On gamma matrices of local zeta functions associated with homogeneous cones

TL;DR

This paper analyzes the structure of coefficient matrices in functional equations of zeta functions linked to homogeneous cones, revealing their decomposition properties and conditions for completion forms.

Contribution

It demonstrates that these coefficient matrices can be decomposed into variable-wise matrices independently of cone choice and identifies conditions for zeta function completion forms.

Findings

Coefficient matrices decompose into variable-wise matrices.

Existence of completion forms for associated zeta functions.

Decomposition holds regardless of the specific homogeneous cone.

Abstract

The purpose of this paper is to investigate coefficient matrices of functional equations of zeta functions associated with homogeneous cones, which are given explicitly in the previous paper, in detail. We prove that the coefficient matrix can be decomposed into variable-wise matrices regardless of the choice of homogeneous cones. Moreover, under a suitable condition, we show that the associated zeta functions admit a kind of completion forms.