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 $\mathcal{D}_{G}^{\kappa}$ 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 $\mathcal{W}_{G}^{\kappa}$, defined by Arakawa as the principal quantum Hamiltonian reduction of $\mathcal{D}_{G}^{\kappa}$. We construct a family of simple modules for $\mathcal{W}_{G}^{\kappa}$ 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.