On L -embeddings and double covers of tori over local fields

To a torus T over a local field F and a subset of its character module subject to certain properties, we associate a canonical double cover T(F)_± of the topological group $T(F)$ of F-rational points of T. We further associate an L-group LT_± to this double cover and establish a natural bijection between L-parameters valued in LT_± and genuine characters of T(F)_±. When T is a maximal torus of a connected reductive group G, we show that there is a canonical L-embedding LT_±→ LG. This leads to a canonical factorization of Langlands parameters. We associate to a genuine character of T(F)_± subject to certain conditions a Harish-Chandra character formula and use it to give a conjectural characterization of the supercuspidal local Langlands correspondence for G, subject to a certain condition on p. This generalizes previous work of Adams--Vogan (1992) for F=R, and reinterprets computations of Langlands--Shelstad (1987). We extend these constructions to twisted Levi subgroups of G.