Representations of a Weil Group

Abstract

Let $F = {\mathbb{F}}_{q}(C)$ be the field of functions on a smooth projective absolutely irreducible curve C over ${\mathbb{F}}_{q}$, $\mathbb{A}$ its ring of adèles, $\overline{F}$ a separable algebraic closure of F, G = GL(r), and ∞ a fixed place of F, as in Chap. 2. This section concerns the higher reciprocity law, which parametrizes the cuspidal $G(\mathbb{A})$-modules whose component at ∞ is cuspidal, by irreducible continuous constructible r-dimensional ℓ-adic (ℓ≠p) representations of the Weil group $W(\overline{F}/F)$, or irreducible rank r smooth ℓ-adic sheaves on SpecF which extend to smooth sheaves on an open subscheme of the smooth projective curve whose function field is F, whose restriction to the local Weil group $W({\overline{F}}_{\infty }/{F}_{\infty })$ at ∞ is irreducible. This law is reduced to Theorem 11.1, which depends on Deligne’s conjecture (Theorem 6.8). This reduction uses the Converse Theorem 13.1, and properties of ε-factors attached to Galois representations due to Deligne (SLN 349:501–597, 1973) and Laumon (Publ Math IHES 65:131–210, 1987). We explain the result twice. A preliminary exposition in the classical language of representations of the Weil group, then in the equivalent language of smooth ℓ-adic sheaves, used e.g. in (Deligne and Flicker, Counting local systems with principal unipotent local monodromy. http://www.math.osu.edu/ flicker.1/df.pdf). Note that in this chapter we denote a Galois representation by ρ, as σ is used to denote an element of a Galois group.