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Derived Weil Representation and Relative Langlands Duality

Haoshuo Fu

Abstract

The Weil representation is a particularly significant linear representation of the metaplectic group, used in the study of theta correspondence. In this paper, I introduce a derived category version of the Weil representation in the local field case. For the dual pair $ (\mathrm{GL}_n,\mathrm{GL}_m) $, I give a coherent description of this category, in the philosophy of relative Langlands duality.

Derived Weil Representation and Relative Langlands Duality

Abstract

The Weil representation is a particularly significant linear representation of the metaplectic group, used in the study of theta correspondence. In this paper, I introduce a derived category version of the Weil representation in the local field case. For the dual pair , I give a coherent description of this category, in the philosophy of relative Langlands duality.

Paper Structure

This paper contains 27 sections, 19 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Weil representations
  3. Relative Langlands duality
  4. Connection with Coulomb branches
  5. Definition of the categories
  6. Notations
  7. Schrödinger model
  8. Lattice model
  9. Finite case
  10. Local case
  11. Fourier transform
  12. Irreducible objects
  13. Cotangent space
  14. Singular support
  15. Relevant orbits
  16. ...and 12 more sections

Key Result

Theorem 1

Let $F$ be a local field and ${\mathcal{O}}$ its ring of integers. For a variety $X$, let $X_{\mathcal{O}}$ be its arc space and $X_F$ be its loop space. Let ${\mathrm{Weil}_{G,H}}$ be the category of ${G_{\mathcal{O}}}\times{H_\mathcal{O}}$-equivariant $({V_\mathcal{O}},\psi)$-equivariant sheaves o

Theorems & Definitions (37)

  • Theorem 1
  • Conjecture 1
  • Remark
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 1
  • Proposition 3
  • ...and 27 more