Depth Preservation and Close-Field Transfer in the Local Langlands Correspondence

We introduce a revised notion of depth for Langlands parameters for a connected reductive G defined over a nonarchimedean local field F that restores depth preservation under the local Langlands correspondence (LLC) -- in particular for all tori. We leverage that preservation to derive structural results that, taken together, yield a canonical transfer of broad harmonic-analytic results from characteristic 0 to characteristic p. When F has suitably large positive characteristic, we prove a block-by-block equivalence: each Bernstein block of G (F) is equivalent to a corresponding block for some G' (F') with F' of characteristic 0 -close to F; using this, we show that an LLC in characteristic 0 corresponds canonically to an LLC in characteristic p. For regular supercuspidals we give a direct, more structured construction via Kaletha. Along the way we recover and extend results on -close fields -- introducing a depth-transfer function generalizing the normalized Hasse--Herbrand function, proving truncated isomorphisms for arbitrary tori and parahorics, establishing a depth- and supercuspidality-preserving Kazhdan-type Hecke-algebra isomorphism for arbitrary maximal parahorics of arbitrary connected reductive groups; and a generalized Cartan decomposition for arbitrary maximal parahorics -- thereby subsuming several earlier results in the literature. Collectively, the results let one work in characteristic 0 without loss of generality for a wide swath of harmonic analysis on p-adic groups.