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

Abstract

For every simple Hermitian Lie group G, we consider a certain maximal parabolic subgroup whose unipotent radical N is either abelian (if G is of tube type) or two-step nilpotent (if G is of non-tube type). By the generalized Whittaker Plancherel formula we mean the Plancherel decomposition of L2(G/N,ω), the space of square-integrable sections of the homogeneous vector bundle over G/N associated with an irreducible unitary representation ω of N. Assuming that the central character of ω is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of G into L2(G/N,ω) and show that the multiplicities are equal to the dimensions of the lowest K-types. The construction is in terms of a kernel function which can be explicitly defined using certain projections inside a complexification of G. This kernel function carries all information about the holomorphic discrete series embedding, the lowest K-type as functions on G/N, as well as the associated Whittaker vectors.