Counting conjectures · Counting conjectures and characters of almost simple groups

This proposal has two major goals: to understand the irreducible complex characters of the finite almost simple groups, and to apply this knowledge to prove two longstanding famous conjectures in the representation theory of finite groups: the McKay conjecture and the Alperin Weight Conjecture.The first goal requires the study of the action of outer automorphisms of finite groups of Lie type on their irreducible characters and the solution of extension problems. The determination of the irreducible characters of all almost simple groups is a fundamental task of group theory.For the second goal, we will build on the recent reductions (by the PI and others) of both conjectures to assertions on characters of finite simple groups. To prove these assertions, one needs to construct certain equivariant bijections with respect to outer automorphisms, which will involve the results from the first goal.Furthermore, we propose to extend the reduction of the McKay conjecture to include several refinements, in particular the block-wise version andcongruences of character degrees.The project will involve the interplay of methods from the theory of algebraic groups, character sheaves, block theory and modular character theory.