... a theorem of Chevalley [Hu, §3.15], the second a theorem of Bott [Hu, §8.9]. We shouldmention that Bott’s proof, although extremely elegant and unified, is not completelyelementary, and more ... elementary proofs of some cases of his theorem have recentlyappeared in [BB, BE , EE, ER].We first consider an infinite family of Coxeter systems with linear diagrams. LetWr,snbe the family of Coxeter ... x(1 − t).Remarks.1. The crucial encoding of functions K : {s0, ,sn}→ used in the middle of the preceding proof is a translation and generalization of the “directencoding”usedin[GG,§1] for...