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

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 $G^{\mathrm{\prime }}(F^{\mathrm{\prime }})$ with $F^{\mathrm{\prime }}$ of characteristic $0$ $\mathrm{\ell }$-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 $\mathrm{\ell }$-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.