Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups

Let H subset of G 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 L-2(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 (G) over tilde/(H) over tilde where G subset of (G) over tilde superset of (H) over tilde. We prove that discrete series representations are non-empty for a number of non-symmetric homogeneous spaces such as Sp(2n, R)/Sp(n(0), C) x GL(n(1), C) x ... x GL(n(k), C) (Sigma n(j) = n) and O(4m, n)/U(2m, j) (0 less than or equal to 2j less than or equal to n). (C) 1998 Academic Press.