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Let W be a reflection group generated by a finite set of simple reflections S. We determine suficient and necessary condition for invertibility and positive definitness of the Poincaré series , where ℓ(w)}denotes the algebraic length on W relative to S. Generalized Poincaré 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.