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

TL;DR

This work analyzes the interplay between the second adjointness theorem and the geometric lemma for $p$-adic reductive groups by focusing on filtrations arising from parabolic induction. The authors show that the smooth dual filtration obtained from the geometric lemma on $P\setminus G/Q$ coincides, via the Bernstein-Casselman canonical pairing, with the filtration attached to $P\setminus G/Q^-$, extending Bezrukavnikov-Kazhdan’s explicit second adjointness description. The approach leverages Richardson-variety techniques to study Bruhat-cell intersections and constructs canonical lifts compatible with both filtrations, ultimately establishing a precise matching of filtrations. Applications include descriptions of adjoint maps, derivatives for $\mathrm{GL}$, and dual-variant perspectives on commutative triples, highlighting the broader impact on understanding dualities in $p$-adic representation theory. The results provide a robust geometric and combinatorial framework that connects Bruhat-order data, Jacquet modules, and dualities in a unified manner, with potential extensions to affine and graded Hecke algebras.

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.

Theorems & Definitions (72)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Definition 3.1
• Theorem 3.2