Modular reduction of complex representations of finite reductive groups

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.