Workshop on Representation Theory and Geometry

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.