Note on a theorem of Eng
Let \((W,S)\) be a finite Coxeter system, \(J\) a subset of \(S\), \(W^J\) the distinguished left coset representatives for the parabolic subgroup \(W_J\) in \(W\), and \(w_0\) the longest element of \(W\). \textit{O. D. Eng} [J. Algebr. Comb. 13, No.1, 29-40 (2001; Zbl 1021.20031)], proved the following theorem. \[ [\sum_{w\in W^J}q^{l(w)}]_{q=-1}=\#\{w\in W^J;\;w_0wW_J=wW_J\}. \] Eng proved this by appeal to the classification of finite irreducible Coxeter systems, and asked for a case-free proof. In this paper, the author gives such a proof via geometry in the case where \((W,S)\) is a Weyl group.
scientific article; zbMATH DE number 1887494