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Abstract: Starting with an ADE singularity C^2/Gamma for Gamma a finite subgroup of SL(2,C), one can build various moduli spaces of geometric and representation-theoretic interest as Nakajima quiver varieties. These spaces depend in particular on a stability parameter; quiver varieties at both generic and non-generic stability are of geometric interest. Generating functions of Euler characteristics at different points in stability space are controlled by specialisation formulae of characters of Lie algebra representations. We will explain some of these connections, focusing in particular on the abelian case. Based on joint papers and projects with Craw, Gammelgaard, Gyenge, and Nemethi.