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Positive Riesz distributions on homogeneous cones. (English) Zbl 0954.43003

Let \(\Omega\) be an open convex cone in a finite-dimensional real vector space, containing no line and on which the group \(\text{GL}(\Omega)\) of linear automorphisms acts transitively. Riesz distributions on \(\Omega\), namely relatively invariant distributions supported by the closure \(\overline\Omega\), were studied by S. G. Gindikin [Funct. Anal. Appl. 9, 50-52 (1975; Zbl 0332.32022)]. He introduced functions \(\Delta_s\) \((s\in\mathbb{C}^r)\) on \(\Omega\) which are relatively invariant under a split solvable Lie group \(H\) acting linearly and simply transitively on \(\Omega\), and also the gamma function \(\Gamma_\Omega\) associated to \(\Omega\). Then, \(d\mu\) being the \(H\)-invariant measure on \(\Omega\), \({\mathcal R}_s= \Gamma^{-1}_\Omega(s) \Delta_sd\mu\) are Riesz distributions on \(\Omega\). Though the set \(\Xi\) of the parameter \(s\) for which \({\mathcal R}_s\) is a positive measure is an object of interest, it is merely roughly sketched in Gindikin’s paper except for the case of symmetric cones treated by J. Faraut and A. Korányi [Analysis on symmetric cones (Oxford 1994; Zbl 0841.43002)].
This work is intended to improve the situation just mentioned. Using the structure theory of normal \(j\)-algebras developed by I. I. Pyatetskii-Shapiro [Automorphic functions and the geometry of classical domains (New York 1969; Zbl 0196.09901)], the author decomposes \(\overline\Omega\) in \(H\)-orbits, gives a clear description of \(\Xi\) by relating it to the \(H\)-orbit structure of \(\overline\Omega\) and describes explicitly each positive Riesz distribution as a measure on an orbit in \(\overline\Omega\). He uses fine parametrizations throughout the paper.

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
46F10 Operations with distributions and generalized functions
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