\section{Constant terms and the excursion character}
\label{fl:sec:constant-terms-cuspidality}
\label{fl:sec:constant-term}

An Eisenstein series must have the parameter of the function it
is induced from, composed with the inclusion of dual Levi
subgroups. To prove this in Section~\ref{fl:sec:eisenstein} we
need the constant term to intertwine \emph{all} excursion
operators of $G$ with those of the Levi subgroup, which is
Theorem~\ref{fl:thm:normalized-ct} below. Compatibility with Hecke
operators at unramified places would not be enough there, since
those determine a parameter only up to local conjugacy at each
place.

The theorem is used in Section~\ref{fl:sec:eisenstein}, that is,
for the parameters which are not elliptic in Theorems~B and~C, and
nowhere else. The criterion for cuspidality of
Section~\ref{fl:sec:arithmetic-cuspidality}, on which Theorem~A
and the two constructions \cite{HuII,HuIII}
depend, uses only the elementary
Lemma~\ref{rw:lem:ct-spherical} at the end of this section.

Fix a split maximal torus and a Borel subgroup containing it, in
the split model of $G$ over the integers. Let $P=MU$ be a standard
parabolic subgroup with its standard Levi subgroup, and let
$i:\widehat M\hookrightarrow\widehat G$ be the inclusion of the
dual Levi subgroup. For an excursion datum
$e=(I,f,\boldsymbol\gamma)$ of $G$ put
$i^*e=(I,f\circ i^{\times I},\boldsymbol\gamma)$, an excursion
datum of $M$. Fix a Haar measure on $U(\mathbb A)$, and let $du$
be the measure which it induces on $U(F)\backslash U(\mathbb A)$.
Every element of $G(\mathbb A)$ is of the form $umg_0$ with
$u\in U(\mathbb A)$, $m\in M(\mathbb A)$ and
$g_0\in G(\prod_v\mathcal O_v)$; this is the Iwasawa
decomposition. The normalized constant term of a
function $h$ is
\[
 \mathsf{CT}^{\mathrm{nor}}_P h(mg_0)
 =\delta_P(m)^{-1/2}
       \int_{U(F)\backslash U(\mathbb A)}h(umg_0)\,du .
\]
The integral exists for every smooth function of finite level,
since $U(F)\backslash U(\mathbb A)$ is compact. For fixed $g_0$ we
regard $m\mapsto\mathsf{CT}^{\mathrm{nor}}_Ph(mg_0)$ as a function
on $M(F)\backslash M(\mathbb A)$ and call it the \emph{slice} of
the constant term at $g_0$. A function of finite level has
finitely many different slices.

\begin{theorem}[Constant terms and excursion operators]
\label{fl:thm:normalized-ct}
Let the Satake isomorphisms of $G$ and of $M$ be normalized by the
same square root of $q$ and by geometric Frobenius elements. For
every smooth function of finite level and every excursion datum
$e$ with $\boldsymbol\gamma\in W_F^I$,
\begin{equation}\label{fl:eq:normalized-ct-excursion}
 \mathsf{CT}^{\mathrm{nor}}_P S_e^{G,\mathrm{fun}}
   =S_{i^*e}^{M,\mathrm{fun}}\mathsf{CT}^{\mathrm{nor}}_P ,
\end{equation}
where the operator on the right is applied to each slice. If the
function is invariant under a subgroup of $T_G(\mathbb A)$, so are
both sides.
\end{theorem}

On the level of cohomology of shtukas the comparison is due to
Xue (Input~\ref{rw:in:xue}(4)): there is a constant-term morphism
$\mathsf{CT}^{\mathrm{geo}}_P$ from the cohomology for $G$ to that
for $M$, which for $I=\varnothing$ is the sum over the fibres of
$\Bun_P\to\Bun_M$ with the weights of the stack
\cite[Theorem~3.2.6, Proposition~3.5.3, Example~3.5.15 and
(3.44)]{XueCusp2020},
\cite[Proposition~4.1.3]{XueHecke2020}. What remains is to compare
normalizations, and this is the content of the section. Three
factors have to be accounted for.
\begin{enumerate}
\item The passage from the counting measure of the stack to Haar
measure on $U(F)\backslash U(\mathbb A)$
(Lemma~\ref{fl:lem:constant-term-haar-measure}).
\item The shift and Tate twist in the restriction of Satake sheaves
from $G$ to $M$
(Lemma~\ref{fl:lem:constant-term-arithmetic-restriction}).
\item The degree of a Frobenius map on a torsor of jets that occurs
when a partial Frobenius is compared for $P$ and for $M$
(Proposition~\ref{fl:lem:constant-term-frobenius-normalization}). This
factor depends on an auxiliary choice, the order of the jets, which
must cancel. Its computation is independent of the rest of the
section and is given in Appendix~\ref{rw:app:partial-frobenius}.
\end{enumerate}
The theorem is proved in
Section~\ref{rw:sec:ct-completion}.

\subsection{Measure and Satake normalization}

We fix a level $N$ and put
\[
 R_U=K_N\cap U(\mathbb A),\qquad
 c_U=\operatorname{vol}(R_U),\qquad
 a_P=\det\bigl(\operatorname{Ad}_M|_{\operatorname{Lie}U}\bigr),
\]
so that $a_P$ is a character of $M$ and
$\delta_P(m)=|a_P(m)|$. For an $M$-bundle of degree
$\nu\in\pi_1(M)$ write
\[
 \delta_P(\nu)=q^{\langle\nu,a_P\rangle},
\]
the pairing being that of $a_P$ with the image of $\nu$ in the
cocharacter group of $M/M_{\mathrm{der}}$. Two conventions are in
force. The degree of the bundle attached to an adele is the
negative of its valuation, so that $\delta_P(m)=\delta_P(\nu)$ if
$\nu$ is the degree of the bundle attached to $m$. And a
modification of relative position $\mu$, from its source to its
target, changes the degree from $\nu$ to $\nu-\mu$; for a line
bundle and $\mu=r$ the target is the source twisted by
$\mathcal O(-rx)$. This is the convention of
Section~\ref{rw:sec:satake-convention}, in which the position of
the source relative to the target is bounded by the weights of the
representation. So $\mu$ is also the weight by which the centre of
$\widehat M$ acts on the corresponding summand of a
representation.

\begin{lemma}[The Haar factor]
\label{fl:lem:constant-term-haar-measure}
For $I=\varnothing$, the morphism $\mathsf{CT}^{\mathrm{geo}}_P$ and
the normalized constant term of functions are related by
\begin{equation}\label{fl:eq:geometric-test-haar}
 \mathsf{CT}^{\mathrm{nor}}_P
        =c_U\delta_P^{1/2}\mathsf{CT}^{\mathrm{geo}}_P.
\end{equation}
\end{lemma}

\begin{proof}
Let $h$ be a function of level $N$ and consider the point $mg_0$.
Since $g_0$ normalizes $K_N$, the function $u\mapsto h(umg_0)$ on
$U(F)\backslash U(\mathbb A)$ is invariant under right translation
by $mR_Um^{-1}$. The double cosets in
$U(F)\backslash U(\mathbb A)/mR_Um^{-1}$ are the points of the
fibre of $\Bun_{P,N}\to\Bun_{M,N}$ over the point defined by $m$,
and the automorphism group of the point defined by $u$, relative
to $\Bun_{M,N}$, is
\[
 \Gamma_u=U(F)\cap u(mR_Um^{-1})u^{-1}.
\]
The morphism $\mathsf{CT}^{\mathrm{geo}}_P$ gives this point the
weight $|\Gamma_u|^{-1}$. In the integral, the same double coset
has volume
$\operatorname{vol}(mR_Um^{-1})|\Gamma_u|^{-1}
=c_U\delta_P(m)|\Gamma_u|^{-1}$. So the integral is
$c_U\delta_P(m)$ times the value of
$\mathsf{CT}^{\mathrm{geo}}_Ph$, and multiplication by
$\delta_P(m)^{-1/2}$ gives \eqref{fl:eq:geometric-test-haar}.
Dividing by a lattice in $T_G(\mathbb A)$ changes neither the
groups $\Gamma_u$ nor the character $a_P$, so the formula holds
with a central lattice as well.
\end{proof}

\begin{lemma}[Restriction of Satake sheaves with their Frobenius]
\label{fl:lem:constant-term-arithmetic-restriction}
Let $\mathsf{Sat}_G(V)$ be the Satake sheaf of a representation
$V$ of $\widehat G^I$, normalized to be pure of weight zero. Let
$i^0_\mu$ be the map which forgets the reduction to $P$ on the
space of local modifications of $P$-bundles of relative position
$\mu$ in $\pi_1(M)$, and let $\pi^0_\mu$ be the map from this space
to the modifications of $M$-bundles. We write
$n_\mu=\langle\mu,a_P\rangle$; the pairing depends only on the image
of $\mu$ in $X_*(M/M_{\mathrm{der}})$.
There is an isomorphism
\begin{equation}\label{fl:eq:geometric-test-satake}
 \bigoplus_\mu(\pi^0_\mu)_!(i^0_\mu)^*\mathsf{Sat}_G(V)
       [-n_\mu](-n_\mu/2)
       \simeq\mathsf{Sat}_M(\operatorname{Res}V),
\end{equation}
compatible with the actions of Frobenius and with fusion. Here
$\operatorname{Res}V$ is the restriction of $V$ to $\widehat M^I$.
\end{lemma}

\begin{proof}
Without the Frobenius structures the isomorphism, with its
compatibility with tensor products, is
\cite[Theorem~3.2.6]{XueCusp2020}, and the compatibility with
fusion is \cite[Lemma~4.1.5]{XueHecke2020}. The commutativity
constraint of the Satake category is the one modified by the
Koszul sign \cite[(5.2.8) and Theorem~5.2.9]{Zhu2017}, and the
geometric Satake equivalence with integral coefficients is
understood with the correction of \cite{MVErratum2018}.

We determine the sign of the shift with the conventions of
Section~\ref{rw:sec:satake-convention}. In the local model the
target is trivialized, and a modification is given by the lattice
of its source. A source of relative degree $\mu$ is the lattice
$t^{-\mu}$, so the semi-infinite orbit $U((t))\,t^\lambda$ which
contains it has $\lambda=-\mu$, and by \eqref{rw:eq:satake-loop}
the sheaf is the one attached by $\mathsf{Sat}^{\rm lp}_G$ to
$V\circ\epsilon_G$, in which the weight $\mu$ of $V$ has become the
weight $\lambda$. For restriction to a maximal torus, the cohomology
with compact supports of a Satake sheaf on this orbit is
concentrated in degree $\langle2\rho_G,\lambda\rangle$
\cite[Theorems~3.5 and~3.6]{MV2007}; for $\ell$-adic sheaves in
positive characteristic this is
\cite[Theorem~5.3.9(2)]{Zhu2017}. So the shift which brings it to
degree zero is $[-\langle2\rho_G,\mu\rangle]$. The restriction to
the torus is the composite of the restriction from $G$ to a Levi
subgroup $M$ and of the restriction from $M$ to the torus
\cite[Remark~5.3.16]{Zhu2017}: the shift from
$G$ to $M$ is the difference of the shifts from $G$ and from $M$
to the torus, that is, $[-n_\mu]$. For sheaves normalized to
weight zero the Tate twist is half the shift.
In \cite[(3.17)]{XueCusp2020} the shift and twist are written
$[n](n/2)$; we use the isomorphism with the shift and twist just
determined, and the following example is a check of the sign.
Take $G=\mathrm{GL}_2$, with $P$
the group of upper triangular matrices and $V$ the standard
representation. Then $a_P=(1,-1)$, and a reduction of a bundle to
$P$ is a line subbundle $L_1$ with quotient $L_2$. In the local
model the target $\mathcal G_1$ of the modification is trivialized,
with its reduction, and the source varies
\cite[Definition~1.1.11]{XueCusp2020}. So the modifications are the
lattices $\mathcal G_0\supset\mathcal G_1$ with quotient of length
one; they form a projective line, on which the Satake sheaf is
$E[1](1/2)$. For $\mu=(1,0)$, that is, when $L_1$ changes, the
lattice $\mathcal G_0$ is the sum of $\mathcal G_1$ and the larger
line, so the fibre of $\pi^0_\mu$ is a point and the restriction is
$E[1](1/2)$; here $n_\mu=1$. For $\mu=(0,1)$ the fibre is an affine
line and the restriction is $E[-1](-1/2)$; here $n_\mu=-1$. In both
cases the shift and twist by $[-n_\mu](-n_\mu/2)$ gives the sheaf
$E$ in degree zero with trivial Frobenius, which is the Satake
sheaf of the torus. (If the source were trivialized instead, the two
fibres would be exchanged, and so would the roles of the unipotent
groups of the source and of the target in the quotient by jets. The
quotient stacks are the same. We use the local model as stated
throughout, in particular in
Lemmas~\ref{fl:lem:constant-term-rotation-square}
and~\ref{fl:lem:constant-term-rotation-trace}.)

It remains to see that the isomorphism commutes with Frobenius.
Both sides are additive in $V$, so we may assume that $V$ is an
exterior tensor product of representations of the factors. On the
open subset where the legs are distinct both sides are then
exterior products, and it suffices to consider one leg. Let $T$ be
a maximal torus of $M$. The restriction functor from $M$ to $T$,
normalized in the same way, is faithful, and its composite with
the restriction from $G$ to $M$ is the restriction from $G$ to
$T$. For the intersection complexes normalized to weight zero,
Frobenius acts trivially on every weight space of the restriction
to $T$
\cite[Remark~5.5.15 and proof of Lemma~5.6.3]{Zhu2017}. Apply the
restriction to $T$ to the two sides of
\eqref{fl:eq:geometric-test-satake}. The Frobenius endomorphisms
of both become the identity, so the isomorphism commutes with
them after restriction to $T$, and by faithfulness it commutes
with them. The decomposition according to $\mu$ is the
decomposition according to the weights of the centre of
$\widehat M$, and is respected. By fusion the compatibility
extends from the open subset to the diagonals.
\end{proof}

\subsection{The partial Frobenius}
\label{rw:sec:ct-partial-frobenius}

The third factor is the subject of the following proposition.

\begin{proposition}[Partial Frobenius and the constant term]
\label{fl:lem:constant-term-frobenius-normalization}
Let $\mu_i$ be the relative position at the leg $i$ and
$n_i=\langle\mu_i,a_P\rangle$. Let $F_i^G$ and $F_i^M$ be the
partial Frobenius maps on the cohomology of the shtukas of $G$ and
of $M$, for Satake sheaves normalized to weight zero, and
$\mathsf{CT}^{\mathrm{geo}}_\nu$ the component of the
constant-term morphism with values in the degree $\nu$ of $M$.
Then
\[
 \mathsf{CT}^{\mathrm{geo}}_\nu F_i^G
   =q^{-n_i/2}F_i^M
                \mathsf{CT}^{\mathrm{geo}}_{\nu-\mu_i},
\]
and the morphism which is
$c_U\delta_P(\nu)^{1/2}\mathsf{CT}^{\mathrm{geo}}_\nu$ in degree
$\nu$ commutes with the partial Frobenius maps.
\end{proposition}

The proof is given in Appendix~\ref{rw:app:partial-frobenius}. It
compares the partial Frobenius morphisms of $P$ and of $M$ on a
torsor under a group of jets. A Frobenius map of this torsor has
degree $q^{m_i}$, where $m_i$ depends on the order of the jets,
which is an auxiliary choice
(Lemmas~\ref{fl:lem:constant-term-rotation-square}
and~\ref{fl:lem:constant-term-rotation-trace}). This factor cancels
against the purity isomorphisms, and $q^{-n_i/2}$ remains. A
compatibility of the constant-term morphism with the partial
Frobenius morphisms is stated in \cite[Lemma~4.1.7]{XueHecke2020},
without the normalization of the sheaves on $M$ at each leg and
the indexing of the components by the degree; for this reason we
prove the proposition from the construction of the constant-term
morphism. The appendix also explains the signs which occur when
Satake sheaves are replaced by functions
(Remark~\ref{rw:rem:signed-satake}), and checks the power of $q$ by
counting in the case of $\operatorname{GL}_2$
(Remark~\ref{rw:rem:ct-normalization-check}).

\subsection{Completion of the comparison}
\label{rw:sec:ct-completion}

We say that a linear operator between spaces of functions on
discrete sets has \emph{finite rows} if the value of the image at
each point depends on the values of the function at finitely many
points only.

\begin{lemma}[Operators with finite rows]
\label{fl:lem:finite-row-testing}
Two operators with finite rows which agree on the characteristic
functions of points agree on all functions. A composite of
operators with finite rows has finite rows.
\end{lemma}

\begin{proof}
The value of the difference of the two operators at a point is a
finite linear combination of values of the function, and the
coefficients are the values of the difference on characteristic
functions of points. For a composite, the value at a point
depends on finitely many values of the intermediate function,
each of which depends on finitely many values of the given one.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{fl:thm:normalized-ct}]
\step{1}{functions with compact support} The cohomology of the
shtukas of $M$ appears in the constant-term morphism as a limit
over truncations defined by the Harder--Narasimhan polygons of
$G$. In a fixed degree of $M$ these truncations and those defined
by the polygons of $M$ are cofinal in each other, since the
inequalities for the roots differ by the central component of the
degree, which is fixed \cite[\S1.5.17, (1.21)]{XueCusp2020}. So
the target of $\mathsf{CT}^{\mathrm{geo}}_P$ is the cohomology of
the shtukas of $M$ considered in
Section~\ref{fl:sec:compact-foundations}, degree by degree. Infinitely
many degrees of $M$ can occur, so the constant-term morphism takes
its values in the product of the components of given degree.

By Lemma~\ref{fl:lem:constant-term-haar-measure} and
Proposition~\ref{fl:lem:constant-term-frobenius-normalization} the
morphism $c_U\delta_P^{1/2}\mathsf{CT}^{\mathrm{geo}}_P$ is the
normalized constant term for $I=\varnothing$ and commutes with
the partial Frobenius maps and their inverses. It commutes with
the action of the fundamental group and with specialization, as
any morphism of sheaves, and with creation and annihilation by
the compatibility with fusion in
Lemma~\ref{fl:lem:constant-term-arithmetic-restriction}; the
restriction to $\widehat M^I$ of the vectors $x$ and $\xi$
defines the function $f\circ i^{\times I}$. An excursion operator
is a composite of finitely many of these maps. For a given
degree at the end, only finitely many shifts of the degree occur
in between, and every class which enters is represented at a
finite bound of Harder--Narasimhan. Hence
\[
 \mathsf{CT}^{\mathrm{nor}}_PS^G_ec
        =S^M_{i^*e}\mathsf{CT}^{\mathrm{nor}}_Pc
\]
for a function $c$ with compact support, where the right-hand
side is understood as follows. The constant term of $c$ need not
have compact support, but its restriction to each degree of $M$
has \cite[Lemma~1.5.19]{XueCusp2020}. By
Proposition~\ref{fl:prop:canonical-compact}(4) the operator
$S^M_{i^*e}$ shifts the degree by finitely many values. So its
value at a point involves the constant term in finitely many
degrees only, and it coincides with the value of
$S^{M,\mathrm{fun}}_{i^*e}$.

\step{2}{all functions} At a fixed level the normalized constant
term has finite rows, because
$U(F)\backslash U(\mathbb A)$ is compact. The excursion operators
have finite rows by
Proposition~\ref{fl:prop:compact-excursion-transpose}. So both
sides of \eqref{fl:eq:normalized-ct-excursion} have finite rows,
and by Lemma~\ref{fl:lem:finite-row-testing} the identity extends
from functions with compact support to all functions of the given
level. The slice at $g_0$ of the constant term of $h$ is the
slice at $1$ of the constant term of a right translate of $h$,
and excursion operators commute with right translations; this
gives the identity on all slices. Finally the character
$\delta_P$ is trivial on $T_G(\mathbb A)$ and all the maps
commute with translations by this group, which gives the last
assertion.
\end{proof}

\begin{corollary}[Constant terms and finite Hecke quotients]
\label{fl:cor:finite-model-ct}
Let $h$ lie in the dual of a finite Hecke quotient for $G$ without
a central lattice. Then every slice of its normalized constant term
along $P$ lies in the dual of a finite Hecke quotient for $M$
without a central lattice, and
\eqref{fl:eq:normalized-ct-excursion} holds on $h$ for all tuples
$\boldsymbol\gamma\in\Gamma_F^I$.
\end{corollary}

\begin{proof}
Let $h$ be annihilated by $\mathfrak a^\star$, for a cofinite
ideal $\mathfrak a$ of $\mathscr H_{G,u,\mathcal O}$. We use the
finitely many slices at points $g_0$ whose component at $u$ is
one. Let
$\operatorname{Sat}_P:\mathscr H_{G,u,\mathcal O}\to
\mathscr H_{M,u,\mathcal O}$ be the normalized Satake
homomorphism. As in the proof of
Proposition~\ref{fl:prop:one-place-automorphicity}, the second
algebra is a finitely generated module over the first: the
algebra $\mathcal O[X_*(T)]$ is finitely generated as a module
over its Noetherian subring of invariants of $W_G$, and the ring
of invariants of $W_M$ is a submodule. Therefore the ideal
\[
 \mathfrak a_M=\operatorname{Sat}_P(\mathfrak a)
                     \mathscr H_{M,u,\mathcal O}
\]
is cofinite. The homomorphism $\operatorname{Sat}_P$ commutes
with $\star$, since on both algebras $\star$ is induced by the
inversion of $X_*(T)$. By Lemma~\ref{rw:lem:ct-spherical}, each slice is annihilated by
$\mathfrak a_M^\star$, so it lies in the dual of the quotient for
$M$ defined by $\mathfrak a_M$. On these duals both sides of
\eqref{fl:eq:normalized-ct-excursion} are continuous in the
tuple, by Theorem~\ref{fl:thm:finite-integral-excursion}, and
they agree on the dense subset $W_F^I$.
\end{proof}

\subsection{Spherical Hecke operators: the classical formula}

For Hecke operators at an unramified place the compatibility of
Theorem~\ref{fl:thm:normalized-ct} is elementary. We give the
proof, because Section~\ref{fl:sec:arithmetic-cuspidality} uses
only this case; nothing from the rest of the present section
enters it.

Let $u$ be a place. Haar measures on $G(F_u)$, $M(F_u)$ and
$U(F_u)$ are normalized so that the groups of points with values
in $\mathcal O_u$ have volume one. The spherical Hecke algebra
$\mathscr H_{G,u}$ consists of the functions with compact support
on $G(F_u)$ which are invariant on both sides under
$G(\mathcal O_u)$, and $f\in\mathscr H_{G,u}$ acts on functions
invariant on the right under $G(\mathcal O_u)$ by
\[
 (T_fh)(g)=\int_{G(F_u)}h(gx)f(x)\,dx .
\]
Put
\[
 \operatorname{Sat}_P(f)(m)
   =\delta_P(m)^{1/2}\int_{U(F_u)}f(mn)\,dn
 \qquad(m\in M(F_u)).
\]
Let $B=TN$ be a Borel subgroup contained in $P$, with
$T\subset M$. The Satake transform of $G$ is
$f\mapsto\bigl(t\mapsto\delta_B(t)^{1/2}\int_{N(F_u)}f(tn)\,dn\bigr)$
\cite[\S4.2]{Cartier1979}. Since $N=(N\cap M)U$ and
$\delta_B=\delta_{B\cap M}\delta_P$ on $T$, it is the composite of
$\operatorname{Sat}_P$ with the Satake transform of $M$. So
$\operatorname{Sat}_P$ is a homomorphism of algebras
$\mathscr H_{G,u}\to\mathscr H_{M,u}$, and the Satake isomorphisms
of $G$ and of $M$, normalized by the same square root of $q$,
identify it with the inclusion
$k[X_*(T)]^{W_G}\subset k[X_*(T)]^{W_M}$. On the dual side it is
the restriction of representations from $\widehat G$ to
$\widehat M$.

\begin{lemma}[Constant terms and spherical Hecke operators]
\label{rw:lem:ct-spherical}
Let $h$ be a smooth function of finite level on
$G(F)\backslash G(\mathbb A)$, invariant on the right under
$G(\mathcal O_u)$, and let $g_0\in G(\prod_v\mathcal O_v)$ have
component one at $u$. For $f\in\mathscr H_{G,u}$ the slice at
$g_0$ of $\mathsf{CT}^{\mathrm{nor}}_P(T_fh)$ is the image under
$T_{\operatorname{Sat}_P(f)}$ of the slice at $g_0$ of
$\mathsf{CT}^{\mathrm{nor}}_Ph$.
\end{lemma}

\begin{proof}
By the Iwasawa decomposition every $x\in G(F_u)$ is of the form
$x=m'n'k$ with $m'\in M(F_u)$, $n'\in U(F_u)$ and
$k\in G(\mathcal O_u)$, and $dx=dm'\,dn'\,dk$, because
$dm'\,dn'$ is a left Haar measure on $P(F_u)$. The elements of
$G(F_u)$ commute with $g_0$, and $h$ is invariant under $k$. So
for $n\in U(\mathbb A)$ and $m\in M(\mathbb A)$
\[
 (T_fh)(nmg_0)=\int_{M(F_u)}\int_{U(F_u)}
    h(nmm'n'g_0)\,f(m'n')\,dn'\,dm' .
\]
We integrate over $n\in U(F)\backslash U(\mathbb A)$. Since
$nmm'n'=n\cdot mm'n'(mm')^{-1}\cdot mm'$ and the measure on
$U(F)\backslash U(\mathbb A)$ is invariant under translations, the
integral of $h(nmm'n'g_0)$ is
$\delta_P(mm')^{1/2}\,\mathsf{CT}^{\mathrm{nor}}_Ph(mm'g_0)$,
whatever $n'$ is. Multiplying by $\delta_P(m)^{-1/2}$ we get
\[
 \mathsf{CT}^{\mathrm{nor}}_P(T_fh)(mg_0)
   =\int_{M(F_u)}\mathsf{CT}^{\mathrm{nor}}_Ph(mm'g_0)\,
      \delta_P(m')^{1/2}\!\int_{U(F_u)}f(m'n')\,dn'\,dm' ,
\]
which is the assertion.
\end{proof}

If the Hecke algebra is made to act through $f(x^{-1})$ instead of
$f(x)$, the lemma holds in the same form, because
$\operatorname{Sat}_P$ commutes with the involutions
$f\mapsto f(x^{-1})$ of the two algebras: on both, the involution
is induced by the inversion of $X_*(T)$.