Asymptotics of Schwartz functions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866916991589154816 |
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| author | Hsu, Chun-Hsien |
| author_facet | Hsu, Chun-Hsien |
| contents | Let $G$ be a split, simply connected, almost simple algebraic group, and let $P$ be a maximal parabolic subgroup of $G$. Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure $X_P$ of $P^{\mathrm{der}}\backslash G$. An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. When $G$ is a classical group or $G_2$, we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. Along the way, we give an alternative construction of the ring of differential operators on $X_P$ using the Fourier theory. We also establish the Poisson summation formulae in these cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_02403 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Asymptotics of Schwartz functions Hsu, Chun-Hsien Number Theory Representation Theory 11F70 (Primary), 11F55, 11F85 (Secondary) Let $G$ be a split, simply connected, almost simple algebraic group, and let $P$ be a maximal parabolic subgroup of $G$. Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure $X_P$ of $P^{\mathrm{der}}\backslash G$. An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. When $G$ is a classical group or $G_2$, we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. Along the way, we give an alternative construction of the ring of differential operators on $X_P$ using the Fourier theory. We also establish the Poisson summation formulae in these cases. |
| title | Asymptotics of Schwartz functions |
| topic | Number Theory Representation Theory 11F70 (Primary), 11F55, 11F85 (Secondary) |
| url | https://arxiv.org/abs/2112.02403 |