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

Abstract

We introduce a revised notion of depth for Langlands parameters for tori defined over a nonarchimedean local field \(F\) that restores depth preservation under the local Langlands correspondence (LLC). 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 \(0\) \(\ell\)-close to \(F\); using this, we show that a LLC in characteristic \(0\) corresponds canonically to a 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 \(\ell\)-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.