The geometric McKay correspondence

Let $\Gamma$ be a finite subgroup of $SU(n)$, let $M=\mathbb{C}^n/\Gamma$, and let $\widetilde{M}$ be the crepant resolution introduced above. The relevant objects are the cohomology and homology of $\widetilde{M}$ and the representation theory of $\Gamma$. Geometric McKay correspondence. There are bijections

$$\text{bases of }H^*(\widetilde{M},\mathbb{Z})\longleftrightarrow\{\text{irreducible representations of }\Gamma\},$$ $$\text{bases of }H_*(\widetilde{M},\mathbb{Z})\longleftrightarrow\{\text{conjugacy classes of }\Gamma\},$$

and an identification between the two ring structures. The source notes that substantial cases were already proved, including finite subgroups of $SU(2)$ and abelian subgroups of $SU(3)$, while presenting this formulation as a general geometric correspondence.