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

2025/09/05 by Mishra, Manish · 1 citation
#11F70 #11S37 #20C08 #20G25 #22E50 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2509.04997

Abstract

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 ℓ-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 ℓ-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.

Citations

• Langlands parameters for Moy-Prasad types
• On parameters of Hecke algebras for p-adic groups
• Construction of tame supercuspidal representations in arbitrary residue characteristic
• Structure of Hecke algebras arising from types
• Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms
• On congruent isomorphisms for tori
• Rigid inner forms over global function fields
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• Construction of Tame Types
• Regular supercuspidal representations
• The local Langlands correspondence for GSp₄ over local function fields
• Rigid inner forms of real and p-adic groups
• Langlands parameters associated to special maximal parahoric spherical representations
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• The Satake isomorphism for special maximal parahoric Hecke algebras
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• Distinguished tame supercuspidal representations

Cited by

• Normalized Indexing for Ramification Subgroups

Related

• Depth preserving property of the local Langlands correspondence for quasi-split classical groups in a large residual characteristic
• Note on the non-preservation of depth
• On depth-zero characters of p-adic groups
• Preservation of depth in local geometric Langlands correspondence
• Comparison of the depths on both sides of the local Langlands\n correspondence for Weil-restricted groups (with appendix by Jessica Fintzen)