Geometric Construction of the McKay-Slodowy Correspondence

TL;DR

This paper develops a geometric framework for the McKay-Slodowy correspondence, extending the classical case to pairs of groups and their representations, without relying on case-by-case proofs.

Contribution

It introduces a geometric construction that generalizes the McKay correspondence to pairs of groups, linking induced representations to components of resolved quotient varieties.

Findings

Established a bijection between induced representations and exceptional components.

Extended the classical McKay correspondence to group pairs with normal subgroups.

Provided a proof not based on case-by-case analysis.

Abstract

This paper presents a geometric construction of the McKay-Slodowy correspondence, which extends the classical McKay correspondence. The classical McKay correspondence says: for a finite subgroup G of SL_2(C), there is a bijection between the set of nontrivial irreducible representations of G and the irreducible components of the exceptional locus of the minimal resolution of the quotient variety C^2/G. We generalizes it to a pair of groups: when G is a finite subgroup of SL_2(C) with a normal subgroup H, the set of induced nontrivial irreducible representations from H to G corresponds one-to-one to the set of pushing-forward of components of the exceptional locus of the minimal resolution of C^2/H under the quotient by G/H-action. Our proof is not given by case-by-case verification.