On the Beem-Nair Conjecture

For a simple linear algebraic group $G$, the chiral universal centralizer $\mathbf{I}_{G,k}$ is a vertex operator algebra, which is the chiralization of the universal centralizer $\mathfrak{Z}_G$. The variety $\mathfrak{Z}_G$ is identified with the spectrum of the equivariant Borel-Moore homology of the affine Grassmannian of the Langlands dual group of $G$. Beem and Nair conjectured that an open symplectic immersion from $\mathrm{KT}_G$, the Kostant-Toda lattice associated to a simple group $G$, to $\mathfrak{Z}_G$ gives rises to a free field realization of the chiral universal centralizer at the critical level. In this paper, we construct a free field realization of $\mathbf{I}_{G,k}$ at any level, which coincides with the one conjectured by Beem and Nair at the critical level. We give an explicit description of this construction in $SL_2(\mathbb{C})$-case.