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A description of discrete series for semisimple symmetric spaces. II. (English) Zbl 0719.22003
Representations of Lie groups: analysis on homogeneous spaces and representations of Lie groups, Proc. Symp., Kyoto/Jap. and Hiroshima/Jap. 1986, Adv. Stud. Pure Math. 14, 531-540 (1988).
[For the entire collection see Zbl 0694.00014. For Part I see T. Oshima and the author, ibid. 4, 331-390 (1984; Zbl 0577.22012).]
In [Ann. Math., II. Ser. 111, 253-311 (1980; Zbl 0462.22006)], M. Flensted-Jensen constructed countably many discrete series for a semisimple symmetric space G/H when (1) \(rank(G/H)=rank(K/K\cap H)\). Conversely, part I proved that (1) holds if there exist discrete series for G/H. Moreover part I constructed Harish-Chandra modules \(B^ j_{\lambda}\) which parametrize all the discrete series for G/H, where j runs through finite indices and \(\lambda\) runs through lattice points contained in a positive Weyl chamber. In this paper, we give a necessary condition for j and \(\lambda\) so that the module \(B^ j_{\lambda}\) is nontrivial. In a subsequent paper [Part III (to appear)], we will prove that the condition also assures \(B^ j_{\lambda}\neq \{0\}\). We remark that our results also cover “limits of discrete series” for G/H. In the appendix, we give a certain simplification of the proof of a main result in part I.

22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces
22E30 Analysis on real and complex Lie groups