Special McKay correspondence

There are many generalizations of the McKay correspondence for higher dimensional Gorenstein quotient singularities and there are some applications to compute the topological invariants today. But some of the invariants are completely different from the classical invariants, in particular for non-Gorenstein cases. In this paper, we would like to discuss the McKay correspondence for 2-dimensional quotient singularities via "special" representations which gives the classical topological invariants and give a new characterization of the special representations for cyclic quotient singularities in terms of combinatorics.