keep the notations of the previous proposition. By positivity of the
coefficients of the anti-spherical Kazdhan-Lusztig polynomials, the fact that
$n_{Br,A}(1)=1$ for all $r\in W_{\mathbf{g}}$ means that the polynomial
$n_{Br,A}$ is always a non-zero monomial.

######  Corollary 5.30.

Let $\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ be a non-special facet,
$\mathbf{h}\in\mathcal{A}^{+}_{\mathbf{g}}$ be such that
$\mathbf{h}\subset\rho^{\vee}+\overline{\mathscr{C}^{+}_{0}}$,
$B\subset\rho^{\vee}+\mathscr{C}^{+}_{0}$ be an alcove such that
$\mathbf{h}\subset\overline{B}$, and let
$\mathbf{v}\subset\rho^{\vee}+\overline{\mathscr{C}^{+}_{0}}$ be a special
facet such that $\mathbf{v}\subset\overline{B}$. If we let $u\in
W_{\mathbf{v}}$ be such that $B^{\prime}:=uB$ is included in
$\rho^{\vee}+\overline{\mathscr{C}^{+}_{0}}$ and denote by
$\mathbf{h}^{\prime}$ the element of $\mathcal{A}^{+}_{\mathbf{g}}$ included
in $\overline{B^{\prime}}$, then we have $w^{\prime}\sim_{\mathbf{g}}w$, where
$w^{\prime},w\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ are such that
$\mathbf{h}=w\mathbf{g}$ and $\mathbf{h}^{\prime}=w^{\prime}\mathbf{g}$.

###### Proof.

Denote by $A$ the alcove which is maximal in
$A^{+}_{\mathbf{v}}W_{\mathbf{g}}$ for $\leq$, and by
$\mathbf{h}^{\prime\prime}$ the element of $\mathcal{A}_{\mathbf{g}}$ included
in $\overline{A}$. Recall that $A\subset\Pi_{\mathbf{v}}$ thanks to
Proposition 5.28, so that in particular
$\mathbf{h}^{\prime\prime}\in\mathcal{A}^{+}_{\mathbf{g}}$. So if we let
$w^{\prime\prime}$ be the element of ${{}_{\mathrm{f}}W}^{\mathbf{g}}$ which
satisfies $w^{\prime\prime}\mathbf{g}=\mathbf{h}^{\prime\prime}$ and $r\in
W_{\mathbf{g}}$ be the element such that $Br$ is maximal in $BW_{\mathbf{g}}$
for $\leq$ (so that $w\mathbf{a}_{1}=Br$ and
$w^{\prime\prime}\mathbf{a}_{1}=A$), then Proposition 5.28 tells us that

$n_{Br,A}(1)=1,$

from which we deduce that $w\sim_{\mathbf{g}}w^{\prime\prime}$ thanks to
Corollary 4.7. The same reasoning (considering this time the alcove
$B^{\prime}$ containing both $\mathbf{h}^{\prime}$ and $\mathbf{v}$ in its
closure) shows that $w^{\prime}\sim_{\mathbf{g}}w^{\prime\prime}$, so finally
$w\sim_{\mathbf{g}}w^{\prime}$ by transitivity. ∎

We can finally complete the proof of Theorem 5.21.

######  Theorem 5.31.

Assume that $\mathfrak{R}^{\vee}$ is irreducible and let
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ be a non-special facet. Then
${{}_{\mathrm{f}}W}^{\mathbf{g}}$ consists of a single class for the
equivalence relation $\sim_{\mathbf{g}}$.

###### Proof.

Thanks to Theorem 5.21, we may and will assume that $\mathbf{g}$ is a point.
With the notation of the proof of Theorem 5.21, recall that we want to show
that $\mathcal{A}^{+}_{\mathbf{g}}$ consists of a single equivalence class for
$\sim_{\mathbf{g}}$. Also recall from the proof of Theorem 5.21 that it is
enough to show that all the elements of $\mathcal{A}^{+}_{\mathbf{g}}$ lying
inside of $\rho^{\vee}+\overline{\mathscr{C}^{+}_{0}}$ are in relation.

Pick $\mathbf{h}$ inside of $\rho^{\vee}+\overline{\mathscr{C}^{+}_{0}}$ and
let $A\subset\rho^{\vee}+\mathscr{C}^{+}_{0}$ be an alcove containing
$\mathbf{h}$ in its closure. Let also $A_{r},A_{r-1},\cdots,A_{1},A_{0}:=A$ be
alcoves as in Lemma 5.19, and denote by $s_{i}\in S$ the reflection such that
$A_{i}s_{i}=A_{i-1}$.

Since $\mathfrak{R}$ is irreducible, there exists a unique simple reflection
$\sigma\in S$ which does not belong to the finite Weyl group $W_{0}$.
Moreover, because $\overline{\mathbf{a}_{1}}$ is a simplex, there is only one
facet inside $\overline{\mathbf{a}_{1}}$ which is a point and which is not
included in the hyperplane fixed by $\sigma$: this is the special facet
$\\{0\\}$. Thus, since $\mathbf{g}$ is not special, it must be included in the
wall fixed by $\sigma$, or in other words we have $\sigma\in W_{\mathbf{g}}$.

For $i\in\llbracket 0,\cdots,r\rrbracket$, denote by $w_{i}$ the element of
${}_{\mathrm{f}}W$ such that $w_{i}\mathbf{a}_{1}=A_{i}$ and put
$\mathbf{h}_{i}:=w_{i}\mathbf{g}$. Notice that
$A_{i}s_{i}=w_{i}s_{i}w_{i}^{-1}A_{i}$ and that
$\mathbf{h}_{i}\subset\overline{A_{i}}$. By construction, we also have

$\mathbf{h}_{i-1}=w_{i}s_{i}w_{i}^{-1}\mathbf{h}_{i}~{}\forall i\in\llbracket
1,\cdots,r\rrbracket~{}\text{and}~{}\mathbf{h}_{0}=\mathbf{h}.$

Fix $i\in\llbracket 1,\cdots,r\rrbracket$. Now, if $s_{i}=\sigma$, then
$w_{i}s_{i}w_{i}^{-1}$ fixes $\mathbf{h}_{i}$, so that
$\mathbf{h}_{i}=\mathbf{h}_{i-1}$. So assume that $s_{i}\neq\sigma$, which
implies that $s_{i}\in W_{0}$, and put $\mathbf{v}_{i}:=w_{i}\\{0\\}$. Then we
see that $\mathbf{v}_{i}$ is a special facet included in $\overline{A_{i}}$,
which thus satisfies the inclusion
$\mathbf{v}_{i}\subset\rho^{\vee}+\overline{\mathscr{C}^{+}_{0}}$, with
$w_{i}s_{i}w_{i}^{-1}\in W_{\mathbf{v}_{i}}$. Therefore we may apply Corollary
5.30 to see that $\mathbf{h}_{i}\sim_{\mathbf{g}}\mathbf{h}_{i-1}$. By
transitivity of the relation we get
$\mathbf{h}\sim_{\mathbf{g}}\mathbf{h}_{r}$, concluding the proof. ∎

######  Remark 5.32.

Take back the notations of the previous proof. Corollary 5.30 (when
$\mathbf{g}$ is a point) and Lemma 5.20 (when $\mathbf{g}$ is not a point)
actually allow us to be a bit more precise: for all $i\in\llbracket
1,\cdots,r\rrbracket$, there exists an element
$u_{i}\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ (which is the maximal element in
$W_{\mathbf{v}_{i}}w_{i}$ in Corollary 5.30, and the maximal element in
$w_{i}W_{\mathbf{q}_{i}}$ in Lemma 5.20) such that $n_{w_{i},u_{i}}(1)\neq 0$
and $n_{w_{i-1},u_{i}}(1)\neq 0$.

One of the advantages of our proof is that it gives an explicit way of linking
an element $w\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ to the unique element $v$ of
${{}_{\mathrm{f}}W}^{\mathbf{g}}$ satisfying
$v\mathbf{g}\subset\rho^{\vee}+\overline{\mathbf{a}_{1}}$. This allows us to
give the following result.

######  Corollary 5.33.

Assume that $\mathfrak{R}^{\vee}$ is irreducible. Let
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ be a non-special facet and
$w,w^{\prime}$ be elements of ${{}_{\mathrm{f}}W}^{\mathbf{g}}$ which are in
the same equivalence class for $\sim_{\mathbf{g}}$. Denote by $A_{w}$ (resp.
$A_{w^{\prime}}$) the alcove which contains $w\mathbf{g}$ (resp.
$w^{\prime}\mathbf{g}$) in its closure and which is minimal in
$A_{w}W_{\mathbf{g}}$ (resp. $A_{w^{\prime}}W_{\mathbf{g}}$) for $\preceq$.
Then there exists a positive integer $s$ and a chain of elements of
${{}_{\mathrm{f}}W}^{\mathbf{g}}$

$w_{s}=w,w_{s-1},\cdots,w_{0}=w^{\prime}$

such that, for all $i\in\llbracket 0,s-1\rrbracket$, there exists an element
$u_{i}\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ satisfying

${n}_{w_{i},u_{i}}(1)\neq 0~{}\text{and}~{}{n}_{w_{i+1},u_{i}}(1)\neq 0,$

and such that

$s\leq 2+d(\hat{A}_{w}-\rho^{\vee})+d(\hat{A}_{w^{\prime}}-\rho^{\vee}).$

###### Proof.

Denote by $v$ the unique element of ${{}_{\mathrm{f}}W}^{\mathbf{g}}$ such
that $v\mathbf{g}\subset\rho^{\vee}+\overline{\mathbf{a}_{1}}$. Let us also
denote by $\hat{w}$ the element of ${{}_{\mathrm{f}}W}^{\mathbf{g}}$ such that
$\hat{w}\mathbf{g}\subset\overline{\hat{A}_{w}}$ and
$A_{0}:=\hat{A}_{w},A_{r-1},\cdots,A_{r}:=\rho^{\vee}+\mathbf{a}_{1}$ be the
alcoves of Lemma 5.19. By Remark 5.32, we know that there exist elements
$w_{i},u_{i}\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ such that
$(w_{r},w_{0})=(v,\hat{w})$ and

$w_{i}\mathbf{g}\subset\overline{A_{i}},~{}n_{w_{i},u_{i}}(1)\neq
0~{}\text{and}~{}n_{w_{i+1},u_{i}}(1)\neq 0,~{}\forall i\in\llbracket
0,r-1\rrbracket.$

Therefore, the chain $w,\hat{w},w_{1},\cdots,w_{r}=v$ linking $w$ to $v$ is of
length equal to $1$ plus the number of hyperplanes separating $\hat{A}_{w}$
from $\mathbf{a}_{1}+\rho^{\vee}$, and this last number is equal to the number
of hyperplanes separating $\hat{A}_{w}-\rho^{\vee}$ from $\mathbf{a}_{1}$,
i.e. to $d(\hat{A}_{w}-\rho^{\vee})$. Repeating the same process for
$w^{\prime}$, we find a chain linking $w^{\prime}$ to $v$ of length equal to
$1+d(\hat{A}_{w^{\prime}}-\rho^{\vee})$, so that the concatenation of these
two chains yields the desired result. ∎

##  6\. Smith-Treumann theory and consequences

###  6.1. Geometric Satake equivalence

The affine Grassmanian $\mathrm{Gr}$ associated with $G$ can be defined as the
fppf-quotient $LG/L^{+}G$, which can be shown to be an ind-projective ind-
scheme over $\mathbb{F}$. In particular, $\mathrm{Gr}$ coincides with the
partial affine flag variety $\mathrm{Fl}_{\\{0\\}}$ (cf. subsection 3.2). For
any $\lambda\in\mathbb{X}^{\vee}$, we denote by $z^{\lambda}\in
T(\mathcal{O})$ the image of $z$ by
$\lambda:\mathbb{G}_{\mathrm{m}}(\mathcal{K})\to T(\mathcal{K})$, and define
$[\lambda]:=z^{\lambda}L^{+}G$. The orbit $L^{+}G\cdot[\lambda]$ will be
denoted $X_{\lambda}$. Using the Cartan decomposition, one can show that the
action of $L^{+}G$ on $\mathrm{Gr}$ induces the following equality

$(\mathrm{Gr})_{\mathrm{red}}=\bigsqcup_{\lambda\in\mathbb{X}^{\vee}_{+}}X_{\lambda},$

where $(\mathrm{Gr})_{\mathrm{red}}$ is the reduced ind-scheme associated with
$\mathrm{Gr}$. Moreover, each orbit $X_{\lambda}$ is a smooth
$\mathbb{F}$-scheme of finite type and

$\overline{X_{\lambda}}=\bigsqcup_{\begin{subarray}{c}\mu\leq\lambda\\\
\mu\in\mathbb{X}^{\vee}_{+}\end{subarray}}X_{\mu}$

is a projective $\mathbb{F}$-scheme, where $\mu\leq\lambda$ means as usual
that $\lambda-\mu$ is a sum of positive coroots.

For any $X$ which is a locally closed finite union of $L^{+}G$-orbits, one can
show that the $L^{+}G$-action on $X$ factors through a quotient group of
finite type $J$ (cf. [Cia21, Lemma 3.2]); this quotient may and will be chosen
so that the kernel of the map $L^{+}G\to J$ is contained in
$\mathrm{ker}(L^{+}G\to G)$. We can then consider the $J$-equivariant derived
category of constructible étale $\mathbf{k}$-sheaves
$D^{b}_{J}(X,\mathbf{k})$, which does not depend on our choice of $J$ (see
[Cia21, §3.3]). We are now allowed to define the category
$D^{b}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$ as the direct limit of the categories
$D^{b}_{J}(X,\mathbf{k})$, indexed by finite and closed unions of orbits $X$
which are ordered by inclusion and where the transition maps are given by
direct images. This category admits a canonical perverse $t$-structure, and we
denote by $\mathrm{Perv}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$ its heart.

For $\lambda\in\mathbb{X}^{\vee}_{+}$, let
$i^{\lambda}:X_{\lambda}\hookrightarrow\mathrm{Gr}$ denote the inclusion and
define

$\Delta^{\mathrm{sph}}_{\lambda}:={{}^{p}\mathrm{H}}^{0}(i^{\lambda}_{!}\underline{\mathbf{k}}_{X_{\lambda}}[\mathrm{dim}(X_{\lambda})]),\qquad\nabla^{\mathrm{sph}}_{\lambda}:={{}^{p}\mathrm{H}}^{0}(i^{\lambda}_{*}\underline{\mathbf{k}}_{X_{\lambda}}[\mathrm{dim}(X_{\lambda})]).$

The complex $\Delta^{\mathrm{sph}}_{\lambda}$, resp.
$\nabla^{\mathrm{sph}}_{\lambda}$, is called the standard perverse sheaf, resp
the costandard perverse sheaf attached to $\lambda$. By construction, there is
a canonical map
$\Delta^{\mathrm{sph}}_{\lambda}\to\nabla^{\mathrm{sph}}_{\lambda}$, and we
define the intersection cohomology complex
$\mathrm{IC}_{\lambda}^{\mathrm{sph}}$ associated with $\lambda$ as the image
of this map. Intersection cohomology complexes cover (up to isomorphism) all
of the simple objects of the category
$\mathrm{Perv}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$ when $\lambda$ runs through
$\mathbb{X}^{\vee}_{+}$.

######  Proposition 6.1 (Proposition 12.4, [BR18]).

The category $\mathrm{Perv}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$ is a highest
weight category, with weight poset $\mathbb{X}^{\vee}_{+}$, standard objects
$\\{\Delta^{\mathrm{sph}}_{\lambda},~{}\lambda\in\mathbb{X}^{\vee}_{+}\\}$ and
costandard objects
$\\{\nabla^{\mathrm{sph}}_{\lambda},~{}\lambda\in\mathbb{X}^{\vee}_{+}\\}$.

The category $D^{b}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$ admits a monoidal
product $\star$, which restricts to a monoidal product on
$\mathrm{Perv}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$. In the following crucial
result, we denote by $G^{\vee}$ the Langlands dual group of $G$ over
$\mathbf{k}$, and by $\mathrm{Rep}_{\mathbf{k}}(G^{\vee})$ the category of
algebraic finite dimensional $\mathbf{k}$-representations of $G^{\vee}$.

######  Theorem 6.2 (Theorem 14.1, [MV04]).

There is an equivalence of monoidal categories

$(\mathrm{Perv}_{L^{+}G}(\mathrm{Gr},\mathbf{k}),\star)\xrightarrow{\sim}(\mathrm{Rep}_{\mathbf{k}}(G^{\vee}),\otimes_{\mathbf{k}}).$

######  Remark 6.3.

Let $\lambda\in\mathbb{X}^{\vee}_{+}$. It is well known that the category
$\mathrm{Rep}_{\mathbf{k}}(G^{\vee})$ also admits a highest weight structure,
and we denote by $\Delta_{\lambda}$, resp. $\nabla_{\lambda}$, resp.
$L_{\lambda}$, the standard object, resp. the costandard object, resp. the
simple module, associated with $\lambda$. Then, the previous equivalence of
categories sends $\nabla_{\lambda}^{\mathrm{sph}}$, resp.
$\Delta_{\lambda}^{\mathrm{sph}}$, resp.
$\mathrm{IC}_{\lambda}^{\mathrm{sph}}$, on $\nabla_{\lambda}$, resp.
$\Delta_{\lambda}$, resp. $L_{\lambda}$ (cf. [MV04, Proposition 13.1]). We
will say that this equivalence is an equivalence of highest weight categories.

###  6.2. Iwahori-Whittaker variant

Recall the construction of $D^{b}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$ from
subsection 3.3. If we denote by $Y_{\lambda}$ the orbit of $[\lambda]$ under
the action of $\mathrm{Iw}_{u}^{+}$, then we have a decomposition

$(\mathrm{Gr})_{\mathrm{red}}=\bigsqcup_{\lambda\in\mathbb{X}^{\vee}}Y_{\lambda},$

where each $Y_{\lambda}$ is an affine space over $\mathbb{F}$. One can show
that an orbit $Y_{\lambda}$ supports a non-zero Iwahori-Whittaker local system
iff $\lambda$ is strictly dominant, and that in this case there exists exactly
one (up to isomorphism) such local system of rank one on $Y_{\lambda}$, which
we will denote by $\mathcal{L}^{\lambda}_{\mathrm{AS}}$.

Once again, the category $D^{b}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$ admits
a canonical perverse $t$-structure, and we will denote by
$\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$ its heart. Thanks to
the fact that the $\mathrm{Iw}_{u}^{+}$-orbits are affine spaces over
$\mathbb{F}$, this category of perverse sheaves admits a transparent highest
weight structure (cf. [BGM+19, Corollary 3.6]). Namely, the weight poset is
given by $\mathbb{X}^{\vee}_{++}$, and the standard, costandard and simple
objects associated with some $\lambda\in\mathbb{X}^{\vee}_{++}$ are
respectively given by

$\Delta^{\mathcal{IW}}_{\lambda}:=j^{\lambda}_{!}\mathcal{L}^{\lambda}_{\mathrm{AS}}[\mathrm{dim}(Y_{\lambda})],\qquad\nabla^{\mathcal{IW}}_{\lambda}:=j^{\lambda}_{*}\mathcal{L}^{\lambda}_{\mathrm{AS}}[\mathrm{dim}(Y_{\lambda})],\qquad\mathrm{IC}_{\lambda}^{\mathcal{IW}},$

where $j^{\lambda}:Y_{\lambda}\hookrightarrow\mathrm{Gr}$ is the inclusion,
and $\mathrm{IC}_{\lambda}^{\mathcal{IW}}$ is obtained as the image of the
canonical morphism
$\Delta^{\mathcal{IW}}_{\lambda}\to\nabla^{\mathcal{IW}}_{\lambda}$. In the
sequel, we will denote by
$\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$ the category of tilting
objects associated with this highest weight category, and by
$\mathscr{T}^{\mathcal{IW}}_{\lambda}$ the indecomposable tilting object of
highest weight $\lambda\in\mathbb{X}^{\vee}_{++}$.

Although the category $D^{b}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$ is not
endowed with a canonical convolution product making it a monoidal category, it
admits a right action of the monoidal category
$D^{b}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$. The following result was found by
the authors of [BGM+19], and gives another incarnation of the category
$\mathrm{Rep}_{\mathbf{k}}(G^{\vee})$. Note that by our assumptions on $G$,
the element $\rho^{\vee}$, defined as the half-sum of positive coroots,
belongs to $\mathbb{X}^{\vee}$.

######  Theorem 6.4.

The functor
$\mathcal{F}\mapsto\nabla^{\mathcal{IW}}_{\rho^{\vee}}\star\mathcal{F}$
induces an equivalence of highest weight categories

$\mathrm{Perv}_{L^{+}G}(\mathrm{Gr},\mathbf{k})\xrightarrow{\sim}\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k}).$

######  Remark 6.5.

At this stage, we can note that the action of $\mathbb{G}_{\mathrm{m}}$ on
$\mathrm{Gr}$ by rescaling $z$ stabilizes each $\mathrm{Iw}_{u}^{+}$-orbit.
This allows us to consider the loop rotation equivariant Iwahori-Whittaker
derived category of $\mathbf{k}$-sheaves

$D^{b}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}(\mathrm{Gr},\mathbf{k}),$

which comes with a natural $t$-exact forgetful functor to
$D^{b}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$ (cf. [RW20, §5.2]). It is then
not difficult to show (cf. [RW20, Lemma 5.2]) that the forgetful functor

(6.1)
$\mathrm{Perv}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}(\mathrm{Gr},\mathbf{k})\to\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$

is an equivalence of categories. These considerations will become useful in
section 6.

###  6.3. Fixed points of the affine Grassmannian and connected components

As in subsection 3.1, we fix an integer $n\geq 1$. We denote by
$\mathbf{\mu}_{n}$ the finite group scheme of $n$-th roots of unity, which
acts on $L^{+}G$ and $LG$ by rescaling the indeterminate $z$; in particular,
$\mu_{n}$ acts on $\mathrm{Gr}$. The following fact, which will only be used
with $n=\ell$, is one of the fundamental tools used by the authors of [RW20]
(cf. [RW20, Proposition 4.7]). For any
$\lambda\in\overline{\mathbf{a}_{n}}\cap\mathbb{X}^{\vee}$, we will denote by
$\mathbf{g}_{\lambda}\subset\overline{\mathbf{a}_{n}}$ the facet which
contains $\lambda$.

######  Proposition 6.6.

For any $\lambda\in\overline{\mathbf{a}_{n}}\cap\mathbb{X}^{\vee}$, the map
$g\mapsto g\cdot[\lambda]$ factors through an open and closed embedding

$\mathrm{Fl}^{n,\circ}_{\mathbf{g}_{\lambda}}\hookrightarrow(\mathrm{Gr})^{\mu_{n}}$

and the induced map

$\bigsqcup_{\lambda\in\overline{\mathbf{a}_{n}}\cap\mathbb{X}^{\vee}}\mathrm{Fl}^{n,\circ}_{\mathbf{g}_{\lambda}}\to(\mathrm{Gr})^{\mu_{n}}$

is an isomorphism of ind-schemes.

If we denote by $\mathrm{Iw}_{u,\ell}^{+}$ the inverse image of $U^{+}$ under
the evaluation map $L_{\ell}^{+}G\to G,~{}z^{\ell}\mapsto 0$, then we get
$(\mathrm{Iw}_{u}^{+})^{\mu_{\ell}}=\mathrm{Iw}_{u,\ell}^{+}$, and the orbits
of $(\mathrm{Gr})^{\mu_{\ell}}$ under the action of $\mathrm{Iw}_{u,\ell}^{+}$
are still parametrized by $\mathbb{X}^{\vee}$ (cf. [RW20, Lemma 4.8]). Thus,
using the morphism $\mathrm{Iw}_{u,\ell}^{+}\to\mathbb{G}_{\mathrm{a}}$
induced by $\chi$, we can take back the constructions of subsection 6.2 to
define the category $D^{b}_{\mathcal{IW}_{\ell}}(Y,\mathbf{k})$, where
$Y\subset(\mathrm{Gr})^{\mu_{\ell}}$ is a finite locally closed union of
$\mathrm{Iw}_{u,\ell}^{+}$-orbits. The theory of parity complexes also adapts
to the present context: a complex $\mathcal{F}\in
D^{b}_{\mathcal{IW}_{\ell}}(Y,\mathbf{k})$ is called $*$-even, resp. $!$-even,
if for any131313Notice that, if $\lambda\notin\mathbb{X}^{\vee}_{++}$, the
restriction and co-restriction of $\mathcal{F}$ to
$(Y_{\lambda})^{\mu_{\ell}}$ is zero (because this orbit supports an Iwahori-
Whittaker local system only when $\lambda\in\mathbb{X}^{\vee}_{++}$), so that
we can restrict ourselves to the case where
$\lambda\in\mathbb{X}^{\vee}_{++}$. $\lambda\in\mathbb{X}^{\vee}_{++}$ such
that $(Y_{\lambda})^{\mu_{\ell}}\subset Y$ the complex
$(j_{\mu_{\ell}}^{\lambda})^{*}\mathcal{F}$, resp.
$(j_{\mu_{\ell}}^{\lambda})^{!}\mathcal{F}$, is concentrated in even degrees,
where $j_{\mu_{\ell}}^{\lambda}:(Y_{\lambda})^{\mu_{\ell}}\hookrightarrow Y$
is the inclusion. The definition for $*$-odd and $!$-odd complexes is similar
and one says that a complex is even, resp. odd, if it is both $*$-even and
$!$-even, resp. $*$-odd and $!$-odd. As usual, a complex is called parity if
it is a direct sum of even and odd complexes.

Defining the category
$D^{b}_{\mathcal{IW}_{\ell}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})$ (resp.
$D^{b}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\mathbf{g}},\mathbf{k})$
for a facet $\mathbf{g}\subset\overline{\mathbf{a}_{\ell}}$) as a direct
limit, it then makes sense to talk about even and odd objects in this
category, and we will denote by
$\mathrm{Par}_{\mathcal{IW}_{\ell}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})$
(resp.
$\mathrm{Par}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\mathbf{g}},\mathbf{k})$)
the additive full subcategory consisting of parity objects. By Proposition
6.6, it is clear that the additive category
$\mathrm{Par}_{\mathcal{IW}_{\ell}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})$
splits into a direct sum of subcategories of the form
$\mathrm{Par}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\mathbf{g}},\mathbf{k})$,
where $\mathbf{g}$ runs through the facets inside
$\overline{\mathbf{a}_{\ell}}$.

It is important to note that we have a bijection
$\mathbf{g}\mapsto\ell\cdot\mathbf{g}$ between facets inside
$\overline{\mathbf{a}_{1}}$ associated with the action $\mbox{$\Box$}_{1}$ of
$W$ on $E$ and facets inside $\overline{\mathbf{a}_{\ell}}$ associated with
the action $\mbox{$\Box$}_{\ell}$ of $W$ on $E$, and that a simple change of
variable (namely, replacing $z$ with $z^{\ell}$) induces canonical
isomorphisms of ind-schemes
$\mathrm{Fl}^{\circ}_{\mathbf{g}}:=\mathrm{Fl}^{1,\circ}_{\mathbf{g}}\simeq\mathrm{Fl}^{\ell,\circ}_{\ell\cdot\mathbf{g}}$,
$\mathrm{Iw}_{u}^{+}\simeq\mathrm{Iw}_{u,\ell}^{+}$, sending
$\mathrm{Iw}_{u}^{+}$-orbits in $\mathrm{Fl}^{\circ}_{\mathbf{g}}$ to
$\mathrm{Iw}_{u,\ell}^{+}$-orbits in
$\mathrm{Fl}^{\ell,\circ}_{\ell\cdot\mathbf{g}}$. For any facet
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$, we thus have a canonical
equivalence of categories

$D^{b}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})\simeq
D^{b}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\ell\cdot\mathbf{g}},\mathbf{k}),$

restricting to an equivalence between the corresponding categories of parity
complexes:

(6.2)
$\mathrm{Par}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})\simeq\mathrm{Par}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\ell\cdot\mathbf{g}},\mathbf{k}).$

In the sequel, we will denote by $\mathcal{E}_{\ell,w}^{\ell\cdot\mathbf{g}}$
the indecomposable parity complex in
$\mathrm{Par}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\ell\cdot\mathbf{g}},\mathbf{k})$
corresponding to $\mathcal{E}^{\mathbf{g}}_{w}$ via the equivalence (6.2). It
is clear that, up to a shift, all of the indecomposable parity complexes in
$\mathrm{Par}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\ell\cdot\mathbf{g}},\mathbf{k})$
(coming from Proposition 3.1) arise in this way.

###  6.4. Smith category and the linkage principle

Since the action of $\mathbb{G}_{\mathrm{m}}$ on $\mathrm{Gr}$ stabilizes the
fixed points $(\mathrm{Gr})^{\mu_{\ell}}$, we can take back the construction
recalled in Remark 6.5 to define the category
$D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(Y,\mathbf{k})$, where
$Y\subset(\mathrm{Gr})^{\mu_{\ell}}$ is a locally closed finite union of
$\mathrm{Iw}_{u,\ell}^{+}$-orbits. We will denote by
$D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(Y,\mathbf{k})_{\mu_{\ell}-\text{perf}}$
the full subcategory whose objects are the $\mathcal{F}$ for which the object
$\mathrm{Res}^{\mathbb{G}_{\mathrm{m}}}_{\mu_{\ell}}(\mathcal{F})$ has perfect
geometric stalks in the sense of [RW20, §3.3]. The Smith category
$\mathrm{Sm}_{\mathcal{IW}}(Y,\mathbf{k})$ on $Y$ is by definition the Verdier
quotient

$D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(Y,\mathbf{k})/D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(Y,\mathbf{k})_{\mu_{\ell}-\text{perf}}.$

Let $X\subset\mathrm{Gr}$ be a locally closed finite union of
$\mathrm{Iw}_{u}^{+}$-orbits, and $i_{X}:(X)^{\mu_{\ell}}\hookrightarrow X$
denote the inclusion. We define the functor

$i_{X}^{!*}:\mathrm{Perv}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}(X,\mathbf{k})\to\mathrm{Sm}_{\mathcal{IW}}((X)^{\mu_{\ell}},\mathbf{k})$

as the composition of the inverse image $i_{X}^{*}$ with the canonical
quotient map

$Q:D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}((X)^{\mu_{\ell}},\mathbf{k})\to\mathrm{Sm}_{\mathcal{IW}}((X)^{\mu_{\ell}},\mathbf{k}).$

A crucial result says that taking $i_{X}^{!}$ instead of $i_{X}^{*}$ in the
previous construction gives isomorphic functors (cf. [RW20, §6.2]), whence the
notation. It then takes a bit more work (cf. [RW20, Lemma 6.1]) to prove that,
for two locally closed finite unions of $\mathrm{Iw}_{u,\ell}^{+}$-orbits
$Z\subset Y$, the canonical functor
$f^{\mathrm{Sm}}_{*}:\mathrm{Sm}_{\mathcal{IW}}(Z,\mathbf{k})\to\mathrm{Sm}_{\mathcal{IW}}(Y,\mathbf{k})$
induced by the direct image in the corresponding derived categories is fully
faithful, and fits into the following commutative diagram

$\textstyle{D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(Z,\mathbf{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(Y,\mathbf{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sm}_{\mathcal{IW}}(Z,\mathbf{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\mathrm{Sm}}_{*}}$$\textstyle{\mathrm{Sm}_{\mathcal{IW}}(Y,\mathbf{k}),}$

where the vertical arrows are the quotient maps. One can thus define the
category $\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})$
as a direct limit indexed by finite closed unions of
$\mathrm{Iw}_{u,\ell}^{+}$-orbits, and consider the functor

$i^{!*}_{\mathrm{Gr}}:\mathrm{Perv}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}(\mathrm{Gr},\mathbf{k})\to\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k}).$

The following statement is [RW20, Theorem 7.4].

######  Theorem 6.7.

The composition of functors

$\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})\xrightarrow{\text{Remark}~{}\ref{rem1}}\mathrm{Perv}_{\mathcal{IW},\mathbb{G}_{\mathrm{m}}}(\mathrm{Gr},\mathbf{k})\xrightarrow{i^{!*}_{\mathrm{Gr}}}\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})$

restricts to a fully faithful functor
$\Phi:\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})\to\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})$.

We can now reformulate the proof of the linkage principle from [RW20, Theorem
8.5]. Thanks to the decomposition of $(\mathrm{Gr})^{\mu_{\ell}}$ into its
connected components from Proposition 6.6, we deduce from Theorem 6.7 above
that two strictly dominant weights $\lambda$ and $\mu$ which are not in the
same orbit for the box action cannot be in relation for $\mathscr{R}_{2}$
(seen as a relation on the weight poset $\mathbb{X}^{\vee}_{++}$ of the
highest weight category
$\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$, cf. section 2).
Indeed, let $\delta$, resp. $\gamma$, be the element of
$W\mbox{$\Box$}_{\ell}\lambda\cap\overline{\mathbf{a}_{\ell}}$, resp. of
$W\mbox{$\Box$}_{\ell}\lambda\cap\overline{\mathbf{a}_{\ell}}$. We have

$\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}_{\lambda}^{\mathcal{IW}},\mathscr{T}_{\mu}^{\mathcal{IW}})\simeq\mathrm{Hom}_{\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})}(\Phi(\mathscr{T}_{\lambda}^{\mathcal{IW}}),\Phi(\mathscr{T}_{\mu}^{\mathcal{IW}})),$

and since the object $\Phi(\mathscr{T}_{\lambda}^{\mathcal{IW}})$, resp.
$\Phi(\mathscr{T}_{\mu}^{\mathcal{IW}})$, is indecomposable (thanks to the
full faithfulness of $\Phi$), its support must be contained in a single
connected component of $(\mathrm{Gr})^{\mu_{\ell}}$, which is easily seen to
be the parametrized by $\delta$, resp. $\gamma$, in the isomorphism of
Proposition 6.6 applied to the case $n=\ell$.

Thanks to Theorem 6.4, this implies that two dominant weights
$\lambda^{\prime},\mu^{\prime}$ which are not in the same orbit for the dot
action cannot be in relation for $\mathscr{R}_{2}$ (where we now consider the
highest weight category $\mathrm{Perv}_{L^{+}G}(\mathrm{Gr},\mathbf{k})$).
Finally, the geometric Satake equivalence (Theorem 6.2) and Theorem 2.3 allow
to prove the linkage principle for $G^{\vee}$ (cf. [Jan03, Corollary 6.17,
Part II]):

(6.3)
$\forall\lambda^{\prime},\mu^{\prime}\in\mathbb{X}^{\vee}_{+}~{}\text{such
that}~{}W\bullet_{\ell}\lambda^{\prime}\neq
W\bullet_{\ell}\mu^{\prime},~{}\text{we
have}~{}\mathrm{Ext}_{G^{\vee}}^{1}(L_{\lambda^{\prime}},L_{\mu^{\prime}})=0.$

###  6.5. Consequences on equivalence relations

Recall the definition of the “dot” action of $W$, which acts on
$\mathbb{X}^{\vee}$ by

$w\bullet_{\ell}\mu:=w\mbox{$\Box$}_{\ell}(\mu+\rho^{\vee})-\rho^{\vee},$

for any $w\in W$, $\mu\in\mathbb{X}^{\vee}$. Let
$\mu\in\mathbb{X}^{\vee}\cap\overline{\mathbf{a}_{\ell}}$ and
$\mathbf{g}_{\mu}$ be the facet (for $\mbox{$\Box$}_{\ell}$) containing $\mu$.
Therefore, $\ell^{-1}\cdot\mathbf{g}_{\mu}\subset\overline{\mathbf{a}_{1}}$ is
a facet for $\mbox{$\Box$}_{1}$. The assignment $w\mapsto
w\mbox{$\Box$}_{\ell}\mu$, resp. $w\mapsto w\bullet_{\ell}(\mu-\rho^{\vee})$,
induces a bijection

(6.4)
${{}_{\mathrm{f}}W}^{\ell^{-1}\cdot\mathbf{g}_{\mu}}\xrightarrow{\sim}W\mbox{$\Box$}_{\ell}\mu\cap\mathbb{X}^{\vee}_{++},~{}\text{resp.}~{}{{}_{\mathrm{f}}W}^{\ell^{-1}\cdot\mathbf{g}_{\mu}}\xrightarrow{\sim}W\bullet_{\ell}(\mu-\rho^{\vee})\cap\mathbb{X}^{\vee}_{+}$

(this follows from the fact that, if we denote by
$W_{\mathbf{g}_{\mu},\mbox{$\Box$}_{\ell}}\subset W$ the stabilizer of
$\mathbf{g}_{\mu}$ for $\mbox{$\Box$}_{\ell}$, then we have an equality
$W_{\mathbf{g}_{\mu},\mbox{$\Box$}_{\ell}}=W_{\ell^{-1}\cdot\mathbf{g}_{\mu}}$).

In the sequel, we will want to determine an exact description of the blocks in
$\mathbb{X}^{\vee}_{++}$ (seen as the weight poset of
$\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$) for $\mathscr{R}_{2}$
(cf. subsection 2.2). The following results will help us to do so (cf. the
proof of [RW20, Theorem 8.9]).

######  Proposition 6.8.

Let $\mu\in\mathbb{X}^{\vee}\cap\overline{\mathbf{a}_{\ell}}$,
$\mathbf{g}_{\mu}$ be the facet (for $\mbox{$\Box$}_{\ell}$) containing $\mu$,
and $w,w^{\prime}$ be elements of
${{}_{\mathrm{f}}W}^{\ell^{-1}\cdot\mathbf{g}_{\mu}}$. We have an isomorphism

$\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{w\mbox{$\Box$}_{\ell}\mu},\mathscr{T}^{\mathcal{IW}}_{w^{\prime}\mbox{$\Box$}_{\ell}\mu})\simeq\mathrm{Hom}^{\bullet}_{\mathrm{Par}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\mathbf{g}_{\mu}},\mathbf{k})}(\mathcal{E}_{\ell,w}^{\mathbf{g}_{\mu}},\mathcal{E}_{\ell,w^{\prime}}^{\mathbf{g}_{\mu}}).$

###### Proof.

This is a direct consequence of [RW20, Proposition 8.11] and of the
equivalence of highest weight categories
$\mathrm{Rep}_{\mathbf{k}}(G^{\vee})\simeq\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$,
which sends the indecomposable tilting module of highest weight
$w\bullet_{\ell}(\mu-\rho^{\vee})$, resp.
$w^{\prime}\bullet_{\ell}(\mu-\rho^{\vee})$, to
$\mathscr{T}^{\mathcal{IW}}_{w\mbox{$\Box$}_{\ell}\mu}$, resp.
$\mathscr{T}^{\mathcal{IW}}_{w^{\prime}\mbox{$\Box$}_{\ell}\mu}$. ∎

Let $\lambda\in\overline{\mathbf{a}_{\ell}}\cap\mathbb{X}^{\vee}$. Note that,
thanks to the equivalence (6.2), we have

$\mathrm{Hom}^{\bullet}_{\mathrm{Par}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\ell^{-1}\cdot\mathbf{g}_{\lambda}},\mathbf{k})}(\mathcal{E}^{\ell^{-1}\cdot\mathbf{g}_{\lambda}}_{w},\mathcal{E}^{\ell^{-1}\cdot\mathbf{g}_{\lambda}}_{w^{\prime}})\simeq\mathrm{Hom}^{\bullet}_{\mathrm{Par}_{\mathcal{IW}_{\ell}}(\mathrm{Fl}^{\ell,\circ}_{\mathbf{g}_{\lambda}},\mathbf{k})}(\mathcal{E}^{\mathbf{g}_{\lambda}}_{\ell,w},\mathcal{E}^{\mathbf{g}_{\lambda}}_{\ell,w^{\prime}}).$

Therefore Proposition 6.8 yields

(6.5) $\forall
w,w^{\prime}\in{{}_{\mathrm{f}}W}^{\ell^{-1}\cdot\mathbf{g}_{\lambda}},\qquad
w\mathscr{R}_{\ell^{-1}\cdot\mathbf{g}_{\lambda}}w^{\prime}\Longleftrightarrow(w\mbox{$\Box$}_{\ell}\lambda)\mathscr{R}_{2}(w^{\prime}\mbox{$\Box$}_{\ell}\lambda).$

###  6.6. Dilating weights by $\ell$

Smith-Treumann theory allows us to understand the effect of dilating dominant
weights by $\ell$ on the equivalence relation $\sim$.

######  Proposition 6.9.

Let $\lambda,\mu\in\mathbb{X}^{\vee}_{++}$. We have an isomorphism

$\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda},\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu})\simeq\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{\lambda},\mathscr{T}^{\mathcal{IW}}_{\mu}).$

###### Proof.

First recall that, by full-faithfulness of $\Phi$, we have an isomorphism

(6.6)
$\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda},\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu})\simeq\mathrm{Hom}_{\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})}(\Phi(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda}),\Phi(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu})).$

Next, notice that we have an embedding

$L_{\ell}G/L^{+}_{\ell}G\hookrightarrow(\mathrm{Gr})^{\mu_{\ell}}$

thanks to [RW20, Remark 4.8], and that the left-hand side identifies (via
Proposition 6.6) with the union of the connected components
$\mathrm{Fl}^{\ell,\circ}_{\mathbf{g}_{\lambda}}$, with $\lambda$ running
through $\overline{\mathbf{a}_{\ell}}\cap\ell\cdot\mathbb{X}^{\vee}$.
Moreover, the $L^{+}_{\ell}G$-orbits in $L_{\ell}G/L^{+}_{\ell}G$ are
parametrized by $\ell\cdot\mathbb{X}_{+}^{\vee}$, and a simple change of
variable $z\mapsto z^{\ell}$ together with (6.1) induce an equivalence of
categories

$R:\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})\simeq\mathrm{Tilt}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(L_{\ell}G/L^{+}_{\ell}G,\mathbf{k}),$

which sends $\mathscr{T}^{\mathcal{IW}}_{\lambda}$ (resp.
$\mathscr{T}^{\mathcal{IW}}_{\mu}$) to the indecomposable tilting object
associated with $\ell\cdot\lambda$ (resp. $\ell\cdot\mu$). Therefore, the
indecomposable tilting objects in
$\mathrm{Tilt}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}(L_{\ell}G/L^{+}_{\ell}G,\mathbf{k})$
are parity complexes (see [BGM+19, Proposition 4.12]), so that we can take
back the same arguments as in the proof of [RW20, Theorem 7.4] to show that
restricting the functor

$Q:D^{b}_{\mathcal{IW}_{\ell},\mathbb{G}_{\mathrm{m}}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})\to\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})$

to $L_{\ell}G/L^{+}_{\ell}G$ and composing it with $R$ yields a fully-faithful
functor

$Q\circ
R:\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})\to\mathrm{Sm}_{\mathcal{IW}}(L_{\ell}G/L^{+}_{\ell}G,\mathbf{k}),$

which satisfies the following isomorphisms:

$Q\circ
R(\mathscr{T}^{\mathcal{IW}}_{\lambda})\simeq\Phi(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda})~{}\text{and}~{}Q\circ
R(\mathscr{T}^{\mathcal{IW}}_{\mu})\simeq\Phi(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu}).$

The desired result thus follows from comparing (6.6) with the following
isomorphism

$\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{\lambda},\mathscr{T}^{\mathcal{IW}}_{\mu})\simeq\mathrm{Hom}_{\mathrm{Sm}_{\mathcal{IW}}((\mathrm{Gr})^{\mu_{\ell}},\mathbf{k})}(Q\circ
R(\mathscr{T}^{\mathcal{IW}}_{\lambda}),Q\circ
R(\mathscr{T}^{\mathcal{IW}}_{\mu})).$

∎

######  Corollary 6.10.

For any $\gamma\in\mathbb{X}^{\vee}_{++}$, denote by $\overline{\gamma}$ the
equivalence class of $\gamma$ for the equivalence relation $\sim$ on
$\mathbb{X}^{\vee}_{++}$ (seen as the weight poset of
$\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$). Then we have

$\forall\lambda\in\mathbb{X}^{\vee}_{++},\quad\ell\cdot\overline{\lambda}\subset\overline{\ell\cdot\lambda}.$

###### Proof.

This is a direct consequence of Proposition 6.9 and of the fact that $\sim$ is
generated by $\mathscr{R}_{2}$ (cf. section 2). ∎

######  Corollary 6.11.

Let $\lambda,\mu\in\mathbb{X}^{\vee}_{++}$. We have an equality

$(\mathscr{T}^{\mathcal{IW}}_{\lambda}:\nabla^{\mathcal{IW}}_{\mu})=(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda}:\nabla^{\mathcal{IW}}_{\ell\cdot\mu}).$

###### Proof.

For any $\gamma\in\mathbb{X}^{\vee}_{++}$, let us denote by $n(\gamma)$ the
number of elements $\nu\in\mathbb{X}^{\vee}_{++}$ such that $\nu<\gamma$. We
may and will assume that $\mu\leq\lambda$ to prove the desired equality, and
proceed by induction on $n(\ell\cdot\mu)$. Standard arguments on highest
weight categories (and the fact that Verdier duality sends
$\Delta^{\mathcal{IW}}_{\nu}$, resp. $\mathscr{T}^{\mathcal{IW}}_{\nu}$, to
$\nabla^{\mathcal{IW}}_{\nu}$, resp. $\mathscr{T}^{\mathcal{IW}}_{\nu}$, for
all $\nu\in\mathbb{X}^{\vee}_{++}$; namely, we use the same arguments as the
one used in [AR16, §6.2]) imply that we have

(6.7)
$(\mathscr{T}^{\mathcal{IW}}_{\lambda}:\nabla^{\mathcal{IW}}_{\mu})=\mathrm{dim}_{\mathbf{k}}\mathrm{Hom}(\mathscr{T}^{\mathcal{IW}}_{\lambda},\mathscr{T}^{\mathcal{IW}}_{\mu})-\sum_{\nu<\mu,~{}\nu\in\mathbb{X}^{\vee}_{++}}(\mathscr{T}^{\mathcal{IW}}_{\lambda}:\nabla^{\mathcal{IW}}_{\nu})\cdot(\mathscr{T}^{\mathcal{IW}}_{\mu}:\nabla^{\mathcal{IW}}_{\nu}).$

Now when $n(\ell\cdot\mu)=0$, we get that $n(\mu)=0$ and so

$(\mathscr{T}^{\mathcal{IW}}_{\lambda}:\nabla^{\mathcal{IW}}_{\mu})=\mathrm{dim}_{\mathbf{k}}\mathrm{Hom}(\mathscr{T}^{\mathcal{IW}}_{\lambda},\mathscr{T}^{\mathcal{IW}}_{\mu})=\mathrm{dim}_{\mathbf{k}}\mathrm{Hom}(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda},\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu})=(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda}:\nabla^{\mathcal{IW}}_{\ell\cdot\mu}),$

where the second equality is due to Proposition 6.9. This proves the first
step of the induction. The induction step still follows from using (6.7) and
Proposition 6.9, together with the fact that for any
$\nu\in\mathbb{X}^{\vee}_{++}$ such that
$(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu}:\nabla^{\mathcal{IW}}_{\nu})\neq
0$, we have that $\nu\in\ell\cdot\mathbb{X}^{\vee}_{++}$ (this is because we
must have $\nu\in W\mbox{$\Box$}_{\ell}(\ell\cdot\mu)$ by the linkage
principle). ∎

###  6.7. The general case for equivalence relations on
${{}_{\mathrm{f}}W}^{\mathbf{g}}$

In this subsection, we apply the equivalence (6.5) and Proposition 6.9 to
finish the study of section 5 by treating the case of special facets. The
arguments that we have used until now do not apply when $\mathbf{g}$ is a
special facet (notice that Corollary 5.27 is false in this case). With good
reason: we are going to see that the set ${{}_{\mathrm{f}}W}^{\mathbf{g}}$
splits into infinitely many classes when some component of $\mathbf{g}$ is a
special facet. As in subsections 5.4 and 5.5, we are first going to deal with
the case where the root system is irreducible before generalizing.

Let $\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ be a facet,
$w\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$, and recall that if $\mathbf{g}$
special, then $w\mathbf{g}$ can be written as $w\mathbf{g}=\\{\lambda\\}$ for
a unique coweight $\lambda$. We will denote by $\overline{w}$ the equivalence
class of $w$ for $\sim_{\mathbf{g}}$, and define $r_{\mathbf{g}}(w)$ to be
equal to $-1$ if $\mathbf{g}$ is not special, and to be the unique (non-
negative) integer such that

$\lambda\in\ell^{r_{\mathbf{g}}(w)}\mathbb{X}^{\vee}\backslash~{}\ell^{r_{\mathbf{g}}(w)+1}\mathbb{X}^{\vee},~{}\text{with}~{}w\mathbf{g}=\\{\lambda\\}$

if $\mathbf{g}$ is special. For any positive integer $r$, we also define the
group $W^{(r)}:=W_{0}\ltimes\ell^{r}\mathbb{Z}\mathfrak{R}^{\vee}$, which is a
subgroup of $W$ isomorphic to it, with $W^{(0)}=W$. Finally, notice that for
any $\lambda\in\mathbb{X^{\vee}}$ and positive integer $r$, we have
$\ell^{r}\cdot
W\mbox{$\Box$}_{\ell}\lambda=W^{(r)}\mbox{$\Box$}_{\ell}(\ell^{r}\cdot\lambda)$.
We start with an easy lemma.

######  Lemma 6.12.

Let $\nu\in\overline{\mathbf{a}_{1}}\cap\mathbb{X}^{\vee}$,
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ be the facet containing $\nu$,
$w\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ and $r$ be a positive integer. If we set
$\lambda:=w\mbox{$\Box$}_{1}\nu$, then the application
$W\to\mathbb{X}^{\vee},~{}u\mapsto u\mbox{$\Box$}_{\ell}(\ell\cdot\nu)$
induces a bijection

$(W^{(r)}wW_{\mathbf{g}})\cap{{}_{\mathrm{f}}W}^{\mathbf{g}}\simeq
W^{(r)}\mbox{$\Box$}_{\ell}(\ell\cdot\lambda)\cap\mathbb{X}^{\vee}_{++}.$

###### Proof.

Notice that $\ell\cdot\lambda=w\mbox{$\Box$}_{\ell}(\ell\cdot\nu)$, so that
$W^{(r)}\mbox{$\Box$}_{\ell}(\ell\cdot\lambda)=W^{(r)}w\mbox{$\Box$}_{\ell}(\ell\cdot\nu)$.
The desired isomorphism then follows from the fact that $W_{\mathbf{g}}$ is
the stabilizer of $\ell\cdot\nu$ for $\mbox{$\Box$}_{\ell}$, and from the
well-known fact that $u\mapsto u\mbox{$\Box$}_{\ell}(\ell\cdot\nu)$ induces a
bijection

${{}_{\mathrm{f}}W}^{\mathbf{g}}\simeq
W\mbox{$\Box$}_{\ell}(\ell\cdot\nu)\cap\mathbb{X}^{\vee}_{++}.$

∎

######  Proposition 6.13.

Assume that $\mathfrak{R}^{\vee}$ is irreducible. For any facet
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ and
$w\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$, we have

$\overline{w}=\begin{cases}\\{w\\}&\text{if $\mathbf{g}$ is special and}\
\mathrm{char}(\mathbf{k})=0\\\
(W^{(r_{\mathbf{g}}(w)+1)}wW_{\mathbf{g}})\cap{{}_{\mathrm{f}}W}^{\mathbf{g}}&\text{otherwise}.\end{cases}$

###### Proof.

If $\mathbf{g}$ is not special, then we have $r_{\mathbf{g}}(w)=-1$ by
definition and

$(W^{(r_{\mathbf{g}}(w)+1)}wW_{\mathbf{g}})\cap{{}_{\mathrm{f}}W}^{\mathbf{g}}=W\cap{{}_{\mathrm{f}}W}^{\mathbf{g}}={{}_{\mathrm{f}}W}^{\mathbf{g}},$

which does coincide with $\overline{w}$ thanks to Theorem 5.31. So we assume
from now on that $\mathbf{g}$ is a special facet. We will first treat the case
where $\mathrm{char}(\mathbf{k})=\ell$.

Let $w,v\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$,
$\nu\in\mathbb{X}^{\vee}\cap\overline{\mathbf{a}_{1}}$ be such that
$\mathbf{g}=\\{\nu\\}$, and put $w\mathbf{g}=\\{\lambda\\}$,
$v\mathbf{g}=\\{\mu\\}$. Notice that
$w\mbox{$\Box$}_{\ell}(\ell\cdot\nu)=\ell\cdot\lambda$,
$v\mbox{$\Box$}_{\ell}(\ell\cdot\nu)=\ell\cdot\mu$, and recall that, thanks to
Proposition 6.8 coupled with (6.2), we have

$\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda},\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu})\simeq\mathrm{Hom}^{\bullet}_{\mathrm{Par}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})}(\mathcal{E}_{w}^{\mathbf{g}},\mathcal{E}_{v}^{\mathbf{g}}),$

which implies that
$w\mathscr{R}_{\mathbf{g}}v\Leftrightarrow(\ell\cdot\lambda)\mathscr{R}_{2}(\ell\cdot\mu)$.
Moreover, recall that by the geometric linkage principle ([RW20, Theorem 8.5])
we have

$\forall\gamma\in\mathbb{X}^{\vee}_{++},~{}(\ell\cdot\lambda)\mathscr{R}_{2}\gamma\Rightarrow\gamma\in
W\mbox{$\Box$}_{\ell}(\ell\cdot\lambda),$

so that in particular

(6.8)
$(\ell\cdot\lambda)\mathscr{R}_{2}\gamma\Rightarrow\gamma\in\ell\cdot\mathbb{X}^{\vee}_{++}.$

Therefore, Lemma 6.12 (applied to $r=r_{\mathbf{g}}(w)+1$) implies that our
claim on $\overline{w}$ is equivalent to showing that

$\overline{\ell\cdot\lambda}=W^{(r_{\mathbf{g}}(w)+1)}\mbox{$\Box$}_{\ell}(\ell\cdot\lambda)\cap\mathbb{X}^{\vee}_{++},$

where $\overline{\ell\cdot\lambda}$ denotes the equivalence class of
$\ell\cdot\lambda$ for the equivalence relation $\sim$ on
$\mathbb{X}^{\vee}_{++}$.

By definition of $r_{\mathbf{g}}(w)$ and $r_{\mathbf{g}}(v)$, we know that
there exist $\lambda^{\prime},\mu^{\prime}\in\mathbb{X}^{\vee}_{++}$ such that
$\ell^{r_{\mathbf{g}}(w)}\cdot\lambda^{\prime}=\lambda$ and
$\ell^{r_{\mathbf{g}}(v)}\cdot\mu^{\prime}=\mu$. Assume (without any loss of
generality, up to switching the roles of $\lambda$ and $\mu$) that
$r_{\mathbf{g}}(v)\geq r_{\mathbf{g}}(w)$, so that Proposition 6.9 yields

(6.9)
$\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\lambda},\mathscr{T}^{\mathcal{IW}}_{\ell\cdot\mu})\simeq\mathrm{Hom}_{\mathrm{Tilt}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathscr{T}^{\mathcal{IW}}_{\lambda^{\prime}},\mathscr{T}^{\mathcal{IW}}_{\ell^{r_{\mathbf{g}}(v)-r_{\mathbf{g}}(w)}\cdot\mu^{\prime}}).$

If we denote by $\mathbf{g}^{\prime}\subset\overline{\mathbf{a}_{\ell}}$ the
facet (for $\mbox{$\Box$}_{\ell}$) containing the element of
$W\mbox{$\Box$}_{\ell}\lambda^{\prime}\cap\overline{\mathbf{a}_{\ell}}$, then
$\ell^{-1}\cdot\mathbf{g}^{\prime}$ is a facet (for $\mbox{$\Box$}_{1}$)
contained in $\overline{\mathbf{a}_{1}}$ which is not special (because
otherwise we could write
$\mathbf{g}^{\prime}=\\{\ell\cdot\lambda^{\prime\prime}\\}$ for some weight
$\lambda^{\prime\prime}$, and
$\lambda^{\prime}=w^{\prime\prime}\mbox{$\Box$}_{\ell}(\ell\cdot\lambda^{\prime\prime})=\ell\cdot(w^{\prime\prime}\mbox{$\Box$}_{1}\lambda^{\prime\prime})$
for some $w^{\prime\prime}\in W$, contradicting the definition of
$r_{\mathbf{g}}(w)$). So thanks to (6.5) and Theorem 5.31 we get

$\overline{\lambda^{\prime}}=W\mbox{$\Box$}_{\ell}\lambda^{\prime}\cap\mathbb{X}^{\vee}_{++}.$

In particular, if $r_{\mathbf{g}}(v)>r_{\mathbf{g}}(w)$, then we easily see
that $\ell^{r_{\mathbf{g}}(v)-r_{\mathbf{g}}(w)}\cdot\mu^{\prime}\notin
W\mbox{$\Box$}_{\ell}\lambda^{\prime}$, so the right-hand side in (6.9) is
non-zero only if $r_{\mathbf{g}}(v)=r_{\mathbf{g}}(w)$. This observation,
together with (6.9) and (6.8), allows us to deduce that
$\overline{\ell^{r_{\mathbf{g}}(w)+1}\cdot\lambda^{\prime}}\subset\ell^{r_{\mathbf{g}}(w)+1}\cdot\overline{\lambda^{\prime}}$.
The reversed inclusion
$\ell^{r_{\mathbf{g}}(w)+1}\cdot\overline{\lambda^{\prime}}\subset\overline{\ell^{r_{\mathbf{g}}(w)+1}\cdot\lambda^{\prime}}$
is obtained by applying Corollary 6.10. We finally get the desired equality:

$\overline{\ell\cdot\lambda}=\overline{\ell^{r_{\mathbf{g}}(w)+1}\cdot\lambda^{\prime}}=\ell^{r_{\mathbf{g}}(w)+1}\cdot\overline{\lambda^{\prime}}=\ell^{r_{\mathbf{g}}(w)+1}\cdot
W\mbox{$\Box$}_{\ell}\lambda^{\prime}\cap\mathbb{X}^{\vee}_{++}=W^{(r_{\mathbf{g}}(w)+1)}\mbox{$\Box$}_{\ell}(\ell\cdot\lambda)\cap\mathbb{X}^{\vee}_{++}.$

Now we pass to the case $\mathrm{char}(\mathbf{k})=0$. We want to show that
for any $w^{\prime}\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ such that
$w^{\prime}\neq w$, we have

(6.10)
$\mathrm{Hom}^{\bullet}_{D^{b}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})}(\mathcal{E}^{\mathbf{g}}_{w^{\prime}},\mathcal{E}^{\mathbf{g}}_{w})=0.$

We claim that there exists $\omega\in\Omega$ such that
$\omega\\{0\\}=\mathbf{g}$. Indeed, the map $u\mapsto u\\{0\\}$ is actually a
bijection from $\Omega$ to $\overline{\mathbf{a}_{1}}\cap\mathbb{X}^{\vee}$
thanks to the first point of the remark in [Bou, Ch. VI, §2.3]. We then have
an isomorphism $L^{+}P_{\mathbf{g}}\simeq\dot{\omega}L^{+}G\dot{\omega}^{-1}$,
and conjugation by $\dot{\omega}$ yields an isomorphism

$\mathrm{Fl}^{\circ}_{\\{0\\}}\simeq\mathrm{Fl}^{\circ}_{\mathbf{g}}.$

Since $\dot{\omega}$ belongs to $\Omega$, conjugation by $\dot{\omega}$
preserves $\mathrm{Iw}_{u}^{+}$, so that we get an equivalence of categories

(6.11)
$D^{b}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})\simeq
D^{b}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\\{0\\}},\mathbf{k}).$

This equivalence implies that we only need to prove (6.10) with $\mathbf{g}$
replaced by $\\{0\\}$. Now, recall that
$\mathrm{Fl}_{\\{0\\}}\simeq\mathrm{Gr}$ and let
$\lambda,\lambda^{\prime}\in\mathbb{X}^{\vee}_{++}$ be such that
$w\mathbf{g}=\\{\lambda\\}$, $w^{\prime}\mathbf{g}=\\{\lambda^{\prime}\\}$. By
[BGM+19, Remark 3.5], the complex $\mathcal{E}_{w}$ is isomorphic to
$\mathrm{IC}^{\mathcal{IW}}_{\lambda}$ when $\mathrm{char}(\mathbf{k})=0$
(which denotes the intersection cohomology complex on $\mathrm{Gr}$ associated
with $\lambda$, cf. subsection 6.2). Therefore in characteristic zero we have
an isomorphism

$\mathrm{Hom}^{\bullet}_{D^{b}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\\{0\\}},\mathbf{k})}(\mathcal{E}_{w^{\prime}},\mathcal{E}_{w})\simeq\mathrm{Hom}^{\bullet}_{D^{b}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})}(\mathrm{IC}^{\mathcal{IW}}_{\lambda^{\prime}},\mathrm{IC}^{\mathcal{IW}}_{\lambda}).$

But we know from [BGM+19, §3.2] that we have an equivalence of triangulated
categories $D^{b}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})\simeq
D^{b}\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$, and that
$\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$ is semi-simple when
$\mathrm{char}(\mathbf{k})=0$ thanks to [BGM+19, Corollary 3.6]. Therefore the
right-hand side above is zero when $\lambda\neq\lambda^{\prime}$ and
$\mathrm{char}(\mathbf{k})=0$, which is equivalent to $w\neq w^{\prime}$.
141414Instead of using the equivalence (6.11), we could have argued by saying
that $\mathrm{Fl}^{\circ}_{\mathbf{g}}$ is a connected component of
$\mathrm{Gr}$.∎

This last result coupled with Proposition 5.18 now enables us to deal with the
case where $\mathfrak{R}^{\vee}$ splits into irreducible root systems,
inducing decompositions $W=W_{1}\times\cdots\times W_{r}$ and
$\mathbf{g}=\mathbf{g}_{1}\times\cdots\times\mathbf{g}_{r}$ (see the beginning
of subsection 5.4).

######  Theorem 6.14.

Assume that $\mathrm{char}(\mathbf{k})=\ell$. For any facet
$\mathbf{g}\subset\overline{\mathbf{a}_{1}}$ and
$w=(w_{1},\cdots,w_{r})\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$, we have

$\overline{w}=\prod_{i=1}^{r}W_{i}^{(r(w_{i})+1)}w_{i}W_{\mathbf{g}_{i}}\cap{{}_{\mathrm{f}}W_{i}}^{\mathbf{g}_{i}}.$

When $\mathrm{char}(\mathbf{k})=0$ and $r(w_{i})\geq 0$ for some $i$, one
replaces the $i$’th component in the above product by $\\{w_{i}\\}$.

##  7\. Applications to representation theory

###  7.1. Preliminaries on alcoves

Just as in subsection 5.3, the action $\bullet_{\ell}$ of $W$ on $E$ defines a
hyperplane arrangement $\mathscr{H}^{\prime}$, where the shift by
$\rho^{\vee}$ induces a bijection between $\mathscr{H}^{\prime}$ and
$\mathscr{H}$. We will call a connected component of
$E\backslash\mathscr{H}^{\prime}$ an alcove for $\bullet_{\ell}$, and a facet
contained in the closure of an alcove for $\bullet_{\ell}$ will be called a
facet for $\bullet_{\ell}$; once again shifting by $\rho^{\vee}$ induces a
bijection between facets for $\bullet_{\ell}$ and facets for
$\mbox{$\Box$}_{\ell}$, and composing this bijection with the dilation by
$\ell^{-1}$ gives a bijection with facets for $\mbox{$\Box$}_{1}$. We put
$C_{\ell}:=\mathbf{a}_{\ell}-\rho^{\vee}$. The bijection between sets of
alcoves allows us to define right and left actions of $W$ on these sets,
together with the Bruhat order $\leq$ and the periodic order $\preceq$, by
transport of structure from the set of alcoves for $\mbox{$\Box$}_{1}$ (or
equivalently, by seeing every alcove for $\bullet_{\ell}$, resp. for
$\mbox{$\Box$}_{\ell}$, as a $W$-translate of $C_{\ell}$, resp. of
$\mathbf{a}_{\ell}$). Note that if $A,B$ are two alcoves for $\bullet_{\ell}$,
then we have $A\preceq B\Leftrightarrow A\uparrow B$, where the order
$\uparrow$ is defined in [Jan03, §6]: this is an immediate consequence of
[Jan03, II, §6.6, (4)] (which says that $\uparrow$ is invariant under
translation) together with [AR18, II, Lemma 10.1] (which implies that
$\uparrow$ coincides with the Bruhat order inside of
$\mathscr{C}^{+}_{0}-\rho^{\vee}$, and thus with $\preceq$). For any alcove
$C$ for $\bullet_{\ell}$, we define the alcove $\hat{C}$ for $\bullet_{\ell}$,
resp. the integer $d(C)$, by transport of structure using once again the
bijection between alcoves for $\bullet_{\ell}$ and $\mbox{$\Box$}_{1}$ (notice
that $d(C)$ is then the number of hyperplanes of $\mathscr{H}^{\prime}$
separating $C$ from $C_{\ell}$).

For any facet $\mathbf{h}$ for $\bullet_{\ell}$, we will denote by
$W_{\mathbf{h},\bullet_{\ell}}\subset W$ the stabilizer of $\mathbf{h}$ for
$\bullet_{\ell}$. One can easily check that we have

$W_{\mathbf{h},\bullet_{\ell}}=W_{\mathbf{h}+\rho^{\vee},\mbox{$\Box$}_{\ell}}=W_{\ell^{-1}\cdot(\mathbf{h}+\rho^{\vee})},$

where the second and third sets denote stabilizers for $\mbox{$\Box$}_{\ell}$
and $\mbox{$\Box$}_{1}$ respectively.

Let $\lambda\in\mathbb{X}_{+}^{\vee}$ and $\mathbf{h}\subset E$ be the facet
for $\bullet_{\ell}$ containing $\lambda$, which is of the form

$\mathbf{h}=\\{\mu\in E~{}|~{}\langle\mu+\rho^{\vee},\alpha\rangle=\ell\cdot
n_{\alpha}~{}\forall\alpha\in\mathfrak{R}^{0}_{+}(\mathbf{h}),~{}\ell\cdot(n_{\alpha}-1)<\langle\mu+\rho^{\vee},\alpha\rangle<\ell\cdot
n_{\alpha}~{}\forall\alpha\in\mathfrak{R}^{1}_{+}(\mathbf{h})\\}$

for suitable integers $n_{\alpha}$ and a partition
$\mathfrak{R}_{+}=\mathfrak{R}^{0}_{+}(\mathbf{h})\sqcup\mathfrak{R}^{1}_{+}(\mathbf{h})$.
If we let $C_{\lambda}$ be the alcove for $\bullet_{\ell}$ defined by the
integers $(n_{\alpha})_{\alpha\in\mathfrak{R}_{+}}$, then $C_{\lambda}$ is the
only alcove satisfying

$\lambda\in\widetilde{C}_{\lambda}:=\\{\mu\in
E~{}|~{}\ell\cdot(n_{\alpha}-1)<\langle\mu+\rho^{\vee},\alpha\rangle\leq\ell\cdot
n_{\alpha}~{}\forall\alpha\in\mathfrak{R}_{+}\\}.$

The set $\widetilde{C}_{\lambda}$ (which is denoted by $\widehat{C}_{\lambda}$
in [Jan03]) is called the upper closure of $C_{\lambda}$. The alcove
$C_{\lambda}$ can be characterized as the only alcove containing $\mathbf{h}$
in its closure which is minimal in
$W_{\mathbf{h},\bullet_{\ell}}\bullet_{\ell}C_{\lambda}$ for the order
$\preceq$ (cf. [Jan03, §6.11]). Now, let us denote by $\mu$ the element of
$W\bullet_{\ell}\lambda$ contained in $\overline{C_{\ell}}$, by
$\mathbf{g}^{\prime}\subset\overline{\mathbf{a}_{\ell}}$ the facet containing
$\mu+\rho^{\vee}$ and by $w$ the element of
${{}_{\mathrm{f}}W}^{\ell^{-1}\cdot\mathbf{g}^{\prime}}$ such that
$w\mbox{$\Box$}_{\ell}(\mu+\rho^{\vee})=\lambda+\rho^{\vee}$ (cf. the first
isomorphism of (6.4)). Then the alcove $A:=\rho^{\vee}+C_{\lambda}$ is the
only alcove for $\mbox{$\Box$}_{\ell}$ containing $\lambda+\rho^{\vee}$ in its
closure which is minimal in

$W_{\mathbf{h}+\rho^{\vee},\mbox{$\Box$}_{\ell}}A=W_{w\mathbf{g}^{\prime},\mbox{$\Box$}_{\ell}}A=AW_{\mathbf{g}^{\prime},\mbox{$\Box$}_{\ell}}$

for $\preceq$. Likewise, the alcove $A_{w}:=\ell^{-1}\cdot A$ is the only
alcove (for $\mbox{$\Box$}_{1}$) containing
$\ell^{-1}\cdot(\mathbf{h}+\rho^{\vee})$ in its closure which is minimal in
$A_{w}W_{\mathbf{g}}$ for $\preceq$, where
$\mathbf{g}:=\ell^{-1}\cdot\mathbf{g}^{\prime}$. By definition of the
operation $\hat{}$ on the set of alcoves for $\bullet_{\ell}$, we have
$\hat{C}_{\lambda}=\ell\cdot\hat{A}_{w}-\rho^{\vee}$, and so
$\hat{C}_{\lambda}-\ell\cdot\rho^{\vee}=\ell\cdot(\hat{A}_{w}-\rho^{\vee})-\rho^{\vee}$.
In particular (once again by definition of the application $d(\cdot)$ on the
set of alcoves for $\bullet_{\ell}$), we have

(7.1) $d(\hat{C}_{\lambda}-\ell\cdot\rho^{\vee})=d(\hat{A}_{w}-\rho^{\vee}).$

###  7.2. A new proof of Donkin’s Theorem

As an application of the study that was made in the previous sections, we can
give a new proof of the description of blocks of
$\mathrm{Rep}_{\mathbf{k}}(G^{\vee})$. Unless specified otherwise, $G$ is
assumed to be a semi-simple algebraic group of adjoint type over $\mathbb{F}$
(so $G^{\vee}$ is simply connected). For any $\lambda\in\mathbb{X}^{\vee}$, we
define $r(\lambda)$ to be the unique non-negative integer such that

$\lambda\in\ell^{r(\lambda)}\cdot\mathbb{X}^{\vee}\backslash~{}\ell^{r(\lambda)+1}\cdot\mathbb{X}^{\vee}.$

If we let $\lambda^{\prime}$ be the $W$-conjugate (for the action
$\mbox{$\Box$}_{\ell}$) of $\lambda$ contained in
$\overline{\mathbf{a}_{\ell}}$,
$\mathbf{g}_{\lambda^{\prime}}\subset\overline{\mathbf{a}_{\ell}}$ be the
facet containing $\lambda^{\prime}$, put
$\mathbf{g}:=\ell^{-1}\cdot\mathbf{g}_{\lambda^{\prime}}$ and let $w$ be the
unique element of ${{}_{\mathrm{f}}W}^{\mathbf{g}}$ such that
$w\mbox{$\Box$}_{\ell}\lambda^{\prime}=\lambda$ (cf. (6.4)), then one can
easily check that

$r(\lambda)=r_{\mathbf{g}}(w)+1,$

where $r_{\mathbf{g}}(w)$ is as it was defined in subsection 6.7.

We take back the setting of the beginning of subsection 5.4 for the next
statement: since $G^{\vee}$ is semi-simple and simply connected, we get a
decomposition $G^{\vee}=G^{\vee}_{1}\times\cdots\times G_{t}^{\vee}$ into
simply connected simple algebraic groups, each $G^{\vee}_{i}$ admitting
$\mathfrak{R}^{\vee}_{i}$ as a root system; this decomposition induces a
decomposition of the root system, affine Weyl group and dominant characters
attached to $G^{\vee}$. Moreover, for every $i$, we will denote by
$\rho_{i}^{\vee}\in\mathbb{X}^{\vee}_{i,+}$ the half sum of positive coroots
relative to $G^{\vee}_{i}$ and, for any positive integer $r$, we will denote
by $W_{i}^{(r)}$ the subgroup of $W_{i}$ whose translation part has been
dilated by $\ell^{r}$ (cf. subsection 6.7 for the precise definition).

######  Theorem 7.1.

Let
$\mu=(\mu_{1},\cdots,\mu_{r})\in\mathbb{X}^{\vee}_{+}=\prod_{i}\mathbb{X}^{\vee}_{i,+}$,
and denote by $\overline{\mu}$ the equivalence class of $\mu$ for the
equivalence relation $\sim$ (cf. section 2) on $\mathbb{X}^{\vee}_{+}$ (seen
as the weight poset of $\mathrm{Rep}_{\mathbf{k}}(G^{\vee})$). We have

$\overline{\mu}=\prod_{i=1}^{r}W_{i}^{(r(\mu_{i}+\rho^{\vee}_{i}))}\bullet_{\ell}\mu_{i}\cap\mathbb{X}^{\vee}_{i,+}.$

###### Proof.

Recall that thanks to the geometric Satake equivalence (Theorem 6.2) and
Theorem 6.4, we have an equivalence of highest weight categories

(7.2)
$\mathrm{Rep}_{\mathbf{k}}(G^{\vee})\xrightarrow{\sim}\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$

sending a tilting module $T_{\mu}$ to the tilting Iwahori-Whittaker perverse
sheaf $\mathscr{T}^{\mathcal{IW}}_{\lambda}$, where
$\lambda:=\mu+\rho^{\vee}$. Therefore, proving the claim is equivalent to
showing that for any
$\lambda\in\mathbb{X}^{\vee}_{++}=\mathbb{X}^{\vee}_{+}+\rho^{\vee}=\prod_{i}(\mathbb{X}^{\vee}_{i,+}+\rho^{\vee}_{i})$,
we have the equality

(7.3)
$\overline{\lambda}=\prod_{i=1}^{r}W_{i}^{(r(\lambda_{i}))}\mbox{$\Box$}_{\ell}\lambda_{i}\cap\mathbb{X}^{\vee}_{i,++},$

where $\overline{\lambda}$ now denotes the equivalence class of
$\lambda=(\lambda_{1},\cdots,\lambda_{r})$ for the equivalence relation $\sim$
on $\mathbb{X}^{\vee}_{++}$, seen as the weight poset of
$\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k})$.

Denote by $\lambda^{\prime}$ the unique $W$-conjugate (for the dilated box
action $\mbox{$\Box$}_{\ell}$) of $\lambda$ which is contained in
$\overline{\mathbf{a}_{\ell}}$, by
$\mathbf{g}_{\lambda^{\prime}}\subset\overline{\mathbf{a}_{\ell}}$ the facet
containing $\lambda^{\prime}$, put
$\mathbf{g}:=\ell^{-1}\cdot\mathbf{g}_{\lambda^{\prime}}\subset\overline{\mathbf{a}_{1}}$
and let $w$ the element of ${{}_{\mathrm{f}}W}^{\mathbf{g}}$ such that
$w\mbox{$\Box$}_{\ell}\lambda^{\prime}=\lambda$. Recall that
${{}_{\mathrm{f}}W}^{\mathbf{g}}\mbox{$\Box$}_{\ell}\lambda^{\prime}=W\mbox{$\Box$}_{\ell}\lambda^{\prime}\cap\mathbb{X}^{\vee}_{++}$.
By (6.5), we have

$(w\mbox{$\Box$}_{\ell}\lambda^{\prime})\sim(w^{\prime}\mbox{$\Box$}_{\ell}\lambda^{\prime})\Longleftrightarrow
w\sim_{\mathbf{g}}w^{\prime},~{}\forall
w,w^{\prime}\in{{}_{\mathrm{f}}W}^{\mathbf{g}},$

and by Proposition 5.18 we have

$w\sim_{\mathbf{g}}w^{\prime}\Longleftrightarrow
w_{i}\sim_{\mathbf{g}_{i}}w^{\prime}_{i}~{}\forall i,$

where $w=(w_{1},\cdots,w_{r})$,
$w^{\prime}=(w^{\prime}_{1},\cdots,w^{\prime}_{r})$.

On the other hand, the linkage principle tells us that

$(w\mbox{$\Box$}_{\ell}\lambda^{\prime})\sim\mu\Rightarrow\mu\in
W\mbox{$\Box$}_{\ell}\lambda^{\prime},$

from which we deduce (also using Lemma 6.12) that proving (7.3) is equivalent
to proving that

$\overline{w}=\prod_{i=1}^{r}W_{i}^{(r(\lambda_{i}))}w_{i}W_{\mathbf{g}_{i}}\cap{{}_{\mathrm{f}}W_{i}}^{\mathbf{g}_{i}}.$

But this last equality was proved in Theorem 6.14 (because
$r(\lambda_{i})=r_{\mathbf{g}_{i}}(w_{i})+1$ for all $i$). ∎

We can also apply our Corollary 5.33 to give a bound on the length of a
minimum chain linking two weights in the same block. We start with a lemma.

######  Lemma 7.2.

Let $\lambda,\mu\in\mathbb{X}_{+}^{\vee}$, $r\in\mathbb{Z}_{\geq 0}$, and put
$\lambda^{\prime}:=\ell^{r}\cdot(\lambda+\rho^{\vee})-\rho^{\vee}$,
$\mu^{\prime}:=\ell^{r}\cdot(\mu+\rho^{\vee})-\rho^{\vee}$. We have

$(T_{\lambda^{\prime}}:\nabla_{\mu^{\prime}})=(T_{\lambda}:\nabla_{\mu}).$

###### Proof.

By the geometric Satake equivalence (Theorem 6.2) coupled with Theorem 6.4, we
have an equivalence of highest weight categories

$\mathrm{Rep}_{\mathbf{k}}(G^{\vee})\xrightarrow{\sim}\mathrm{Perv}_{\mathcal{IW}}(\mathrm{Gr},\mathbf{k}),$

sending $T_{\gamma}$ to $\mathscr{T}^{\mathcal{IW}}_{\gamma+\rho^{\vee}}$ for
every $\gamma\in\mathbb{X}^{\vee}_{+}$. We can thus apply Corollary 6.11:

$(T_{\lambda^{\prime}}:\nabla_{\mu^{\prime}})=(\mathscr{T}^{\mathcal{IW}}_{\ell^{r}\cdot(\lambda+\rho^{\vee})}:\nabla^{\mathcal{IW}}_{\ell^{r}\cdot(\mu+\rho^{\vee})})\overset{\ref{red
to non
special2}}{=}(\mathscr{T}^{\mathcal{IW}}_{\lambda+\rho^{\vee}},\nabla^{\mathcal{IW}}_{\mu+\rho^{\vee}})=(T_{\lambda}:\nabla_{\mu}).$

∎

######  Proposition 7.3.

Assume that $G^{\vee}$ is a simple and simply connected algebraic group over
$\mathbf{k}$. Let $\lambda,\lambda^{\prime}$ be two elements of
$\mathbb{X}^{\vee}_{+}$ in the same equivalence class for $\sim$, and denote
by $\tilde{\lambda}$ (resp. $\tilde{\lambda^{\prime}}$) the unique element of
$\mathbb{X}^{\vee}_{+}$ which satisfies
$\lambda+\rho^{\vee}=\ell^{r}\cdot(\tilde{\lambda}+\rho^{\vee})$ (resp.
$\lambda^{\prime}+\rho^{\vee}=\ell^{r}\cdot(\tilde{\lambda^{\prime}}+\rho^{\vee})$),
where $r=r(\lambda+\rho^{\vee})$ (notice that $\lambda=\tilde{\lambda}$ when
$r=0$). Also denote by $C_{\tilde{\lambda}}$ (resp.
$C_{\tilde{\lambda^{\prime}}}$) the alcove containing ${\tilde{\lambda}}$
(resp. ${\tilde{\lambda^{\prime}}}$) in its upper closure. Then there exists a
chain of dominant characters

$\lambda_{s}=\lambda,\lambda_{s-1},\cdots,\lambda_{0}=\lambda^{\prime}$

such that, for all $i\in\llbracket 0,s-1\rrbracket$, there exists an
indecomposable $G^{\vee}$-module $M_{i}$ satisfying

$[M_{i}:L_{\lambda_{i}}]\neq 0~{}\text{and}~{}[M_{i}:L_{\lambda_{i+1}}]\neq 0$

and such that

$s\leq
2+d(\hat{C}_{\tilde{\lambda}}-\ell\cdot\rho^{\vee})+d(\hat{C}_{\tilde{\lambda^{\prime}}}-\ell\cdot\rho^{\vee}).$

###### Proof.

Denote by $\mu$ the $W$-conjugate (for $\bullet_{\ell}$) of $\lambda$ which is
included in $\overline{C_{\ell}}$. We will first deal with the case where
$r=0$, which means that the facets (for $\mbox{$\Box$}_{\ell}$) containing
$\lambda+\rho^{\vee}$ and $\lambda^{\prime}+\rho^{\vee}$ are not special
(notice that, since $\lambda$ and $\lambda^{\prime}$ are in the same
equivalence class, we must have
$r(\lambda+\rho^{\vee})=r(\lambda^{\prime}+\rho^{\vee})$ thanks to Theorem
7.1).

We denote by $\mathbf{g}_{\mu}\subset\overline{\mathbf{a}_{\ell}}$ the facet
containing $\mu+\rho^{\vee}$, put $\mathbf{g}:=\ell^{-1}\cdot\mathbf{g}_{\mu}$
and let $w,w^{\prime}$ be the elements of ${{}_{\mathrm{f}}W^{\mathbf{g}}}$
satisfying $w\mbox{$\Box$}_{\ell}(\mu+\rho^{\vee})=\lambda+\rho^{\vee}$,
$w^{\prime}\mbox{$\Box$}_{\ell}(\mu+\rho^{\vee})=\lambda^{\prime}+\rho^{\vee}$.
The facet $\mathbf{g}$ for $\mbox{$\Box$}_{1}$ is a non-special facet included
in the closure of $\overline{\mathbf{a}_{1}}$. Let us also denote by $A_{w}$,
resp. $A_{w^{\prime}}$, the alcove containing $w\mathbf{g}$, resp.
$w^{\prime}\mathbf{g}$, in its closure and which is minimal in
$A_{w}W_{\mathbf{g}}$, resp. $A_{w^{\prime}}W_{\mathbf{g}}$, for the order
$\preceq$. Thus, we can apply Corollary 5.33, and pick a chain of elements of
${{}_{\mathrm{f}}W}^{\mathbf{g}}$

$w_{s}=w,w_{s-1},\cdots,w_{0}=w^{\prime}$

as in this corollary, with $s\leq
2+d(\hat{A}_{w}-\rho^{\vee})+d(\hat{A}_{w^{\prime}}-\rho^{\vee})$. Thanks to
the equation (7.1), we have that
$d(\hat{A}_{w}-\rho^{\vee})=d(\hat{C}_{\lambda}-\ell\cdot\rho^{\vee})$ and
$d(\hat{A}_{w^{\prime}}-\rho^{\vee})=d(\hat{C}_{\lambda^{\prime}}-\ell\cdot\rho^{\vee})$,
so that $s$ is bounded by the desired integer. This same corollary tells us
that for all $i\in\llbracket 0,s-1\rrbracket$, there exists an element
$u_{i}\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$ satisfying

$n_{w_{i},u_{i}}(1)\neq 0~{}\text{and}~{}n_{w_{i+1},u_{i}}(1)\neq 0.$

By Proposition 4.4, we can replace $n$ with ${{}^{\ell}n}$ in the above, so
that the character formula of tilting modules given in [RW20, Theorem 8.9]
yields

$(T_{u_{i}\bullet_{\ell}\mu}:\nabla_{w_{i}\bullet_{\ell}\mu})\neq
0~{}\text{and}~{}(T_{u_{i}\bullet_{\ell}\mu}:\nabla_{w_{i+1}\bullet_{\ell}\mu})\neq
0,$

which implies that

$[T_{u_{i}\bullet_{\ell}\mu}:L_{w_{i}\bullet_{\ell}\mu}]\neq
0~{}\text{and}~{}[T_{u_{i}\bullet_{\ell}\mu}:L_{w_{i+1}\bullet_{\ell}\mu}]\neq
0.$

So we get the result by putting $\lambda_{i}=w_{i}\bullet_{\ell}\mu$ and
$M_{i}=T_{u_{i}\bullet_{\ell}\mu}$.

Now we assume that $r>0$. Since
$r(\tilde{\lambda}+\rho)=r(\tilde{\lambda^{\prime}}+\rho)=0$, we can apply the
previous step and find two sequences of dominant characters
$(\tilde{\lambda}_{i}),~{}(\tilde{\nu}_{i})$ of the desired length such that
$\tilde{\lambda}_{s}=\tilde{\lambda},~{}\tilde{\lambda}_{0}=\tilde{\lambda^{\prime}}$
and

$(T_{\tilde{\nu}_{i}}:\nabla_{\tilde{\lambda}_{i}})\neq
0~{}\text{and}~{}(T_{\tilde{\nu}_{i}}:\nabla_{\tilde{\lambda}_{i+1}})\neq 0.$

By Lemma 7.2, we see that if we put
$\nu_{i}:=\ell^{r}\cdot(\tilde{\nu}_{i}+\rho)-\rho$ and
$\lambda_{i}:=\ell^{r}\cdot(\tilde{\lambda}_{i}+\rho)-\rho$, we get:

$(T_{\nu_{i}}:\nabla_{\lambda_{i}})=(T_{\tilde{\nu}_{i}}:\nabla_{\tilde{\lambda}_{i}})\neq
0~{}\text{and}~{}(T_{\nu_{i}}:\nabla_{\lambda_{i+1}})=(T_{\tilde{\nu}_{i}}:\nabla_{\tilde{\lambda}_{i+1}})\neq
0,$

from which we deduce that

$[T_{\nu_{i}}:L_{\lambda_{i}}]\neq
0~{}\text{and}~{}[T_{\nu_{i}}:L_{\lambda_{i+1}}]\neq 0.$

This concludes the proof. ∎

We will now use the process described in [Jan03, §II.7.3] to deduce from
Theorem 7.1 the block decomposition of $\mathrm{Rep}_{\mathbf{k}}(G^{\vee})$
in the general case where $G^{\vee}$ is a reductive group151515We warn the
reader that the statement (3) given in [Jan03, §II.7.3] is wrong, since one
needs to assume the semi-simple group of loc. cit. to be simple for it to be
correct., starting with a lemma.

######  Lemma 7.4.

Let $H_{1},H_{2}$ be reductive algebraic groups over $\mathbf{k}$, with
$T_{1}\subset B_{1}$, resp. $T_{2}\subset B_{2}$, a maximal torus and a Borel
subgroup of $H_{1}$, resp. of $H_{2}$, and $\varphi:H_{1}\to H_{2}$ a central
isogeny such that $\varphi(T_{1})=T_{2}$, $\varphi(B_{1})=B_{2}$. Denote by
$\mathbb{X}(T_{1})_{+}\subset\mathbb{X}(T_{1})$ (resp.
$\mathbb{X}(T_{2})_{+}\subset\mathbb{X}(T_{2})$) the dominant characters and
characters associated with $T_{1}\subset B_{1}$ (resp. $T_{2}\subset B_{2}$).

Then $\varphi$ induces an injective morphism
$\mathbb{X}(T_{2})_{+}\hookrightarrow\mathbb{X}(T_{1})_{+}$, and the blocks of
$H_{2}$ are the blocks of $H_{1}$ contained in $\mathbb{X}(T_{2})_{+}$.

###### Proof.

Since $\varphi$ is a central isogeny, the morphism induced on root data
identifies the root systems of $H_{1}$ and $H_{2}$, (cf. [Jan03, §II.1.17]).
Therefore, the injection $\mathbb{X}(T_{2})\hookrightarrow\mathbb{X}(T_{1})$
(induced by $\varphi|_{T_{1}}$) induces an injection
$\mathbb{X}(T_{2})_{+}\hookrightarrow\mathbb{X}(T_{1})_{+}$. On the other
hand, pulling back by $\varphi$ induces a fully-faithful functor
$\varphi^{*}:\mathrm{Rep}_{\mathbf{k}}(H_{2})\to\mathrm{Rep}_{\mathbf{k}}(H_{1})$,
which sends an indecomposable tilting module of highest weight $\lambda$ to
the indecomposable tilting module of same highest weight (cf. [Jan03, §E.7]),
and such that there exists no non-zero morphism between an indecomposable
tilting module in the essential image of $\varphi^{*}$ and an indecomposable
tilting module of $\mathrm{Rep}_{\mathbf{k}}(H_{1})$ which is not in this
essential image. The conclusion follows easily, thanks to the description of
blocks via tilting modules (cf. Theorem 2.3). ∎

######  Corollary 7.5.

Assume that $G$ is a general reductive group. Denote by $\mathcal{D}G^{\vee}$
the derived subgroup of $G^{\vee}$, by $T_{2}$ the reduced part of the neutral
connected component of the centre of $G^{\vee}$ and let $H_{1},\cdots,H_{t}$
be the simply connected covers of the minimal closed connected normal
subgroups of positive dimension of $\mathcal{D}G^{\vee}$. For each
$i\in\llbracket 1,t\rrbracket$, also denote by $T^{\vee}_{i}$ the split
maximal torus of $H_{i}$ determined by the split maximal torus of $G^{\vee}$
which is Langlands dual to $T$, by $W_{i}$ (resp. by
$\mathbb{X}^{\vee}_{i,+}$, resp. by $\rho^{\vee}_{i}$) the affine Weyl group
(resp. the set of dominant characters of $T^{\vee}_{i}$ determined by
$\mathfrak{R}_{+}^{\vee}$, resp. the half-sum of positive roots) associated
with $H_{i}$, seen as a subgroup of the affine Weyl group associated with
$\mathcal{D}G^{\vee}$, and by $\mathbb{X}(T_{2})$ the group of characters of
$T_{2}$. Then there is a central isogeny

$\varphi:\mathcal{D}G^{\vee}\times T_{2}\to G^{\vee},$

which induces an injection
$\mathbb{X}^{\vee}_{+}\hookrightarrow\prod_{i}\mathbb{X}^{\vee}_{i,+}\times\mathbb{X}(T_{2})$
and such that, for every

$\lambda=(\lambda_{1},\cdots,\lambda_{t},\lambda_{t+1})\in\mathbb{X}^{\vee}_{+},$

we have

$\overline{\lambda}=\prod_{i=1}^{t}W_{i}^{(r(\lambda_{i}+\rho_{i}^{\vee}))}\bullet_{\ell}\lambda_{i}\cap\mathbb{X}^{\vee}_{i,+}\times\\{\lambda_{t+1}\\}.$

###### Proof.

The fact that $\varphi$ exists follows from [Jan03, II.1.18]. Moreover, we
know that $\mathcal{D}G^{\vee}$ is a semi-simple algebraic group, so that we
have a central isogeny $H^{\prime}_{1}\times\cdots\times
H^{\prime}_{t}\to\mathcal{D}G^{\vee}$, where
$H^{\prime}_{1},\cdots,H^{\prime}_{t}$ are the minimal closed normal subgroups
of positive dimension of $\mathcal{D}G^{\vee}$, which are simple algebraic
groups. We then get a central isogeny $H_{1}\times\cdots\times
H_{t}\to\mathcal{D}G^{\vee}$ by replacing each $H_{i}^{\prime}$ with its
simply connected cover. Therefore, Lemma 7.4 and Theorem 7.1 tell us that for
any dominant character $\lambda^{\prime}$ of the maximal torus
$T\cap\mathcal{D}G^{\vee}$ of $\mathcal{D}G^{\vee}$, the associated block of
$\mathrm{Rep}_{\mathbf{k}}(\mathcal{D}G^{\vee})$ is equal to

$\prod_{i=1}^{t}W_{i}^{(r(\lambda^{\prime}_{i}+\rho^{\vee}_{i}))}\bullet_{\ell}\lambda^{\prime}_{i}\cap\mathbb{X}^{\vee}_{i,+},$

where we see $\lambda^{\prime}$ as the element
$(\lambda^{\prime}_{1},\cdots,\lambda^{\prime}_{t})$ of
$\prod_{i}\mathbb{X}^{\vee}_{i,+}$ (notice that the above set is included in
the set of dominant characters of $T\cap\mathcal{D}G^{\vee}$, because each
$W_{i}$ is a subgroup of the affine Weyl group associated with
$\mathcal{D}G^{\vee}$).

Finally, since $T_{2}$ is a torus (in particular, the category
$\mathrm{Rep}_{\mathbf{k}}(T_{2})$ is semi-simple), one can easily check that
the block of $\lambda$ in $\mathcal{D}G^{\vee}\times T_{2}$ is equal to the
product of blocks associated to $(\lambda_{1},\cdots,\lambda_{t})$ and
$\lambda_{t+1}$ in $\mathrm{Rep}_{\mathbf{k}}(\mathcal{D}G^{\vee})$ and
$\mathrm{Rep}_{\mathbf{k}}(T_{2})$ respectively, i.e. to

$\prod_{i=1}^{t}W_{i}^{(r(\lambda_{i}+\rho^{\vee}_{i}))}\bullet_{\ell}\lambda_{i}\cap\mathbb{X}^{\vee}_{i,+}\times\\{\lambda_{t+1}\\}.$

This concludes the proof by Lemma 7.4, since the above set is included in
$\mathbb{X}^{\vee}_{+}$. ∎

######  Remark 7.6.

We take back the context of Corollary 7.5, and let $\lambda,\lambda^{\prime}$
be dominant characters in the same equivalence class. The result of
Proposition 7.3 can also be generalized to the case where $G^{\vee}$ is a
general reductive group. Indeed, using the central isogeny
$\mathcal{D}G^{\vee}\times T_{2}\to G^{\vee}$, one is reduced to proving it
for $\mathcal{D}G^{\vee}$, and since $\mathcal{D}G^{\vee}$ is a central
isogeny of the product of $t$ simple subgroups (because $\mathcal{D}G^{\vee}$
is semi-simple), we can further reduce to the case where $G^{\vee}$ is a
product of simply connected simple groups $H_{1}\times\cdots\times H_{t}$.
Finally, the equality (5.4) of Proposition 5.18 allows us to bound the length
of a minimum chain linking two weights $(\lambda_{1},\cdots,\lambda_{t})$ and
$(\lambda_{1}^{\prime},\cdots,\lambda_{t}^{\prime})$ in the same block by
$\mathrm{max}\\{s_{i},~{}i\in\llbracket 1,t\rrbracket\\}$, where $s_{i}$ is
the bound obtained in Proposition 7.3 for the length of a minimum chain
linking $\lambda_{i}$ and $\lambda^{\prime}_{i}$.

###  7.3. Block decomposition for a quantum group

In this subsection, $G$ is semi-simple of adjoint type. We assume that
$\mathrm{char}(\mathbf{k})=0$, and that there exists a primitive $\ell$-th
root of unity $q$ in $\mathbf{k}$. Moreover, we assume that $\ell$ is odd,
greater than the Coxeter number of $\mathfrak{R}^{\vee}$ and not equal to $3$
if $\mathfrak{R}^{\vee}$ has a component of type $G_{2}$. We then denote by
$U_{q,\mathbf{k}}$ Lusztig’s quantized enveloping algebra specialized at $q$
and associated with $G^{\vee}$ (we take the conventions of [Jan03, §H]). The
category of finite dimensional representations of $U_{q,\mathbf{k}}$, which we
will denote by $\mathrm{Rep}(U_{q,\mathbf{k}})$, has many features in common
with the category $\mathrm{Rep}(G^{\vee}_{\mathbf{k}^{\prime}})$, where
$G^{\vee}_{\mathbf{k}^{\prime}}$ denotes the Langlands dual group of $G$ over
some field $\mathbf{k}^{\prime}$ of characteristic $\ell$. In particular,
$\mathrm{Rep}(U_{q,\mathbf{k}})$ is a highest weight category with weight
poset $\mathbb{X}^{\vee}_{+}$; for any $\lambda\in\mathbb{X}^{\vee}_{+}$, we
will denote by $T_{q}(\lambda)$, resp. $\nabla_{q}(\lambda)$, the
indecomposable tilting module, resp. the costandard object, with highest
weight $\lambda$.

We fix a weight $\lambda\in\overline{C}_{\ell}\cap\mathbb{X}^{\vee}$, let
$\mathbf{g}^{\prime}$ be the facet for $\mbox{$\Box$}_{\ell}$ containing
$\lambda+\rho^{\vee}$ and $w,w^{\prime}$ be elements of
${{}_{\mathrm{f}}W}^{\mathbf{g}}$, where
$\mathbf{g}:=\ell^{-1}\cdot\mathbf{g}^{\prime}$ (recall the bijections (6.4)).
Most importantly for us, the multiplicity of costandard objects in tilting
objects is known.

######  Proposition 7.7.

We have

$(T_{q}(w\bullet_{\ell}\lambda):\nabla_{q}(w^{\prime}\bullet_{\ell}\lambda))=n_{w^{\prime},w}(1).$

Moreover, we have that
$(T_{q}(w\bullet_{\ell}\lambda):\nabla_{q}(\lambda^{\prime}))=0$ whenever
$\lambda^{\prime}\notin W\bullet_{\ell}\lambda$.

###### Proof.

The second claim is due to the linkage principle for quantum groups, see
[APK91, §8].

Let us denote by $W_{(0)}$ the subset of $W$ consisting of elements $w$ which
are minimal in $wW_{0}$, and recall that the action of $\mathrm{Iw}$ by left
multiplication on $\mathrm{Fl}_{\mathbf{a}_{1}}^{\circ}$, resp. on
$\mathrm{Gr}^{\circ}$, yields stratifications

$(\mathrm{Fl}_{\mathbf{a}_{1}}^{\circ})_{\mathrm{red}}=\bigsqcup_{u\in
W}X_{u},\qquad(\mathrm{Gr}^{\circ})_{\mathrm{red}}=\bigsqcup_{v\in
W_{(0)}}Y_{v},$

where $Y_{u}:=\mathrm{Iw}\cdot\dot{u}$, resp.
$X_{v}:=\mathrm{Iw}\cdot\dot{v}$, is an affine $\mathbb{F}$-space of dimension
$l(u)$, resp. $l(v)$. Moreover, the canonical projection
$\pi:\mathrm{Fl}_{\mathbf{a}_{1}}^{\circ}\to\mathrm{Gr}^{\circ}$ is ind-
proper, $\mathrm{Iw}$-equivariant and satisfies
$\pi^{-1}(Y_{w})=\bigsqcup_{z\in W_{0}}X_{wz}$ for every $w\in W_{(0)}$. We
denote by
$\mathrm{Perv}_{(\mathrm{Iw})}(\mathrm{Fl}_{\mathbf{a}_{1}}^{\circ},\mathbf{k})$,
resp. $\mathrm{Perv}_{(\mathrm{Iw})}(\mathrm{Gr}^{\circ},\mathbf{k})$, the
category of perverse sheaves on $\mathrm{Fl}_{\mathbf{a}_{1}}^{\circ}$, resp.
$\mathrm{Gr}^{\circ}$, constant along the $\mathrm{Iw}$-orbits, and by
$\mathrm{Rep}_{\overline{0}}(U_{q,\mathbf{k}})$ the Serre subcategory of
$\mathrm{Rep}(U_{q,\mathbf{k}})$ generated by the simple objects whose highest
weight belongs to $W\bullet_{\ell}0\cap\mathbb{X}^{\vee}_{+}$ (the fact that
$\mathrm{Rep}_{\overline{0}}(U_{q,\mathbf{k}})$ is truly a block will follow
from Proposition 7.10).

Thanks to [ABG04], we have an equivalence of highest weight categories

$\mathrm{Rep}_{\overline{0}}(U_{q,\mathbf{k}})\simeq\mathrm{Perv}_{(\mathrm{Iw})}(\mathrm{Gr}^{\circ},\mathbf{k}),$

sending $T_{q}(x\bullet_{\ell}0)$ (resp. $\nabla_{q}(y\bullet_{\ell}0)$) to
the indecomposable tilting object $T^{(\mathrm{Iw})}_{x^{-1}}$ (resp. to the
costandard object $\nabla^{(\mathrm{Iw})}_{y^{-1}}$) associated with $x$, for
any $x\in{{}_{\mathrm{f}}W}$ (resp. $y\in{{}_{\mathrm{f}}W}$)161616Notice that
$w\mapsto w^{-1}$ induces a bijection ${{}_{\mathrm{f}}W}\simeq W_{(0)}$..

For any $u\in W$, let us denote by
$\mathscr{T}_{u}\in\mathrm{Perv}_{(\mathrm{Iw})}(\mathrm{Fl}_{\mathbf{a}_{1}}^{\circ},\mathbf{k})$
the indecomposable tilting object associated with $u$; by [Yun09, Proposition
3.4.1], we have an isomorphism $\pi_{!}\mathscr{T}_{w}\simeq
T^{(\mathrm{Iw})}_{w}$ for any $w\in W_{(0)}$. Therefore, combining [Yun09,
(3.4.1)] with [Yun09, Theorem 5.3.1], we get

$(T^{(\mathrm{Iw})}_{u}:\nabla^{(\mathrm{Iw})}_{v})=\sum_{z\in
W_{0}}(-1)^{l(z)}h_{vz,u}(1)~{}\forall u,v\in W_{(0)},$

where $(h_{x,y},~{}x,y\in W)$ denotes the ordinary Kazhdan-Lusztig polynomials
associated with $W$, as described in [Soe97]. Using [Soe97, Proposition 3.4]
together with the fact that $h_{x,y}=h_{x^{-1},y^{-1}}$ for all $x,y\in W$
(which is a consequence of the fact that the anti-automorphism $i$ from the
proof of [Soe97, Theorem 2.7] commutes with the involution
$d:\mathcal{H}\to\mathcal{H}$ defined in loc. cit.) show that the right-hand
side in the above equation coincides with $n_{v^{-1},u^{-1}}(1)$. So we get

$(T_{q}(w\bullet_{\ell}0):\nabla_{q}(w^{\prime}\bullet_{\ell}0))=(T^{(\mathrm{Iw})}_{w^{-1}}:\nabla^{(\mathrm{Iw})}_{w^{\prime-1}})=n_{w^{\prime},w}(1),$

which proves our claim in the case where $\lambda=0$. Using the second point
of [Soe97, Remark 7.2], one can deduce from this the general case. ∎

######  Remark 7.8.

The character formula of Proposition 7.7 was originally stated as a conjecture
in [Soe97, Conjecture 7.1], and proved in [Soe98] for $\ell>33$.

In particular, the second assertion of Proposition 7.7 tells us that
$\overline{\mu}\subset W\bullet_{\ell}\mu$ for any
$\mu\in\mathbb{X}^{\vee}_{+}$.

######  Corollary 7.9.

We have an equality

$\mathrm{dim}_{\mathbf{k}}\mathrm{Hom}_{\mathrm{Rep}(U_{q,\mathbf{k}})}(T_{q}(w\bullet_{\ell}\lambda),T_{q}(w^{\prime}\bullet_{\ell}\lambda))=\mathrm{dim}_{\mathbf{k}}\mathrm{Hom}^{\bullet}_{\mathrm{Par}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})}(\mathcal{E}_{w}^{\mathbf{g}},\mathcal{E}_{w^{\prime}}^{\mathbf{g}}).$

###### Proof.

Proposition 7.7 and standard arguments (see [AR16, §6.2]) show that we have

(7.4)
$\mathrm{dim}_{\mathbf{k}}\mathrm{Hom}_{\mathrm{Rep}(U_{q,\mathbf{k}})}(T_{q}(w\bullet_{\ell}\lambda),T_{q}(w^{\prime}\bullet_{\ell}\lambda))=\sum_{y\in{{}_{\mathrm{f}}W}^{\mathbf{g}}}n_{y,w}(1)\cdot
n_{y,w^{\prime}}(1).$

But we also know that

$\mathrm{dim}_{\mathbf{k}}~{}\mathrm{Hom}^{\bullet}_{D^{b}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})}(\mathcal{E}_{x}^{\mathbf{g}},\nabla^{\mathbf{g}}_{y})=n_{y,x}(1)~{}\forall
x,y\in{{}_{\mathrm{f}}W}^{\mathbf{g}}$

thanks to (4.8) together with Lemma 4.3. So by [JMW14, Proposition 2.6] the
right-hand side in (7.4) coincides with

$\mathrm{dim}_{\mathbf{k}}\mathrm{Hom}^{\bullet}_{\mathrm{Par}_{\mathcal{IW}}(\mathrm{Fl}^{\circ}_{\mathbf{g}},\mathbf{k})}(\mathcal{E}_{w}^{\mathbf{g}},\mathcal{E}_{w^{\prime}}^{\mathbf{g}}).$

∎

Therefore, we get a very similar situation as the one in subsection 6.5:

$w\bullet_{\ell}\lambda\sim w^{\prime}\bullet_{\ell}\lambda\Longleftrightarrow
w\sim_{\mathbf{g}}w^{\prime}.$

For any $\mu\in\mathbb{X}^{\vee}_{+}$, recall the definition of $r(\mu)$ from
subsection 7.2, and put

$\delta(\mu):=\begin{cases}1&\text{if}\ r(\mu)=0\\\
0&\text{othewise.}\end{cases}$

We define $W^{\delta}$ to be equal to $W$ when $\delta=1$, and to
$\\{\mathrm{id}\\}$ otherwise. The next result is obtained from Theorem 6.14
in the same way that Theorem 7.1 was, from which we take back the notations.

######  Proposition 7.10.

Let
$\mu=(\mu_{1},\cdots,\mu_{r})\in\mathbb{X}^{\vee}_{+}=\prod_{i}\mathbb{X}^{\vee}_{i,+}$,
and denote by $\overline{\mu}$ the equivalence class of $\mu$ in
$\mathbb{X}^{\vee}_{+}$ (seen as the weight poset of
$\mathrm{Rep}(U_{q,\mathbf{k}})$) for the equivalence relation $\sim$ from
section 2. We have

$\overline{\mu}=\prod_{i=1}^{r}W_{i}^{\delta(\mu_{i}+\rho^{\vee}_{i})}\bullet_{\ell}\mu_{i}\cap\mathbb{X}^{\vee}_{i,+}.$

######  Remarks 7.11.

  1. (1)

The proof of the previous proposition does not require Smith-Treumann theory
(because the proof of Theorem 6.14 does not require it when
$\mathrm{char}(\mathbf{k})=0$).

  2. (2)

One can use Corollary 5.33 to give a bound for the length of a minimal chain
linking two weights in the same block for $\mathrm{Rep}(U_{q,\mathbf{k}})$, of
the same kind as Proposition 7.3.

  3. (3)

Another proof of Proposition 7.10 was found in [Tha94] (under the same
assumptions on $\ell$). However, the results of loc. cit. uses the proof of
Donkin from [Don80], so it does not allow to give a bound as in the previous
point.

## References

  * [ABG04] Sergey Arkhipov, Roman Bezrukavnikov, and Victor Ginzburg.  Quantum groups, the loop Grassmannian, and the Springer resolution.  J. Amer. Math. Soc., 17(3):595–678, 2004.
  * [Ach21] Pramod N. Achar.  Perverse Sheaves and Applications to Representation Theory, volume 258.  American Mathematical Soc., 2021.
  * [ACR18] Pramod N. Achar, Nicholas Cooney, and Simon Riche.  The parabolic exotic t-structure.  Épijournal Géom. Algébrique, Volume 2, November 2018.  URL: https://epiga.episciences.org/4984, doi:10.46298/epiga.2018.volume2.4520.
  * [And86] Henning Haahr Andersen.  An inversion formula for the Kazhdan-Lusztig polynomials for affine Weyl groups.  Adv. Math., 60(2):125–153, 1986.
  * [APK91] Henning Haahr Andersen, Patrick Polo, and Wen Kexin.  Representations of quantum algebras.  Invent. Math., 104(1):1–59, 1991.
  * [AR16] Pramod N. Achar and Simon Riche.  Modular perverse sheaves on flag varieties I: tilting and parity sheaves.  In Ann. Sci. Éc. Norm. Supér., volume 49, pages 325–370, 2016.  With a joint appendix by Geordie Williamson.
  * [AR18] Pramod N. Achar and Simon Riche.  Reductive groups, the loop Grassmannian, and the Springer resolution.  Invent. Math., 214(1):289–436, 2018.
  * [AR21] Pramod N. Achar and Simon Riche.  A geometric Steinberg formula.  2021\.  arXiv:2109.11980.
  * [AR22] Pramod N. Achar and Simon Riche.  A geometric model for blocks of Frobenius kernels.  2022\.  arXiv:2203.03530.
  * [BBDG18] Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and Ofer Gabber.  Faisceaux pervers.  Astérisque, 100, 2018.
  * [BGM+19] Roman Bezrukavnikov, Dennis Gaitsgory, Ivan Mirković, Simon Riche, and Laura Rider.  An Iwahori-Whittaker model for the Satake category.  J. É. polytech. — Math., 6:707–735, 2019.  doi:10.5802/jep.104.
  * [Bou] Nicolas Bourbaki.  Groupes et algèbres de Lie: Chapitres 4 à 6.  Springer Berlin Heidelberg, Berlin, Heidelberg.
  * [BR18] Pierre Baumann and Simon Riche.  Notes on the geometric Satake equivalence.  In D. Prasad V. Heiermann, editor, Relative aspects in Representation Theory, Langlands Functoriality and Automorphic Forms, volume 2221 of Lecture Notes in Mathematics, pages 1–134. Springer, 2018\.  URL: https://hal.archives-ouvertes.fr/hal-01491529, doi:10.1007/978-3-319-95231-4\\_1.
  * [Cia21] Joshua Ciappara.  Hecke category actions via Smith-Treumann theory, 2021.  arXiv:2103.07091.
  * [Don80] Stephen Donkin.  The blocks of a semisimple algebraic group.  J. Algebra, 67(1):36–53, 1980.
  * [DV08] Maud De Visscher.  On the blocks of semisimple algebraic groups and associated generalized Schur algebras.  J. Algebra, 319:952–965, 02 2008.  doi:10.1016/j.jalgebra.2007.11.015.
  * [GG82] Robert Gordon and Edward L Green.  Graded artin algebras.  J. Algebra, 76(1):111–137, 1982.
  * [Gib] Joel Gibson.  Lievis.  https://www.jgibson.id.au/lievis/.
  * [HJ78] James Edward Humphreys and Jens Carsten Jantzen.  Blocks and indecomposable modules for semisimple algebraic groups.  J. Algebra, 54(2):494–503, 1978.
  * [Hum75] James Edward Humphreys.  Linear Algebraic Groups.  Graduate Texts in Mathematics. Springer, 1975.
  * [Jan03] Jens Carsten Jantzen.  Representations of Algebraic Groups.  Math. Surveys Monogr. American Mathematical Society, 2003.
  * [JMW14] Daniel Juteau, Carl Mautner, and Geordie Williamson.  Parity sheaves.  J. Amer. Math. Soc, 27(4):1169–1212, May 2014.  URL: http://dx.doi.org/10.1090/S0894-0347-2014-00804-3, doi:10.1090/s0894-0347-2014-00804-3.
  * [Knu91] Max-Albert Knus.  Quadratic and Hermitian forms over rings, volume 294.  Springer-Verlag, 1991.  With a foreword by I. Bertuccioni.
  * [Lus80] George Lusztig.  Hecke algebras and Jantzen’s generic decomposition patterns.  Adv. Math., 37(2):121–164, 1980.  URL: https://www.sciencedirect.com/science/article/pii/0001870880900316, doi:https://doi.org/10.1016/0001-8708(80)90031-6.
  * [Mil17] James S. Milne.  Algebraic groups: the theory of group schemes of finite type over a field, volume 170.  Cambridge University Press, 2017.
  * [MV04] Ivan Mirković and Kari Vilonen.  Geometric Langlands duality and representations of algebraic groups over commutative rings.  Ann. of Math., 166:95–143, 2004.
  * [PR08] George Pappas and Michael Rapoport.  Twisted loop groups and their affine flag varieties.  Adv. Math., 219(1):118–198, 2008.  URL: https://www.sciencedirect.com/science/article/pii/S0001870808001205, doi:https://doi.org/10.1016/j.aim.2008.04.006.
  * [Ric16] Simon Riche.  Geometric Representation Theory in positive characteristic.  Habilitation à diriger des recherches, Université Blaise Pascal (Clermont Ferrand 2), 2016.  URL: https://tel.archives-ouvertes.fr/tel-01431526.
  * [RW18] Simon Riche and Geordie Williamson.  Tilting modules and the p-canonical basis.  Astérisque, 397, 2018.
  * [RW20] Simon Riche and Geordie Williamson.  Smith-Treumann theory and the linkage principle.  2020\.  arXiv:2003.08522.
  * [Soe97] Wolfgang Soergel.  Kazhdan-Lusztig polynomials and a combinatoric for tilting modules.  Represent. Theory, 1:83–114, 1997.
  * [Soe98] Wolfgang Soergel.  Character formulas for tilting modules over kac-moody algebras.  Represent. Theory 2, 2(13):432–448, 1998.
  * [Sta22] The Stacks project authors.  The stacks project.  https://stacks.math.columbia.edu, 2022.
  * [Tha94] Lars Thams.  The blocks of a quantum algebra.  Comm. Algebra, 22(5):1617–1628, 1994.
  * [Wil12] Geordie Williamson.  Modular intersection cohomology complexes on flag varieties.  Math. Z, 272(3):697–727, 2012.  With an appendix by T.Braden.
  * [Yun09] Zhiwei Yun.  Weights of mixed tilting sheaves and geometric Ringel duality.  Selecta Math., 14(2):299–320, 2009.