The inductive blockwise Alperin weight condition for type $\mathsf C$ and the prime $2$

TL;DR

This paper proves the inductive blockwise Alperin weight condition for simple symplectic groups of Lie type C at prime 2, advancing understanding of modular representation theory for these groups.

Contribution

It introduces a labelling set for irreducible 2-Brauer characters of symplectic groups and establishes a Jordan decomposition for weights, key steps in the proof.

Findings

Established the inductive blockwise Alperin weight condition for type C groups at prime 2

Developed a labelling set for irreducible 2-Brauer characters of Sp_{2n}(q)

Proved a Jordan decomposition for weights

Abstract

We establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$. As a main step, we derive a labelling set for the irreducible $2$-Brauer characters of the finite symplectic groups Sp$_{2n}(q)$ (with odd $q$), together with the action of automorphisms. As a further important ingredient we prove a Jordan decomposition for weights.