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

Kai, Chifune

J. Math. Soc. Japan 60, No. 4, 1107-1134 (2008).

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 \(H\).

Reviewer: Jacques Faraut (Paris)

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

References:

[1]	H. Asano, On the irreducibility of homogeneous convex cones, J. Fac. Sci. Univ. Tokyo Sect. I, 15 (1968), 201-208. · Zbl 0186.40502
[2]	J. E. D’Atri and I. Dotti Miatello, A characterization of bounded symmetric domains by curvature, Trans. Amer. Math. Soc., 276 (1983), 531-540. · Zbl 0517.32014
[3]	J. Dorfmeister, Inductive construction of homogeneous cones, Trans. Amer. Math. Soc., 252 (1979), 321-349. · Zbl 0424.17012
[4]	J. Dorfmeister, Homogeneous Siegel domains, Nagoya Math. J., 86 (1982), 39-83. · Zbl 0445.32025
[5]	J. Faraut and A. Korányi, Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994. · Zbl 0841.43002
[6]	O. Güler, Hyperbolic polynomials and interior point methods for convex programming, Math. Oper. Res., 22 (1997), 350-377. · Zbl 0883.90099
[7]	C. Kai, A characterization of symmetric Siegel domains by convexity of Cayley transform images, Tohoku Math. J., 59 (2007), 101-118. · Zbl 1201.32011
[8]	C. Kai and T. Nomura, A characterization of symmetric cones through pseudoinverse maps, J. Math. Soc. Japan, 57 (2005), 195-215. · Zbl 1086.32021
[9]	T. Nomura, On Penney’s Cayley tranform of a homogeneous Siegel domain, J. Lie Theory, 11 (2001), 185-206. · Zbl 0977.32015
[10]	T. Nomura, Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms, Diff. Geom. Appl., 18 (2003), 55-78. · Zbl 1023.22007
[11]	T. Nomura, Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains, J. Funct. Anal., 198 (2003), 229-267. · Zbl 1026.32046
[12]	R. Penney, The Harish-Chandra realization for non-symmetric domains in \(\bm{C}^n\), Topics in Geometry, ed. S. Gindikin, Progr. Nonlinear Differential Equations Appl., 20 , Birkhäuser Boston, Boston, MA, 1996, pp.,295-313. · Zbl 0881.32002
[13]	I. Satake, Algebraic Structures of Symmetric Domains, Iwanami Shoten, Tokyo; Princeton Univ. Press, Princeton, 1980.
[14]	H. Shima, A differential geometric characterization of homogeneous self-dual cones, Tsukuba J. Math., 6 (1982), 79-88. · Zbl 0521.53047
[15]	T. Tsuji, A characterization of homogeneous self-dual cones, Tokyo J. Math., 5 (1982), 1-12. · Zbl 0503.53037
[16]	T. Tsuji, On connection algebras of homogeneous convex cones, Tsukuba J. Math., 7 (1983), 69-77. · Zbl 0535.17011
[17]	E. B. Vinberg, The theory of convex homogeneous cones, Trudy Moskov Mat. Obšč., 12 (1963), 303-358; Trans. Moscow Math. Soc., 12 (1963), 340-403. · Zbl 0138.43301
[18]	E. B. Vinberg, The structure of the group of automorphisms of a homogeneous convex cone, Trudy Moskov Mat. Obšč., 13 (1965), 56-83; Trans. Moscow Math. Soc., 13 (1967), 63-93. · Zbl 0311.17008