Local Langlands correspondence for covering groups of tori, and the packet-indexing groups

We determine the finite group $\mathcal S$ parametrizing a packet in the local Langlands correspondence for a Brylinski-Deligne covering group of an algebraic torus, under some assumption on ramification. Especially, this work generalizes Weissman’s result on covering groups of tori that split over an unramified extension of the base field.

💡 Research Summary

The paper studies the local Langlands correspondence for covering groups of algebraic tori that arise from the Brylinski‑Deligne framework. Let (F) be a non‑archimedean local field of characteristic zero, (T) an algebraic torus defined over (F), and (\mu_n\to\widetilde T\to T) a central extension obtained by pushing out a Brylinski‑Deligne (K_2)-extension via the Hilbert symbol. Such a covering is encoded by a Galois‑invariant quadratic form (Q:Y\to\mathbb Z) on the cocharacter lattice (Y=\operatorname{Hom}(\mathbb G_m,T)); the associated bilinear form (B) is defined by (B(y,y’)=Q(y+y’)-Q(y)-Q(y’)).

For any subgroup (Y’\subset Y) one defines \