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 |
Keywords:
unitary representation of Lie group; invariant measures; convex cone; formally real Jordan algebra; structure group; Wallach set; unitarization problem; analytic continuation of discrete seriesReferences:
| [1] | Bourbaki, N.: Intégration. Paris: Hermann 1965 |
| [2] | Braun, H., Koecher, M.: Jordan algebren. Berlin-Heidelberg-New York: Springer 1966 · Zbl 0145.26001 |
| [3] | Koecher, M.: Jordan algebras and their applications. Lect. Notes University of Minnesota, Minneapolis, 1962 · Zbl 0128.03101 |
| [4] | Muller, I., Rubenthaler, H., Schiffmann, G.: Structure des espaces préhomogènes associés à certaines algèbres de Lie graduées. Math. Ann.274, 95-123 (1986) · Zbl 0568.17007 · doi:10.1007/BF01458019 |
| [5] | Resnikoff, H.: On a class of linear differential equations for automorphic forms in several complex variables. Am. J. Math.95, 321-332 (1973) · Zbl 0276.32023 · doi:10.2307/2373788 |
| [6] | Rossi, H., Vergne, M.: Analytic continuation of the holomorphic discrete series of a semi-simple Lie group. Acta math.136, 1-59 (1976) · Zbl 0356.32020 · doi:10.1007/BF02392042 |
| [7] | Satake, I.: Algebraic structures of symmetric domains. Princeton Univ. Press, 1980 · Zbl 0483.32017 |
| [8] | Satake, I.: Special values of zeta functions associated with self-dual homogeneous cones. In: Manifolds and Lie groups, pp. 359-384. Boston: Birkhäuser 1981 · Zbl 0497.14008 |
| [9] | Satake, I.: A formula in simple Jordan algebras. Tohoku Math. J.36, 611-622 (1984) · Zbl 0576.17014 · doi:10.2748/tmj/1178228766 |
| [10] | Satake, I., Faraut, J.: The functional equation of zeta distributions associated with formally real Jordan algebras. Tohoku Math. J.36, 469-482 (1984) · Zbl 0566.10016 · doi:10.2748/tmj/1178228811 |
| [11] | Upmeier, H.: Jordan algebras and harmonic analysis on symmetric spaces. Am. J. Math.108, 1-25 (1986) · Zbl 0603.46055 · doi:10.2307/2374466 |
| [12] | Wallach, N.: The analytic continuation of the discrete series II. Trans. Am. Math. Soc.251, 19-37 (1979) · Zbl 0419.22018 · doi:10.1090/S0002-9947-79-99965-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
