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The Dirac operator on homogeneous spaces and representations of reductive Lie groups. II, III. (English) Zbl 0669.22003

Am. J. Math. 109, No. 3, 499-520 (1987); 110, No. 3, 451-471 (1988).
[Part I, cf. ibid. 283-301 (1987; Zbl 0649.58031).]
In the theory of representations of Lie groups there is the important problem to construct the discrete series. Some geometrical ideas and methods of differential equations are fruitful here. [See R. Parthasarathy, Ann. Math., II. Ser. 96, 1-30 (1972; Zbl 0249.22003); M. F. Atiyah and W. Schmid, Invent. Math. 42, 1-62 (1977; Zbl 0373.22001).]
The papers under review make a contribution to the problem mentioned above. In Part II the author investigates the spaces of harmonic spinors in terms of some Dirac operators on Riemannian connected homogeneous spaces. Using these results in Part III in a more general situation the author obtains the discrete series of representations for a reductive Lie group.
Reviewer: A.Venkov

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A85 Harmonic analysis on homogeneous spaces
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
22E30 Analysis on real and complex Lie groups
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