Coefficient systems on the A_2 Bruhat-Tits building

Adam Jones

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arXiv · 2510.01594

Coefficient systems on the A_2 Bruhat-Tits building

Adam Jones

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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.

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Adam Jones. 2025-10-02. Coefficient systems on the A_2 Bruhat-Tits building. https://arxiv.org/abs/2510.01594

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