Coefficient systems on the A_2 Bruhat-Tits building
Adam Jones
TL;DR
The work addresses a central obstruction (sur) in the mod-$p$ representation theory of reductive p-adic groups by linking it to the exactness of an oriented chain complex attached to a coefficient system on the Bruhat-Tits building. It develops a rank-2, type-\\tilde{A}_2 framework, focusing on G = $\\mathrm{SL}_3(K)$, and introduces new combinatorial tools—peaks, summits, crowns, and extended crowns—to analyze the geometric realization of the $\\widetilde{A}_2$ building. A key contribution is a strategy that reduces the global exactness question to local, region-by-region exactness and orbit-by-orbit control on boundary regions, supported by a crucial lemma that produces a single $A$-orbit on boundary edges after a suitable shift. The results provide strong evidence for exactness in the $\\widetilde{A}_2$ setting and lay a robust groundwork with a detailed combinatorial and homological framework that can be extended toward a full proof for higher ranks, with significant implications for the mod-$p$ local Langlands program.
Abstract
We address a conjecture (referred to as sur in the literature) in the representation theory of a reductive p-adic Lie group G which has important implications for the relationship between mod-p smooth representations and pro-p Iwahori-Hecke modules, and is currently only known for G of rank 1. We prove that sur follows from exactness of the associated oriented chain complex of a coefficient system, when restricted to a local region of the Bruhat-Tits building for G. Our main result gives strong evidence towards this exactness in the case where G=SL_3(K) for K a totally ramified extension of Q_p. We also develop new combinatorial techniques for analysing the geometric realisation of the A_2 Bruhat-Tits building, which are fundamental to the proof of our main result, and which we hope will inspire further investigation in Bruhat-Tits theory.