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

2024/06/19 by Chan, Kei Yuen
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2406.13546

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∖ 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.

Related

• The Second Adjointness Theorem for reductive p-adic groups
• The Geometrical Lemma for Smooth Representations in Natural Characteristic
• Geometry of second adjointness for p-adic groups
• Geometric structure in the representation theory of p-adic groups
• Some results on the admissible representations of non-connected reductive p-adic groups