Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909979306360832 |
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| author | Chan, Kei Yuen |
| author_facet | Chan, Kei Yuen |
| contents | Let $P,Q$ be standard parabolic subgroups of a $p$-adic reductive group $G$. We study the smooth dual of the filtration on a parabolically induced module arising from the geometric lemma associated to the cosets $P\setminus G/Q$. We prove that the dual filtration coincides with the filtration associated to the cosets $P\setminus G/Q^-$ via the Bernstein-Casselman canonical pairing from the second adjointness of parabolic induction. This result generalizes a result of Bezrukavnikov-Kazhdan on the explicit description in the second adjointness. Along the way, we also study some group theoretic results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups Chan, Kei Yuen Representation Theory Number Theory Let $P,Q$ be standard parabolic subgroups of a $p$-adic reductive group $G$. We study the smooth dual of the filtration on a parabolically induced module arising from the geometric lemma associated to the cosets $P\setminus G/Q$. We prove that the dual filtration coincides with the filtration associated to the cosets $P\setminus G/Q^-$ via the Bernstein-Casselman canonical pairing from the second adjointness of parabolic induction. This result generalizes a result of Bezrukavnikov-Kazhdan on the explicit description in the second adjointness. Along the way, we also study some group theoretic results. |
| title | Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2406.13546 |