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

Manish Mishra

TL;DR

The paper resolves depth-preservation issues in the local Langlands correspondence by introducing a revised depth for Langlands parameters of $F$-tori and a global depth-transfer function that jointly control torus and root data. Using Yu's minimal congruence filtration and Deligne–Kazhdan close-field techniques, it constructs canonical, blockwise transfers of harmonic-analysis data, including generalized Cartan decompositions and parahoric Hecke-algebra isomorphisms, across characteristics. It then extends LLC from characteristic 0 to positive characteristic for a wide class of groups, notably via Kaletha’s regular supercuspidal framework, and provides a practical, canonical recipe to obtain LLC in positive characteristic from the characteristic-0 theory. The results enable working in characteristic 0 without loss of generality for a broad range of $p$-adic harmonic analysis, and they unify several prior congruence-transfer results under a coherent depth-theoretic umbrella.

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.

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

TL;DR

The paper resolves depth-preservation issues in the local Langlands correspondence by introducing a revised depth for Langlands parameters of -tori and a global depth-transfer function that jointly control torus and root data. Using Yu's minimal congruence filtration and Deligne–Kazhdan close-field techniques, it constructs canonical, blockwise transfers of harmonic-analysis data, including generalized Cartan decompositions and parahoric Hecke-algebra isomorphisms, across characteristics. It then extends LLC from characteristic 0 to positive characteristic for a wide class of groups, notably via Kaletha’s regular supercuspidal framework, and provides a practical, canonical recipe to obtain LLC in positive characteristic from the characteristic-0 theory. The results enable working in characteristic 0 without loss of generality for a broad range of -adic harmonic analysis, and they unify several prior congruence-transfer results under a coherent depth-theoretic umbrella.

Abstract

We introduce a revised notion of depth for Langlands parameters for tori defined over a nonarchimedean local field 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 to characteristic . When 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 of characteristic -close to ; using this, we show that a LLC in characteristic corresponds canonically to a LLC in characteristic . 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 without loss of generality for a wide swath of harmonic analysis on -adic groups.

Paper Structure

This paper contains 46 sections, 40 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Depth-r local Langlands for tori
  3. Truncated isomorphisms over $\ell$-close fields
  4. Close field Hecke algebra isomorphisms.
  5. Local Langlands correspondence in positive characteristic.
  6. Cartan decomposition and Hecke algebras
  7. Filtrations on a torus
  8. Naïve valuation filtration
  9. Moy-Prasad filtration
  10. Minimal congruence filtration on tori
  11. Definition: Minimal congruence filtration
  12. The Hasse-Herbrand function
  13. Depth comparison under LLC for tori
  14. Standard depth of a parameter and its generalization
  15. Depth preservation for induced torus
  16. ...and 31 more sections

Key Result

Theorem 1

Let $T/F$ be an arbitrary torus and let $r\ge0$. The local Langlands correspondence for tori $\mathcal{L}_T\colon\mathop{\mathrm{Hom}}\nolimits\!\bigl(T(F),\mathbb{C}^\times\bigr)\xrightarrow{\sim} H^1(W_F,\widehat{T})$ preserves depth, i.e., for every character $\chi$ of $T(F)$, we have, where the depth of a Langlands parameter is as defined in Definition def:depth-parameter.

Theorems & Definitions (90)

  • Theorem 1: Depth-$r$ LLC for tori
  • Theorem 2: Truncated-torus close field isomorphisms
  • Theorem 3: Truncated-parahoric close field isomorphisms
  • Theorem 4: Hecke-algebra isomorphism for Bernstein blocks over $\ell$--close fields
  • Theorem 5
  • Theorem 6
  • Theorem 7: Generalised Cartan decomposition
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 80 more