Double covers of tori and the local Langlands correspondence
Given a maximal torus T of a connected reductive group G over a local field F, there does not exist a canonical embedding of the L-group of T into the L-group of G. Generalizing work of Adams and Vogan in the case F=R, we will construct a natural double cover of the topological group T(F) and an associated L-group that does have a canonical embedding into the L-group of G. This leads to a natural bijection between discrete Langlands parameters for G emanating from the Weil group of F, and certain pairs consisting of a maximal torus and a genuine character of its double cover. The associated L-packet can be characterized by its stable character, and we will give a formula for it in terms of the genuine character of the double cover. We will then discuss an explicit construction of this L-packet in the non-archimedean case. Certain assumptions on p will be required.