The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups
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| Format: | Preprint |
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2024
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| author | Frahm, Jan Ólafsson, Gestur Ørsted, Bent |
| author_facet | Frahm, Jan Ólafsson, Gestur Ørsted, Bent |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups Frahm, Jan Ólafsson, Gestur Ørsted, Bent Representation Theory Primary 22E46, Secondary 43A85 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. |
| title | The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups |
| topic | Representation Theory Primary 22E46, Secondary 43A85 |
| url | https://arxiv.org/abs/2401.06427 |