Morita equivalences and the inductive blockwise Alperin weight condition for type $\mathsf A$

TL;DR

This paper verifies the inductive blockwise Alperin weight condition for simple groups of Lie type A, using Jordan decomposition and reduction to unipotent blocks, advancing the proof of the conjecture for these groups.

Contribution

It provides a proof of the inductive blockwise Alperin weight condition for Lie type A groups, utilizing Jordan decomposition to reduce to unipotent blocks.

Findings

Verification of the inductive condition for all simple groups of Lie type A.

Reduction of the problem to unipotent blocks via Jordan decomposition.

Progress towards the blockwise Alperin weight conjecture.

Abstract

As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Sp\"ath for simple groups of Lie type $\mathsf A$, split or twisted. Key to the proofs is to reduce the verification of the inductive condition to the isolated (that means unipotent) blocks, using the Jordan decomposition for blocks of finite reductive groups given by Bonnaf\'e, Dat and Rouquier.