THE INDUCTIVE BLOCKWISE ALPERIN WEIGHT CONDITION FOR TYPE AND THE PRIME

Abstract

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 $\operatorname {Sp}_{2n}(q)$ (with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.