Functional equations of zeta functions associated with homogeneous cones

Abstract

In this paper, we focus on solvable prehomogeneous vector spaces associated with homogeneous cones, and consider the associated zeta functions in several variables. We discuss $\mathbb{Q}$-structures of these prehomogeneous vector spaces, and give explicit formulas of functional equations of the zeta functions by using the data of homogeneous cones. The associated $b$-functions are also described explicitly.