Weil–Deligne representation

Definition

Let $F$ be a nonarchimedean local field. A Weil–Deligne representation on a finite-dimensional complex vector space $V$ is a pair $(r,N)$ , where

r:W_F\longrightarrow\operatorname{GL}(V)

is a representation of the Weil group, continuous with open kernel on the inertia subgroup, $N\in\operatorname{End}(V)$ is nilpotent, and

r(w)Nr(w)^{-1}=|w|_F N \qquad(w\in W_F).

Morphisms intertwine both $r$ and $N$ . This is the representation-theoretic form of a representation of the Weil–Deligne group.

Frobenius semisimplification

The pair is Frobenius-semisimple if $r(\operatorname{Fr}_F)$ is semisimple. Replacing the Frobenius action by its semisimple part gives the Frobenius semisimplification. Local Langlands for $\operatorname{GL}_n$ is normally stated using isomorphism classes of Frobenius-semisimple Weil–Deligne representations.

Local factor

With geometric Frobenius and inertia $I_F$ , the standard local factor is

L(s,r,N)= \det\!\left( 1-q_F^{-s}r(\operatorname{Fr}_F) \mid(\ker N)^{I_F} \right)^{-1}.

Changing to arithmetic Frobenius changes the displayed convention, not the underlying representation.

The corresponding epsilon factor and gamma factor enter the local functional equation.

Monodromy-free case

The condition $N=0$ gives an ordinary Weil-group representation. If inertia also acts trivially, the representation is unramified and is determined by the semisimple conjugacy class of Frobenius.

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.
Michael Harris, “On the local Langlands correspondence,” 2003, §2. arXiv.