  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.
