\section{Realization for orthogonal and spin groups}
\label{rw:sec:orthogonal-realization}

We return to arbitrary positive characteristic and collect the
results of
Sections~\ref{fl:sec:common-descent}--\ref{fl:sec:char2-d}. The
Eisenstein residue and the global Fourier coefficients have
produced a nonzero generic descent. Two things remain: the descent
must be cuspidal, and its excursion parameter, not only its
unramified Hecke eigenvalues, must be $\sigma$. Both follow from
ellipticity through
Proposition~\ref{fl:prop:prelim-full-parameter}.

Recall the situation of Section~\ref{fl:sec:even-residual}. The
parameter $\sigma$ takes values in $\operatorname{GSp}_{2n}(k)$ or
$\operatorname{GSO}_{2n}(k)$; it is continuous, finitely ramified,
semisimple, elliptic modulo scalars and strongly pure, and its
multiplier $\chi$ has finite order. The residual representation
attached to $\sigma$ lives on $\operatorname{GSpin}_{4n}$ or
$\operatorname{GSpin}_{4n+1}$, and its descent is a space of
functions on split $\operatorname{GSpin}_{2n+1}$ or
$\operatorname{GSpin}_{2n}$, respectively.

\begin{lemma}[The descent is cuspidal]
\label{fl:lem:orthogonal-descent-cuspidality}
Every function in the descent of the residual representation of
Section~\ref{fl:sec:even-residual} to the split group, and every
right translate of such a function, is cuspidal.
\end{lemma}
\begin{proof}
Let $h$ be a function in the descent; it is complex valued, and we
consider $\iota^{-1}h$. Let $u_v$ be an element of the descent
radical at a place $v$. Right translation by $u_v$ multiplies the
descent integral by $\psi_{{\rm desc},v}(u_v)$: at a point of the
target group one conjugates $u_v$ by this point and changes the
variable of integration. So, when the components at the places
other than $v$ are fixed, the descent factors through the twisted
Jacquet module at $v$, and the resulting map is equivariant for
the target group at $v$. At a place where $h$ is spherical and
the residual representation is unramified, the local component of
the latter is a subquotient of the induced representation of
Proposition~\ref{fl:prop:uniform-b-jacquet}. The twisted Jacquet
functor and the functor of invariants under a hyperspecial
subgroup are exact. It follows that
$\iota^{-1}h$ satisfies the Hecke equations of $\sigma$ for the
standard representation and for the multiplier, possibly in
generalized form. This holds also when some of the inducing
characters coincide, and for both summands in the second case of
the proposition. Since $\sigma$ is elliptic,
Proposition~\ref{fl:prop:prelim-full-parameter} shows that
$\iota^{-1}h$ is cuspidal. A constant term at a fixed argument is
a finite sum, so $h$ is cuspidal as well. A right translate of $h$
is spherical outside a larger finite set of places, and the same
argument applies to it.
\end{proof}

\begin{theorem}[Orthogonal realization with multiplier]
\label{fl:thm:orthogonal-realization}
Let $G$ be split $\operatorname{GSpin}_{2n+1}$ with $n\ge1$, or
split $\operatorname{GSpin}_{2n}$ with $n\ge2$, over a global
function field $F$ of any characteristic. Let $\sigma$ be a
continuous, finitely ramified, semisimple parameter with values in
the dual group $\operatorname{GSp}_{2n}(k)$ or
$\operatorname{GSO}_{2n}(k)$, elliptic modulo scalars and strongly
pure, whose multiplier $\chi$ has finite order. Then the
excursion summand of $\nu_\sigma$ in the space of cusp forms on
$G$ contains a generic irreducible cuspidal automorphic
representation. Its connected central character is
$\operatorname{CFT}(\chi)$.
\end{theorem}
\begin{proof}
\step{1}{a nonzero generic descent} For
$\operatorname{GSpin}_3=\operatorname{GL}_2$ the theorem is
Input~\ref{rw:in:llafforgue}. Otherwise take the residual
representation of Section~\ref{fl:sec:even-residual}, which has a
nonzero Siegel--Whittaker coefficient by
Proposition~\ref{fl:lem:residual-analytic}. For the odd target,
Proposition~\ref{fl:prop:uniform-b-nonzero} gives a nonzero generic
descent of length $n-1$. For the even target, a nonzero generic
descent of length $n$ to the split group over $F$ is given by
Proposition~\ref{fl:prop:even-generic-nonzero} in odd
characteristic, and by
Proposition~\ref{fl:prop:char2-direct-generic} and
Corollary~\ref{fl:cor:char2-descent-split} in characteristic two.
In all cases the connected centre acts through
$\omega_\sigma=\operatorname{CFT}(\chi)$.

\step{2}{cuspidality and the parameter} By
Lemma~\ref{fl:lem:orthogonal-descent-cuspidality} the descent is
cuspidal. Take a function in it with nonzero Whittaker coefficient
and transport it to $k$ by $\iota^{-1}$;
Lemma~\ref{fl:lem:prelim-coeff} shows that it is defined over a
finite extension of $\mathbb Q_\ell$, at a fixed level and for a
fixed central lattice. It satisfies the standard and multiplier
Hecke equations of $\sigma$, so
Proposition~\ref{fl:prop:prelim-full-parameter} provides a function
in the excursion summand of $\nu_\sigma$ whose Whittaker
coefficient is still nonzero. In the even case that proposition
may first require replacing $\sigma$ by its conjugate under the
outer automorphism; applying the corresponding rational outer
automorphism of $G$ to the function removes this ambiguity.

\step{3}{an irreducible representation} Transport the function
back to $\mathbb C$. By
Lemma~\xref{I}{fl:lem:unitary-admissible-semisimplicity} the
representation it generates is a finite sum of irreducible
cuspidal representations, and the Whittaker functional is nonzero
on one of them. Excursion operators are given by finite sums at
each fixed level, so they commute with the transport of
coefficients; and right translations preserve excursion summands.
This representation therefore lies in the excursion summand of
$\nu_\sigma$.
\end{proof}

\begin{theorem}[Spin groups]
\label{fl:thm:odd-orthogonal-spin-realization}
Let $G$ be split $\operatorname{Spin}_{2n+1}$ with $n\ge1$ or split
$\operatorname{Spin}_{2n}$ with $n\ge2$, in any characteristic, and
let $\sigma$ be a continuous, finitely ramified, semisimple,
elliptic and strongly pure parameter with values in
$\operatorname{PSp}_{2n}(k)$ or $\operatorname{PSO}_{2n}(k)$,
respectively. Then the excursion summand of $\nu_\sigma$ in the
space of cusp forms on $G$ is nonzero.
\end{theorem}
\begin{proof}
By Lemma~\ref{fl:lem:central-torus-lift}, $\sigma$ lifts to a
parameter with values in $\operatorname{GSp}_{2n}(k)$ or
$\operatorname{GSO}_{2n}(k)$ with multiplier of finite order, and
the lift is again semisimple, elliptic modulo scalars and strongly
pure. Theorem~\ref{fl:thm:orthogonal-realization} realizes the lift
on $\operatorname{GSpin}$. Take a nonzero function in that
realization, translate it on the right so that its value at the
identity is not zero, and restrict it to $\operatorname{Spin}$. The
restriction is cuspidal, because the two groups have the same
unipotent radicals of parabolic subgroups. By
Lemma~\xref{I}{fl:lem:arithmetic-central-pullback} and the invariance
of excursion operators under right translation
\cite[Remark 12.4]{VLa18}, the restriction lies in the excursion summand of the image of the lift in the dual group of
$\operatorname{Spin}$, which is $\sigma$. The cases $D_2$ and $D_3$
are included.
\end{proof}

\begin{corollary}[Special orthogonal groups]
\label{fl:cor:char2-odd-orthogonal-realization}
Let $H$ be split $\operatorname{SO}_{2n+1}$ with $n\ge1$ or split
$\operatorname{SO}_{2n}$ with $n\ge2$, in any characteristic, and
let $\sigma$ be a continuous, finitely ramified, semisimple
elliptic parameter with values in $\operatorname{Sp}_{2n}(k)$ or
$\operatorname{SO}_{2n}(k)$, respectively, whose standard
representation is pure of weight zero for every embedding of the
coefficients into $\mathbb C$. Then the excursion summand of
$\nu_\sigma$ in the space of cusp forms on $H$ contains a generic
irreducible cuspidal automorphic representation.
\end{corollary}
\begin{proof}
Regard $\sigma$ as a parameter with values in
$\operatorname{GSp}_{2n}(k)$ or $\operatorname{GSO}_{2n}(k)$ with
trivial multiplier. The irreducible summands of its standard
representation have determinants of finite order, and the purity
theorem of Section~\xref{I}{fl:sec:arithmetic-purity} gives the
hypotheses of the residual construction.
Theorem~\ref{fl:thm:orthogonal-realization} gives a generic
cuspidal representation of $\operatorname{GSpin}$ with trivial
connected central character. Its functions are invariant under the
central torus $\mathbb G_m$, which is smooth and split in every
characteristic, including two, and so they descend to the quotient
$H$. Lemma~\xref{I}{fl:lem:cuspidal-central-quotient} identifies the
excursion summand containing the descended representation as
that of $\nu_\sigma$, and the Whittaker coefficient is unchanged.
\end{proof}