Local Langlands Conjectures - Local Langlands Conjectures For GL_n

Local Langlands Conjectures For GL_n

The local Langlands conjectures for general linear groups state that there are unique bijections π ↔ ρ_π from equivalence classes of irreducible admissible representations π of GL_n(F) to equivalence classes of continuous Frobenius semisimple complex n-dimensional Weil–Deligne representations ρ_π of the Weil group of F, that preserve L-functions and ε-factors of pairs of representations, and coincide with the Artin map for 1-dimensional representations. In other words,

• L(s,ρ_π⊗ρ_π') = L(s,π×π')
• ε(s,ρ_π⊗ρ_π',ψ) = ε(s,π×π',ψ)

Laumon, Rapoport & Stuhler (1993) proved the local Langlands conjectures for the general linear group GL_n(K) for positive characteristic local fields K. Carayol (1992) gave an exposition of their work.

Richard Taylor and Michael Harris (2001) proved the local Langlands conjectures for the general linear group GL_n(K) for characteristic 0 local fields K. Henniart (2001) gave another proof. Carayol (2000) and Wedhorn (2008) gave expositions of their work.