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Representation theory of the principal equivariant $\mathcal{W}$-algebra and Langlands duality

Damien Simon

Abstract

The object of this article is to study some aspects of the quantum geometric Langlands program in the language of vertex algebras. We investigate the representation theory of the vertex algebra of chiral differential operators on a reductive group $\mathcal{D}_{G}^κ$ for generic levels. For instance we show that the geometric Satake equivalence degenerates. Then, we study the representation theory of the equivariant affine $\mathcal{W}$-algebra $\mathcal{W}_{G}^κ$, defined by Arakawa as the principal quantum Hamiltonian reduction of $\mathcal{D}_{G}^κ$. We construct a family of simple modules for $\mathcal{W}_{G}^κ$ whose combinatorics matches that of the representation theory of the Langlands dual group. To put this construction in perspective we propose a vertex-algebraic formulation of the fundamental local equivalence of Gaitsgory and Lurie. Finally, we give a proof when the group is an algebraic torus or is simple adjoint of classical simply laced type.

Representation theory of the principal equivariant $\mathcal{W}$-algebra and Langlands duality

Abstract

The object of this article is to study some aspects of the quantum geometric Langlands program in the language of vertex algebras. We investigate the representation theory of the vertex algebra of chiral differential operators on a reductive group for generic levels. For instance we show that the geometric Satake equivalence degenerates. Then, we study the representation theory of the equivariant affine -algebra , defined by Arakawa as the principal quantum Hamiltonian reduction of . We construct a family of simple modules for whose combinatorics matches that of the representation theory of the Langlands dual group. To put this construction in perspective we propose a vertex-algebraic formulation of the fundamental local equivalence of Gaitsgory and Lurie. Finally, we give a proof when the group is an algebraic torus or is simple adjoint of classical simply laced type.

Paper Structure

This paper contains 31 sections, 64 theorems, 427 equations.

Table of Contents

  1. Conventions and notations
  2. Reductive groups
  3. Affine vertex algebras
  4. Chiral differential operators on an algebraic group
  5. General results
  6. The conformal structure
  7. The case of an abelian Lie algebra
  8. The case of a simple Lie algebra
  9. The case of a reductive Lie algebra
  10. Chiral Peter--Weyl decomposition
  11. Degeneration of the geometric Satake equivalence
  12. Spectral flow in the theory of vertex algebras
  13. Generalities
  14. The spectral flow group of $V^{\kappa}(\mathfrak{g})$
  15. The spectral flow group of $\mathcal{D}_{G}^{\kappa}$
  16. ...and 16 more sections

Key Result

Theorem A

There exists a family of pairwise non-equivalent simple $\mathcal{W}_{G}^{\kappa}$-modules labeled by the dominant cocharacters of $G$. They are obtained as quantum Hamiltonian reduction of spectral flow twists of the adjoint module $\mathcal{D}_{G}^{\kappa}$.

Theorems & Definitions (131)

  • Theorem A: Theorem \ref{['theorem simplicity of modules for equivariant w algebra']}
  • Theorem B: Theorem \ref{['theorem degeneration of the geometric Satake correspondence']}, Corollary \ref{['corollaire of theorem fle tore']}
  • Conjecture A
  • Theorem C: Theorem \ref{['theorem FLE pour le tore']} and Theorem \ref{['theorem FLE for adjoint type ADE']}
  • Theorem 1.1: Theorems 22.2 and 22.42 of Milne2017
  • Corollary 1.2
  • Proposition 1.3
  • proof
  • Definition 1.4
  • Remark 1.5
  • ...and 121 more