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

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.