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Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups. (English) Zbl 0937.22008
Let \(G\) be a real reductive Lie group and \(H\) be a closed subgroup that is reductive in \(G\). An irreducible representation of \(G\) realized as a closed \(G\)-invariant subspace of \(L^2(G/H)\) is called a discrete series representation for \(G/H\). The condition for the existence of discrete series representations for \(G/H\) was previously known for semisimple symmetric spaces through the work of M. Flensted-Jensen [Ann. Math. (2) 111, 253-311 (1980; Zbl 0462.22006)] and T. Oshima and T. Matsuki [in: Group representations and systems of differential equations (Tokyo, 1982), 331-390, Adv. Stud. Pure Math. 4, North-Holland, Amsterdam (1984; Zbl 0577.22012)].
The author gives a sufficient condition for the existence of discrete series representations for \(G/H\) in the setting in which \(G/H\) is a submanifold of \(\widetilde{G}/\widetilde{H}\) where \(G\subset\widetilde{G}\supset \widetilde{H}\) and harmonic analysis on \(\widetilde{G}/\widetilde{H}\) is well understood. The difficulty of the restriction of \(L^2\)-functions to a submanifold is overcome by assuming the admissibility of the restriction of the unitary representation with respect to a reductive subgroup, which was developed by the author in [Invent. Math. 117, 181-205 (1994; Zbl 0826.22015), Invent. Math. 131, 229-256 (1998; Zbl 0907.22016)].
The author gives several new examples of \(G/H\) that have discrete series representations such as \[ \text{Sp}(2n,\mathbb{R})/\text{Sp}(n_0,\mathbb{C})\times \text{GL}(n_1,{\mathbb{C}})\times\cdots\times\text{GL}(n_k,{\mathbb{C}})\quad \Bigl(\sum n_j=n\Bigr) \] and \(\text{O}(4m,n)/\text{U}(2m,j)\) \((0\leq 2j\leq n)\).

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E30 Analysis on real and complex Lie groups
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