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 of the topological group T(F). We further associate an L-group to this double cover and establish a natural bijection between L-parameters valued in this L-group and genuine characters of the double cover. When T is a maximal torus of a connected reductive group G, we show that there is a canonical L-embedding from the L-group of the double cover of T to the L-group of G. This leads to a canonical factorization of Langlands parameters. We associate to a genuine character of the double cover 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 and Vogan F=R, and reinterprets computations of Langlands and Shelstad.