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

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 G/Q. We prove that the dual filtration coincides with the filtration associated to the cosets P 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.