Completion of local zeta functions associated with a certain class of homogeneous cones

TL;DR

This paper extends the theory of local zeta functions linked to homogeneous cones by demonstrating their possible completion forms, analogous to the completed Riemann zeta function, for a specific class of cones.

Contribution

It introduces a new completion framework for local zeta functions associated with certain homogeneous cones, expanding the understanding of their functional equations.

Findings

Local zeta functions admit completion forms for specific homogeneous cones.

The completed local zeta functions satisfy symmetric functional equations.

Extension of classical zeta function concepts to geometric structures.

Abstract

It is well known that the Riemann zeta function can be completed to the Riemann xi function $\xi(s)$ in the sense that its functional equation has a higher symmetric form $\xi(1-s)=\xi(s)$. In the previous paper (Tohoku Math. J. 72 (2020), 349--378), we give an explicit formula of functional equations between local and global zeta functions associated with a homogeneous cone and with its dual cone. In this paper, we consider a completion of these local zeta functions and show that, for a certain class of homogeneous cones, the associated local zeta functions admit a kind of completion forms.