Eigenvalues of Bethe vectors in the Gaudin model

According to the Feigin–Frenkel–Reshetikhin theorem, the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors can be found using the center of an affine vertex algebra at the critical level. We recently calculated explicit Harish-Chandra images of the generators of the center in all classical types. Combining these results leads to explicit formulas for the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors. The Harish-Chandra images can be interpreted as elements of classical $\mathcal{W}$-algebras. By calculating classical limits of the corresponding screening operators, we elucidate a direct connection between the rings of $q$-characters and classical $\mathcal W$-algebras.