\section{An Eisenstein residue with a nonzero Fourier coefficient}
\label{fl:sec:common-descent}

Let $\sigma$ be an elliptic symplectic or even orthogonal
parameter whose multiplier $\chi$ has finite order. By
Proposition~\ref{fl:prop:prelim-block} its standard representation
is a sum of distinct blocks $\varrho_i$, each carrying a form of the
same kind with multiplier $\chi$. Since
$(\det\varrho_i)^2=\chi^{\dim\varrho_i}$, the determinants have finite
order, and L.~Lafforgue's theorem gives unitary cuspidal
representations $\Pi_i$ of $\mathrm{GL}_{m_i}(\mathbb A_F)$ with
parameters $\varrho_i$. They satisfy
$\Pi_i^\vee\otimes\omega_\sigma\simeq\Pi_i$, where $\omega_\sigma$
is the character corresponding to $\chi$ by class field theory.

The descent starts from an Eisenstein series on a group
$S=\mathrm{GSpin}_{2m}$ or $\mathrm{GSpin}_{2m+1}$, induced from
$\boxtimes_i\Pi_i$ on a Levi subgroup contained in the Siegel Levi
subgroup. This section proves what is needed about it
(Proposition~\ref{fl:lem:residual-analytic}): its iterated residue is a
nonzero, irreducible, square-integrable automorphic representation,
and a specific Fourier coefficient, denoted $\mathcal{SW}$, does
not vanish on it. This is the function-field form of the first
step of the descent method of Ginzburg, Rallis and Soudry
\cite{GRS1999,GRS2011}.

The proof has the usual shape. The constant terms of the
Eisenstein series are sums over Weyl elements of intertwining
operators, whose unramified parts are ratios of $L$-functions; the
poles of these ratios at the point of interest come from
Lemma~\ref{fl:lem:prelim-pole}. In the constant term along the
Siegel parabolic subgroup exactly one term has a pole of maximal
order, and its Whittaker coefficient along the Levi subgroup is an
explicit nonzero product of $L$-values; this shows that the
leading coefficient is not zero. Separately, every exponent of
every cuspidal constant term of the leading coefficient is
strictly negative; this shows that it is square integrable. At
the ramified places nothing is computed: we exhibit test vectors
for which the local coefficient is $1$.

\subsection{Whittaker functionals on general linear groups}
\label{fl:sec:gl-whittaker}

Let $\psi_d(n)=\psi(\sum_{i<d}n_{i,i+1})$ on the group $U_d$ of
upper triangular unipotent matrices in $\mathrm{GL}_d$, and
$W_f(g)=\int_{[U_d]}f(ng)\psi_d(n)^{-1}\,dn$.

\begin{lemma}[Whittaker functionals on general linear groups]\label{fl:lem:gl-whittaker}
\begin{enumerate}
\item For a cusp form $f$ on $\mathrm{GL}_d(\mathbb A_F)$,
\[
 f(g)=\sum_{\gamma\in U_{d-1}(F)\backslash\mathrm{GL}_{d-1}(F)}
 W_f\!\left(\begin{pmatrix}\gamma&0\\0&1\end{pmatrix}g\right).
\]
In particular $W_f\ne0$ if $f\ne0$.
\item Over a nonarchimedean local field of any characteristic, the
space of Whittaker functionals on an irreducible smooth
representation of $\mathrm{GL}_d$ has dimension at most one.
\item The Whittaker functional $f\mapsto W_f(1)$ on a cuspidal
representation $\otimes'_v\Pi_v$ is a product of local Whittaker
functionals.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Expand $f$ along the last column of $U_d$, a vector group on
which $\mathrm{GL}_{d-1}(F)$ acts. The term of the trivial
character is a constant term along a proper parabolic subgroup and
vanishes. The nontrivial characters form one orbit, and the
stabilizer of a point is the mirabolic subgroup of
$\mathrm{GL}_{d-1}$. Repeating the argument inside the successive
mirabolic subgroups gives the formula; at each stage the term of
the trivial character is a constant term of $f$ integrated over a
unipotent group, and for fixed $g$ all the sums are finite.

(2) By \cite[Theorem~2.5]{BZ77} an irreducible representation
embeds in a representation induced from supercuspidal
representations of a Levi subgroup. The space of Whittaker
functionals of a representation of $\mathrm{GL}_d$ is the dual of
its derivative of order $d$ in the sense of Bernstein and
Zelevinsky, the derivatives are exact functors
\cite[Proposition~1.9(a)]{BZ77}, and the derivative of order $d$ of
such an induced representation is one-dimensional
\cite[Theorem~4.4, Lemma~4.5 and Corollary~4.6]{BZ77}. These
results hold over any nonarchimedean local field.

(3) Choose a pure tensor on which the functional is nonzero. For a
finite set of places, the restriction of the functional to the
tensors that agree with the chosen one outside that set is, by
(2), the product of the local functionals normalized to take the
value $1$ on the chosen components, times the value on the chosen
tensor. These descriptions are compatible when the set grows.
\end{proof}

\begin{lemma}[Whittaker functional of an induced representation]\label{fl:lem:gl-induced-whittaker}
Let $v$ be a place and consider the representation of
$\mathrm{GL}_m(F_v)$ induced from
$\boxtimes_i\Pi_{i,v}|\det|^{s_i}$, with the Whittaker functional
given by the integral over the open cell of the Whittaker
functionals of the $\Pi_{i,v}$.
\begin{enumerate}
\item If all the data are unramified, the value of this functional
on the spherical vector, normalized by the spherical Whittaker
vectors of the $\Pi_{i,v}$, is
\[
 \prod_{i<j}L_v(1+s_i-s_j,\Pi_i\times\Pi_j^\vee)^{-1}.
\]
\item In general there is a section supported in the open cell,
with compact support there, on which the functional is $1$ for
all $(s_i)$.
\end{enumerate}
\end{lemma}
\begin{proof}
The functional is defined on the whole induced representation and
is compatible with induction in stages
\cite[Theorems~1.4 and 1.6, Corollary~1.7]{CS80}. For a principal
series with Satake parameters $z_a$ its value on the spherical
vector is $\prod_{a<b}(1-q_v^{-1}z_az_b^{-1})$
\cite[Theorem~5.4]{CS80}; the normalization is fixed by the case
of rank one, where the integral over $\mathcal O_v$ is $1$, the
next shell contributes $-q_v^{-1}z_1z_2^{-1}$ and the others
contribute zero. Writing each $\Pi_{i,v}$ as a subquotient of a
principal series and dividing by the factors with $a,b$ in the
same block gives (1), first for distinct parameters and then in
general because both sides are rational functions. For (2) take a
section supported on a small compact open subset of the open cell,
with values a vector on which the Whittaker functional of the
inducing representation is nonzero, and with the dependence on the
unipotent variable chosen to cancel the character.
\end{proof}

\subsection{Regularized values of Eisenstein series}

Let $S$ be a split GSpin group and $P_{\mathrm S}$ its Siegel
parabolic subgroup, with Levi subgroup $\mathrm{GL}_m\times\Gm$ and
unipotent radical $U_{P_{\mathrm S}}$. The coefficient
$\mathcal{SW}$ is the Whittaker coefficient, along the maximal
unipotent subgroup $U_{\mathrm{GL},m}$ of $\mathrm{GL}_m$, of the
constant term along $P_{\mathrm S}$:
\[
 \mathcal{SW}(f)(g)=
 \int_{[U_{\mathrm{GL},m}]}\int_{[U_{P_{\mathrm S}}]}
       f(ung)\psi_m(n)^{-1}\,du\,dn.
\]

The Eisenstein series we need has a pole of order equal to the
number of blocks, along several hyperplanes at once. The next
lemma defines its leading term without choosing an order in which
to take residues.

\begin{lemma}[Regularized Eisenstein value]
\label{fl:lem:regularized-eisenstein-family}
Let $E(f,x)$ be a family of Eisenstein series induced from
cuspidal data with unitary central character, depending on a
parameter $x$ near $0$ in a vector space, through unramified twists
which are trivial on the centre of the ambient group; so the centre
acts on all the members of the family through one unitary
character. Suppose that every
cuspidal constant term of $E(f,x)$ is a sum of terms of the form
$u(x)/\prod_{j\in J}L_j(x)$, with $u$ holomorphic and
$J\subset\{1,\ldots,N\}$, where $L_1,\ldots,L_N$ are distinct
linear forms. Put $\delta=\prod_{j=1}^NL_j$.
\begin{enumerate}
\item $\delta(x)E(f,x)$ is holomorphic at $x=0$.
\item Its value $\mathcal R(f)$ at $0$ depends only on the value of
the section $f$ at $x=0$, and $\mathcal R$ is equivariant.
\item A term of a constant term contributes to the constant term
of $\mathcal R(f)$ only if $J=\{1,\ldots,N\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
At a fixed level the Eisenstein series is meromorphic in $x$, and
its values at all $g$ have a common denominator locally
\cite[\S9.5]{G2BernsteinLapid24}; for a holomorphic family of
sections, take the product of the denominators for a basis of the
inducing space of the given level, in its compact realization.
Suppose that $\delta E$ has a pole. Choose a smooth point of an
irreducible component of its polar divisor which lies on no other
component, and at which the leading coefficient of some value of
$\delta E$ is not zero. In a coordinate transverse to the divisor,
this leading coefficient, as a function of $g$,
is an automorphic function all of whose cuspidal constant terms
vanish, by the hypothesis on the denominators. The centre acts on
it through the same unitary character, and it is orthogonal to the
cusp forms on which the centre acts through that character: in the
region of convergence, the
pairing of an Eisenstein series with a cusp form unfolds to an
integral of a constant term of the cusp form, which is zero, and
this persists by analytic continuation. An automorphic function
with these two properties is zero
\cite[\S9.1, equation (9.2), and \S9.5]{G2BernsteinLapid24}. This
proves (1). Two sections with the same value at $0$ differ by a
sum $\sum_ix_if_i(x)$, on which the regularized value vanishes;
and the equivariance of $E$ is an identity of meromorphic
functions.
This is (2). In (3), a term with $J\ne\{1,\ldots,N\}$ becomes,
after multiplication by $\delta$, divisible by a linear form and
vanishes at $0$.
\end{proof}

\begin{lemma}[Square integrability of the regularized value]\label{fl:lem:residual-image-square-integrable}
In the situation of Lemma~\ref{fl:lem:regularized-eisenstein-family},
where the centre acts through a unitary character, suppose that the exponents of all the terms contributing to the
constant terms of $\mathcal R(f)$ are strictly negative on the
relevant cones of reduction theory. Then the image of $\mathcal R$
consists of functions that are square integrable modulo the
centre. If in addition the coefficient $\mathcal{SW}$ of $E(f,x)$
is $u(x)/\delta(x)$ with $u$ holomorphic and $u(0)\ne0$ for some
$f$, then $\mathcal R\ne0$, and $\mathcal{SW}(\mathcal R(f))=u(0)$.
\end{lemma}
\begin{proof}
The first statement is the criterion for square integrability in
terms of constant terms. A function of finite level agrees with
its constant terms far in the cusps
\cite[Chapter~I, \S5.9]{Morris1982}, so a strictly negative
exponent bounds the square norm on a Siegel set by a sum
$C\sum_{d\in\mathbb Z_{\ge0}^r}(1+\|d\|)^aq^{-\epsilon\|d\|}$ with
$\epsilon>0$. The coefficient $\mathcal{SW}$ is an integral over a
compact set, so it commutes with multiplication by $\delta$ and
evaluation at $0$.
\end{proof}

\begin{corollary}[Irreducibility]\label{fl:cor:residual-image-irreducible}
Suppose moreover that at every place the local induced
representation at $x=0$ has an irreducible cosocle $J_v$, that is,
a unique maximal proper subrepresentation with quotient $J_v$, and
that at almost all places the spherical vector has nonzero
image in it. Then the image of $\mathcal R$ is the irreducible
representation $\bigotimes'_vJ_v$.
\end{corollary}
\begin{proof}
The image is a nonzero quotient of the restricted tensor product
of the local induced representations, and it is unitary, being
square integrable. Apply
Proposition~\xref{I}{fl:prop:restricted-tensor-unitary-image}.
\end{proof}

\subsection{The residue attached to self-dual blocks}

\begin{proposition}[The residual representation]\label{fl:lem:residual-analytic}
Let $m=\sum_im_i$, and let $\Pi_i$ be the distinct cuspidal
representations described at the beginning of the section. Let
$S=\mathrm{GSpin}_{2m}$ if the blocks are alternating and
$S=\mathrm{GSpin}_{2m+1}$ if they are symmetric, and consider the
Eisenstein series $E(f_t,t)$ on $S$ induced from
$\boxtimes_i\Pi_i|\det|^{t/2}\boxtimes\omega_\sigma$ on the
parabolic subgroup contained in $P_{\mathrm S}$ with Levi subgroup
$\prod_i\mathrm{GL}_{m_i}\times\Gm$. Let $N$ be the number of
blocks.
\begin{enumerate}
\item The limit $\mathcal R(f)=\lim_{t\to1}(t-1)^NE(f_t,t)$ exists,
and $\mathcal R$ is a nonzero equivariant map to functions that are
square integrable modulo the centre.
\item The image of $\mathcal R$ is irreducible, and the connected
centre acts on it through $\omega_\sigma$.
\item The coefficient $\mathcal{SW}$ is not identically zero on the
image of $\mathcal R$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $S_{\rm ram}$ be a finite set of places containing those where
some $\Pi_i$ or $\psi$ is ramified, and put
\[
 L_i(z)=L^{S_{\rm ram}}(z,\Pi_i,r\otimes\omega_\sigma^{-1}),\qquad
 L_{ij}(z)=L^{S_{\rm ram}}(z,\Pi_i\times\Pi_j\otimes\omega_\sigma^{-1}),
\]
where $r=\Lambda^2$ in the alternating case and
$r=\operatorname{Sym}^2$ in the symmetric case. These are the
$L$-functions of the representations of the dual Levi subgroup on
the Lie algebra of the dual Siegel radical, whose weights are
$z_az_b/c$ with $a<b$, respectively $a\le b$.

The twists $|\det|^{s_i}$ which occur below are trivial on the
centre of $S$, as Lemma~\ref{fl:lem:regularized-eisenstein-family}
requires. Write an element of the maximal torus of the Levi
subgroup $\mathrm{GL}_m\times\Gm$ as $(t_1,\ldots,t_m;t_0)$. In
type $B_m$ the roots $e_i$ vanish on the centre, which forces
$t_i=1$. In type $D_m$ the roots $e_i\pm e_j$ force
$t_1=\cdots=t_m=\varepsilon$ with $\varepsilon^2=1$. In both cases
$|\det|$ is one on the centre of each block, and the centre of $S$
acts through the character $t\mapsto\omega_\sigma(t_0)
\prod_i\omega_{\Pi_i}(\varepsilon)$, where $\omega_{\Pi_i}$ is the
central character of $\Pi_i$ and $\varepsilon=1$ in type $B$. It
does not depend on the twists, and it is unitary, because
$\omega_\sigma$ has finite order and $\varepsilon^2=1$.

\step{1}{the poles} We first let the exponents vary independently,
inducing from $\boxtimes_i\Pi_i|\det|^{s_i}$. The constant terms
are sums of intertwining operators, products of operators of rank
one. The unramified parts of the latter are
\[
 \frac{L_{ij}(s_i+s_j)}{L_{ij}(1+s_i+s_j)},\qquad
 \frac{L_i(2s_i)}{L_i(1+2s_i)},
\]
and similar ratios in the differences $s_i-s_j$. By
Lemma~\ref{fl:lem:prelim-pole} and the remark following it,
\[
 a_i=\operatorname*{Res}_{z=1}L_i(z)\ne0,
 \qquad L_i(2)L_{ij}(1)L_{ij}(2)\ne0\quad(i\ne j).
\]
We need the operators of rank one only near the real point at
which all the $s_i$ are equal to $1/2$; there the differences are
$0$ and the sums are $1$. An operator in a difference is
evaluated at the real point of the unitary axis, where the
operators of rank one are holomorphic
\cite[Chapter~III, Theorem~5.2(i)]{Morris1982}. Consider an
operator in a sum $s_i+s_j$ with $i\ne j$, as a function of
$z=s_i+s_j$. Its unramified part $L_{ij}(z)/L_{ij}(1+z)$ is
holomorphic at $z=1$, by Lemma~\ref{fl:lem:prelim-pole} and the
remark which follows it; this is where
$\Pi_j^\vee\otimes\omega_\sigma\simeq\Pi_j\not\simeq\Pi_i$ is used.
At the places of $S_{\rm ram}$ the local operators are given by
integrals which converge for $\Re z>0$, because the
representations $\Pi_{i,v}$ are tempered
(Input~\ref{rw:in:llafforgue}(3))
\cite[Proposition~IV.2.1]{Waldspurger2003}. So this operator is
holomorphic at $z=1$. The remaining operators, in $2s_i$, have at
most simple poles
\cite[Chapter~III, Theorem~5.2(i)--(ii)]{Morris1982}. So the
hypothesis of Lemma~\ref{fl:lem:regularized-eisenstein-family}
holds with the $N$ linear forms $2s_i-1$, and restricting to
$s_i=t/2$ gives the existence of
$\mathcal R(f)=\lim_{t\to1}(t-1)^NE(f_t,t)$.

\step{2}{the exponents} We verify the hypothesis of
Lemma~\ref{fl:lem:residual-image-square-integrable}. As explained
at the end of this step, it suffices to consider the cuspidal
components of the constant terms along the standard parabolic
subgroups $P'=M'U'$. Let $M$ be the inducing Levi subgroup. Since
the inducing datum is cuspidal, the cuspidal component of the
constant term along $P'$ is a sum over the Weyl elements $w$ with
$wMw^{-1}=M'$ \cite[(6.17) and (6.18)]{G2BernsteinLapid24}; the
terms induced from smaller Levi subgroups of $M'$ are orthogonal to
cusp forms of $M'$. By
Lemma~\ref{fl:lem:regularized-eisenstein-family}(3) a Weyl element
$w$ contributes to a remaining constant term only if it produces
all $N$ poles. Factor the intertwining operator of $w$ along a
reduced sequence of reflections in relative roots. The factors
attached to two different blocks are holomorphic at the point, and
the factor attached to the relative root $\beta_i$ of the $i$th
block which is fixed by the inducing datum has at most a simple
pole; so $w$ must make every $\beta_i$ negative. The coroots which
restrict to $\beta_i$ on the centre of the inducing Levi subgroup
are the $e_a+e_b$ with $a,b$ in the block $i$, including $a=b$ when
the block is of symmetric-square type. For such a coroot
$\alpha^\vee$, the restriction of $w\alpha^\vee$ to the centre of
$M'=wMw^{-1}$ is the relative root $w\beta_i$, which is negative.
The coroots outside $M'$ which are positive are those of the
unipotent radical of $P'$, and their restrictions to the centre of
$M'$ are positive relative roots; so $w\alpha^\vee$ is negative. Thus $w$ makes negative all the coroots
$e_a+e_b$ with $a,b$ in the same block. Let
\[
 \gamma_i=\frac{2}{m_i}\sum_{a\text{ in block }i}e_a,
 \qquad \lambda=\frac12\sum_{a=1}^me_a
              =\sum_i\frac{m_i}{4}\gamma_i ,
\]
so that $\lambda$ is the exponent at $t=1$ and the $\gamma_i$ span
the space of exponents fixed by the Weyl group of the inducing Levi
subgroup, modulo the central direction of the ambient group. Each
$\gamma_i$ is a positive combination of these coroots:
\[
 \gamma_i=\frac{2}{m_i(m_i-1)}\sum_{a<b}(e_a+e_b)
 \qquad\text{or}\qquad
 \gamma_i=\frac{2}{m_i(m_i+1)}\sum_{a\le b}(e_a+e_b),
\]
with $a,b$ in the block $i$, according to whether the block is of
alternating-square type, where $m_i\ge2$, or of symmetric-square
type; each coordinate occurs $m_i-1$ times in the first sum and
$m_i+1$ times in the second. For a contributing $w$, each
$w\gamma_i$ is therefore a positive combination of negative
coroots, hence a combination with coefficients $\le0$ of the simple
coroots. Consider a simple root that does not belong to the Levi
subgroup of the constant term. The coefficient of its coroot in
$w\lambda$ is a sum of nonpositive numbers, and it is not zero:
otherwise the coefficient would vanish in every $w\gamma_i$,
whereas the $w\gamma_i$ span the space of exponents fixed by the
Weyl group of $M'$, modulo the central direction of the ambient
group. For an inner product on the real space of coroots which is
invariant under the Weyl group, this space is the orthogonal
complement of the span of the coroots of $M'$, and it maps
isomorphically onto the quotient by that span. So the coefficient
of a simple coroot outside $M'$ does not vanish on the whole
space.
Hence $\langle w\lambda,d\rangle<0$ for all $d\ne0$ in the positive
cone, which is the negativity to be proved; the half-sum of roots
in the normalized induction
cancels the modulus character in the integral of the square. It
suffices to consider cuspidal constant terms: an exponent of an
arbitrary constant term is the restriction of an exponent of a
cuspidal constant term along a smaller parabolic subgroup, by the
transitivity of constant terms, and the positive cone of the
larger parabolic subgroup lies in the closed positive cone of the
smaller. The regularized value is a Laurent coefficient, so its
constant terms can contain polynomials in the degree multiplied by
the same exponentials; a polynomial does not affect the
convergence of the sum in the proof of
Lemma~\ref{fl:lem:residual-image-square-integrable}, in which the
factor $(1+\|d\|)^a$ already appears.

\step{3}{the coefficient $\mathcal{SW}$} This is the main point. In
the constant term along $P_{\mathrm S}$, only the terms with
$wMw^{-1}$ contained in the Siegel Levi subgroup remain, because
the inducing datum is cuspidal. For such a term the roots which
are differences of two indices of one block force all the indices
of the block to have the same sign. If the sign of a block is
positive, the roots which are sums of two of its indices remain
positive, and the block contributes no pole. So a term which
produces all the poles changes the sign of every block. Modulo
the Weyl group of the Siegel Levi subgroup there is one such
term, the one making the whole Siegel
radical negative. Its term is a function on $\mathrm{GL}_m$,
induced from $\boxtimes_i\Pi_i$ with shifted exponents, and its
Whittaker coefficient is computed on the Bruhat cells of
$\mathrm{GL}_m$: on a cell other than the open one the character
$\psi_m$ is nontrivial on a simple root subgroup contained in the
stabilizer, so that cell contributes zero. On the open cell, at
the places outside $S_{\rm ram}$ and for spherical data, the
intertwining operator and
Lemma~\ref{fl:lem:gl-induced-whittaker}(1) give the factor
\[
 \prod_i\frac{L_i(t)}{L_i(t+1)}
 \prod_{i<j}\frac{L_{ij}(t)}{L_{ij}(t+1)L_{ij}(1)},
\]
and $(t-1)^N$ times this tends to
$\prod_ia_iL_i(2)^{-1}\prod_{i<j}L_{ij}(2)^{-1}\ne0$.

At a place of $S_{\rm ram}$ we choose the section so that the
local factor is $1$ for all values of the parameter. The cell in
question is the product of the opposite Siegel radical, with
coordinate $z_P$, and of the open cell of the Whittaker
functional in $\mathrm{GL}_m$, with coordinate $z_A$. The
corresponding Weyl elements, whose inverses are the representatives
which occur in the integral, satisfy
\[
 w_P=w_{0,S}w_{0,L_{\mathrm S}},\qquad
 w_A=w_{0,M}w_{0,L_{\mathrm S}},\qquad w_Aw_P=-w_{0,M},
\]
with additive lengths, where $M$ is the inducing Levi subgroup and
$w_{0,S}=-1$; in the even orthogonal case this holds because $m$
is even. In the two successive integrations the product of the
representatives is
$w_P^{-1}u_Pw_A^{-1}u_A=(w_Aw_P)^{-1}(w_Au_Pw_A^{-1})u_A$. The
root coordinates of the two unipotent factors are disjoint, and
together they parametrize the open cell for the inducing parabolic
subgroup and the Borel subgroup of the ambient group. Choose a
small compact open neighbourhood $\Omega_v$
of the identity in these coordinates and a vector in the inducing
representation whose Whittaker function is $1$ at the identity,
and let the section be
$\operatorname{vol}(\Omega_v)^{-1}\psi_A(z_A)$ times this vector
on $\Omega_v$, extended by zero. On $\Omega_v$ the Levi components
in the Iwasawa decomposition are units, so the exponents do not
enter, and the local integral is $1$.

Thus for suitable $f$ the coefficient $\mathcal{SW}(E(f_t,t))$ is
$u(t)/(t-1)^N$ with $u(1)\ne0$, and
Lemma~\ref{fl:lem:residual-image-square-integrable} gives (1) and
(3).

\step{4}{irreducibility} Each $\Pi_{i,v}$ is tempered
\cite[Theorem~VI.10(i)]{LLa02}. The representation of
$\mathrm{GL}_m(F_v)$ induced from $\boxtimes_i\Pi_{i,v}$ is
irreducible and tempered: in terms of segments, those attached to
tempered representations are centred on the unitary axis, and two
such segments on the same cuspidal line are nested, hence not
linked \cite[Theorem~9.7(a)]{Zel80}. By induction in stages the
local induced representation at $t=1$ is induced from the Siegel
parabolic subgroup, from this tempered representation twisted by
a positive exponent. By the Langlands classification it is a
standard module, and a standard module has an irreducible cosocle
\cite[Theorem~A and \S2]{SolleveldStandard2025}; at an
unramified place the image of the spherical vector in it is
nonzero. Corollary~\ref{fl:cor:residual-image-irreducible} gives
(2); the central character is that of the inducing data.
\end{proof}

\begin{remark}
Purity of the blocks enters through the poles in step~1. Square
integrability comes from the signs of the exponents in step~2. The
test vectors at the ramified places in step~3 are needed because
the Euler product over the unramified places says nothing about
the remaining local factors of the coefficient.
\end{remark}