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Algèbre de Jordan et ensemble de Wallach. (Jordan algebra and Wallach set). (French) Zbl 0622.22008
The main theme of this paper is to formulate the problem of determining the relatively invariant measures on a closure of a convex cone in a formally real Jordan algebra and to solve it. Let A be a formally real Jordan algebra whose unit is E. Let \({\mathbb{G}}\) be the connected component of the structure group of A containing the neutral element and let \(\Omega\) be the \({\mathbb{G}}\)-orbit of \(e\in A\). Then the closure \({\bar \Omega}\) decomposes into a finite number of \({\mathbb{G}}\)-orbits. The author proves that each orbit in \(\Omega\) has relatively invariant measures and determines all of them. This contributes to the determination of the Wallach set, which is defined in relation to the unitarization problem of analytic continuation of discrete series.
Reviewer: M.Muro

MSC:
22E30 Analysis on real and complex Lie groups
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17C10 Structure theory for Jordan algebras
22D10 Unitary representations of locally compact groups
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