Applications of the trace formalism to Deligne-Lusztig theory

TL;DR

The paper develops a trace-theoretic framework for Deligne–Lusztig theory by exploiting the free monodromic Hecke category, producing a categorical Jordan decomposition for $\mathbf{G}^{\mathrm{F}}$-representations and a spectral description of endomorphisms of the Gelfand–Graev representation. It establishes endoscopy for free monodromic Hecke categories, proves $t$-exactness and $\,\ell$-inversion properties, and shows compatibility with Deligne–Lusztig induction, thereby linking geometric and rational series to unipotent representations via endoscopic equivalences. The trace formalism yields a canonical decomposition of the representation category into geometric and rational series, and provides a modular framework that recovers and extends classical results (e.g., Dudas, Shotton–Li) through geometric and categorical means. Collectively, the results offer a unified, geometric and categorical approach to Jordan decomposition, endoscopy, and endomorphism algebras in finite groups of Lie type, with potential impact on decomposition matrices and Lusztig’s classification in modular settings.

Abstract

This paper is a continuation of previous work of the author. We use the categorical trace formalism to give a construction of the categorical Jordan decomposition for representations of finite groups of Lie type. As a second application, we study the endomorphism algebra of the Gelfand-Graev representation and recover a result of Li and Shotton-Li.

Theorems & Definitions (202)

• Theorem 1.1.1: EteveFreeMonodromic
• Definition 1.2.1
• Theorem 1.2.2: Endoscopy for Hecke categories
• Theorem 1.2.3: Endoscopy for Hecke categories, Theorem \ref{['thm:EndoscopyHeckeCat']}
• Definition 1.3.1
• Theorem 1.3.2: Jordan decomposition
• Remark 1.3.3
• Remark 1.3.4
• Theorem 1.3.5: Theorem \ref{['thm:LinearityForReps']}
• Remark 1.3.6