  The Schur limit of the superconformal index of four-dimensional $\mathcal
N=2$ superconformal field theories has been shown to equal the supercharacter
of the vacuum module of their associated chiral algebra. Applying localization
techniques to the theory suitably put on $S^3\times S^1$, we obtain a direct
derivation of this fact. We also show that the localization computation can be
extended to calculate correlation functions of a subset of local operators,
namely of the so-called Schur operators. Such correlators correspond to
insertions of chiral algebra fields in the trace-formula computing the
supercharacter. As a by-product of our analysis, we show that the standard lore
in the localization literature stating that only off-shell supersymmetrically
closed observables are amenable to localization, is incomplete, and we
demonstrate how insertions of fermionic operators can be incorporated in the
computation.
