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Orbital symmetric spaces and finite multiplicity. (English) Zbl 0843.22018
The author studies algebraic symmetric spaces. Let $$G$$ be a real algebraic group, $$\sigma$$ an involution of $$G$$ and let $$H$$ be the set of fixed points of $$\sigma$$. In terms of the orbit method the author gives a necessary and sufficient condition for the quasi-regular representation $$\tau = \text{ind}^G_N 1$$ to be irreducible and to be a finite sum of irreducibles. Then he proves that any algebraic symmetric space has finite multiplicity (i.e. $$\tau$$ has finite multiplicity) and examines the boundedness of the multiplicity. To set up the inductive scheme, he slightly generalizes the situation. Let $$Z$$ be a unipotent connected $$\sigma$$-invariant central subgroup of $$G$$, and let $$\chi$$ be a unitary character of $$HZ$$ satisfying $$\sigma \chi = \overline{\chi}$$. Then $$\tau$$ is replaced by the induced representation $$\tau = \text{ind}^G_{HZ} \chi$$. – The method of proof is an amalgam of the Mackey machine and the orbit method based on the works of L. Auslander and B. Kostant [Invent. Math. 14, 255-354 (1971; Zbl 0233.22005)], L. Corwin, F. P. Greenleaf and G. Grélaud [Trans. Am. Math. Soc. 304, 549-583 (1987; Zbl 0629.22005)], M. Duflo [Ann. Sci. Éc. Norm. Supér., IV. Sér. 5, 71-120 (1972; Zbl 0241.22030)] and on a series of the author’s previous researches [Pac. J. Math. 140, 117-147 (1989; Zbl 0645.43010); ibid. 159, No. 2, 351-377 (1993; Zbl 0798.22005); “Representation theory of Lie groups and Lie algebras”, 120-139, World Scientific, Singapore (1992)].

##### MSC:
 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 43A85 Harmonic analysis on homogeneous spaces
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