Exercise 24, § 2

Let G be a group and B a Tits subgroup of G (Exerc. 3, of which we use the notation). Assume that, for all s ∈ S, the double coset C(s) is the union of a finite number q_s of left cosets with respect to B. We use the notation of Exerc. 22 and put a_w = a_{C(w)} for all w ∈ W.

a) Show that conditions (i) and (ii) of Exerc. 22 are satisfied. We can therefore speak of the Hecke algebra H_k(G, B) (k being a commutative ring), of which (a_w)_{w ∈ W} is a basis.

b) Let s ∈ S and w ∈ W. Show that a_s \* a_s = (q_s - 1)a_s + q_s; show that if l(sw) > l(w) then a_s \* a_w = a_{sw}.

c) Show that the linear map from the algebra E_k((q_s - 1), (q_s)) associated to the Coxeter system (W, S) (Exerc. 23) to H_k(G, B) that sends e_w to a_w for all w ∈ W is an isomorphism of algebras.