Cohomology of Deligne-Lusztig varieties for groups of type A

TL;DR

This paper investigates the cohomology of certain algebraic varieties related to groups of type A, demonstrating that a geometric conjecture and a specific formula hold for all unipotent blocks if they hold for the principal block.

Contribution

It proves that Broué's conjecture and Craven's formula for unipotent blocks follow from their validity for the principal Phi_1-block in type A groups.

Findings

Broué's conjecture holds for all unipotent blocks if it holds for the principal Phi_1-block.

Craven's formula is valid for unipotent blocks under the same condition.

The results connect geometric properties of Deligne-Lusztig varieties to block theory in representation theory.

Abstract

We study the cohomology of parabolic Deligne-Lusztig varieties associated to unipotent blocks of GLn(q). We show that the geometric version of Brou\'e's conjecture over Q_\ell, together with Craven's formula, holds for any unipotent block whenever it holds for the principal Phi_1-block, that is for the variety X(\pi).