A canonical anti-isomorphism of matrix Hecke rings

Abstract

In the case of abstract Hecke rings D(Γ S) associated with a Hecke pair (Γ S) for a multiplicative group ([2], § 3.1), there exists a canonical anti-isomorphism taking a double coset Γg Γ for g in S to the double coset Γ g ^-1Γ. Anti-automorphisms in Hecke rings are of interest, since, under suitable conditions, they imply the commutativity of these rings. In a recent paper [1] on the multiplicative properties of integral representations of quadratic forms by quadratic forms, Andrianov has defined an abstract matrix Hecke ring, motivated by his concept of the ring of classes of automorphs of a given system of quadratic forms. The object of this note is to seek an answer to a natural question raised by him on the existence of a canonical anti-isomorphism in the case of these abstract matrix Hecke rings.