Weil–Deligne group

Definition

Let $F$ be a nonarchimedean local field. The Weil–Deligne group is the semidirect-product group scheme

W'_F=W_F\ltimes\mathbb G_a, \qquad w x w^{-1}=|w|_F x,

where $W_F$ is the Weil group and $|\cdot|_F$ is normalized using geometric Frobenius. A finite-dimensional representation of $W'_F$ is equivalently a Weil–Deligne representation $(r,N)$ .

The

W_F\times\mathrm{SL}_2

presentation

For complex Langlands parameters one often writes the local Langlands group as

L_F=W_F\times\mathrm{SL}_2(\mathbb C).

An algebraic action of $\mathrm{SL}_2(\mathbb C)$ packages the nilpotent monodromy operator $N$ . Passing between the two presentations includes a standard $|w|_F^{1/2}$ -twist, so the associated Weil action is not obtained by merely restricting a parameter to $W_F$ .

Scope

For $F=\mathbb R$ or $\mathbb C$ , local Langlands parameters use the archimedean Weil group itself; the auxiliary $\mathrm{SL}_2$ factor is a nonarchimedean feature. Arthur parameters introduce a different $\mathrm{SL}_2$ factor and must not be conflated with Deligne monodromy.

References

Pierre Deligne, “Les constantes des équations fonctionnelles des fonctions $L$ ,” in Modular Functions of One Variable II, Lecture Notes in Mathematics 349, 1973. DOI.
Jayce R. Getz, An Introduction to Automorphic Representations, §10.2. Author notes.