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

TL;DR

This paper studies the structure of Hecke algebras associated with unimodular and modular groups over hermitian fields and quaternion algebras, revealing their complex decomposition properties and providing generators and operator laws.

Contribution

It provides new insights into the structure of these Hecke algebras, including generator sets and conditions for Siegel operator interchange laws.

Findings

No general decomposition into primary components.

Explicit generators for the Hecke algebras.

Special cases with interchange laws for Siegel $\

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.