\section{The partial Frobenius and the constant term}
\label{rw:app:partial-frobenius}

This appendix proves
Proposition~\ref{fl:lem:constant-term-frobenius-normalization}. The
proposition enters the proof of Theorem~\ref{fl:thm:normalized-ct},
and through it the realization of the parameters which are not
elliptic; it is not used for elliptic parameters. The proof is a
computation of normalizations: it determines a power of $q$ and
shows that an auxiliary choice cancels. It is independent of the
rest of the paper, which is why it is placed here. The notation is
that of Section~\ref{fl:sec:constant-term}.

\subsection{The Frobenius of the torsor of jets}


Let $Y\subset X^I$ be an open subset on which the legs, and the
finitely many translates of the legs by powers of Frobenius which
occur in the composite of maps under consideration, are pairwise
distinct. Choose an integer $d$ such that, for all the bounds on
modifications which occur, the action of the groups of jets on
the bounded affine Grassmannians factors through the jets of
order $d$. For $H\in\{G,P,M\}$ let $H_{i,d}$ be the group of jets
of order $d$ at the leg $i$, and put
\[
 U_{i,d}=\ker(P_{i,d}\to M_{i,d}),\qquad
 m_i=\dim U_{i,d},\qquad m=\sum_{i\in I}m_i .
\]
Fix a leg $i$. For $a=1,2$ let $\operatorname{Cht}^a_H$ be the
stack of shtukas of $H$ whose modifications are taken in the order
given by the partition $(\{i\},I\setminus\{i\})$ for $a=1$ and
$(I\setminus\{i\},\{i\})$ for $a=2$. The partial Frobenius
morphism at $i$ maps $\operatorname{Cht}^1_H$ to
$\operatorname{Cht}^2_H$. We write
$\operatorname{LocMod}^a_H$ for the corresponding stacks of local
modifications, the quotients of the bounded affine Grassmannians
by the groups of jets.

The constant-term morphism of Xue is defined by means of the stack
\[
 \operatorname{Cht}^{a,\mathrm{corr}}_M
  =\operatorname{Cht}^a_M
       \times_{\operatorname{LocMod}^a_M}\operatorname{LocMod}^a_P ,
\]
which adds to a shtuka of $M$ the local data of a modification of
$P$-bundles. The map
$\pi_{a,d}:\operatorname{Cht}^a_P\to
\operatorname{Cht}^{a,\mathrm{corr}}_M$ is smooth of relative
dimension $m$ \cite[Lemma~3.1.8 and (3.9)]{XueCusp2020}. The
partial Frobenius morphisms of $P$ and of $M$ are compatible with
these maps, and define
\[
 c:\operatorname{Cht}^1_P\longrightarrow
 \mathcal B'_i=
 \operatorname{Cht}^{1,\mathrm{corr}}_M
   \times_{\operatorname{Fr}^{\mathrm{corr}}_{M,i},
                 \operatorname{Cht}^{2,\mathrm{corr}}_M}
                   \operatorname{Cht}^2_P .
\]
The map $c$ sends a shtuka of $P$ to its image in the corrected
stack for $M$ together with its image under the partial
Frobenius.

\begin{lemma}[Rotation and its relative Frobenius]
\label{fl:lem:constant-term-rotation-square}
The map $c$ is a universal homeomorphism, finite and flat of
degree $q^{m_i}$.
\end{lemma}

\begin{proof}
All fibre products in this proof are taken with their scheme
structures, and the degree of a relative Frobenius morphism is
computed on its fibres as schemes. The \'etale site, and with it
the sheaves, are the same for a scheme and for its reduction.

The proof has two steps. The first recalls, for each of the groups
$G$, $P$ and $M$, the description of a partial Frobenius morphism
as a base change of the relative Frobenius of a torsor under a
group of jets, due to V.~Lafforgue. The second compares these
descriptions for $P$ and for $M$, and this is where the degree
$q^{m_i}$ appears: it is the order of the kernel of Frobenius on a
form of a group of jets of the unipotent radical.

\step{1}{the rotation square} Let $H$ be one of the three groups.
The objects introduced in this step concern the leg $i$ only: they
live over the curve $X-N$ in which this leg varies, and not over
$Y$. Let $Z_H$ be the bounded space of local modifications at one
point, with its action of $H_{i,d}$, and
\[
 \operatorname{LocMod}_{H,i}=[H_{i,d}\backslash Z_H],\qquad
 \mathcal B_{H,i}=\Bun_{H,N}\times\operatorname{LocMod}_{H,i}.
\]
A point of $\operatorname{LocMod}_{H,i}$ is a point $x$ of $X-N$,
a torsor $\mathcal P$ under $H_{i,d}$ and an equivariant map
$\mathcal P\to Z_H$. So $\mathcal B_{H,i}$ carries two torsors
under $H_{i,d}$: the torsor $\mathcal P$, and the torsor
$\mathcal T$ of trivializations of the restriction of the universal
bundle to the divisor $d\,\Gamma_x$. Let $\operatorname{Mod}_{H,i}$
be the stack of modifications $c_1:E_0\dashrightarrow E_1$ at one
point of $X-N$, with the given bound, where $E_1$ has a structure
of level $N$. Sending $c_1$ to the pair formed by $E_1$ and the
local modification of $c_1$ defines a map
\[
 \operatorname{loc}_H:\operatorname{Mod}_{H,i}
   \longrightarrow\mathcal B_{H,i},
\]
and a modification with a given image is the same as an
isomorphism of torsors $\mathcal P\simeq\mathcal T$. So
$\operatorname{loc}_H$ is a torsor under the group scheme
$\mathcal A_H=\underline{\operatorname{Aut}}(\mathcal P)$ over
$\mathcal B_{H,i}$, an inner form of $H_{i,d}$
\cite[Construction~1.8 and Remark~1.11]{VLa18}. This group scheme
is smooth of relative dimension $\dim H_{i,d}$. Let
\[
 F_{\operatorname{Mod}/\mathcal B}:\operatorname{Mod}_{H,i}
 \longrightarrow\operatorname{Mod}_{H,i}^{(q)}
 =\mathcal B_{H,i}\times_{F,\mathcal B_{H,i}}\operatorname{Mod}_{H,i}
\]
be the Frobenius of $\operatorname{Mod}_{H,i}$ relative to
$\mathcal B_{H,i}$, where $F$ is the Frobenius endomorphism of
$\mathcal B_{H,i}$. It is equivariant with respect to the relative
Frobenius of $\mathcal A_H$, a faithfully flat homomorphism whose
kernel $\mathcal A_H[F]$ is finite, locally free of rank
$q^{\dim H_{i,d}}$ and infinitesimal. So
$F_{\operatorname{Mod}/\mathcal B}$ is a torsor under
$\mathcal A_H[F]$.

For a bundle $E$ over $X\times S$, with $S$ a scheme over
$\mathbb F_q$, write
${}^{\tau}E=(\operatorname{id}_X\times\operatorname{Frob}_{q,S})^*E$.
A point of $\operatorname{Cht}^1_H$ consists of legs
$(x_j)_{j\in I}$ in $Y$ and a pair of modifications
\[
 c_1:E_0\dashrightarrow E_1,
 \qquad c_2:E_1\dashrightarrow{}^\tau E_0 ,
\]
the first at the leg $x_i$ and the second at the other legs. Its
image under the partial Frobenius is the pair $(c_2,{}^\tau c_1)$;
the leg $i$ has become $\operatorname{Frob}(x_i)$ and the other legs
are unchanged. Accordingly $\operatorname{Cht}^2_H$ is considered
over the open subset of $X^I$ which is the image of $Y$ under the
Frobenius of the factor $i$. Let
\[
 \mathcal T'_H=\operatorname{Cht}^2_H
   \times_{\operatorname{LocMod}_{H,i},F}\operatorname{LocMod}_{H,i},
\]
where the map from $\operatorname{Cht}^2_H$ is the local
modification of the second modification, the one at the leg $i$,
and $F$ is the Frobenius endomorphism of
$\operatorname{LocMod}_{H,i}$. So a point of $\mathcal T'_H$ is a
point of $\operatorname{Cht}^2_H$, whose second modification we
call $c_{\mathrm{mov}}$, a point $z$ of
$\operatorname{LocMod}_{H,i}$, and an isomorphism between the local
modification of $c_{\mathrm{mov}}$ and $F(z)$. Only the leg $i$ is
involved in $z$; the other legs belong to the point of
$\operatorname{Cht}^2_H$ and are not touched. If $E_1$ is the first
bundle of the point of $\operatorname{Cht}^2_H$, then $b_0=(E_1,z)$
is a point of $\mathcal B_{H,i}$, the target of $c_{\mathrm{mov}}$
is ${}^\tau E_1$, and the data identify
$\operatorname{loc}_H(c_{\mathrm{mov}})$ with
$F(b_0)=({}^\tau E_1,F(z))$. This defines a map from
$\mathcal T'_H$ to $\operatorname{Mod}_{H,i}^{(q)}$. The square
\begin{equation}\label{fl:eq:ct-framed-rotation-square}
\begin{tikzcd}[column sep=large]
 \operatorname{Cht}^{1}_H\arrow[r,"\operatorname{rot}_i"]
    \arrow[d] & \mathcal T'_H\arrow[d]\\
 \operatorname{Mod}_{H,i}
   \arrow[r,"F_{\operatorname{Mod}/\mathcal B}"']
   &\operatorname{Mod}_{H,i}^{(q)}
\end{tikzcd}
\end{equation}
in which the left vertical map is $(c_1,c_2)\mapsto c_1$, the
upper map is the partial Frobenius with $z$ the local
modification of $c_1$, and the lower map is the relative
Frobenius, is $2$-Cartesian. Indeed, a point of the fibre product
consists of $c_1$, of a point $(c_2,c_{\mathrm{mov}})$ of
$\operatorname{Cht}^2_H$ and of an isomorphism
$c_{\mathrm{mov}}\simeq{}^\tau c_1$ which induces the given
identification of local modifications. The target of $c_1$ is the
first bundle $E_1$ of the second datum, and
$c_2:E_1\dashrightarrow{}^\tau E_0$ is then the second modification
of a shtuka in $\operatorname{Cht}^1_H$, whose legs lie in $Y$
because their image under the Frobenius of the factor $i$ lies in
the image of $Y$. This construction is inverse to the map from
$\operatorname{Cht}^1_H$, on objects and on isomorphisms. The
statement is the variant, with trivializations at the target, of
\cite[Proposition~3.4 and Remark~3.5]{VLa18}; the argument does
not use that the group is reductive.

It follows that
$\operatorname{rot}_i:\operatorname{Cht}^1_H\to\mathcal T'_H$ is a
base change of $F_{\operatorname{Mod}/\mathcal B}$. So it is
representable, finite and locally free of degree
$q^{\dim H_{i,d}}$, and it is a torsor under the pullback of
$\mathcal A_H[F]$.

\step{2}{the map $c$} A point of $\mathcal B'_i$ consists of a
shtuka $(c_1^M,c_2^M)$ of $M$ in $\operatorname{Cht}^1_M$, of local
modifications of $P$-bundles at the legs which lift those of this
shtuka, of a shtuka $s'$ of $P$ in $\operatorname{Cht}^2_P$, and of
an identification of the images of these data in
$\operatorname{Cht}^{2,\mathrm{corr}}_M$. Let $z_P$ be the local
modification of $P$-bundles at the leg $i$, and $z_M$ its image.
Then $(s',z_P)$ is a point of $\mathcal T'_P$, and by step~1 for
$H=P$ the shtukas of $P$ with image $(s',z_P)$ under
$\operatorname{rot}_i$ form a torsor $\mathcal R_P$ under
$\mathcal A_P[F]$, the fibre of
$F_{\operatorname{Mod}/\mathcal B}$ for $P$. In the same way the
image of $(s',z_P)$ in $\mathcal T'_M$ defines a torsor
$\mathcal R_M$ under $\mathcal A_M[F]$, of which the shtuka
$(c_1^M,c_2^M)$ is a point. The fibre of $c$ over the given point
of $\mathcal B'_i$ is the fibre over this point of the map
$\mathcal R_P\to\mathcal R_M$ induced by $P\to M$: a shtuka of $P$
in $\operatorname{Cht}^1_P$ lies over the given point of
$\mathcal B'_i$ if and only if its image under
$\operatorname{rot}_i$ is $(s',z_P)$ and its image in
$\operatorname{Cht}^1_M$ is $(c_1^M,c_2^M)$.

The map $\mathcal R_P\to\mathcal R_M$ is equivariant with respect
to the homomorphism $\mathcal A_P[F]\to\mathcal A_M[F]$. Since
$P_{i,d}$ is the semidirect product of $U_{i,d}$ and $M_{i,d}$,
compatibly with Frobenius, the homomorphism
$\mathcal A_P\to\mathcal A_M$ is smooth and surjective, its kernel
$\mathcal U$ is a form of $U_{i,d}$, and
\[
 1\longrightarrow\mathcal U[F]\longrightarrow\mathcal A_P[F]
 \longrightarrow\mathcal A_M[F]\longrightarrow1
\]
is an exact sequence of finite locally free group schemes: locally
for the flat topology it is the sequence of the Frobenius kernels
of $U_{i,d}$, $P_{i,d}$ and $M_{i,d}$. So the fibres of
$\mathcal R_P\to\mathcal R_M$ are torsors under $\mathcal U[F]$,
which is finite, locally free of rank $q^{\dim U_{i,d}}=q^{m_i}$,
and infinitesimal. This holds over every scheme mapping to
$\mathcal B'_i$. Therefore $c$ is representable, finite and locally
free of degree $q^{m_i}$, and a universal homeomorphism.

As a check, the composite of $c$ with the map
$\mathcal B'_i\to\mathcal T'_P$ which sends a point as above to
$(s',z_P)$ is $\operatorname{rot}_i$ for $P$, of degree
$q^{\dim P_{i,d}}$; and $\mathcal B'_i\to\mathcal T'_P$ is a base
change of $\operatorname{rot}_i$ for $M$, of degree
$q^{\dim M_{i,d}}$. The quotient of the two degrees is $q^{m_i}$.
\end{proof}

\begin{lemma}[The trace of the map $c$]
\label{fl:lem:constant-term-rotation-trace}
Let $\pi_0=\pi_{1,d}$, and let
$\pi_1:\mathcal B'_i\to\operatorname{Cht}^{1,\mathrm{corr}}_M$ be
the base change of $\pi_{2,d}$, so that $\pi_0=\pi_1c$. Under the
purity isomorphisms for the smooth maps $\pi_0$ and $\pi_1$, the
trace map of $\pi_0$ composed with the pullback by $c$ is
$q^{m_i}$ times the trace map of $\pi_1$. The same holds after any
base change.
\end{lemma}

\begin{proof}
Both $\pi_0$ and $\pi_1$ are smooth of relative dimension $m$.
The constant-term morphism uses the trace map
\[
 (\pi_0)_!(\pi_0)^*[2m](m)
       \simeq(\pi_0)_!(\pi_0)^!
       \xrightarrow{\epsilon_{\pi_0}}\operatorname{id},
\]
as in \cite[(3.22)--(3.23)]{XueCusp2020}. Write $\pi_0=\pi_1c$.
For a sheaf $\mathcal K$ pulled back from
$\operatorname{Cht}^{1,\mathrm{corr}}_M$ to $\mathcal B'_i$, the
purity isomorphisms of $\pi_0$ and $\pi_1$ identify the
comparison between the two trace maps with
\begin{equation}\label{fl:eq:ct-finite-flat-coefficient-trace}
 c_!c^*\mathcal K\xrightarrow{c_!\operatorname{cl}_c}
 c_!c^!\mathcal K\xrightarrow{\epsilon_c}\mathcal K,
\end{equation}
where $\operatorname{cl}_c$ is the fundamental class of the
finite flat map $c$; this follows from the rule
$\epsilon_{\pi_0}=\epsilon_{\pi_1}\circ(\pi_1)_!\epsilon_c$ for
the composition of trace maps. For a finite flat map, the
adjunction map $\mathcal K\to c_*c^*\mathcal K$ followed by
\eqref{fl:eq:ct-finite-flat-coefficient-trace} is multiplication
by the degree \cite[Theorem~0.2.1 and (0.3.2)]{Laumon1976}, which
is $q^{m_i}$ by
Lemma~\ref{fl:lem:constant-term-rotation-square}. For the
constant sheaf this is an identity between maps of sheaves in
degree zero, because $c$ is a finite universal homeomorphism. The
projection formula gives it for every $\mathcal K$, complexes
included. Therefore
\[
 \operatorname{tr}_{\pi_0}\circ c^{*}
       =q^{m_i}\operatorname{tr}_{\pi_1}.
\]
Only the leg $i$ contributes, since the maps on the
trivializations at the other legs are the same on the two sides.
Base change preserves the purity isomorphisms and the trace maps,
and so does an isomorphism of the trivializations.
\end{proof}

\subsection{Proof of the proposition}

\begin{proof}[Proof of Proposition~\ref{fl:lem:constant-term-frobenius-normalization}]
Step~1 fixes the notation. The computation is in step~2. The square
formed by the constant-term maps and the partial Frobenius
morphisms is not Cartesian, and the factor $q^{-n_i/2}$ of
\eqref{fl:eq:ct-coefficient-rotation-identity} is the product of
the factor $q^{m_i}$ of
Lemma~\ref{fl:lem:constant-term-rotation-trace} and of the twist
$q^{-n_i/2-m_i}$ of the sheaves. Step~3 passes to cohomology.

\step{1}{notation, and the map of coefficients} We work over $Y$
and over its image under the Frobenius of the factor $i$. For
$H\in\{G,P,M\}$ let $\operatorname{LocMod}_H$ be the stack of local
modifications at all the legs, which over these open subsets is
the product over $j\in I$ of the stacks
$\operatorname{LocMod}_{H,j}$ of the proof of
Lemma~\ref{fl:lem:constant-term-rotation-square}. Let
\[
 \epsilon^a_H:\operatorname{Cht}^a_H\to\operatorname{LocMod}_H,
 \qquad
 \iota:\operatorname{LocMod}_P\to\operatorname{LocMod}_G,
 \qquad
 \rho:\operatorname{LocMod}_P\to\operatorname{LocMod}_M
\]
be the maps which take local modifications and which are induced
by $P\subset G$ and $P\to M$, and let $\Phi_H$ be the endomorphism
of $\operatorname{LocMod}_H$ which is the Frobenius endomorphism of
the factor $i$ and the identity on the other factors. The maps
$\iota$ and $\rho$ commute with $\Phi$, and
\begin{equation}\label{rw:eq:ct-local-frobenius}
 \epsilon^2_H\circ\operatorname{Fr}_{H,i}
   =\Phi_H\circ\epsilon^1_H ,
\end{equation}
because the local modification of ${}^\tau c_1$ is the image under
Frobenius of that of $c_1$.

Let $\mathcal S_G$ be the Satake sheaf of $V$ on
$\operatorname{LocMod}_G$, an exterior product over the legs, and
$\mathcal S_M=\bigoplus_\mu\mathcal S_{M,\mu}$ that of
$\operatorname{Res}V$, decomposed according to the weights
$\mu=(\mu_j)_{j\in I}$ of the centre of $\widehat M^I$. Every sheaf
$\mathcal K$ on a stack over $\mathbb F_q$ has a canonical
isomorphism with its pullback by the Frobenius endomorphism of the
stack, and these isomorphisms are compatible with pullbacks, with
direct images with proper supports and with base change. Applied to
the factor $i$ they give isomorphisms
\[
 \phi_G:\Phi_G^*\mathcal S_G\simeq\mathcal S_G,\qquad
 \phi_M:\Phi_M^*\mathcal S_M\simeq\mathcal S_M .
\]
By \eqref{rw:eq:ct-local-frobenius} they induce isomorphisms
$\operatorname{Fr}_{H,i}^*\mathcal F^2_H\simeq\mathcal F^1_H$ for
the sheaves $\mathcal F^a_H=(\epsilon^a_H)^*\mathcal S_H$ on the
stacks of shtukas, and the partial Frobenius map $F^H_i$ on
cohomology is the pullback by $\operatorname{Fr}_{H,i}$ followed by
this isomorphism \cite[\S\S3 and~4.3]{VLa18}. For the constant
sheaf $E(a)$ on a stack over $\mathbb F_q$ the canonical
isomorphism is the action of the geometric Frobenius, that is,
multiplication by $q^{-a}$ when $E(a)$ is identified with its
pullback as a sheaf coming from $\operatorname{Spec}\mathbb F_q$.

Lemma~\ref{fl:lem:constant-term-arithmetic-restriction}, together
with the fact that the cohomology with compact supports of the
classifying stack of $U_{j,d}$ is $E[2m_j](m_j)$
\cite[(3.18)]{XueCusp2020}, gives an isomorphism of sheaves on
$\operatorname{LocMod}_M$
\begin{equation}\label{rw:eq:ct-local-iso}
 \lambda:\rho_!\iota^*\mathcal S_G\simeq
 \bigoplus_\mu\mathcal S_{M,\mu}\otimes\mathcal L_\mu,
 \qquad
 \mathcal L_\mu=\bigotimes_{j\in I}
     E[n_{\mu_j}+2m_j]\bigl(\tfrac{n_{\mu_j}}2+m_j\bigr),
\end{equation}
which is an exterior product over the legs of isomorphisms defined
over $\mathbb F_q$. Let
$\widetilde\rho_a:\operatorname{Cht}^{a,\mathrm{corr}}_M\to
\operatorname{Cht}^a_M$ and
$\widetilde\epsilon_a:\operatorname{Cht}^{a,\mathrm{corr}}_M\to
\operatorname{LocMod}_P$ be the two projections, so that the map
$\pi^a:\operatorname{Cht}^a_P\to\operatorname{Cht}^a_M$ is
$\widetilde\rho_a\pi_{a,d}$ and
$\epsilon^a_P=\widetilde\epsilon_a\pi_{a,d}$. The map of
coefficients which defines the constant-term morphism
\cite[(3.18)--(3.23)]{XueCusp2020} is the composite
\begin{equation}\label{rw:eq:ct-coefficient-map}
\begin{aligned}
 \kappa_a:\ (\pi^a)_!(\epsilon^a_P)^*\iota^*\mathcal S_G
 &=(\widetilde\rho_a)_!(\pi_{a,d})_!\pi_{a,d}^*
       \widetilde\epsilon_a^*\iota^*\mathcal S_G\\
 &\xrightarrow{\ \operatorname{tr}\ }
   (\widetilde\rho_a)_!\widetilde\epsilon_a^*\iota^*
        \mathcal S_G[-2m](-m)\\
 &\xrightarrow{\ \sim\ }
   (\epsilon^a_M)^*\rho_!\iota^*\mathcal S_G[-2m](-m)\\
 &\xrightarrow{\ \lambda\ }
   \bigoplus_\mu\mathcal F^a_{M,\mu}[n_\mu]\bigl(\tfrac{n_\mu}2\bigr),
\end{aligned}
\end{equation}
where the first map is the trace map of the smooth map
$\pi_{a,d}$, the second is the base change isomorphism of the
Cartesian square which defines
$\operatorname{Cht}^{a,\mathrm{corr}}_M$, and in the third the
twist $(m)$ of $\mathcal L_\mu$ is cancelled against $(-m)$; here
$n_\mu=\sum_jn_{\mu_j}$. The component of
$\mathsf{CT}^{\mathrm{geo}}$ on the summand $\mu$ is obtained by
identifying $E(n_\mu/2)$ with $E$; these identifications are chosen
once and for all, compatibly with tensor products.

\step{2}{the identity of maps of sheaves} We claim that, on the
summand of weight $\mu$,
\begin{equation}\label{fl:eq:ct-coefficient-rotation-identity}
 \kappa_1\circ(\pi^1)_!(\epsilon^1_P)^*\iota^*(\phi_G)
      \circ\mathrm{BC}
 =q^{-n_i/2}\,\phi_M\circ\operatorname{Fr}_{M,i}^*(\kappa_2),
 \qquad n_i=n_{\mu_i},
\end{equation}
as maps from
$\operatorname{Fr}_{M,i}^*(\pi^2)_!(\epsilon^2_P)^*\iota^*\mathcal S_G$
to $\mathcal F^1_{M,\mu}[n_\mu]$. Here $\mathrm{BC}$ is the base
change map of the commutative square formed by $\pi^1$, $\pi^2$
and the partial Frobenius morphisms of $P$ and of $M$: the pullback
of sheaves along the map from $\operatorname{Cht}^1_P$ to the fibre
product. The square is not Cartesian, and this is the point of the
computation. We compare the three maps of
\eqref{rw:eq:ct-coefficient-map} for $a=1$ and $a=2$ in turn.

(a) \emph{The trace map.} The square formed by $\pi_{1,d}$,
$\pi_{2,d}$ and the partial Frobenius morphisms factors through
the Cartesian square with corner $\mathcal B'_i$ and the map $c$.
The trace map of $\pi_{2,d}$ is compatible with the base change to
$\mathcal B'_i$, and by
Lemma~\ref{fl:lem:constant-term-rotation-trace} the trace map of
$\pi_{1,d}=\pi_1c$ composed with the pullback by $c$ is $q^{m_i}$
times the trace map of $\pi_1$. Hence, for every sheaf $\mathcal K$
on $\operatorname{Cht}^{2,\mathrm{corr}}_M$,
\[
 \operatorname{tr}_{\pi_{1,d}}\circ\mathrm{BC}_d
   =q^{m_i}\,(\operatorname{Fr}^{\mathrm{corr}}_{M,i})^*
        (\operatorname{tr}_{\pi_{2,d}})
\]
as maps from
$(\operatorname{Fr}^{\mathrm{corr}}_{M,i})^*(\pi_{2,d})_!\pi_{2,d}^*
\mathcal K$ to
$(\operatorname{Fr}^{\mathrm{corr}}_{M,i})^*\mathcal K[-2m](-m)$,
where $\mathrm{BC}_d$ is the base change map of the square. In this
identity the sheaf $E(-m)$ is identified with its pullback as a
sheaf coming from $\operatorname{Spec}\mathbb F_q$; the partial
Frobenius morphisms are morphisms of stacks over $\mathbb F_q$.

(b) \emph{The base change isomorphism.} The Cartesian squares
which define $\operatorname{Cht}^{1,\mathrm{corr}}_M$ and
$\operatorname{Cht}^{2,\mathrm{corr}}_M$ are the two ends of a
commutative cube whose other edges are
$\operatorname{Fr}^{\mathrm{corr}}_{M,i}$,
$\operatorname{Fr}_{M,i}$, $\Phi_P$ and $\Phi_M$. Base change maps
for composite squares are composites of base change maps. So the
second map of \eqref{rw:eq:ct-coefficient-map} transforms the base
change map of $\widetilde\rho$ with respect to the partial
Frobenius morphisms into the pullback by $\epsilon^1_M$ of the
base change map of $\rho$ with respect to $\Phi_P$ and $\Phi_M$.

(c) \emph{The local isomorphism.} Since the canonical isomorphisms
with Frobenius pullbacks are compatible with $\iota^*$, with
$\rho_!$ and with base change, the isomorphism
$\rho_!\iota^*(\phi_G)$, composed with the base change map of
$\rho$ with respect to $\Phi$, is the canonical isomorphism of the
sheaf $\rho_!\iota^*\mathcal S_G$ for the factor $i$. The
isomorphism $\lambda$ of \eqref{rw:eq:ct-local-iso} is an exterior
product of isomorphisms defined over $\mathbb F_q$, so it
transforms this canonical isomorphism into that of the right-hand
side, which on the summand $\mu$ is the tensor product of $\phi_M$
and of the canonical isomorphism of the factor $i$ of
$\mathcal L_\mu$. By step~1 the latter is multiplication by
\[
 q^{-n_i/2-m_i}.
\]

Combining (a), (b) and (c): the composite on the left of
\eqref{fl:eq:ct-coefficient-rotation-identity} is
$\phi_M\circ\operatorname{Fr}_{M,i}^*(\kappa_2)$ multiplied by
\[
 q^{m_i}\cdot q^{-n_i/2-m_i}=q^{-n_i/2},
\]
in which the auxiliary dimension $m_i$ has cancelled, in agreement
with the fact that the constant-term morphism does not depend on
$d$. The shifts do
not enter an identity between maps of sheaves; their effect on
trace functions is discussed in Remark~\ref{rw:rem:signed-satake}.

It remains to pass to cohomology with compact supports. Fix the
degree of the $M$-bundle at the target and use the bounded
correspondences of \cite[\S3.5]{XueCusp2020}. On the dense open
subset of Proposition~3.5.3 of that paper the pullback from $G$ to
$P$ is the pullback along a proper map of the correspondence.
Without a central lattice the same proper maps are obtained on the
open and closed components of given degree of the correspondence
with a central lattice. For the partial Frobenius maps take the inverse
image of the bounded stack at the target, as in
\cite[\S4.3]{VLa18}, and extend by zero to a larger bound which
contains it. The comparison of trace maps and base change maps
above is compatible with these restrictions and with the
extensions by zero when the bound increases
\cite[\S3.5.9]{XueCusp2020}. Hence \eqref{fl:eq:ct-coefficient-rotation-identity} passes to
the limit. This is an identity on the generic stalk, which is
where the partial Frobenius maps enter the construction of the
action of $W_F^I$. Over the diagonals no Frobenius is involved:
there the constant-term morphism is compatible with fusion by
Lemma~\ref{fl:lem:constant-term-arithmetic-restriction} and
\cite[Lemma~4.1.5]{XueHecke2020}, and on the diagonal the summand
of weight $\mu$ lies in the summand whose weight is the sum of the
$\mu_j$, with $n$ additive.

\step{3}{normalization} The constant-term morphism on cohomology
is the pullback from the shtukas of $G$ to those of $P$, which
commutes with the partial Frobenius maps, followed by the direct
image of $\kappa_a$. The partial Frobenius morphism of $M$ maps the
shtukas of degree $\nu$ whose modification at the leg $i$ has
weight $\mu_i$ to shtukas of degree $\nu-\mu_i$. So taking
cohomology in \eqref{fl:eq:ct-coefficient-rotation-identity} gives
the first formula of the proposition. With the factor of
Lemma~\ref{fl:lem:constant-term-haar-measure},
\[
 \begin{aligned}
 c_U\delta_P(\nu)^{1/2}
       \mathsf{CT}^{\mathrm{geo}}_\nu F_i^G
 &=F_i^M c_U\delta_P(\nu)^{1/2}q^{-n_i/2}
                  \mathsf{CT}^{\mathrm{geo}}_{\nu-\mu_i}\\
 &=F_i^M c_U\delta_P(\nu-\mu_i)^{1/2}
                  \mathsf{CT}^{\mathrm{geo}}_{\nu-\mu_i},
 \end{aligned}
\]
because $\delta_P(\nu-\mu_i)=q^{-n_i}\delta_P(\nu)$. This is the
second assertion. Note finally that for a shtuka the sum of the
$\mu_i$ has image zero in the cocharacter group of
$M/M_{\mathrm{der}}$, because the pullback by Frobenius does not
change the degree of a bundle. So the sum of the shifts $[-n_i]$
is zero, and the constant-term morphism preserves the
cohomological degree.
\end{proof}

\begin{remark}
The map $c$ of Lemma~\ref{fl:lem:constant-term-rotation-square}
is a universal homeomorphism, so it induces an equivalence of
\'etale sites. Its trace map is nevertheless multiplication by
its degree $q^{m_i}$ and not the identity, and this is the factor
that cancels the dependence on the order of the jets.
\end{remark}

\begin{remark}[Signs of trace functions]\label{rw:rem:signed-satake}
Proposition~\ref{fl:lem:constant-term-frobenius-normalization} is an
identity between maps of cohomology groups, and no sign occurs in
it. Signs occur when Satake sheaves are replaced by functions. The
shift by $n_i$ multiplies the trace function of
Section~\ref{fl:app:conventions} by $(-1)^{n_i}$, and a negative
half Tate twist cannot absorb this sign at all places, since at a
place of degree $d$ it would contribute $(-1)^d$. The function
attached to a Satake sheaf $\mathcal S$ supported on a connected
component $c$ of the affine Grassmannian is therefore taken with
the sign of \cite[before Remark~1.21]{VLa18}:
\[
 h_{\mathcal S}=(-1)^{p_G(c)}t_{\mathcal S},\qquad
 p_G(c)=\langle2\rho_G,\lambda\rangle\bmod2
 \quad\text{for }\lambda\in c .
\]
This sign is multiplicative for convolution, and with it the
function of the sheaf attached to $V$ is the element of the
spherical Hecke algebra whose Satake transform is the character of
$V$. For example, for $\mathrm{PGL}_2$ and the minuscule coweight
the intersection complex has trace $-q_v^{-1/2}$ on its support,
and the function is $q_v^{-1/2}$ there; here $q_v$ is the
cardinality of the residue field of the place at which the trace
is taken. In the
restriction at the leg $i$, the component $\mu_i$ of the
modifications of $M$ lies over a component $c$ of the
modifications of $G$, and $n_i\equiv p_G(c)-p_M(\mu_i)\pmod2$ by
the definition of $n_i$. So the three signs $(-1)^{p_G(c)}$,
$(-1)^{n_i}$ and $(-1)^{p_M(\mu_i)}$ cancel, and the comparison of
functions has the factor $q_v^{-n_i/2}$ and no sign.
For a semisimple simply connected group, in particular in
\cite{HuII}, all coweights lie in the coroot
lattice and $p_G=0$.
\end{remark}

\begin{remark}[A check on the normalization]
\label{rw:rem:ct-normalization-check}
In Proposition~\ref{fl:lem:constant-term-frobenius-normalization} the
scalar $q^{-n_i/2}$ appears when the
Satake sheaves of $M$ are normalized to weight zero at each leg
separately, and the degree changes from $\nu$ to $\nu-\mu_i$
because the partial Frobenius replaces the first bundle of a shtuka
by the second. Both can be checked on Hecke operators. For
an excursion operator which is a Hecke operator at an unramified
place, Theorem~\ref{fl:thm:normalized-ct} says that
$\mathsf{CT}^{\mathrm{nor}}_P$ intertwines a spherical Hecke
operator of $G$ with its image under the homomorphism of Hecke
algebras which corresponds to the restriction of representations
from $\widehat G$ to $\widehat M$. That homomorphism is the
Satake transform with the factor $\delta_P^{1/2}$, and the
statement is the classical one: if $\varphi$ lies in a
representation induced from $\pi\otimes\delta_P^{1/2}$, then
$m\mapsto\delta_P(m)^{-1/2}\varphi(mg)$ transforms according to
$\pi$. With another power of $q$ in
Proposition~\ref{fl:lem:constant-term-frobenius-normalization} the
exponent of $\delta_P$ would be different.

Here is the check in the simplest case, by counting. Let
$G=\mathrm{GL}_2$ and $P$ the upper triangular subgroup, so that
$\mathsf{CT}^{\mathrm{geo}}h(L_1,L_2)$ is the sum of $h(E)$ over
the extensions $E$ of $L_2$ by $L_1$, divided by
$|\operatorname{Hom}(L_2,L_1)|$, and
$\delta_P(L_1,L_2)=q^{\deg L_1-\deg L_2}$. Let $v$ be a place of
degree one, and $T_v$ the Hecke operator of the double coset which
enlarges lattices; it is the transpose of the one which shrinks
them. It sums a function over the
$q+1$ bundles $E'\supset E$ with $E'/E$ of length one at $v$. For
an extension $E$, one of these bundles is the pushout of $E$ along
$L_1\to L_1(v)$, and the other $q$ are extensions of $L_2(v)$ by
$L_1$. The pushout map
$\operatorname{Ext}^1(L_2,L_1)\to\operatorname{Ext}^1(L_2,L_1(v))$
is surjective with kernel of order
$q\,|\operatorname{Hom}(L_2,L_1)|/|\operatorname{Hom}(L_2,L_1(v))|$,
by the cohomology sequence of
$0\to L_2^{-1}L_1\to L_2^{-1}L_1(v)\to k_v\to0$. And the pairs
$E\subset E'$ of the second kind correspond, by pullback along
$L_2\to L_2(v)$, to the extensions $E'$ of $L_2(v)$ by $L_1$, with
the same automorphisms. Hence
\[
 \mathsf{CT}^{\mathrm{geo}}(T_vh)(L_1,L_2)
 =q\,\mathsf{CT}^{\mathrm{geo}}h(L_1(v),L_2)
  +\mathsf{CT}^{\mathrm{geo}}h(L_1,L_2(v)).
\]
Since
$\delta_P(L_1(v),L_2)=q\,\delta_P(L_1,L_2)$ and
$\delta_P(L_1,L_2(v))=q^{-1}\delta_P(L_1,L_2)$, the two
coefficients are $q^{1/2}(\delta_P(m')/\delta_P(m))^{1/2}$, where
$m'$ is the pair on the right. So
$\delta_P^{1/2}\mathsf{CT}^{\mathrm{geo}}$ intertwines
$q^{-1/2}T_v$ with the sum of the two translations, as the proposition
says: the coefficient of the term in which the degree of the
$M$-bundle is changed by $-\mu$ is $q^{-n_\mu/2}$ times the
symmetric one.
\end{remark}