We define a new cohomology set H 1 (u → W, Z → G) for an affine algebraic group G and a finite central subgroup Z, both defined over a local field of characteristic zero, which is an enlargement of the usual first Galois cohomology set of G. We show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered L-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the case of real groups, we show that this description is correct using Shelstad's work.

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p p -adic group G G arise from pairs ( S , θ ) (S,\theta ) , where S S is a tame elliptic maximal torus of G G , and θ \theta is a character of S S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p p -adic groups.

We study the cohomology of certain Galois gerbes over number fields. This cohomology provides a bridge between refined local endoscopy, as introduced in [Kal16], and classical global endoscopy. As particular applications, we express the canonical adelic transfer factor that governs the stabilization of the Arthur-Selberg trace formula as a product of normalized local transfer factors, we give an explicit constriction of the pairing between an adelic L-packet and the corresponding S-group (based on the conjectural pairings in the local setting) that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group, and we give a proof of some expectations of Arthur.

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We prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. The proof holds for tempered representations of quasi-split real K-groups and quasi-split p-adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.