Modular reduction of complex representations of finite reductive groups
Roman Bezrukavnikov, Michael Finkelberg, David Kazhdan, Calder Morton-Ferguson
TL;DR
This work provides an explicit, $p$‑independent construction of the Lusztig‑type expansion for the Brauer reduction of irreducible unipotent representations of finite reductive groups. By introducing the KL–Steinberg basis $\{f_w\}$ and defining $M_w$ via a precise pairing, it proves that $\underline{\rho} = \sum_{w\in \mathcal{J}} (\rho:R_{\alpha_w}) M_w$ for any irreducible unipotent $\rho$, thereby giving a concrete reduction formula for all unipotent representations. The paper also analyzes structural properties of the $M_w$, identifies limits of conjectured symmetries and positivity, and explores their uniqueness; it further links these reductions to exceptional collections in derived categories, with computational confirmations in several Lie types. Overall, the results enhance our understanding of modular reductions in finite groups of Lie type and open avenues for categorification via coherent sheaves on partial flag varieties.
Abstract
Given a complex representation of a finite group, Brauer and Nesbitt defined in 1941 its reduction mod p, obtaining a representation over the algebraic closure of $\mathbb{F}_p$. In 2021, Lusztig studied the characters obtained by reducing mod p an irreducible unipotent representation of a finite reductive group over $\mathbb{F}_p$. He gave a conjectural formula for this character as a linear combination of terms which had no explicit definition and were only known in some small-rank examples. In this paper we provide an explicit formula for these terms and prove Lusztig's conjecture, giving a formula for the reduction mod p of any unipotent representation of $G(\mathbb{F}_q)$ for q a power of p. We also propose a conjecture linking this construction to the full exceptional collection in the derived category of coherent sheaves on a partial flag variety constructed recently by Samokhin and van der Kallen.