The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

Jan Frahm*, Gestur Ólafsson, Bent Ørsted

*Corresponding author af dette arbejde

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avisTidsskriftartikelForskningpeer review

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,ψ), where N is the unipotent radical of the Siegel parabolic subgroup and ψ 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,ψ) 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,ψ) as boundary values of holomorphic functions on a domain Ξ in a complexification G C of G forming a Hardy type space H 2(Ξ,ψ).

OriginalsprogEngelsk
Artikelnummer110333
TidsskriftJournal of Functional Analysis
Vol/bind286
Nummer6
Antal sider41
ISSN0022-1236
DOI
StatusUdgivet - mar. 2024

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