Stack of local L-parameters

Let $E$ be a nonarchimedean local field, let $W_E$ be its Weil group, and let the dual group $\widehat G$ carry the pinned $W_E$ -action determined by a reductive $E$ -group $G$ . The stack of local $L$ -parameters is the quotient stack

\operatorname{LocSys}_{\widehat G}(W_E) = [Z^1(W_E,\widehat G)/\widehat G],

where $Z^1(W_E,\widehat G)$ is the moduli scheme of suitably continuous $1$ -cocycles and $\widehat G$ acts by conjugation.

A cocycle $\phi$ is equivalently a homomorphism $W_E\to{}^LG$ into the $L$ -group whose projection to $W_E$ is the identity.

Why a stack

The stabilizer of a point is the centralizer

S_\phi=\operatorname{Cent}_{\widehat G}(\phi).

Keeping this automorphism group is necessary for packet enhancements and categorical actions. The coarse conjugacy-class set loses it, and the invariant-function spectrum can identify nonclosed orbits with their semisimplifications.

Algebraic construction

For $\ell\neq p$ and a coefficient ring $\Lambda$ , continuity must be formulated uniformly in families. Fargues–Scholze use condensed $1$ -cocycles. The resulting $Z^1(W_E,\widehat G)$ is a union of open and closed affine schemes indexed by sufficiently deep wild-inertia quotients; each finite-depth piece is a flat local complete intersection under their hypotheses.

Parameter content

This stack records Weil-group cocycles. In the Fargues–Scholze semisimplified correspondence it does not retain a separate monodromy operator or Deligne $\operatorname{SL}_2$ . Thus it should not be identified without qualification with the moduli of full Weil–Deligne parameters.

Spectral role

Global functions on the stack form the spectral Bernstein center, while Perfect complexes on it act on sheaves on $\operatorname{Bun}_G$ through the spectral action.

References

Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” Chapters VIII and I.8. arXiv.
Jean-François Dat, David Helm, Robert Kurinczuk, and Gilbert Moss, “Moduli of Langlands parameters,” 2020. arXiv.