Discrete Series Representations for the Orbit Spaces Arising from Two Involutions of Real Reductive Lie Groups

LetH⊂Gbe real reductive Lie groups. A discrete series representation for a homogeneous spaceG/His an irreducible representation ofGrealized as a closedG-invariant subspace ofL2(G/H). The condition for the existence of discrete series representations forG/Hwas not known in general except for reductive symmetric spaces. This paper offers a sufficient condition for the existence of discrete series representations forG/Hin the setting thatG/His a homogeneous submanifold of a symmetric space G/HwhereG⊂G⊃H. We prove that discrete series representations are non-empty for a number of non-symmetric homogeneous spaces such asSp(2n, R)/Sp(n0, C)×GL(n1, C)×…×GL(nk, C) (∑ nj=n) andO(4m, n)/U(2m, j) (0⩽2j⩽n).