Schubert cells and Whittaker functionals for $\text{GL}(n,\mathbb{R})$ part II: Existence via integration by parts
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913869179387904 |
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| author | Kim, Doyon |
| author_facet | Kim, Doyon |
| contents | We give a new proof of the existence of Whittaker functionals for principal series representation of $\text{GL}(n,\mathbb{R})$, utilizing the analytic theory of distributions. We realize Whittaker functionals as equivariant distributions on $\text{GL}(n,\mathbb{R})$, whose restriction to the open Schubert cell is unique up to a constant. Using a birational map on the Schubert cells, we show that the unique distribution on the open Schubert cell extends to a distribution on the entire space $\text{GL}(n,\mathbb{R})$. This technique gives a proof of the analytic continuation of Jacquet integrals via integration by parts. We briefly discuss an application of the method to the Bessel functions on $\text{GL}(n,\mathbb{R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schubert cells and Whittaker functionals for $\text{GL}(n,\mathbb{R})$ part II: Existence via integration by parts Kim, Doyon Representation Theory Number Theory 11F70, 46F10 We give a new proof of the existence of Whittaker functionals for principal series representation of $\text{GL}(n,\mathbb{R})$, utilizing the analytic theory of distributions. We realize Whittaker functionals as equivariant distributions on $\text{GL}(n,\mathbb{R})$, whose restriction to the open Schubert cell is unique up to a constant. Using a birational map on the Schubert cells, we show that the unique distribution on the open Schubert cell extends to a distribution on the entire space $\text{GL}(n,\mathbb{R})$. This technique gives a proof of the analytic continuation of Jacquet integrals via integration by parts. We briefly discuss an application of the method to the Bessel functions on $\text{GL}(n,\mathbb{R})$. |
| title | Schubert cells and Whittaker functionals for $\text{GL}(n,\mathbb{R})$ part II: Existence via integration by parts |
| topic | Representation Theory Number Theory 11F70, 46F10 |
| url | https://arxiv.org/abs/2411.09862 |