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arxiv: 2606.00983 · v2 · pith:3GKG2M4Onew · submitted 2026-05-31 · 🧮 math.NT · math.AG· math.RT

The categorical local Langlands conjecture

David Hansen , Lucas Mann This is my paper

Pith reviewed 2026-06-28 16:54 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.RT
keywords categorical local Langlands conjectureWhittaker coefficient functorEisenstein seriesp-adic groupsGL_ninduction principleind-coherent sheavesspectral action
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The pith

Compatibility of the enhanced Whittaker coefficient functor with Eisenstein series implies the full categorical local Langlands conjecture for GL_n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out a program to establish the categorical local Langlands conjecture for quasisplit p-adic groups under the assumption that Fargues-Scholze L-parameters match the semisimplification of known automorphic parametrizations. For GL_n it proves that one working hypothesis, the compatibility of the enhanced Whittaker coefficient functor with Eisenstein series, is already sufficient to obtain the entire conjecture. For general groups it supplies an induction principle that reduces the conjecture for G to the conjecture for all proper Levi subgroups plus a limited amount of additional data on G itself. The same principle applies directly to many classical groups with existing tools. Foundational results on finiteness for spectral constant term functors, a spectral version of Bernstein's finite global dimension theorem, admissible ind-coherent sheaves, and duality for the spectral action are developed to support these reductions and to define an explicit functor t_ψ from the spectral side to the automorphic side.

Core claim

For GL_n the compatibility of the enhanced Whittaker coefficient functor c_ψ with Eisenstein series alone implies the full categorical local Langlands conjecture. For a general quasisplit p-adic group G an induction principle reduces the conjecture for G to the conjecture for all proper Levi subgroups together with a very small amount of information about G. Using these reductions and the new foundational results the paper unconditionally constructs an explicit functor t_ψ from the spectral side to the automorphic side that is defined on enough ind-coherent sheaves to control the conjecture.

What carries the argument

The enhanced Whittaker coefficient functor c_ψ together with its compatibility with Eisenstein series; the induction principle that reduces CLLC for G to CLLC for proper Levi subgroups plus limited data on G.

If this is right

  • The full CLLC for GL_n follows from the single compatibility hypothesis on c_ψ.
  • CLLC for many classical groups follows unconditionally from current technology via the induction principle.
  • An explicit functor t_ψ from the spectral side to the automorphic side is defined on a large enough class of ind-coherent sheaves to control the conjecture.
  • Spectral constant term functors satisfy a strong finiteness theorem.
  • A duality theorem holds for the spectral action.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the compatibility hypothesis is verified, the categorical conjecture becomes directly tied to classical constructions of Eisenstein series.
  • The induction principle suggests that the conjecture for larger groups can be settled by checking only the base cases for their Levi subgroups.
  • The theory of admissible ind-coherent sheaves may supply a general tool for duality statements on other derived stacks arising in representation theory.
  • The explicit functor t_ψ could be used to compute L-parameters in additional families once the hypothesis is established.

Load-bearing premise

The working hypothesis that the enhanced Whittaker coefficient functor c_ψ is compatible with Eisenstein series.

What would settle it

A counterexample showing that the compatibility of c_ψ with Eisenstein series holds for GL_n yet the categorical local Langlands conjecture fails for GL_n.

read the original abstract

We formulate a program to prove the categorical local Langlands conjecture (CLLC) of Fargues-Scholze, for all quasisplit $p$-adic groups where the Fargues-Scholze $L$-parameters agree with the semisimplification of a known "automorphic" local Langlands parametrization. A key working hypothesis - which we expect to prove elsewhere jointly with Hamann - is the compatibility of the enhanced Whittaker coefficient functor $c_\psi$ with Eisenstein series. For $\mathrm{GL}_n$, we show that this hypothesis alone implies the full CLLC. For more general groups $G$, we prove an induction principle which reduces CLLC for $G$ to CLLC for all proper Levi subgroups together with a very small amount of information about $G$. This principle applies unconditionally to many classical groups with current technology. Along the way, we establish many foundational results. In particular: - We prove a very strong finiteness theorem for spectral constant term functors. - We prove a spectral analogue of Bernstein's finite global dimension theorem for $p$-adic Hecke algebras. - We introduce and develop the theory of admissible ind-coherent sheaves and admissible duality on derived stacks. - We prove a duality theorem for the spectral action. Using all of these results, we unconditionally define a new and explicit functor $t_{\psi}$ from the spectral side to the automorphic side, which is defined on enough ind-coherent sheaves to control the entire conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript formulates a program to prove the categorical local Langlands conjecture (CLLC) of Fargues-Scholze for quasisplit p-adic groups where the Fargues-Scholze L-parameters agree with the semisimplification of a known automorphic parametrization. A central working hypothesis (to be proved jointly with Hamann) is the compatibility of the enhanced Whittaker coefficient functor c_ψ with Eisenstein series; under this hypothesis the paper proves the full CLLC for GL_n. It also establishes an induction principle reducing CLLC for general G to CLLC for proper Levi subgroups plus limited data on G (unconditionally applicable to many classical groups). Foundational results proved unconditionally include a strong finiteness theorem for spectral constant term functors, a spectral analogue of Bernstein's finite global dimension theorem, the theory of admissible ind-coherent sheaves and admissible duality on derived stacks, and a duality theorem for the spectral action; these are used to define an explicit functor t_ψ from the spectral side to the automorphic side on a sufficiently large class of ind-coherent sheaves.

Significance. If the working hypothesis holds, the paper would constitute a substantial advance by reducing the CLLC to a single explicit compatibility statement while supplying an explicit functor t_ψ and a practical induction principle. The unconditional foundational results on finiteness, spectral Bernstein theory, admissible ind-coherent sheaves, and spectral-action duality are valuable contributions in their own right and may find applications beyond the CLLC.

minor comments (2)
  1. [Abstract] The abstract and introduction could more sharply distinguish the scope of the unconditional foundational theorems from the conditional implication for GL_n (e.g., by adding a short table or enumerated list of which statements require the working hypothesis).
  2. Notation for admissible ind-coherent sheaves and the functor t_ψ is introduced in the foundational sections; a brief comparison table with classical coherent sheaves would improve readability for readers outside the immediate spectral Langlands literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring rebuttal or revision at this stage. We will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claim for GL_n is an explicit implication from an external working hypothesis (compatibility of c_ψ with Eisenstein series, to be proved jointly with Hamann) to the full CLLC; this hypothesis is not derived within the paper. Foundational results (strong finiteness for spectral constant term functors, spectral Bernstein theorem, admissible ind-coherent sheaves and duality, spectral action duality) are stated as proved unconditionally and used to define t_ψ without reduction to the target conjecture. The induction principle reduces CLLC for G to Levis plus limited data on G, with unconditional applicability to classical groups. Reliance on Fargues-Scholze L-parameters is cited as prior independent work. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The program rests on one explicit working hypothesis plus background results from Fargues-Scholze; new conceptual tools (admissible ind-coherent sheaves, admissible duality) are introduced without independent falsifiable evidence outside the paper.

axioms (1)
  • ad hoc to paper Compatibility of the enhanced Whittaker coefficient functor c_ψ with Eisenstein series
    Stated as the key working hypothesis expected to be proved elsewhere.
invented entities (2)
  • admissible ind-coherent sheaves no independent evidence
    purpose: Develop theory of admissible duality on derived stacks for the spectral action
    New notion introduced to support the duality theorem and functor t_ψ
  • admissible duality on derived stacks no independent evidence
    purpose: Support duality theorem for the spectral action
    Introduced as part of the foundational results

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