We apply tilting theory over preprojective algebras Λ to a study of moduli space of Λ-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using tilting modules over Λ. Moreover we prove that the equivalence induces an isomorphism of algebraic varieties between moduli spaces. In particular, we study in the case when the moduli spaces related to the Kleinian singularity. We generalize a result of Crawley-Boevey which is known another proof of the McKay correspondence of Ito-Nakamura type.

Abstract. Let G be a polyhedral group G ⊂ SO(3) of types Z/nZ, D2n and T. We prove that there exists a one-to-one correspondence between flops of G-Hilb(C 3 ) and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities C 3 /G, which are constructed as moduli spaces MC of quivers with potential for some chamber C in the space Θ of stability conditions. In addition, we study the relation between the exceptional locus in MC with the corresponding quiver QC, and we describe explicitly the part of the chamber structure in Θ where every such resolution can be found.

Let G be a finite subgroup of SL(n, C), then the quotient C n /G has a Gorenstein canonical singularity. If n = 2 or 3, it is known that there exist crepant resolutions of the quotient singularity. In higher dimension, there are many results which assume existence of crepant resolutions. However, few examples of crepant resolutions are known. In this paper, we will show several trials to obtain crepant resolutions and give a conjecture on existence of crepant resolutions.

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