Kostant's generating functions, Ebeling's theorem and McKay's observation relating the Poincare series

Abstract

We generalize B. Kostant's construction of generating functions to the case of multiply-laced diagrams and we prove for this case W. Ebeling's theorem which connects the Poincare series [P_G(t)]_0 and the Coxeter transformations. According to W. Ebeling's theorem [P_G(t)]_0 = \frac{X(t^2)}{\tilde{X}(t^2)}, where X is the characteristic polynomial of the Coxeter transformation and \tilde{X} is the characteristic polynomial of the corresponding affine Coxeter transformation. We prove McKay's observation relating the Poincare series [P_G(t)]_i: (t+t^{-1})[P_G(t)]_i = \sum\limits_{i \leftarrow j}[P_G(t)]_j, where j runs over all vertices adjacent to i.