Global properties of a Hecke ring associated with the Heisenberg Lie algebra

TL;DR

This paper explores the properties of a Hecke ring linked to the Heisenberg Lie algebra, generalizing classical series and deriving explicit formulas for associated zeta functions.

Contribution

It introduces a formal Dirichlet series for Hecke rings related to the Heisenberg Lie algebra and establishes an analog of Shimura's series identity.

Findings

Derived an analog of Shimura's series identity.

Recovered the explicit pro-isomorphic zeta function formula.

Generalized classical series using Hecke rings.

Abstract

This study concerns (not necessarily commutative) Hecke rings associated with certain algebras and describes a formal Dirichlet series with coefficients in the Hecke rings, which can be used to generalize Shimura's series. Considering the case of the Heisenberg Lie algebra, an analog of the identity for Shimura's series derived employing the rationality theorem, presented by Hecke and Tamagawa, is established. Moreover, this analog recovers the explicit formula for the pro-isomorphic zeta function of the Heisenberg Lie algebra shown by Grunewald, Segal and Smith.