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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.02542 |
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| _version_ | 1866917186044428288 |
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| author | Boisseau, Paul |
| author_facet | Boisseau, Paul |
| contents | We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods for non-tempered automorphic representations, which we show compute special values of $L$-functions. The proof relies on shifts of contours of integration à la Langlands. We also establish two technical but crucial results on bounds and singularities for discrete Eisenstein series of $\mathrm{GL}_n$ in the positive Weyl chamber. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02542 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The fine spectral expansion of the Rankin-Selberg period Boisseau, Paul Number Theory Representation Theory We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods for non-tempered automorphic representations, which we show compute special values of $L$-functions. The proof relies on shifts of contours of integration à la Langlands. We also establish two technical but crucial results on bounds and singularities for discrete Eisenstein series of $\mathrm{GL}_n$ in the positive Weyl chamber. |
| title | The fine spectral expansion of the Rankin-Selberg period |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2601.02542 |