The unramified computation of a Shimura integral for $\mathrm{SL}(2)\times \mathrm{GL}(2)$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908816840327168 |
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| author | Yan, Pan |
| author_facet | Yan, Pan |
| contents | In this note, we revisit the Rankin-Selberg integral of Shimura type for generic representations of $\mathrm{SL}_2\times \mathrm{GL}_2$, constructed by Ginzburg, Rallis, and Soudry. We give a different and more ``intrinsic'' proof of the unramified computation. In contrast to their proof we avoid local functional equation for the general linear groups but use the Casselman-Shalika formulas for unramified Whittaker functions for $\mathrm{SL}_2$ and $\mathrm{GL}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_13653 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The unramified computation of a Shimura integral for $\mathrm{SL}(2)\times \mathrm{GL}(2)$ Yan, Pan Number Theory Representation Theory Primary 11F70, Secondary 22E50, 11F55 In this note, we revisit the Rankin-Selberg integral of Shimura type for generic representations of $\mathrm{SL}_2\times \mathrm{GL}_2$, constructed by Ginzburg, Rallis, and Soudry. We give a different and more ``intrinsic'' proof of the unramified computation. In contrast to their proof we avoid local functional equation for the general linear groups but use the Casselman-Shalika formulas for unramified Whittaker functions for $\mathrm{SL}_2$ and $\mathrm{GL}_2$. |
| title | The unramified computation of a Shimura integral for $\mathrm{SL}(2)\times \mathrm{GL}(2)$ |
| topic | Number Theory Representation Theory Primary 11F70, Secondary 22E50, 11F55 |
| url | https://arxiv.org/abs/2210.13653 |