## 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:

 2.2e+48 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 2.2e+31 Analysis on real and complex Lie groups

### Citations:

Zbl 0462.22006; Zbl 0577.22012; Zbl 0826.22015; Zbl 0907.22016
Full Text:

### References:

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