Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Free monodromic Hecke categories and their categorical traces

Arnaud Eteve

TL;DR

This work develops a new construction of free monodromic categories and uses it to streamline the definition and analysis of free monodromic Hecke categories. It proves canonical identifications of categorical traces and centers with representations of finite groups of Lie type and with (free) character-sheaf-like objects, via horocycle correspondences and Deligne–Lusztig geometry. The results yield modular and integral extensions of Deligne–Lusztig theory, provide a conceptual route to proofs of key DL theorems, and connect the trace/center data to completed character-sheaf theories, including a precise description of the free monodromic unit. Overall, the paper offers a unified, geometrically grounded framework for free monodromic phenomena, with broad implications for arithmetic representation theory and the theory of character sheaves.

Abstract

The goal of this paper is to give a new construction of the free monodromic categories defined by Yun. We then use this formalism to give simpler constructions of the free monodromic Hecke categories and then compute the trace of Frobenius and of the identity on them. As a first application of the formalism, we produce new proofs of key theorems in Deligne--Lusztig theory.

Free monodromic Hecke categories and their categorical traces

TL;DR

This work develops a new construction of free monodromic categories and uses it to streamline the definition and analysis of free monodromic Hecke categories. It proves canonical identifications of categorical traces and centers with representations of finite groups of Lie type and with (free) character-sheaf-like objects, via horocycle correspondences and Deligne–Lusztig geometry. The results yield modular and integral extensions of Deligne–Lusztig theory, provide a conceptual route to proofs of key DL theorems, and connect the trace/center data to completed character-sheaf theories, including a precise description of the free monodromic unit. Overall, the paper offers a unified, geometrically grounded framework for free monodromic phenomena, with broad implications for arithmetic representation theory and the theory of character sheaves.

Abstract

The goal of this paper is to give a new construction of the free monodromic categories defined by Yun. We then use this formalism to give simpler constructions of the free monodromic Hecke categories and then compute the trace of Frobenius and of the identity on them. As a first application of the formalism, we produce new proofs of key theorems in Deligne--Lusztig theory.

Paper Structure

This paper contains 43 sections, 73 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. The free monodromic formalism
  3. Hecke categories
  4. Categorical trace, representations and character sheaves
  5. Convention and notations
  6. $\infty$-categories
  7. Geometry
  8. Sheaves
  9. Group
  10. Condensed mathematics
  11. Acknowledgments
  12. The free monodromic formalism
  13. The rings
  14. Categories of sheaves
  15. The universal multiplicative character
  16. ...and 28 more sections

Key Result

Theorem 1.1.3

There is a natural equivalence of categories where the notation on the RHS is from BezrukRiche for the free monodromic unipotent category.

Theorems & Definitions (196)

  • Definition 1.1.1: Section \ref{['sectionTheRings']}
  • Definition 1.1.2
  • Theorem 1.1.3: Theorem \ref{['ThmEquivalenceBR']}
  • Theorem 1.2.1
  • Theorem 1.3.1: Theorem \ref{['thmTraceFrobenius']}
  • Definition 1.3.2
  • Theorem 1.3.3: Theorem \ref{['thmTraceIdentity']}
  • Remark 2.1.1
  • Remark 2.1.2
  • Definition 2.1.3: HemoRicharzScholbach
  • ...and 186 more