Counting conjectures and $e$-local structures in finite reductive groups

TL;DR

This paper advances the understanding of finite reductive groups by linking Brauer--Lusztig blocks to $ ext{l}$-adic cohomology, proposing geometric conjectures for $ ext{l}$-local structures, and reducing these to Clifford theoretic properties.

Contribution

It introduces new descriptions of Brauer--Lusztig blocks via $ ext{l}$-adic cohomology and proposes geometric analogues of local structures, connecting them to counting conjectures.

Findings

New description of Brauer--Lusztig blocks using $ ext{l}$-adic cohomology.

Proposed conjectures relating geometric $ ext{l}$-local structures to counting conjectures.

Reduced conjectures to Clifford theoretic properties of Harish-Chandra series.

Abstract

We prove new results in generalized Harish-Chandra theory providing a description of the so-called Brauer--Lusztig blocks in terms of the information encoded in the $\ell$-adic cohomology of Deligne--Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the $\ell$-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Brou\'e, Fong and Srinivasan between $\ell$-structures and their geometric counterpart. Finally, using the description of Brauer--Lusztig blocks mentioned above, we reduce our conjectures to the verification of Clifford theoretic properties expected from certain parametrisation of generalised Harish-Chandra series.