Hecke C*-algebras and semi-direct products

We analyse Hecke pairs (G,H) and the associated Hecke algebra H when G is a semi-direct product N ⋊ Q and H = M R for subgroups M N and R Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of Hin C*(G) is MoritaRieffel equivalent to a crossed product IQ/R, where I is a certain ideal in the fixed-point algebra C*(N)^R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K², where K = or K = [p¹]. In particular we look at the ax + b group of a quadratic extension of K.