  In this article, we revisit the classical McKay correspondence via
homological mirror symmetry. Specifically, we demonstrate how this
correspondence can be articulated as a derived equivalence between the category
of vanishing cycles associated with a Kleinian surface singularity and the
category of perfect complexes on the corresponding quotient orbifold. We
further illustrate how this equivalence allows for the interpretation of the
spectrum of a Kleinian surface singularity solely in terms of the
representation-theoretic data of the associated binary polyhedral group.
