A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps (Q998936)

The author establishes a new criterion which characterizes the symmetric cones among the homogeneous convex ones. An open convex cone \(\Omega \) in a real vector space \(V\) is said to be homogeneous if the group \(G(\Omega )\) of linear automorphisms of \(\Omega \) acts on \(\Omega \) transitively. Moreover if there exists on \(V\) an inner product such that the cone \(\Omega \) is equal to the dual cone \(\Omega ^*\), then \(\Omega \) is said to be symmetric. Let us recall the definition of the Vinberg map \(x\mapsto x^*\), from a homogeneous cone \(\Omega \) onto its dual \(\Omega ^*\). One defines first the characteristic function \(\varphi \) of \(\Omega \): \[ \varphi (x)=\int _{\Omega ^*} e^{-\langle x,f\rangle } \,df. \] Then \(x^*=\text{grad\;log}\, \varphi (x)\). One considers on \(\Omega \) a partial ordering: \(x\geq y\) if \(x-y\in \overline{\Omega }\), and similarly on \(\Omega ^*\). The author proves that the cone \(\Omega \) is symmetric if and only if it satisfies the following property: for \(x,y\in \Omega \), \(x\geq y\) if and only if \(y^*\geq x^*\). In fact the author proves a more general result for which the characteristic function \(\varphi \) is replaced by a semi-invariant function on \(\Omega \) with respect to a solvable subgroup \(H\) of \(G(\Omega )\) acting transitively on \(\Omega \). The proofs use criterions for a homogeneous convex cone to be symmetric involving the generalized Peirce decomposition of \(V\) related to the clan structure on \(V\) associated to the homogeneous cone \(\Omega \) and the subgroup \(H\).