On Hasse zeta functions of enveloping algebras of solvable Lie algebras. II (Q1357036)

The purpose of this paper is to prove a theorem on functional equations of Hasse zeta functions of enveloping algebras of solvable Lie algebras, which was proposed by the author in her previous paper [cf. ibid. 72, No. 8, 187-188 (1996; Zbl 0868.11040)]. She proved there the theorem under certain conditions on \(p\)-mappings on the algebra. In this paper she succeeds in giving a proof of the theorem without any additional conditions. The theorem is the following. Let \(R\) be a finitely generated commutative ring over the integers \(\mathbb{Z}\). Let \({\mathfrak g}\) be a solvable Lie algebra over \(R\) which is free of finite rank \(n\) as an \(R\)-module, and let \(A\) be the universal enveloping algebra of \({\mathfrak g}\) over \(R\). Then we have \[ \zeta_A(s) =\zeta_R(s-n), \] where \(\zeta_A (s)\) and \(\zeta_R(s)\) are Hasse zeta functions of \(A\) and \(R\), respectively.