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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.

##### MSC:
 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