\section{Introduction}
\label{rw:sec:II-introduction}

Let \(F\) be the function field of a curve over a finite field and
let \(G\) be a split reductive group over \(F\). Vincent Lafforgue
decomposed the space of cusp forms on \(G\) according to semisimple
Langlands parameters
\(\Gamma_F\to\widehat G(\overline{\mathbb Q}_\ell)\). His theorem
attaches a parameter to each summand of the decomposition. It does
not say which parameters occur.

We prove that every semisimple elliptic parameter occurs when
\(G\) is simply connected and the characteristic does not divide
the order of the Weyl group. The proof is geometric: it starts from
the geometric Langlands correspondence over a field of
characteristic zero and ends with a function on the points of
\(\operatorname{Bun}_G\) over the given finite field. This is the
second paper of a series of three. The first \cite{HuI} contains
the arithmetic foundations and deduces from the present theorem,
and from its counterpart for classical groups \cite{HuIII}, the
existence of automorphic realizations of semisimple parameters on all
split reductive groups.

\subsection{The theorem}

Let \(X/\mathbb F_q\) be a smooth projective geometrically connected
curve, let \(F=\mathbb F_q(X)\), and put \(p=\operatorname{char}F\).
Fix a prime \(\ell\ne p\), put \(k=\overline{\mathbb Q}_\ell\), and
write \(\Gamma_F=\operatorname{Gal}(F^{\mathrm{sep}}/F)\). For a
split connected reductive group \(G/F\), the dual group
\(\widehat G\) is taken over \(k\).

\begin{definition}\label{fl:def:intro-parameter}
A \emph{parameter} is a continuous homomorphism
\(\sigma:\Gamma_F\to\widehat G(k)\) unramified at all but finitely
many places. It is \emph{semisimple} if the identity component of
the Zariski closure of its image is reductive. It is \emph{elliptic}
if
\[
 C_{\widehat G}(\operatorname{im}\sigma)/Z(\widehat G)
 \quad\text{is finite}.
\]
\end{definition}

The link between cusp forms and parameters is given by excursion
operators. An excursion datum consists of a finite set \(I\), a
function
\(f\in k[\widehat G\backslash\widehat G^I/\widehat G]\), invariant
under simultaneous left and right multiplication, and a tuple
\(\boldsymbol\gamma\in\Gamma_F^I\). Lafforgue's operator
\(S_{I,f,\boldsymbol\gamma}\) acts on cusp forms: it creates an
invariant vector in the cohomology of shtukas with legs indexed by
\(I\), transports it by \(\boldsymbol\gamma\), and contracts with an
invariant covector representing \(f\)
\cite[\S\S0.3--0.5]{VLa18}. These operators commute, and a cusp
form \(\varphi\) has parameter \(\sigma\) when
\begin{equation}\label{fl:eq:main-full-label}
 S^{\mathrm{fun}}_{I,f,\boldsymbol\gamma}\varphi
 =f\bigl((\sigma(\gamma_i))_{i\in I}\bigr)\varphi
 \qquad\text{for every excursion datum }(I,f,\boldsymbol\gamma).
\end{equation}
We write \(\nu_\sigma(I,f,\boldsymbol\gamma)\) for the scalar on the
right. A nonzero cusp form \(\varphi\) which satisfies
\eqref{fl:eq:main-full-label} is called an \emph{automorphic
realization} of \(\sigma\), and we say that it \emph{realizes}
\(\sigma\). The superscript in \(S^{\mathrm{fun}}\) records that we need
these operators on functions that are not yet known to be cuspidal;
on cusp forms \(S^{\mathrm{fun}}\) is Lafforgue's operator, and the
extension is constructed in
\cite[Section~\xnum{I}{fl:sec:compact-foundations}]{HuI}.

\begin{theorem}\label{rw:thm:II-main}
Let \(G/F\) be split, simply connected and semisimple, and suppose
that \(p\) does not divide the order of the Weyl group of \(G\).
Let \(\sigma:\Gamma_F\to\widehat G(k)\) be a semisimple elliptic
parameter. Then there are an open compact subgroup
\(K\subset G(\mathbb A_F)\) and a nonzero cusp form
\(\varphi_\sigma\) on \(G(F)\backslash G(\mathbb A_F)/K\) satisfying
\eqref{fl:eq:main-full-label}.
\end{theorem}

This is Theorem~\ref{fl:thm:weyl-prime-realization}. Three features
of the result should be stated at once. The cusp form lives over
\(F\) itself: neither the field nor its field of constants is
extended. The parameter may be ramified at finitely many places and
may have disconnected, even finite, monodromy. And the parameter is
recovered through every excursion operator, so the form determines
\(\sigma\) up to conjugacy and not only the conjugacy classes of
its Frobenius elements.

For the types \(G_2\) and \(F_4\) the hypothesis is \(p>3\), for
\(E_6\) it is \(p>5\), and for \(E_7\) and \(E_8\) it is \(p>7\).
For a group of classical type it is a condition which grows with
the rank; these groups are treated in every characteristic in
\cite{HuIII}, by a different method, so the theorem is needed in
\cite{HuI} only for the five exceptional types. The hypothesis
belongs to the method. It supplies nondegenerate invariant forms on
the Lie algebra and its Levi subalgebras, and it keeps wild inertia
inside tori when the parameter is lifted to characteristic zero. We
do not expect it to reflect an obstruction to automorphy.

The function \(\varphi_\sigma\) is a nonzero element of the
excursion summand of \(\sigma\), and it is not unique. We do not
determine multiplicities, the structure of packets, or
compatibility with the local correspondence at the ramified
places.

\subsection{Relation to earlier work}
\label{rw:sec:earlier-work}

\emph{General linear groups and excursion operators.}
Laurent Lafforgue proved the correspondence for irreducible
\(\mathrm{GL}_n\)-parameters with finite-order determinant
\cite[Introduction, Theorem \textup{(i)}$_n$--\textup{(ii)}$_n$]{LLa02}.
For a split reductive group, Vincent Lafforgue constructed the
decomposition of cusp forms by parameters \cite[Theorem~0.1]{VLa18}.
That theorem identifies the parameters of the summands that occur;
that the summand of a prescribed parameter is nonzero is a separate
assertion, and it is the subject of this series of papers. For a split
semisimple group and a parameter which is everywhere unramified
and has Zariski dense image, \cite[Theorem~1.1]{BHKT2019} gives,
after a finite Galois extension of the function field, a cuspidal
representation whose unramified components have the Satake
parameters prescribed by the parameter. Our
conclusion keeps the field fixed.

\emph{Geometric Langlands.}
The geometric input is the restricted geometric Langlands
equivalence of Gaitsgory and Raskin over an algebraically closed
field of characteristic zero \cite[Theorem~1.3.9(ii)]{GR2025},
together with its Whittaker functional. In positive characteristic
their theorem gives an equivalence on a union of connected
components of the spectral side, and whether this union is
everything is their Conjecture~1.3.10. The subsequent preprint
\cite[Theorem~1.4]{OpenAIRestricted2026}, by OpenAI, states that it
is, over \(\overline{\mathbb F}_q\) under the four Lie-theoretic
assumptions of the restricted theory, and over an arbitrary
algebraically closed field when in addition \(p\) is very good and
\(p\nmid|W_G|\); its Corollary~8.2 gives the arithmetic formula for
unramified parameters. We do not use that theorem. Our result
concerns arithmetic parameters over \(F\) that are ramified at
finitely many places and may have finite or disconnected monodromy,
and it produces functions, not sheaves.

Our method does, however, owe several ideas to the three Langlands
preprints
\cite{OpenAIRestricted2026,OpenAIConstructible2026,OpenAIFrobenius2026}
in OpenAI's public \emph{math} repository. Testing a Hecke
eigensheaf against a hypersurface transverse to the nilpotent cone,
letting modification points collide, and working with framed
bundles all appear there and are starting points for
the present paper. The statements of those preprints have
different hypotheses from ours: the tame eigensheaf theorem
\cite[Theorem~1.1]{OpenAIConstructible2026} assumes geometrically
dense monodromy and one tame regular-unipotent puncture, and the
Frobenius theorem \cite[Theorem~1.1]{OpenAIFrobenius2026} treats
\(\mathrm{SL}_n\) with \(p>n\) and allows a finite extension of
constants. None of them is used as an input.

What is new in the geometric argument is the combination of three
steps: lifting the ramification of the parameter to a curve in
characteristic zero on which it becomes unramified; constructing a
coefficient whose Frobenius supertrace is a nonzero determinant;
and comparing the whole category of coefficients, not one object,
through specialization, which reduces the ambiguity in the
Frobenius structure to a single scalar. A trace calculation with a
homogeneous test function then yields a bundle defined over
\(\mathbb F_q\) at which the resulting function does not vanish.
That calculation draws on \cite[Appendix~A]{BBMY2026}; the
Whittaker and spectral comparisons use
\cite{AGKRRV2022,NY2019,GKRV2022,GLCI2024,NT2026}.

\subsection{Organization}

Section~\ref{fl:sec:strategy} explains the construction without
proofs and shows how the main results depend on one another.
Section~\ref{rw:sec:inputs} lists the results we import,
including those of \cite{HuI},
Section~\ref{fl:sec:parameters-parameters} fixes conventions, and
Section~\ref{rw:sec:geometric-route} introduces the curves and the
Hecke families which are used throughout. The construction
itself occupies
Sections~\ref{fl:sec:punctured-spectral-fibre}--\ref{fl:sec:exceptional-assembly}.
A parameter over \(\mathbb F_5(t)\) with image
\(S_3\subset\mathrm{PGL}_2(k)\) is followed through the first
steps in Sections~\ref{fl:sec:low-rank-example}
and~\ref{rw:sec:example-curve}. A list of notation is at the end,
and Appendix~\ref{rw:app:computer} lists the identities which were
also checked by computer.