Exercise 25, § 2

We use the notation of Exerc. 8. Assume in addition that, for all s ∈ S, the index q_s of B ∩ gBg^{-1} in B is finite for all g ∈ BsB.

a) Show that the pair (\tilde{G}, B) satisfies conditions (i) and (ii) of Exerc. 22.

b) Show that, for all γ ∈ Γ, the map x ↦ γxγ^{-1} defines an automorphism σ of the Hecke algebra H_k(G, B) and that σ depends only on the class ω of γ in Ω = Γ/(Γ ∩ B).

c) Let k[Ω] be the group algebra of Ω and (e_ω) its canonical basis. Show that the linear map j from k[Ω] ⊗_k H_k(G, B) to H_k(\tilde{G}, B) defined by j(e_ω ⊗ a_{BwB}) = a_{BωwB} (in the notation of Exerc. 22), for ω ∈ Ω and w ∈ W, is bijective and that

for ω, ω' ∈ Ω and x, y ∈ H_k(G, B).