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

Abstract

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).