Hecke algebras for protonormal subgroups

Abstract

We introduce the term protonormal to refer to a subgroup H of a group G such that for every x in G the subgroups x −1 H x and H commute as sets. If moreover ( G , H ) is a Hecke pair we show that the Hecke algebra H ( G , H ) is generated by the range of a canonical partial representation of G vanishing on H. As a consequence we show that there exists a maximum C*-norm on H ( G , H ) , generalizing previous results by Brenken, Hall, Laca, Larsen, Kaliszewski, Landstad and Quigg. When there exists a normal subgroup N of G, containing H as a normal subgroup, we prove a new formula for the product of the generators and give a very clean description of H ( G , H ) in terms of generators and relations. We also give a description of H ( G , H ) as a crossed product relative to a twisted partial action of the group G / N on the group algebra of N / H . Based on our presentation of H ( G , H ) in terms of generators and relations we propose a generalized construction for Hecke algebras in case ( G , H ) does not satisfy the Hecke condition.