Coxeter Transformations, the McKay correspondence, and the Slodowy correspondence

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This talk was presented at Workshop "Spectral Methods in Representation Theory of Algebras and Applications to the Study of Rings of Singularities", 2008 (Banff, Canada). W. Ebeling established a connection between certain Poincare series, the Coxeter transformation C, and the corresponding affine Coxeter transformation Cₐ (in the context of the McKay correspondence). We consider the generalized Poincare series [PG(t)]₀ for the case of multiply-laced diagrams(in the context of the McKay-Slodowy correspondence) and extend the Ebeling theorem for this case: [PG(t)]₀ = X(t²)/X(t²), where X is the characteristic polynomial of the Coxeter transformation and X is the characteristic polynomial of the corresponding affine Coxeter transformation. We obtain that Poincare series coincide for pairs of diagrams obtained by folding: X (Γ) / X (Γ) = X (Γᶠ) / X (Γᶠ), where Γ is any (A, D, E type) Dynkin diagram, Γ is the extended Dynkin diagram, and the diagrams Γᶠ and Γᶠ are obtained by folding from Γ and Γ, respectively.