Inhomogeneous Kleinian singularities and quivers

International audience; The purpose of this article is to generalize a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of types A_r , D_r , E_6 , E_7 and E_8 to the singularities of inhomogeneous types B_r , C_r , F_4 and G_2 defined in 1978 by P. Slodowy. Let Γ be a finite subgroup of SU_2. Then C^2/ Γ is a simple singularity of type ∆(Γ). By studying the representation space of a quiver defined from Γ via the McKay correspondence , and a well chosen finite subgroup Γ ′ of SU_2 containing Γ as normal subgroup, we will use the symmetry group Ω = Γ′/Γ of the Dynkin diagram ∆(Γ) and explicitly compute the semiuniversal deformation of the singularity (C^2/Γ, Ω) of inhomogeneous type. The fibers of this deformation are all equipped with an induced Ω-action. By quotienting we obtain a deformation of a singularity C^2/Γ′ with some unexpected fibers.