Local Langlands conjecture

In mathematics, the local Langlands conjectures, introduced by Langlands (1967, 1970), are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.

The local Langlands conjectures for GL₁(K) follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL₁(K)= K^* to the abelianization of the Weil group. In particular irreducible smooth representations of GL₁(K) are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL₁(C). This gives the Langlands correspondence between homomorphisms of the Weil group to GL₁(C) and irreducible smooth representations of GL₁(K).

Representations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups. To get a bijection, one has to slightly modify the notion of a representation of the Weil group, to something called a Weil–Deligne representation. This consists of a representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw⁻¹=||w||N, or equivalently a representation of the Weil–Deligne group. In addition the representation of the Weil group should have an open kernel, and should be (Frobenius) semisimple.

For every Frobenius semisimple complex n-dimensional Weil–Deligne representations ρ of the Weil group of F there is an L-function L(s,ρ) and a local ε-factor ε(s,ρ,ψ) (depending on a character ψ of F).

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