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Two Categorifications of the Local Langlands Correspondence for Tori

Ruide Fu

Abstract

The stack of local Langlands parameters for a torus is a Picard stack. In this article, we explicitly determine its Picard dual and show that the Fourier-Mukai transform gives rise to the integral categorical local Langlands correspondence for the torus. This is the categorification of the local Langlands correspondence and answers a conjecture of X. Zhu. Moreover, we establish a geometric version of this correspondence, whose categorical trace reproduces the previous result.

Two Categorifications of the Local Langlands Correspondence for Tori

Abstract

The stack of local Langlands parameters for a torus is a Picard stack. In this article, we explicitly determine its Picard dual and show that the Fourier-Mukai transform gives rise to the integral categorical local Langlands correspondence for the torus. This is the categorification of the local Langlands correspondence and answers a conjecture of X. Zhu. Moreover, we establish a geometric version of this correspondence, whose categorical trace reproduces the previous result.

Paper Structure

This paper contains 12 sections, 46 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Notation I
  3. Lemmas in Homological Algebra
  4. Split Exact Sequence
  5. Duality
  6. Quasi-Coherent Sheaves on Tor
  7. Categorical Local Langlands Correspondence for Tori
  8. Notation II
  9. Stack of Inertial Langlands Parameters
  10. Categorical Trace and Convolution Pattern
  11. Geometric Local Langlands Correspondence for Tori
  12. Lemma in Homological Algebra

Key Result

Theorem 1.4

Let $p$ be invertible in $\Lambda$. The Fourier--Mukai transform via the Poincaré line bundle $\mathcal{L}_T$ gives the equivalence of stable $\infty$-categories

Theorems & Definitions (90)

  • Conjecture 1.1: Zhu21, Conjecture 4.6.4
  • Conjecture 1.3: Zhu21, Conjecture 3.2.2
  • Theorem 1.4
  • Theorem 1.6
  • Proposition 1.7
  • Remark 2.9
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 80 more