The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions. (English) Zbl 0659.10027
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.
Reviewer: Siegfried Böcherer (Mannheim)
MSC:
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
Keywords:
unimodular group; modular group; Hecke algebras; Hurwitz order of integral quaternions; Siegel’s \(\Phi\)-operatorCitations:
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