We describe the Hecke algebra ℋ(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ⊂ Γ/N satisfies S⁻¹ S = Γ/N, we show that ℋ(Γ,Γ₀) is the twisted crossed product of ℋ(N,Γ₀) by S . This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.
Communications in Algebra Vol. 33, Issue 11, p. 4135-4147