Local Langlands correspondence for tempered representations of quasi-split reductive groups

Establish a surjective local Langlands correspondence for tempered irreducible representations of every quasi-split reductive group over a non-archimedean local field of characteristic zero, satisfying compatibility with parabolic induction, twists by unitary unramified characters, enhanced L-parameters, characterization of discrete series, and stability of L-packets.

Background

The paper formulates a conjectural local Langlands correspondence as a surjection from tempered irreducible representations of each standard Levi subgroup to tempered L-parameters. The conjecture includes compatibility with parabolic induction and unramified twists, a parametrization of L-packets by irreducible representations of component groups, the equivalence between discrete series representations and discrete parameters, and stability of the associated character distribution.

The paper proves or cites this correspondence for several groups, including general linear, symplectic, classical similitude, and, with stated exceptions, exceptional groups. In particular, for G2 the cited result does not establish Property (3) for discrete parameters in residual characteristic 3 or Property (5), although the paper does not use those missing properties.