Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones
In this paper, we give necessary and su cient conditions for a homogeneous cone to be symmetric in two ways. One is by using the multiplier matrix of , and the other is in terms of the basic relative invariants of . 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 . This is a generalization of a result of Ishi and Nomura [ ]. Keywords: Homogeneous cones, symmetric cones, tube domains, basic relative invariants MSC : Primary M ; secondary A , E , S Introduction Symmetric cones are typical examples of Riemannian symmetric spaces, and have been studied from various points of view in many areas of mathematics (see [ ], for example). Homogeneous cones are generalizations of symmetric cones from the standpoint of homogeneity, although no longer symmetric spaces in general, and Vinberg [ ] has laid the foundation of the theory. On the other hand, the present author has studied the basic relative invariants of homogeneous cones and the multiplier matrices determined by them (see below for definition) from several perspectives (cf. [ ]). As a natural research stream, we shall give two characterizations of symmetric cones in this paper. The first one is by using the multiplier matrices, not only of homogeneous cones but also of the dual cones of . The second is in terms of the basic relative invariants of and , where both of them are analytically continued to the corresponding tube domains. For the latter discussions, we generalize a result of Ishi and Nomura [ ] in such a way that the real parts of meromorphic rational functions obtained by the basic relative invariants of are positive, and that this property is characteristic of these rational functions. Let us explain the contents of this paper in more detail. Let be a homogeneous cone of rank r and (V, ) the corresponding Vinberg algebra with unit element e . Then, V has the...