\section{Generic cuspidal support and symplectic realization}
\label{fl:sec:classical-double-line}

This section proves the realization theorem for symplectic groups
(Theorem~\ref{fl:thm:symplectic-realization}). Let $\sigma$ be an
elliptic parameter with values in $\operatorname{SO}_{2n+1}$ and
$\varrho=\operatorname{Std}\sigma$. By
Proposition~\ref{fl:prop:prelim-block} the space of invariants of
$\varrho$ has dimension zero or one. If it has dimension one,
$\varrho\ominus1$ is an elliptic even orthogonal parameter of dimension
$2n$, and the theta lift of
Section~\ref{fl:sec:odd-theta} of a generic cusp form with that
parameter has standard parameter $\varrho$ on
$\operatorname{Sp}_{2n}$. If it has dimension zero, we must start
from $\varrho\oplus1$ on $\operatorname{SO}_{2n+2}$, and the lift, on
$\operatorname{Sp}_{2n+2}$, has standard parameter
$\varrho\oplus1^{\oplus2}$. The lift is then not cuspidal, and its
cuspidal support is a representation of
$\operatorname{GL}_1\times\operatorname{Sp}_{2n}$ whose second
factor has parameter $\varrho$. To make this precise we need a version
of cuspidal support which applies to a generalized Hecke
eigenfunction and which keeps track of the Whittaker coefficient
(Proposition~\ref{fl:lem:generic-cuspidal-support}); this lemma is the
main work of the section.

\subsection{Cuspidal support of a generic function}

\begin{lemma}[No doubled line in a cuspidal representation]
\label{fl:lem:generic-double-line-obstruction}
Let $\pi$ be an irreducible cuspidal automorphic representation of
split $\operatorname{Sp}_{2s}$, $s\ge1$, and let $\tau$ be a
continuous semisimple representation of $\Gamma_F$ of dimension
$2s+1$, unramified at almost all places. Suppose that the
eigenvalues of Frobenius on $\tau$ at the unramified places are
algebraic numbers all of whose complex absolute values are one,
and that at almost all places the characteristic polynomial of
Frobenius on $\tau$ is that of the standard Satake parameter of
$\pi$. Then the space of invariants of $\Gamma_F$ in $\tau$ has
dimension at most one.
\end{lemma}
\begin{proof}
We use one Eisenstein series of rank one. Let
$H=\operatorname{Sp}_{2s+2}$, and let $P=MU$ be the stabilizer of
an isotropic line, with $M=\mathbb G_m\times\operatorname{Sp}_{2s}$.
The roots of $U$ are $e_1\pm e_j$, for $2\le j\le s+1$, and $2e_1$.
Let $w_0$ represent the element of the Weyl group which changes
the sign of $e_1$; it normalizes $M$ and maps $U$ to the opposite
unipotent radical. For $z\in\mathbb C$ let $I(z)$ be the space
induced from $|\cdot|^z\otimes\pi$, with the normalization by
$\delta_P^{1/2}$, and
\[
 M(z):I(z)\longrightarrow I(-z),\qquad
 (M(z)\Phi)(g)=\int_{U(\mathbb A)}\Phi(w_0^{-1}ug)\,du ,
\]
the intertwining operator. The integral converges when $\Re z$ is
large, and on the sections of a given level $M(z)$ is a rational
function of $q^{-z}$.

\step{1}{the poles are simple} The parabolic subgroup $P$ is
maximal, and the inducing representation is cuspidal and unitary:
the centre of $\operatorname{Sp}_{2s}$ is finite, so its cusp forms
are square integrable. So a pole of $M(z)$ at a real point $z>0$
is simple
\cite[Chapter~III, Theorem~5.2(i)]{Morris1982}.

\step{2}{the unramified places} Write $\pi=\bigotimes'_v\pi_v$,
and let $S$ be a finite set of places outside of which $\pi$ and
$\tau$ are unramified and the characteristic polynomials agree.
For $\Re z$ large and a pure tensor $\Phi=\bigotimes_v\Phi_v$, the
operator $M(z)$ is the product of the local operators $A_v(z)$
defined by the same integral over $U(F_v)$. Let $v\notin S$, and
let $t_v$ be the Satake parameter of $\pi_v$, an element of
$\operatorname{SO}_{2s+1}$. The representation $\pi_v$ is a
subrepresentation of an unramified principal series of
$\operatorname{Sp}_{2s}$, so $I_v(z)$ is a subrepresentation of an
unramified principal series of $H$, on which $A_v(z)$ is the
intertwining operator of the Weyl element $w_0$. The coroots of
the roots of $U$ are $e_1\pm e_j$ and $e_1$: on them the character
of the torus which defines this principal series takes the values
$q_v^{-z}a$, where $a$ runs over the
eigenvalues of $t_v$ in the standard representation. So the
formula of Gindikin and Karpelevich
\cite[Theorem~3.1]{Casselman80} gives, for the spherical vector
$\Phi_v^0$ of $I_v(z)$ whose value at the identity is the
spherical vector $\xi_v^0$ of $\pi_v$,
\[
 A_v(z)\Phi^0_v
 =\frac{\det(1-q_v^{-1-z}t_v)}{\det(1-q_v^{-z}t_v)}\,\Phi^{0}_{v,-}
 =\frac{L_v(z,\tau)}{L_v(1+z,\tau)}\,\Phi^{0}_{v,-},
\]
where $\Phi^0_{v,-}$ is the spherical vector of $I_v(-z)$ with the
same value at the identity. The formula holds for regular characters
in the region where the integral converges absolutely; both sides
are holomorphic in the character there, so it holds for the
characters above, whose entries along the Levi subgroup may
coincide, and then for all $z$ by meromorphic continuation.

\step{3}{the other places} Let $v\in S$. Choose a nonzero vector
$\xi_v$ of $\pi_v$ and a compact open subgroup $U_v$ of $U(F_v)$,
and let $\Phi_{v,z}\in I_v(z)$ be the section which is supported
in the open subset $P(F_v)w_0^{-1}U_v$ and is given there by
\[
 \Phi_{v,z}(p\,w_0^{-1}u)=\delta_P^{1/2}(p)\,|a|^z\,\pi_v(m)\xi_v
 \qquad(u\in U_v),
\]
where $(a,m)$ is the image of $p$ in $M(F_v)$. The map
$P\times U\to Pw_0^{-1}U$ is bijective, so this is well defined; it
is a smooth vector, and its restriction to a maximal compact
subgroup depends holomorphically on $z$. For every $z$,
\[
 (A_v(z)\Phi_{v,z})(1)=\int_{U(F_v)}\Phi_{v,z}(w_0^{-1}u)\,du
 =\operatorname{vol}(U_v)\,\xi_v ,
\]
an integral over a compact set.

\step{4}{conclusion} Take $\Phi_z$ with the components of step~3
at the places of $S$ and the spherical vectors elsewhere. It is a
holomorphic family of sections of a fixed level, and by steps~2
and~3, first for $\Re z$ large and then for all $z$ by
continuation,
\[
 (M(z)\Phi_z)(1)=\frac{L^S(z,\tau)}{L^S(1+z,\tau)}
 \prod_{v\in S}\operatorname{vol}(U_v)\cdot\xi ,
\]
where $\xi$ is the nonzero vector of $\pi$ with components $\xi_v$
at the places of $S$ and $\xi_v^0$ elsewhere. The
representation $\tau$ defines a semisimple lisse sheaf on the
complement of $S$, pure of weight zero. By
Lemma~\ref{fl:lem:prelim-pole} the function $L^S(z,\tau)$ has a
pole at $z=1$ of order $\dim\tau^{\Gamma_F}$, and
$L^S(2,\tau)$ is finite and not zero. So $M(z)$ has a pole at
$z=1$ of order at least $\dim\tau^{\Gamma_F}$, and by step~1 this
dimension is at most one.
\end{proof}

For generic $\pi$ the functorial transfer to general linear groups
\cite[Theorem~7.2]{Lomeli19}, which is proved under Kim's
Assumption~A \cite[\S4]{DelCastilloLomeli2024}, gives more: $\tau$
is multiplicity free.

The function produced by Corollary~\ref{fl:cor:odd-theta-moment}
is a generalized Hecke eigenfunction with a nonzero Whittaker
coefficient, and it need not be cuspidal. For a Levi subgroup $M$,
write $\mathsf D_M$ for the image of the adelic degree map in
$\operatorname{Hom}(X_F^*(M),\mathbb Z)$.

\begin{proposition}[Generic cuspidal support]
\label{fl:lem:generic-cuspidal-support}
Let $J$ be split connected reductive, and let $h$ be a function of
finite level on $J(F)\backslash J(\mathbb A)$ on which the
spherical Hecke algebras at the unramified places act through a
generalized character $\eta$, and whose principal Whittaker
coefficient is nonzero.
\begin{enumerate}
\item There are a standard Levi subgroup $M$, possibly equal to
$J$, a generic irreducible cuspidal representation $\pi$ of
$M(\mathbb A)$ whose central character has finite order on the
connected centre, and a character $\chi:\mathsf D_M\to k^\times$,
such that the Hecke character of $\pi\otimes(\chi\circ\deg)$
corresponds to $\eta$ under the normalized Satake restriction from
$J$ to $M$.
\item If at one unramified place the eigenvalues given by $\eta$ in
a faithful representation of the dual group are $\ell$-adic units,
then $\chi$ takes values in the units of a finite extension of
$\mathbb Q_\ell$, and $\pi\otimes(\chi\circ\deg)$ has a continuous
parameter.
\end{enumerate}
\end{proposition}
\begin{proof}
The space of functions with compact support is spanned by
pseudo-Eisenstein series built from cuspidal data. We separate
those built from generic cuspidal representations from the others,
show by pairing with a Poincar\'e series that $\eta$ occurs in the
first part, and then specialize the degree variables. The step
which needs care is the third. The image of the Hecke algebra in
the endomorphisms of the functions with compact support does not
act on the module from which the generic part is induced, so the
specialization is made with the abstract Hecke algebra, and with
translations by a central lattice which do not belong to it and
only commute with it.

\step{1}{pseudo-Eisenstein series} Fix the level $K$ and an
unramified place $u$, and let
$\mathcal C=C_c(J(F)\backslash J(\mathbb A)/K,k)$. It is a finitely
generated module over the spherical Hecke algebra $\mathscr H_u$
(Proposition~\xref{I}{fl:prop:classical-one-place-finiteness}). By
\cite[\S9.2, (9.3)]{G2BernsteinLapid24}, $\mathcal C$ is the sum,
over the standard Levi subgroups $M$, of the images of the
pseudo-Eisenstein maps, which send a compactly supported cuspidal
function on $M$, placed on finitely many pieces of the Iwasawa
decomposition, to the sum of its translates under
$P(F)\backslash J(F)$. With the normalization by the half modulus
these maps are compatible with the Satake restriction. The
statement holds over $k$ by transport under $\iota$, because at a
fixed level the sums and the integrals which define constant terms
are finite. The level on the Levi subgroup depends on the piece of
the Iwasawa decomposition: we choose a representative
$g_a\in J(\mathbb A)$ of each double coset in
$P(\mathbb A)\backslash J(\mathbb A)/K$, and let $K_{M,a}$ be the image in
$M(\mathbb A)$ of $P(\mathbb A)\cap g_aKg_a^{-1}$. The
representatives can be chosen with component one at the unramified
places, and there the local Satake restriction acts on each piece.

A compactly supported cuspidal function on $M$ of given level is
a finite sum of functions supported in one degree
$d\in\mathsf D_M$. Such a function is the restriction to degree
$d$ of a cusp form which is periodic under a central lattice
$\Xi$ whose degrees are large enough for the translates of the
support to be disjoint. Periodic cusp forms of fixed level form a
finite-dimensional space
\cite[\S9.2, proof after (9.3)]{G2BernsteinLapid24}, and the
representation which it generates is a finite sum of irreducible
cuspidal representations by
Lemma~\xref{I}{fl:lem:unitary-admissible-semisimplicity}; their
central characters have finite order on the connected centre,
because the connected centre modulo $\Xi$ and the level subgroup
is finite. So the cuspidal functions with compact support are
spanned by the functions $1_{\deg=d}\varphi$, with $\varphi$ in an
irreducible cuspidal representation $\pi$ of this kind. Let
$\mathcal C^{\rm gen}$, respectively $\mathcal C^{\rm nong}$, be
the span of the pseudo-Eisenstein series of those for which $\pi$
is generic, respectively not generic. Then
\[
 \mathcal C=\mathcal C^{\rm gen}+\mathcal C^{\rm nong},
\]
and both are stable under the Hecke operators at all unramified
places: a Hecke operator of $M$ of a given degree shifts $d$, and
the pseudo-Eisenstein maps are compatible with Satake restriction.

\step{2}{the character $\eta$ occurs in the generic part} Choose
$g_0$ at which the Whittaker coefficient of $h$, for the character
$\psi_B$ of the maximal unipotent subgroup $U_B$, is nonzero.
Since $h$ is invariant under $K$, the character is trivial on
$U_B(\mathbb A)\cap g_0Kg_0^{-1}$, and
$\phi(ng_0k)=\psi_B(n)^{-1}$ is a well defined function on
$U_B(\mathbb A)g_0K$. Its Poincar\'e series
$P_0=\sum_{\gamma\in U_B(F)\backslash J(F)}\phi(\gamma\,\cdot)$ has
compact support on $J(F)\backslash J(\mathbb A)$, because the
support of $\phi$ is compact modulo $U_B(\mathbb A)$ and $[U_B]$
is compact. Unfolding gives
\[
 \langle h,P_0\rangle\ne0,\qquad
 \langle E,P_0\rangle=0\quad\text{for }E\in\mathcal C^{\rm nong}.
\]
For the second, unfold along the Bruhat decomposition, with
representatives $w$ of minimal length in their classes modulo the
Weyl group of $M$. If the cell of $w$ is not the open one, there is
a simple root $\alpha$ such that $w\alpha$ is a root of the
unipotent radical $N$ of the parabolic subgroup. The section is
invariant on the left under $N(\mathbb A)$, so its translate by
$w$ is invariant under the root group of $\alpha$, on which
$\psi_B$ is not trivial; the contribution of the cell is zero. The
open cell contributes the Whittaker coefficient on $M$ of the
function from which the section is induced. This function is a
function in the nongeneric representation, restricted to one
degree; the restriction does not affect integrals over unipotent
groups, so the coefficient is zero.

Let $A$ be the image of the Hecke algebra of all unramified
places in $\operatorname{End}_{\mathscr H_u}(\mathcal C)$. It is a
finitely generated module over the noetherian ring $\mathscr H_u$,
being a submodule of a finite sum of copies of $\mathcal C$. An
operator which vanishes on $\mathcal C$ vanishes on all functions
of level $K$, so $\eta$ is a character of $A$. We claim that
\[
 \operatorname{Ann}_A(\mathcal C^{\rm gen})\subset\ker\eta .
\]
Let $\mathsf T$ annihilate $\mathcal C^{\rm gen}$. Its transpose
$\mathsf T^\star$ satisfies
$\langle E,\mathsf T^\star P_0\rangle=\langle\mathsf TE,P_0\rangle=0$
for $E$ in $\mathcal C^{\rm gen}$ and also for $E$ in
$\mathcal C^{\rm nong}$, which is stable under $\mathsf T$. Hence
$\mathsf T^\star P_0=0$ and $\langle\mathsf Th',P_0\rangle=0$ for
every function $h'$ of level $K$. If $\eta(\mathsf T)\ne0$, then
$\mathsf T$ is invertible on the generalized eigenspace of $h$:
with $(\mathsf T-\eta(\mathsf T))^ah=0$ and
\[
 h_0=\eta(\mathsf T)^{-1}\sum_{j=0}^{a-1}
          \bigl(-\eta(\mathsf T)^{-1}(\mathsf T-\eta(\mathsf T))\bigr)^jh
\]
we have $\mathsf Th_0=h$, and $\langle h,P_0\rangle=0$, a
contradiction. This proves the claim. Since
$\mathcal C^{\rm gen}$ is a finitely generated $A$-module whose
annihilator is contained in the maximal ideal $\ker\eta$,
Nakayama's lemma gives
$\mathcal C^{\rm gen}\otimes_Ak_\eta\ne0$.

\step{3}{specializing the degree} Finitely many generic $\pi$
suffice to generate $\mathcal C^{\rm gen}$ over $\mathscr H_u$.
For each of them choose a central lattice $\Xi$, generated by
elements at the place $u$, which acts trivially on $\pi$
\cite[\S6.2]{G2BernsteinLapid24}, and let
$\Lambda=\deg\Xi\subset\mathsf D_M$, a sublattice of finite index,
and $R=k[\Lambda]$. On each of the finitely many pieces of the
Iwasawa decomposition consider the functions $1_{\deg=d}\varphi$
with $\varphi\in\pi^{K_{M,a}}$.
Together they span
\[
 \mathcal D_\pi=\bigoplus_{\bar d\in\mathsf D_M/\Lambda}
                       R\otimes_kV^{\rm cut}_{\bar d},
\]
where $V^{\rm cut}_{\bar d}$ is the direct sum, over these pieces, of
the spaces of restrictions to one degree of the class $\bar d$.
The lattice $\Lambda$ acts through the translations by $\Xi$. So $\mathcal D_\pi$ is a free
$R$-module of finite rank.

Two algebras act on $\mathcal D_\pi$.
Let $\mathscr H^{\rm ur}$ be the Hecke algebra of $J$ at all
unramified places, as an abstract algebra. It acts on $\mathcal C$,
through its image $A$, and on $\mathcal D_\pi$, through the Satake
restriction to $M$; $\mathcal D_\pi$ is stable because a Hecke
operator of $M$ is a sum of operators which shift the degree and
map each $\pi^{K_{M,a}}$ into itself. The pseudo-Eisenstein map
$\mathcal D_\pi\to\mathcal C^{\rm gen}$ is equivariant for
$\mathscr H^{\rm ur}$. The image $A$ itself need not act on
$\mathcal D_\pi$, since this map need not be injective; so we work
with $\mathscr H^{\rm ur}$ and with the ideal
$\mathfrak m_\eta\subset\mathscr H^{\rm ur}$, the kernel of the
character defined by $\eta$. The translations by $\Xi$ are not
elements of $\mathscr H^{\rm ur}$: already for a group of
semisimple rank one, the Satake algebra of the torus is
$k[z,z^{-1}]$, that of the group is $k[z+z^{-1}]$, and translation
is multiplication by $z$. But they commute with the action of
$\mathscr H^{\rm ur}$ on $\mathcal D_\pi$, because $\Xi$ is central
in $M$. Commutation is all that is needed.

By step~2, $\mathcal C^{\rm gen}/\mathfrak m_\eta\mathcal C^{\rm gen}
\ne0$. The sum of the maps $\mathcal D_\pi\to\mathcal C^{\rm gen}$
over the finitely many chosen $\pi$ is surjective and equivariant,
so by right exactness
$Q=\mathcal D_\pi/\mathfrak m_\eta\mathcal D_\pi\ne0$ for one of
them. The submodule $\mathfrak m_\eta\mathcal D_\pi$ is stable
under $R$, because the two actions commute. So $Q$ is a nonzero
finitely generated $R$-module, and by Nakayama's lemma there is a
maximal ideal of $R$, that is, a character
$\zeta:\Lambda\to k^\times$, such that the fibre of $Q$ at $\zeta$
is not zero. Extend $\zeta$ to $\chi:\mathsf D_M\to k^\times$.

The fibre of $\mathcal D_\pi$ at $\zeta$ is described by the
identity
\[
 \sum_{\nu\in\Lambda}\zeta(\nu)^{-1}T_\nu(1_{\deg=d}\varphi)
 =\chi(d)^{-1}1_{\deg\equiv d\bmod\Lambda}\,(\varphi\otimes\chi),
\]
where $a_\nu$ is an element of $\Xi$ of degree $\nu$ and
$T_\nu f(x)=f(xa_\nu)$.
At each point only one term of the sum is not zero. So the fibre
is identified with its image: restricted to one chosen degree in
each class, passing to the quotient by $\Xi$ gives back the corresponding space
of restrictions.
Since
\[
 1_{\deg\equiv d\bmod\Lambda}
   =\frac1{|\mathsf D_M/\Lambda|}
          \sum_{\varepsilon\in(\mathsf D_M/\Lambda)^\vee}
                          \varepsilon(\deg-d),
\]
the fibre is contained in the sum of the cuspidal representations
$\pi\otimes\chi\varepsilon$. They are generic, since twisting by a
character of the degree does not affect unipotent integrals, and
$\mathscr H^{\rm ur}$ acts on their spherical vectors by
characters. As the fibre of $Q$ at $\zeta$ is a nonzero quotient of
this sum on which $\mathscr H^{\rm ur}$ acts through $\eta$, one of
these characters is $\eta$. Replacing $\chi$ by $\chi\varepsilon$ gives
(1).

\step{4}{continuity} Assume the hypothesis of (2). All the data
are defined over a finite extension of $\mathbb Q_\ell$. The
cuspidal representation $\pi$ has a continuous parameter, namely
that of a nonzero excursion summand, and its eigenvalues at $u$ in
a faithful representation $r$ of the dual group of $J$ are units.
On the weight space of a weight $w_i$ of the connected centre of
the dual group of $M$ in $r$, the twist by $\chi$ multiplies the
eigenvalues by the value of $w_i$ on the element of that centre
corresponding to $\chi$, with coordinates $z_j$. By hypothesis
the products are units, so
\[
 \sum_j(w_i)_j\operatorname{val}_\ell(z_j)=0\quad\text{for every }i.
\]
Since $r$ is faithful the $w_i$ span the rational character space
of the centre, and all the $z_j$ are units; this is not affected
by the extraction of a root needed to pass from the degree of $u$
to degree one. A unit of a finite extension of $\mathbb Q_\ell$
generates a subgroup with compact closure, so the character of
$\mathbb Z$ it defines extends continuously to
$\widehat{\mathbb Z}$. This gives (2).
\end{proof}

\begin{remark}
The usual cuspidal support would produce a Levi subgroup and a
cuspidal representation but could lose genericity. Here the
Poincar\'e series $P_0$ of step~2 forces $\eta$ to occur in the
generic part before the degree variables are specialized. The
hypothesis on units is used only in step~4, to make the character
of the degree continuous.
\end{remark}

\subsection{Removing the two trivial lines}

\begin{proposition}[Removing the two trivial lines]
\label{fl:prop:classical-doubled-line-extraction}
Let $\sigma:\Gamma_F\to\operatorname{SO}_{2r+1}(k)$ be continuous,
finitely ramified, semisimple, elliptic and strongly pure, and assume
that $\varrho=\operatorname{Std}\sigma$ has no nonzero invariant
vector, that is, $\varrho^{\Gamma_F}=0$. (A line on which $\Gamma_F$
acts by a nontrivial character is allowed.) Suppose there
is a function of finite level on $\operatorname{Sp}_{2r+2}$ with a
nonzero principal Whittaker coefficient on which the spherical
Hecke algebras act, as a generalized character, with standard
parameter $\varrho\oplus1^{\oplus2}$. Then the excursion summand
of $\sigma$ on $\operatorname{Sp}_{2r}$ contains a generic
irreducible cuspidal representation.
\end{proposition}
\begin{proof}
Proposition~\ref{fl:lem:generic-cuspidal-support} applies: for a
symplectic group the coefficients of the standard characteristic
polynomial generate the spherical Hecke algebra, and the Satake
parameters are unit-valued because $\sigma$ is continuous. It gives
a
Levi subgroup
$M=\prod_i\operatorname{GL}_{d_i}\times\operatorname{Sp}_{2s}$ and
a generic cuspidal representation of it with continuous
parameters $\tau_i$ of the factors $\operatorname{GL}_{d_i}$ and
$\tau_s$ of $\operatorname{Sp}_{2s}$, such that
\[
 \bigoplus_i(\tau_i\oplus\tau_i^\vee)\oplus\operatorname{Std}\tau_s
                      \simeq\varrho\oplus1^{\oplus2}
\]
by Chebotarev. The right side is pure of weight zero, because
$\sigma$ is strongly pure, and it contains the trivial
representation twice. So by
Lemma~\ref{fl:lem:generic-double-line-obstruction} it is not
$\operatorname{Std}\tau_s$ alone: the Levi subgroup is proper. By Proposition~\ref{fl:prop:prelim-block} the irreducible
constituents of $\varrho$ are nontrivial, self-dual and of
multiplicity one. A nontrivial constituent of some $\tau_i$ would
be a constituent of $\varrho$, hence self-dual, and would occur also
in $\tau_i^\vee$, with multiplicity at least two on the left. So
each $\tau_i$ is a sum of trivial representations; being the
parameter of a cuspidal representation of
$\operatorname{GL}_{d_i}$ it is irreducible, so $d_i=1$ and
$\tau_i=1$. Each such factor accounts for two trivial lines, and
there are exactly two. Hence there is one factor
$\operatorname{GL}_1$, $s=r$, and
$\operatorname{Std}\tau_s\simeq\varrho$.

The component on $\operatorname{Sp}_{2r}$ is a generic cuspidal
representation whose Hecke eigenvalues are those of $\varrho$. Every
excursion summand which meets it has a parameter with standard
representation $\varrho$, by Chebotarev, and such a parameter is
conjugate to $\sigma$ by Lemma~\ref{fl:lem:prelim-form} in odd
dimension.
\end{proof}

\begin{theorem}[Symplectic groups]
\label{fl:thm:symplectic-realization}
Every continuous, finitely ramified, semisimple, elliptic and strongly
pure parameter $\sigma:\Gamma_F\to\operatorname{SO}_{2n+1}(k)$
has a nonzero cuspidal excursion summand on split
$\operatorname{Sp}_{2n}$, in every positive characteristic. For
$n\ge3$, if its standard representation has no nonzero invariant
vector, this summand contains a globally generic irreducible
cuspidal representation.
\end{theorem}
\begin{proof}
For $n=1$ this is the case of type $A_1$
(Theorem~\ref{fl:thm:type-a-realization}), and for $n=2$ it
follows from $\operatorname{Sp}_4\simeq\operatorname{Spin}_5$ and
Theorem~\ref{fl:thm:odd-orthogonal-spin-realization}. Let
$n\ge3$, and let $n_0\in\{0,1\}$ be the dimension of the space of
invariants of $\varrho=\operatorname{Std}\sigma$. Put
\[
 \varrho_0=\begin{cases}
          \varrho\ominus1,&m=n,\quad\text{if }n_0=1,\\
          \varrho\oplus1,&m=n+1,\quad\text{if }n_0=0.
        \end{cases}
\]
This is an orthogonal representation of dimension $2m$ and
determinant one whose irreducible constituents are orthogonal and
distinct, hence an elliptic parameter for $\operatorname{SO}_{2m}$
by Proposition~\ref{fl:prop:prelim-block}. By
Corollary~\ref{fl:cor:char2-odd-orthogonal-realization} there is a
generic cuspidal representation of split $\operatorname{SO}_{2m}$
with this parameter, and by
Corollary~\ref{fl:cor:odd-theta-moment} a function on
$\operatorname{Sp}_{2m}$ with nonzero Whittaker coefficient which
is a generalized Hecke eigenfunction with standard parameter
$\varrho_0\oplus1$.

If $n_0=1$ this parameter is $\varrho$, and
Proposition~\ref{fl:prop:prelim-full-parameter} shows that the
function is cuspidal and that the summand of $\sigma$ is nonzero.
If $n_0=0$ it is $\varrho\oplus1^{\oplus2}$ on
$\operatorname{Sp}_{2n+2}$, and
Proposition~\ref{fl:prop:classical-doubled-line-extraction} gives
a generic cuspidal representation in the summand of $\sigma$ on
$\operatorname{Sp}_{2n}$.
\end{proof}