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 adeles, \(\overline{F}\) 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 e-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.