"Case-free" derivation for Weyl groups of the number of reflection   factorisations of a Coxeter element
  Chapuy and Stump have given a nice generating series for the number of factorisations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations. 
Jean Michel (IMJ)
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