Algebraic length and Poincare series on reflection groups with applications to representations theory

Let W be a reflection group generated by a finite set of simple reflections S. We determine sufficient and necessary condition for invertibility and positive definitness of the Poincare series Sigma(w) q(l(w)) w, where l(w) denotes the algebraic length on W relative to S. Generalized Poincare series are defined and similar results for them are proved. In case of finite W, representations are constructed which are canonically associated with the algebraic length. For crystallographic groups (Weyl groups) these representations are decomposed into irreducible components. Positive definitness of certain functions involving generalized lengths on W is proved. The proofs don't make use of the classification of finite reflection groups. Examples are provided.