Euler characteristics of affine ADE Nakajima quiver varieties via collapsing fibers

Abstract

We prove a universal substitution formula that compares generating series of Euler characteristics of Nakajima quiver varieties associated with affine ADE diagrams at generic and at certain nongeneric stability conditions via a study of collapsing fibres in the associated variation of GIT map, unifying and generalising earlier results of the last two authors with N\'emethi and of Nakajima. As a special case, we compute generating series of Euler characteristics of noncommutative Quot schemes of Kleinian orbifolds. In type A and rank 1, we give a second, combinatorial proof of our substitution formula, using torus localisation and partition enumeration. This gives a combinatorial model of the fibers of the variation of GIT map, and also leads to relations between our results and the representation theory of the affine and finite Lie algebras in type A.