Exercise 27, § 2

Let $(G, B, N, S)$ be a Tits system and let $k$ be a commutative field. Assume that $G$ is finite and that the characteristic of $k$ divides neither the order of $G$ nor the order of the Weyl group $W = N/(B \cap N)$. Show that the algebras $H_k(G, B)$ (Exerc. 22) and $k[W]$ are absolutely semi-simple and have the same numerical invariant, and hence are isomorphic when $k$ is algebraically closed (let $q_s$ be the index of $B \cap gBg^{-1}$ in $B$ for $g \in BsB, s \in S$. Consider the algebra $E_{k[X]}((X(q_s - 1)), (1 + X(q_s - 1)))$ constructed as in Exerc. 23 from the Coxeter system $(W, S)$ and the polynomial ring $k[X]$ and use Exerc. 26 a) and b), remarking that, by Maschke’s theorem (Chap. V, Appendix), the bilinear form $\operatorname{Tr}_{k[W]/k}(xy)$ is non-degenerate).