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.
Categories: math.RT