The fundamental local equivalence conjecture for principal equivariant W-algebras

Let $G$ be a reductive group with maximal torus $T$, coweight lattice $X_{*}(T)$, dominant coweights $X_{*}(T)_{+}$, Langlands dual group $\check{G}$, and principal equivariant affine $\mathcal{W}$-algebra $\mathcal{W}_{G}^{\kappa}$ at level $\kappa$. Let $\mathcal{D}_{G}^{\kappa}$ be the corresponding vertex algebra module, let $H^{*}_{DS}$ denote principal quantum Hamiltonian reduction, and let $\mathcal{W}_{G}^{\kappa}\mathrm{-Mod}^{\mathcal{J}_{\infty}G}$ be the full subcategory of $\mathcal{W}_{G}^{\kappa}$-modules integrable with respect to $\mathcal{J}_{\infty}G$. For each $\gamma\in X_{*}(T)_{+}$, write $\gamma\cdot\mathcal{D}_{G}^{\kappa}$ for the corresponding spectral-flow twist, and let $V_{\gamma}$ be the simple $\check{G}$-module of highest weight $\gamma$. The fundamental local equivalence conjecture. For generic levels, the category $\mathcal{W}_{G}^{\kappa}\mathrm{-Mod}^{\mathcal{J}_{\infty}G}$ is semisimple, with simple objects $H^{*}_{DS}(\gamma\cdot\mathcal{D}_{G}^{\kappa})$ for $\gamma\in X_{*}(T)_{+}$. In particular, there is an equivalence of abelian categories

$$\mathcal{W}_{G}^{\kappa}\mathrm{-Mod}^{\mathcal{J}_{\infty}G}\simeq\check{G}\mathrm{-Mod}$$

that sends $H^{*}_{DS}(\gamma\cdot\mathcal{D}_{G}^{\kappa})$ to $V_{\gamma}$ for every $\gamma\in X_{*}(T)_{+}$. This is an abelian-category formulation of the fundamental local equivalence in the quantum geometric Langlands program, identifying the representation theory of the principal equivariant $\mathcal{W}$-algebra at generic level with representations of the Langlands dual group. The conjecture remains unproved in the supplied text.