\section{The realization theorem for classical groups}
\label{fl:sec:CLA-proof}

This section assembles the results of the preceding sections into
the realization theorem for all split semisimple groups with
factors of classical type.

\begin{theorem}[Classical groups]
\label{fl:thm:classical-realization}
Let $G/F$ be split connected semisimple with simple factors of
types $A,B,C,D$, in any positive characteristic, and let
$\sigma:\Gamma_F\to\widehat G(k)$ be a semisimple elliptic
parameter in the sense of Definition~\ref{fl:def:intro-parameter}.
Then the cuspidal excursion summand of $\sigma$ is nonzero at some
finite level. This holds for every group in each isogeny class.
\end{theorem}

This is the conclusion of Theorem~\xref{I}{fl:thm:semisimple-elliptic}
for these groups, and it is Theorem~\ref{rw:thm:III-main}: a nonzero excursion summand at a finite level is
a finite-dimensional space on which the excursion operators commute
and act through the generalized character of $\sigma$, so it
contains a cusp form satisfying \eqref{fl:eq:main-full-label}. In
the statements of this paper the words continuous and finitely
ramified repeat what Definition~\ref{fl:def:intro-parameter}
requires of a parameter.

\begin{proof}
By Corollary~\xref{I}{fl:cor:simple-group-reduction} it suffices to
treat the simply connected simple groups, the passage to products
and to an arbitrary central quotient being made there. The
parameters of the simple factors are again continuous, finitely
ramified, semisimple and elliptic, hence strongly pure by
Theorem~\xref{I}{fl:thm:arithmetic-strong-purity}. For
$\operatorname{SL}_n$ the assertion is
Theorem~\ref{fl:thm:type-a-realization}; for
$\operatorname{Spin}_{2n+1}$ and $\operatorname{Spin}_{2n}$ it is
Theorem~\ref{fl:thm:odd-orthogonal-spin-realization}; for
$\operatorname{Sp}_{2n}$ it is
Theorem~\ref{fl:thm:symplectic-realization}. The isomorphisms in
low rank ($B_1=C_1=A_1$, $D_2=A_1\times A_1$, $D_3=A_3$,
$B_2=C_2$) are included in these statements.

One point in the reduction should be kept in mind. The central
subgroup by which the product of the simply connected factors is
divided need not be a product of subgroups of the factors, and it
need not be reduced when the characteristic divides its order.
The transfer of Corollary~\xref{I}{fl:cor:simple-group-reduction} is
therefore applied once, to the product and the given subgroup, and
not factor by factor.
\end{proof}