Tilting representations of finite groups of Lie type
Arnaud Eteve
TL;DR
This work develops a Deligne--Lusztig categorical framework, introducing the Deligne--Lusztig category $ ext{O}^{ ext{DL}}$ and its tilting objects to study representations of finite groups of Lie type. A horocycle-based functor $ ext{ch}$ produces tilting representations of $oldsymbol{G}^{ ext{F}}$ that generate the representation category and are projective in degree $0$, linking geometric and algebraic perspectives via Hecke categories and Kazhdan–Lusztig theory. The authors prove a key conjecture of Dudas--Malle for large $ extell$, describing how the Alvis--Curtis duality interacts with the mixed and unipotent pieces of the DL framework, and deduce consequences for unipotent decomposition numbers which become independent of $q$ and $ extell$. The work uncovers deep connections between Koszul duality in category $ ext{O}$, Alvis–Curtis duality, and the representation theory of finite groups of Lie type, providing explicit projective generators and a robust toolkit for modular representations.
Abstract
Let $\mathbf{G}$ be a connected reductive group over a finite field $\mathbb{F}_q$ of characteristic $p > 0$. In this paper, we study a category which we call Deligne--Lusztig category $\mathcal{O}$ and whose definition is similar to category $\mathcal{O}$. We use this to construct a collection of representations of $\mathbf{G}(\mathbb{F}_q)$ which we call the tilting representations. They form a generating collection of integral projective representations of $\mathbf{G}(\mathbb{F}_q)$. Finally we compute the character of these representations and relate their expression to previous calculations of Lusztig and we then use this to establish a conjecture of Dudas--Malle.