The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

TL;DR

This paper analyzes the contribution of holomorphic discrete series to the Whittaker Plancherel formula for Hermitian Lie groups of tube type, providing explicit formulas and boundary value interpretations.

Contribution

It explicitly constructs holomorphic discrete series embeddings via generalized Whittaker vectors and derives formulas for their formal dimensions.

Findings

Holomorphic discrete series contribute finitely to the Whittaker Plancherel decomposition.

Explicit formulas for embeddings and formal dimensions are obtained.

Holomorphic discrete series are interpreted as boundary values of holomorphic functions in a Hardy space.

Abstract

For a Hermitian Lie group $G$ of tube type we find the contribution of the holomorphic discrete series to the Plancherel decomposition of the Whittaker space $L^2(G/N,\psi)$, where $N$ is the unipotent radical of the Siegel parabolic subgroup and $\psi$ is a certain non-degenerate unitary character on $N$. The holomorphic discrete series embeddings are constructed in terms of generalized Whittaker vectors for which we find explicit formulas in the bounded domain realization, the tube domain realization and the $L^2$-model of the holomorphic discrete series. Although $L^2(G/N,\psi)$ does not have finite multiplicities in general, the holomorphic discrete series contribution does. Moreover, we obtain an explicit formula for the formal dimensions of the holomorphic discrete series embeddings, and we interpret the holomorphic discrete series contribution to $L^2(G/N,\psi)$ as boundary values of holomorphic functions on a domain $\Xi$ in a complexification $G_{\mathbb{C}}$ of $G$ forming a Hardy type space $\mathcal{H}_2(\Xi,\psi)$.