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The aim of this paper is to study structures of rings of invariant differential operators on homogeneous cones, and give an explicit description of their generators. For the group actions, we consider split solvable Lie groups acting on the cones linearly and simply transitively. As an application, we present Capelli-type identities for generalized Vinberg cones.

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.

The purpose of this paper is to show that it is characteristic of symmetric cones among irreducible homogeneous cones that there exists a non-constant relatively invariant polynomial such that its Laplace transform is the reciprocal of a certain polynomial. To prove our theorem, we need the inductive structure of the basic relative invariants of a homogeneous cone. However, we actually work in a more general setting, and consider the inducing of the basic relative invariants from lower rank cones.

In this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].