The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula
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| Format: | Preprint |
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2022
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| author | Frahm, Jan Ólafsson, Gestur Ørsted, Bent |
| author_facet | Frahm, Jan Ólafsson, Gestur Ørsted, Bent |
| contents | 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 $G_{\mathbb{C}}$ of $G$ forming a Hardy type space $\mathcal{H}_2(Ξ,ψ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_14784 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula Frahm, Jan Ólafsson, Gestur Ørsted, Bent Representation Theory Primary 22E46, Secondary 43A85 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 $G_{\mathbb{C}}$ of $G$ forming a Hardy type space $\mathcal{H}_2(Ξ,ψ)$. |
| title | The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula |
| topic | Representation Theory Primary 22E46, Secondary 43A85 |
| url | https://arxiv.org/abs/2203.14784 |