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.


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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