The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups (2401.06427v1)

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 $L^2(G/N,\omega)$, the space of square-integrable sections of the homogeneous vector bundle over $G/N$ associated with an irreducible unitary representation $\omega$ of $N$. Assuming that the central character of $\omega$ is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of $G$ into $L^2(G/N,\omega)$ 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.