# nLab Weil-Deligne representation

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Definition

Let

• $F$ be a p-adic field with $\kappa$ denoting its residue field;

• $\Vert\sigma\Vert$ denotes the valuation of the corresponding element of $F^{\times}$ under the isomorphism of local class field theory.

###### Definition

(Weil-Deligne representation)
A Weil-Deligne representation is a pair $(\rho_{0},N)$ where

• $\rho_{0} \colon W_{F} \to GL_{n}(\mathbb{C})$ is a linear representation of the Weil group of $F$

and

• $N$ is a nilpotent monodromy operator

satisfying

$\rho_{0}(\sigma)N\rho_{0}(\sigma)^{-1} \;=\; \left\Vert \sigma \right\Vert N$

for all $\sigma\in W_{F}$.

## References

• John Tate, Section 4 in: Number theoretic background, in: Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, RI (ISBN:978-0-8218-3371-1, pdf, pdf)

• Robin Zhang, Weil-Deligne Representations I – Local Langlands seminar (pdf, pdf)