A Pinned Local Langlands Correspondence for Depth-Zero Supercuspidal Representations

Abstract: We construct a pinned local Langlands correspondence for depth-zero supercuspidal representations of a connected reductive group over a non-archimedean local field. After fixing a pinned splitting of the quasi-split inner form, we obtain a canonical bijection between irreducible depth-zero supercuspidal representations and relevant cuspidal enhanced depth-zero Langlands parameters. The construction separates the tame toral part from the unramified unipotent part. The toral part is normalized by the local Langlands correspondence for maximally unramified elliptic tori and by the corresponding canonical (L)-embeddings. The finite representation occurring in a depth-zero type is then passed, through a pinned Jordan decomposition for possibly disconnected finite reductive quotients, to a cuspidal unipotent label on the dual centralizer. The unramified unipotent contribution is supplied by the Feng--Opdam--Solleveld correspondence for supercuspidal unipotent representations. Combining these ingredients gives the enhanced parameter attached to a depth-zero supercuspidal representation, and the inverse map is obtained by reversing the same finite Jordan decomposition. The correspondence is independent of auxiliary choices apart from the fixed pinned normalization. It is compatible with the tame inertial parameter attached to the depth-zero character, with weakly unramified twists, and with central characters via the torus correspondence. Thus the main output is a canonical, pinning-normalized bijection between the two depth-zero supercuspidal sides, together with the finite unipotent bookkeeping needed to make the construction reversible.