## A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps.(English)Zbl 1167.32015

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 $$H$$.

### MSC:

 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces

### Keywords:

homogeneous convex cone; symmetric cone
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### References:

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