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

Abstract

Abstract: 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 $T(F)_\pm$ of the topological group $T(F)$ of $F$-rational points of $T$. We further associate an $L$-group $^LT_\pm$ to this double cover and establish a natural bijection between $L$-parameters valued in $^LT_\pm$ and genuine characters of $T(F)_\pm$. When $T$ is a maximal torus of a connected reductive group $G$, we show that there is a canonical $L$-embedding $^LT_\pm\to {^LG}$. This leads to a canonical factorization of Langlands parameters. We associate to a genuine character of $T(F)_\pm$ 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 $F=\mathbb{R}$, and reinterprets computations of Langlands--Shelstad (1987). We extend these constructions to twisted Levi subgroups of $G$.

The authors' abstract, as published at the source. American Journal of Mathematics, 2026 · DOI ↗