T. Kobayashi,
Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups,
J. Funct. Anal. 152 (1998), 100-135..

Let HG be real reductive Lie groups. A discrete series representation for a homogeneous space G/H is an irreducible representation of G realized as a closed G-invariant subspace of L2(G/H). The condition for the existence of discrete series representations for G/H was not known in general except for reductive symmetric spaces. This paper offers a sufficient condition for the existence of discrete series representations for G/H in the setting that G/H is a homogeneous submanifold of a symmetric space \tilde{G}/\tilde{H} where G ⊂\tilde{G} ⊃\tilde{H}. We prove that discrete series representa-tions are non-empty for a number of non-symmetric homogeneous spaces such as Sp(2n,R)/Sp(n0,CGL(n1,C)×...×GL(nk,C) (nj=n) and O(4m,n)/U(2m,j) (0≤2jn).
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