Hecke Algebras Of Group Extensions

We describe the Hecke algebra H(Gamma,Gamma(0)) of a Hecke pair (Gamma, Gamma(0)) in terms of the Hecke pair (N, Gamma(0)) where N is a normal subgroup of Gamma containing Gamma(0). To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S subset of Gamma/ N satisfies S-1 S = Gamma/N , we show that H(Gamma, Gamma(0)) is the twisted crossed product of (N ,Gamma(0)) by S . This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.