\section{Overview of the construction}
\label{rw:sec:descent-route}

This section describes the proof of Theorem~\ref{rw:thm:III-main}
without giving it. The construction starts
from cusp forms on general linear groups, which exist by the
theorem of L.~Lafforgue, and transports them to \(G\) by integral
operators. Functions are present from the beginning, and the
difficulty is to show that the operators do not annihilate them.

In type \(A\) the parameter is lifted to a general linear group,
and the correspondence of L.~Lafforgue applies
(Theorem~\ref{fl:thm:type-a-realization}). The rest of this
outline is about the symplectic and orthogonal cases.

\emph{From the parameter to general linear groups.} Let
\(\varrho=\operatorname{Std}\sigma\) be the
composite of \(\sigma\) with the standard representation of
\(\widehat G\). Since \(\sigma\) is elliptic, \(\varrho\) is a sum
of pairwise nonisomorphic irreducible representations
\(\varrho_i\), each of which preserves a form of the same type,
symmetric or alternating
(Proposition~\ref{fl:prop:prelim-block}). Each \(\varrho_i\) is
the parameter of a cuspidal representation \(\Pi_i\) of a general
linear group. The problem is to make a function on \(G\) out of
the \(\Pi_i\).

\emph{Orthogonal descent.} As in the work of Ginzburg, Rallis and
Soudry, we regard
the product of these general linear groups as a Levi subgroup of a
classical group \(S\) about twice as large as \(G\), and form the
Eisenstein series on \(S\) induced from the \(\Pi_i\). That
\(\varrho_i\) preserves a form of the given type is reflected in a
pole of an \(L\)-function of \(\Pi_i\), and this pole produces a
pole of the Eisenstein series. Its residue is a space of functions
on \(S(F)\backslash S(\mathbb A)\) which are square integrable
modulo the centre
(Section~\ref{fl:sec:common-descent}). The group \(S\) contains a
unipotent subgroup \(U_{\rm desc}\) with a character
\(\psi_{\rm desc}\) whose stabilizer contains \(G\). The
\emph{descent} of a function \(E\) in the residue is the function
\[
 g\longmapsto\int_{U_{\rm desc}(F)\backslash U_{\rm desc}(\mathbb A)}
     E(ug)\,\psi_{\rm desc}(u)^{-1}\,du,
 \qquad g\in G(\mathbb A).
\]
In words: one integrates \(E\) over the directions of \(S\) which
do not belong to \(G\), against a character, and what remains is a
function on \(G(F)\backslash G(\mathbb A)\).

\emph{Two separate questions.} At an unramified place the local
component of the residue is a subquotient of an explicit induced
representation, and the corresponding local operation, the Jacquet
module twisted by \(\psi_{\rm desc}\), is exact. The twisted
Jacquet module of the induced representation can be computed, and
its unramified constituents have the Satake parameter of
\(\sigma\) (Section~\ref{fl:sec:orthogonal-descent}). In the
smallest case, the descent from \(\operatorname{SO}_5\) to a torus
of dimension one, it is the sum of two characters, inverse to each
other, if the torus is split, and it is zero otherwise
(Section~\ref{rw:sec:smallest-case}). This gives
the Hecke eigenvalues of the descent, provided the descent is not
zero; it says nothing if it is zero. That the descent is not zero
is a global statement. The residue has a nonzero Fourier
coefficient of another kind, which comes from the Whittaker
coefficients of the \(\Pi_i\)
(Proposition~\ref{fl:lem:residual-analytic}). We show that the Whittaker
coefficient of the descent is not zero by transforming one
coefficient into the other. The tool is elementary
(Lemma~\ref{fl:lem:odd-exchange}): when two unipotent subgroups
\(X\) and \(Y\) are put in duality by the commutator and a
character, an integral over \(Y(F)\backslash Y(\mathbb A)\) may be
exchanged for an integral over
\(X(F)\backslash X(\mathbb A)\) without affecting vanishing. A
proof of nonvanishing is a sequence of such exchanges, sometimes a
long one, and its whole difficulty is to keep track of the root
subgroups involved
(Sections~\ref{fl:sec:orthogonal-descent}
and~\ref{fl:sec:odd-symplectic}).
Figures~\ref{rw:fig:interleave}--\ref{rw:fig:char2-cross} show
these subgroups in small cases.

\emph{Small characteristic.} The exchanges rest on the commutator
relations between root subgroups, with their structure constants,
and on Fourier analysis on \(F\backslash\mathbb A\). For an odd
orthogonal target the argument is the same in all
characteristics, once the orthogonal group is defined by a
quadratic form and not by a bilinear form. For an even orthogonal
target one step moves a Fourier coefficient from the trivial
character of a root subgroup to a nontrivial one, and in odd
characteristic it does so by a computation in which a quadratic
exponent is divided by two. In characteristic two we replace it by
the following observation: the function
\(a\mapsto\psi(sa^2)\) is then an additive character of \(a\), so
it is of the form \(\psi(C_\psi(s)a)\), and the map \(C_\psi\),
which is computed by the Cartier operator, is surjective
(Section~\ref{fl:sec:char2-d}).

\emph{Symplectic groups.} For a symplectic group we do not use a
descent but the theta lift from an even orthogonal group, which on
parameters adds a trivial line to the standard representation.
Nonvanishing and the Hecke eigenvalues both follow from one
computation, that of a Jacquet module of the Weil representation
(Sections~\ref{fl:sec:odd-theta}
and~\ref{fl:sec:classical-double-line}).

\emph{From Hecke eigenvalues to the parameter.} In these cases the
outcome is a nonzero function on \(G\) which has the Hecke
eigenvalues of \(\sigma\) at almost all places and a nonzero
Whittaker coefficient. This is less than what the theorem asserts,
which concerns all excursion operators. The gap is closed by the
arithmetic results of \cite{HuI}: such a function is cuspidal because \(\sigma\) is
elliptic, the cusp forms with these Hecke eigenvalues have
parameters whose standard representations are isomorphic to
\(\varrho\), and for the groups in question this determines the
parameter up to conjugacy, or up to the outer automorphism in the
even orthogonal case, which is then dealt with separately
(Proposition~\ref{fl:prop:prelim-full-parameter}).