Branching laws for Stein's complementary series and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$
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| Formato: | Preprint |
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2025
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| _version_ | 1866908481358921728 |
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| author | Ditlevsen, Jonathan Frahm, Jan |
| author_facet | Ditlevsen, Jonathan Frahm, Jan |
| contents | We obtain the explicit direct integral decomposition of Stein's complementary series representations and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$ when restricted to the subgroup $\operatorname{GL}(2n-1, \mathbb{R})$. The decomposition is a direct integral of unitarily induced representations from a maximal parabolic subgroup of $\operatorname{GL}(2n-1, \mathbb{R})$ with Levi factor $\operatorname{GL}(2n-2, \mathbb{R})\times\operatorname{GL}(1, \mathbb{R})$, where the induction data consists of a complementary series or Speh representation of the factor $\operatorname{GL}(2n-2, \mathbb{R})$ with the same parameter as the one of $\operatorname{GL}(2n, \mathbb{R})$ and a character of $\operatorname{GL}(1, \mathbb{R})$. These results are in line with the theory of adduced representations.
The main tools in the proof are two families of symmetry breaking operators between degenerate series representations of $\operatorname{GL}(2n, \mathbb{R})$ and $\operatorname{GL}(2n-1, \mathbb{R})$ whose meromorphic properties are studied in great detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Branching laws for Stein's complementary series and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$ Ditlevsen, Jonathan Frahm, Jan Representation Theory 22E45 (Primary), 22E46 (Secondary) We obtain the explicit direct integral decomposition of Stein's complementary series representations and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$ when restricted to the subgroup $\operatorname{GL}(2n-1, \mathbb{R})$. The decomposition is a direct integral of unitarily induced representations from a maximal parabolic subgroup of $\operatorname{GL}(2n-1, \mathbb{R})$ with Levi factor $\operatorname{GL}(2n-2, \mathbb{R})\times\operatorname{GL}(1, \mathbb{R})$, where the induction data consists of a complementary series or Speh representation of the factor $\operatorname{GL}(2n-2, \mathbb{R})$ with the same parameter as the one of $\operatorname{GL}(2n, \mathbb{R})$ and a character of $\operatorname{GL}(1, \mathbb{R})$. These results are in line with the theory of adduced representations. The main tools in the proof are two families of symmetry breaking operators between degenerate series representations of $\operatorname{GL}(2n, \mathbb{R})$ and $\operatorname{GL}(2n-1, \mathbb{R})$ whose meromorphic properties are studied in great detail. |
| title | Branching laws for Stein's complementary series and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$ |
| topic | Representation Theory 22E45 (Primary), 22E46 (Secondary) |
| url | https://arxiv.org/abs/2508.05442 |