Local Langlands Conjectures For GLn
The local Langlands conjectures for general linear groups state that there are unique bijections π ↔ ρπ from equivalence classes of irreducible admissible representations π of GLn(F) to equivalence classes of continuous Frobenius semisimple complex n-dimensional Weil–Deligne representations ρπ 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,ρπ⊗ρπ') = L(s,π×π')
- ε(s,ρπ⊗ρπ',ψ) = ε(s,π×π',ψ)
Laumon, Rapoport & Stuhler (1993) proved the local Langlands conjectures for the general linear group GLn(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 GLn(K) for characteristic 0 local fields K. Henniart (2001) gave another proof. Carayol (2000) and Wedhorn (2008) gave expositions of their work.
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