Modular reduction of complex representations of finite reductive groups

TL;DR

This paper provides an explicit linear combination description of unipotent representations of finite groups of Lie type, confirming part of Lusztig's conjecture and introducing a new basis for algebraic functions.

Contribution

It introduces the Kazhdan-Lusztig-Steinberg basis and relates virtual representations to geometric objects, advancing understanding of modular representations and their reductions.

Findings

Confirms part of Lusztig's conjecture (2021).

Defines a new basis for algebraic functions called the Kazhdan-Lusztig-Steinberg basis.

Provides a geometric interpretation of virtual representations in the derived category.

Abstract

The main result describes the Brauer-Nesbitt reduction of unipotent representations of a finite group of Lie type, expressing it as an explicit linear combination of the restriction of Weyl modules from the algebraic group to the group of $\mathbb{F}_q$ points. This partly confirms Lusztig's conjecture (2021), which was the main source of motivation for this work. The explicit virtual representations of the algebraic group come from a certain endomorphism of the space ${\mathbb Z}[T]$ of regular functions on the torus which approximates pullback under Frobenius and is linear over the ring ${\mathbb Z}[T]^W$ of $W$-invariant functions. This endomorphism is constructed from a new basis for ${\mathbb Z}[T]$ over ${\mathbb Z}[T]^W$ which we call the Kazhdan-Lusztig-Steinberg basis. We compare this basis to the canonical basis appearing in the study of modular representations of the algebraic group and the related noncommutative Springer resolution. This leads to canonically defined objects in the derived category of $G$-modules representing the above virtual representations and to a geometric interpretation for the resulting lift of the principal series representation $\overline{\mathbb{F}_q} [G/P(\mathbb{F}_q)]$ to a virtual representation of the algebraic group, which comes from a decomposition of diagonal in the equivariant Grothendieck group of the partial flag variety.