The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions
The author determines the structure of the abstract Hecke algebras of \(\mathrm{Gl}_ n\) and \(\mathrm{Sp}_n\) over the Hurwitz order of integral quaternions. The main technical tool is the elementary divisor theory for the Hurwitz order as developed by the author [Linear Multilinear Algebra 21, 325--344 (1987; Zbl 0663.15003)]. In the case \(n=1\) the Hecke algebras fail to be commutative. For \(n>1\) they are commutative and isomorphic to the tensor product of their primary components. Each primary component turns out to be a polynomial ring. In the symplectic case, the relation to Siegel's \(\Phi\)-operator is investigated; the homomorphism of Hecke algebras induced by \(\Phi\) is surjective except for two exceptional weights.
scientific article; zbMATH DE number 4077336
scientific article