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Acyclicity Of Broussous-Schneider coefficient systems

Javier Navarro

TL;DR

This work resolves the Acyclicity Conjecture of Broussous and Schneider for GL_N(F) by establishing the exactness of the augmented chain complex $C_{\bullet}^{\rm or}(X_\pi, \mathcal{C}[\pi]) \to \mathcal{V}$ for every admissible representation within Bernstein blocks tied to simple types. Building on BS2017type, it introduces a key parahoric-intertwining lemma that ensures uniformity of Hom-spaces across intertwining elements, enabling a robust geometric realization of Bushnell–Kutzko simple types via Broussous–Schneider coefficient systems. The paper also clarifies the relation to Dat’s coefficient systems, showing equivalence in GL_N when $p \nmid N$, and proves the acyclicity theorem to yield exact resolutions in the Bernstein block, with projectivity in fixed-central-character categories. The results provide explicit resolutions and pave the way for explicit coefficent data, character formulas, and pseudo-coefficients for discrete series in this setting, strengthening the link between type theory, Hecke algebras, and geometric models of representations.

Abstract

We give a proof of the Acyclicity Conjecture stated by Broussous and Schneider in [BrouSch2017]. As a consequence, we obtain an exact resolution of every admissible representation on each Bernstein block of ${\rm GL}(N)$ associated to a simple type.

Acyclicity Of Broussous-Schneider coefficient systems

TL;DR

This work resolves the Acyclicity Conjecture of Broussous and Schneider for GL_N(F) by establishing the exactness of the augmented chain complex for every admissible representation within Bernstein blocks tied to simple types. Building on BS2017type, it introduces a key parahoric-intertwining lemma that ensures uniformity of Hom-spaces across intertwining elements, enabling a robust geometric realization of Bushnell–Kutzko simple types via Broussous–Schneider coefficient systems. The paper also clarifies the relation to Dat’s coefficient systems, showing equivalence in GL_N when , and proves the acyclicity theorem to yield exact resolutions in the Bernstein block, with projectivity in fixed-central-character categories. The results provide explicit resolutions and pave the way for explicit coefficent data, character formulas, and pseudo-coefficients for discrete series in this setting, strengthening the link between type theory, Hecke algebras, and geometric models of representations.

Abstract

We give a proof of the Acyclicity Conjecture stated by Broussous and Schneider in [BrouSch2017]. As a consequence, we obtain an exact resolution of every admissible representation on each Bernstein block of associated to a simple type.

Paper Structure

This paper contains 7 sections, 2 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Results from hereditary orders and the theory of types
  4. A useful Lemma
  5. Broussous-Schneider Coefficient Sysyems
  6. Relation to Dat's Coefficient Systems
  7. Acyclicity

Key Result

Lemma 1

Let $\bf G$ be $\mathop{\mathrm{GL}}\nolimits_N$ and assume that $p\nmid N$. Let $(J,\lambda)$ be a simple type of $G$. Let $\phi$ be the image in $\Phi(P_F,G)$ of the Langlands parameter $\varphi\in H^1(W_F, \hat{\bf G})$ of any irreducible representation in $\mathcal{R}_{(J,\lambda)}(G)$. Let $I$

Theorems & Definitions (10)

  • proof
  • Remark 3.1
  • Lemma
  • proof
  • Remark 4.1
  • proof
  • proof
  • proof
  • proof
  • Theorem