Decomposition of gamma matrices of local zeta functions associated with homogeneous cones

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, (Tohoku Math. J. 72 (2020), 349--378), in detail. We prove that the coefficient matrix can be decomposed as a product of diagonal matrices each in a single variable and of orthogonal matrices with constant entries for a general homogeneous cone. Moreover, under a suitable condition, we show that the associated zeta functions admit a kind of completion forms.