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

Abstract

We study the structure and the representation theory of a certain class of vertex algebras. Our study was partly motivated by the quantum geometric Langlands program and we explain what some of our results mean in this framework. We begin our investigation with the vertex algebra of chiral differential operators on a reductive group $\mathcal{D}_{G}^{\kappa}$ for generic levels. In particular, we prove that its vertex algebraic structure is essentially unique. We also study its representation theory and show that the geometric Satake equivalence degenerates. The latter leads us to formulate a vertex-algebraic version of the fundamental local equivalence of Gaitsgory and Lurie. In turn, this brings us to study the representation theory of the principal 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. Finally, we establish the fundamental local equivalence when the group is an algebraic torus or is simple adjoint of classical simply laced type.