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

TL;DR

The article extends the holomorphic discrete series contribution to the generalized Whittaker Plancherel formula from tube-type to non-tube-type simple Hermitian groups by establishing a unique standard maximal parabolic subgroup $P=MAN$ with $G o P^+K_C N_C$. It proves that every holomorphic discrete series appears discretely in $L^2(G/N,C)$ with multiplicity equal to the dimension of its highest weight space, and constructs explicit embeddings using a kernel function $Psi_{pi,eta}$ arising from intertwiners; the kernel encodes both the lowest $K$-type and the associated Whittaker vectors. The approach relies on a detailed Harish-Chandra realization of the bounded domain, strategic SU$(1,1)$-embeddings, and a Fock-space model for the nilradical $N$, yielding an explicit, computable description of the embedded holomorphic discrete series inside the generalized Whittaker space $L^2(G/N,C)$. By connecting intertwiners, kernels, and Whittaker functionals, the paper provides a kernel-based framework that clarifies the relationships among holomorphic discrete series, lowest $K$-types, and Whittaker models, and it extends known tube-type results to the broader non-tube-type setting.

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 $L^2(G/N,ω)$, 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 $L^2(G/N,ω)$ 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.

Theorems & Definitions (37)

• Theorem A: see Theorem \ref{['thm:pkn1']}
• Theorem B: see Corollary \ref{['cor:HolDScontribution']}
• Theorem C: see Proposition \ref{['prop:AdjointIntertwiner']}, Lemma \ref{['lem:LKTembedding']} and Proposition \ref{['prop:WhittakerVectorsAsAntiholFcts']}
• Theorem 1.1: Moore's Theorem, first version
• Lemma 1.2
• Lemma 1.3
• Theorem 1.4: Moore's Theorem, second version
• Lemma 1.5: see VR76 and Sa80
• Theorem 2.1: see FOO24
• Corollary 2.2