Hecke algebras related to the unimodular and modular groups over hermitian fields and definite quaternion algebras

Abstract

We investigate the structure of the Hecke algebras related to the unimodular and modular group over hermitian fields and definite quaternion algebras. In particular we show that in general there is no decomposition into primary components. We can give a set of generators and in some special cases we deduce the law of interchange of the Siegel $\Phi$-operator.