The Hecke algebras for the orthogonal group SO(2,3) and the paramodular group of degree 2

Abstract

In this paper, we consider the integral orthogonal group with respect to the quadratic form of signature [Formula: see text] given by [Formula: see text] for square-free [Formula: see text]. The associated Hecke algebra is commutative and also the tensor product of its primary components, which turn out to be polynomial rings over [Formula: see text] in two algebraically independent elements. The integral orthogonal group is isomorphic to the paramodular group of degree [Formula: see text] and level [Formula: see text], more precisely to its maximal discrete normal extension. The results can be reformulated in the paramodular setting by virtue of an explicit isomorphism. The Hecke algebra of the non-maximal paramodular group inside [Formula: see text] fails to be commutative if [Formula: see text].