Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups

Kei Yuen Chan

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arXiv · 2406.13546

Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups

Kei Yuen Chan

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Abstract

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.

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Kei Yuen Chan. 2024-06-19. Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups. https://arxiv.org/abs/2406.13546

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