Bernstein-Zelevinsky duality for locally analytic principal series representations

TL;DR

This work develops a concrete realization of Bernstein-Zelevinsky duality for locally analytic principal series over split $p$-adic groups by constructing dual Koszul resolutions via Kohlhaase-Schraen machinery. It connects these dual complexes to dual Verma modules and Beilinson-Bernstein localization, and then leverages localization and patching techniques to relate the duality to Grothendieck-Serre duality on patched eigenvarieties. The results provide a precise formulation and verification of dualities at the level of complexes and coherent sheaves, including a formalism that unifies representations and coherent-analytic geometry in the $p$-adic setting. The framework paves the way for explicit calculations of duals in integral weight cases and offers a solid pathway to understanding dualities in the patched eigenvariety context with potential implications for the $p$-adic Langlands program.

Abstract

We consider certain dual of the Kohlhaase-Schraen resolutions for locally analytic principal series representations of $p$-adic Lie groups in the case of integral weights. The dual complexes calculate the expected Bernstein-Zelevinsky dual of the locally analytic representations and lead to the Grothendieck-Serre duality of coherent sheaves on patched eigenvarieties.

Theorems & Definitions (97)

• Theorem 1.3
• Theorem 1.4: Theorem \ref{['theoremresolution']} and Theorem \ref{['theoremdualKS']}
• Conjecture 1.5
• Theorem 1.6: Theorem \ref{['theoremBZdual']}
• Theorem 1.9: Theorem \ref{['theoremdualitypatchingmodule']}
• Proposition 1.10
• Lemma 2.1
• proof
• Proposition 2.2
• proof