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

TL;DR

This paper studies the structure of Hecke algebras associated with the orthogonal group of signature (2,3) and its relation to the paramodular group of degree 2, revealing their algebraic properties and differences.

Contribution

It establishes the commutative structure of the Hecke algebra for the orthogonal group and its isomorphism to the paramodular group, highlighting new algebraic insights.

Findings

Hecke algebra is commutative and a tensor product of polynomial rings.

Orthogonal group is isomorphic to the paramodular group of degree 2.

Hecke algebra of non-maximal paramodular group is non-commutative for N > 1.

Abstract

In this paper we consider the integral orthogonal group with respect to the quadratic form of signature $(2,3)$ given by $\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right) \perp \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right) \perp (-2N)$ for squarefree $N\in \mathbb{N}$. The associated Hecke algebra is commutative and the tensor product of its primary components, which turn out to be polynomial rings over $\mathbb{Z}$ in $2$ algebraically independent elements. The integral orthogonal group is isomorphic to the paramodular group of degree $2$ and level $N$, 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 $\mathrm{Sp}(2;\mathbb{Q})$ fails to be commutative if $N> 1$.