The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups

Abstract

For every simple Hermitian Lie group , we consider a certain maximal parabolic subgroup whose unipotent radical is either abelian (if is of tube type) or two-step nilpotent (if is of non-tube type). By the generalized Whittaker Plancherel formula we mean the Plancherel decomposition of , the space of square-integrable sections of the homogeneous vector bundle over associated with an irreducible unitary representation of . Assuming that the central character of is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of into and show that the multiplicities are equal to the dimensions of the lowest -types. The construction is in terms of a kernel function which can be explicitly defined using certain projections inside a complexification of . This kernel function carries all information about the holomorphic discrete series embedding, the lowest -type as functions on , as well as the associated Whittaker vectors.