\section{Results used}
\label{rw:sec:inputs}

We collect the theorems on which the construction rests, in the
form in which they are used. Precise references to the individual
statements are repeated at each point of use.

\subsection{From the literature}

\begin{extinput}[General linear groups]\label{rw:in:llafforgue}
Let \(\varrho\) be a continuous irreducible \(n\)-dimensional
\(\overline{\mathbb Q}_\ell\)-representation of \(\Gamma_F\),
unramified at almost all places, whose determinant has finite order.
\begin{enumerate}
\item There is a cuspidal automorphic representation \(\pi\) of
\(\mathrm{GL}_n(\mathbb A_F)\) whose Satake parameters at the
unramified places are the eigenvalues of geometric Frobenius
elements in \(\varrho\)
\cite[Introduction, Theorem \textup{(i)}$_n$--\textup{(ii)}$_n$]{LLa02}.
\item The Frobenius eigenvalues of \(\varrho\) at the unramified
places are algebraic numbers all of whose complex absolute values
are one \cite[Theorem~VII.6]{LLa02}.
\item The representation \(\pi\) is tempered at every place
\cite[Theorem~VI.10(i)]{LLa02}.
\end{enumerate}
\end{extinput}

In part (1) the Hecke operators act by right translation, as in
Section~\ref{fl:sec:parameters-parameters}. The statement does not
depend on the conventions of \cite{LLa02} for Hecke operators and
for Frobenius elements: a change of either convention replaces
$\varrho$ by its dual and $\pi$ by its contragredient, and the
theorem applies to all irreducible representations.

Parts (1) and (3) are the starting point of the construction; part
(2) is the source of the purity theorem of
\cite[Section~\xnum{I}{fl:sec:arithmetic-purity}]{HuI}.

\begin{extinput}[Eisenstein series, Whittaker functionals and the
Weil representation]\label{rw:in:descent}
We follow the descent method of
Ginzburg, Rallis and Soudry \cite{GRS1999,GRS2011}: one forms the
residue of an Eisenstein series on a larger classical group, induced
from self-dual cuspidal representations of general linear groups,
and takes a Fourier coefficient of it. No theorem is quoted from
those works. The statements we do quote are the following.
\begin{enumerate}
\item Cusp forms of fixed level on a group over a function field
have compact support modulo the centre
\cite[Chapter~I, \S5.3 and Proposition~5.3]{Morris1982}.
\item A function of finite level agrees with its constant terms
far in the cusps \cite[Chapter~I, \S5.9]{Morris1982}; consequently
an automorphic form on which the centre acts through a unitary
character, and whose constant terms have exponents in the negative
chamber, is square integrable modulo the centre (the estimate is
written out
in the proof of
Lemma~\ref{fl:lem:residual-image-square-integrable}). For a maximal
parabolic subgroup and unitary cuspidal data, at a fixed level, the
intertwining operator and the Eisenstein series are rational
functions on the complex torus of unramified characters by which
the data are twisted. Along the real points of this torus, with
parameter $s\in\mathbb R$ and $s=0$ the trivial character, they are
holomorphic at $s=0$ and have at most simple poles for $s>0$
\cite[Chapter~III, Lemmas~3.2 and~3.5, Theorem~5.2]{Morris1982}.
\item Over a nonarchimedean local field of any characteristic,
the intertwining operators for tempered inducing data are given
by convergent integrals in the open positive chamber
\cite[Proposition~IV.2.1]{Waldspurger2003}, and on the spherical
vector of an unramified principal series they are given by the
formula of Gindikin and Karpelevich
\cite[Theorem~3.1]{Casselman80}.
\item The Whittaker functional on a principal series
representation is given by an integral over the open cell
\cite[Theorems~1.4 and~1.6, Corollary~1.7]{CS80}, with the
unramified value of \cite[Theorem~5.4]{CS80}.
\item Eisenstein series over function fields continue
meromorphically, with the constant-term formulas of
\cite[Theorem~2.3, (6.2), (6.4), (6.17)]{G2BernsteinLapid24}; pseudo-Eisenstein
series span the functions of compact support
\cite[\S9.2, (9.3)]{G2BernsteinLapid24}. In the same
work automorphic forms are characterized by finite systems of
difference equations satisfied by the cuspidal projections of
their constant terms
\cite[\S9.1]{G2BernsteinLapid24}.
\item The theta distribution of the Weil representation is
invariant under rational points
\cite[\S\S29--30 and 40--41]{Weil64}. The metaplectic
cocycle in odd characteristic is described in \cite{Rao1993}.
\end{enumerate}
What is proved here, and not taken from the literature, is the
nonvanishing of the descent over $F$, the identification of its
excursion parameter, and every argument in characteristic two.
\end{extinput}

\subsection{From the first paper}
\label{rw:sec:from-I}

The arithmetic part of the argument is contained in \cite{HuI}. It
is applied in Section~\ref{fl:sec:classical-descent}, up to
Proposition~\ref{fl:prop:prelim-full-parameter}, and in the
sections where the realization theorems are proved. We use the
following definitions and statements; a reader who accepts them
does not need the rest of that paper.
\begin{enumerate}
\item \emph{Excursion operators on functions.} For a level $N$ the
excursion operators act on the functions with compact support on
$Y_N=G(F)\backslash G(\mathbb A_F)/K_N$, through the cohomology of
stacks of shtukas, and $S_e^{\mathrm{fun}}$ is defined on all
functions on $Y_N$ by transposition
(Definition~\xref{I}{fl:def:function-excursion-action}). On cusp
forms it is the operator of V.~Lafforgue. The cohomology in
question is finitely generated over the spherical Hecke algebra at
one place
(Proposition~\xref{I}{fl:prop:classical-one-place-finiteness}).
\item \emph{Purity and fields of coefficients.} A semisimple
elliptic parameter with values in a semisimple group is strongly
pure
(Theorem~\xref{I}{fl:thm:arithmetic-strong-purity}). We also use
Lemma~\xref{I}{fl:lem:arithmetic-finite-coefficients} on compact
subgroups of $\mathrm{GL}_n(k)$ and the decomposition of a parameter
into a twist by the constant field and a parameter
$\sigma^{\mathrm{fin}}$
(Lemma~\xref{I}{fl:lem:finite-coefficients}).
\item \emph{From Hecke eigenvalues to the parameter.}
Proposition~\xref{I}{fl:prop:full-label-cuspidality}, which shows
that a function with the Hecke eigenvalues of an elliptic parameter
at almost all places is cuspidal, and
Lemma~\xref{I}{rw:lem:noncuspidal-levi-parameter}, the form of
this statement for a subalgebra of the Hecke algebra;
Lemma~\xref{I}{fl:lem:faithful-satake-model}, by which the Hecke
operators attached to one representation with finite kernel
suffice; and Lemma~\xref{I}{fl:lem:hecke-moment-extraction}, which
extracts an eigenfunction from a pairing.
\item \emph{Irreducible constituents.} An admissible subspace of
smooth vectors in a unitary representation is semisimple
(Lemma~\xref{I}{fl:lem:unitary-admissible-semisimplicity}), and
Proposition~\xref{I}{fl:prop:restricted-tensor-unitary-image} on
images of restricted tensor products.
\item \emph{Isogenies.} The pullback of cusp forms along a
homomorphism with central kernel and normal image
(Lemma~\xref{I}{fl:lem:arithmetic-central-pullback}), the descent
along the quotient by a central torus
(Lemma~\xref{I}{fl:lem:cuspidal-central-quotient}), and the
reduction of the existence of a cusp form with a given elliptic
parameter to simply connected simple groups
(Corollary~\xref{I}{fl:cor:simple-group-reduction}).
\end{enumerate}
Those conventions of
\cite[Section~\xnum{I}{fl:sec:parameters-parameters}]{HuI} which are
used in this paper are repeated in
Section~\ref{fl:sec:parameters-parameters}.