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Categorification of local relative Langlands duality

Yuta Takaya, Milton Lin

TL;DR

The paper advances a local, categorical reformulation of relative Langlands duality by defining an ℓ-adic six-functor framework on small v-stacks and formulating A-side (Bun_G^X, period sheaves) and B-side (Par_LG, L-sheaves) structures. It computes unnormalized periods for the Iwasawa-Tate and Hecke cases on both sides, and establishes normalization via degree and half-cyclotomic twists, drawing a precise parallel between geometrized period data and spectral L-parameters. The work also links the normalized period conjecture to distinction problems, showing how Langlands-Shahidi-type parameters and eigenstructure should sit inside the image of the norm-twisted, normalized L-sheaf under a hypothetical categorical local Langlands correspondence. Overall, it lays out a rigorous, derived-local framework for comparing automorphic and spectral data in the relative Langlands setting and illustrates the normalization and functional-equation features that mirror global conjectures in a local geometric context.

Abstract

We formulate the normalized period conjecture proposed by Ben-Zvi, Sakellaridis and Venkatesh in the framework of the categorical local Langlands correspondence and study its relation to distinction problems. Motivated by the work of Feng and Wang in the geometric setting, we verify the conjecture for the Iwasawa-Tate and Hecke periods, assuming the existence of the categorical local Langlands correspondence for $\mathrm{GL}_2$ with the Eisenstein compatibility.

Categorification of local relative Langlands duality

TL;DR

The paper advances a local, categorical reformulation of relative Langlands duality by defining an ℓ-adic six-functor framework on small v-stacks and formulating A-side (Bun_G^X, period sheaves) and B-side (Par_LG, L-sheaves) structures. It computes unnormalized periods for the Iwasawa-Tate and Hecke cases on both sides, and establishes normalization via degree and half-cyclotomic twists, drawing a precise parallel between geometrized period data and spectral L-parameters. The work also links the normalized period conjecture to distinction problems, showing how Langlands-Shahidi-type parameters and eigenstructure should sit inside the image of the norm-twisted, normalized L-sheaf under a hypothetical categorical local Langlands correspondence. Overall, it lays out a rigorous, derived-local framework for comparing automorphic and spectral data in the relative Langlands setting and illustrates the normalization and functional-equation features that mirror global conjectures in a local geometric context.

Abstract

We formulate the normalized period conjecture proposed by Ben-Zvi, Sakellaridis and Venkatesh in the framework of the categorical local Langlands correspondence and study its relation to distinction problems. Motivated by the work of Feng and Wang in the geometric setting, we verify the conjecture for the Iwasawa-Tate and Hecke periods, assuming the existence of the categorical local Langlands correspondence for with the Eisenstein compatibility.
Paper Structure (59 sections, 92 theorems, 330 equations, 1 table)

This paper contains 59 sections, 92 theorems, 330 equations, 1 table.

Table of Contents

  1. Background
  2. Formulation
  3. The ${\mathcal{A}}$-side
  4. The ${\mathcal{B}}$-side
  5. Main computations
  6. The Iwasawa-Tate period
  7. The Hecke period
  8. Relation to distinction problems
  9. An $\ell$-adic six functor formalism on small $v$-stacks
  10. Preliminaries on six functor formalisms
  11. Motivic six functor formalism on small $v$-stacks
  12. Base change to an $\ell$-adic coefficient
  13. Examples of $!$-able maps
  14. Cohomology of Banach-Colmez spaces
  15. Formulation of the ${\mathcal{A}}$-side
  16. ...and 44 more sections

Key Result

Proposition 1

(prop:BunGXArtin and prop:trivfib) The prestack ${\operatorname{Bun}}_G^X$ is a $v$-stack on ${\operatorname{Perf}}^{\mathrm{std}}$ and can be extended to an Artin $v$-stack on ${\operatorname{Perf}}$. There is an open and closed immersion over ${\operatorname{Bun}}_G^1 \cong [\ast/\underline{G(F)}]$ and it is an isomorphism if $X$ is quasi-affine.

Theorems & Definitions (245)

  • Proposition 1
  • Definition 2
  • Proposition 3
  • Definition 4
  • Conjecture 5
  • Theorem 6
  • Theorem 7
  • Proposition 8
  • Proposition 9
  • Definition 1.1
  • ...and 235 more