The Alperin-McKay conjecture for blocks of finite simple groups of Lie type

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Groups form the mathematical language for symmetry that is applied within mathematics and other sciences. A representation of a group allows the group to be realised as matrices and apply the machinery from linear algebra. A fundamental task is to describe irreducible representations of a group, the building bricks of all representations.The local-global conjectures, such as the McKay conjecture, propose that the number of irreducible objects depends only on similar objects in local subgroups (smaller groups related to the prime power structures of the group). For a prime number p there is a partition of a groups irreducible representations into p-blocks, each of which is associated naturally with a Brauer correspondent, a p-block of a local subgroup.The Alperin-McKay conjecture is a refinement of the McKay conjecture which takes into account this partition into p-blocks. It postulates that there is a bijection between the height zero irreducible representations of a p-block and those of the Brauer correspondent.A reduction theorem by Späth proves that the Alperin-McKay conjecture would follow from the verification of the inductive Alperin-McKay condition on representations of finite simple groups. The open cases occur as finite groups of Lie type and recently a new criterion tailored to these groups has been provided by Späth and the applicant.The proposed project aims to establish the inductive condition by means of this new criterion in all open cases. Groups of Lie type can be seen as finite analogues of algebraic groups. After the groundbreaking work of Deligne and Lusztig, the representation theory of these groups has been intensively studied using machinery from algebraic geometry together with combinatorial tools such as d-Harish-Chandra theory established by Broué, Malle and Michel. The results of Cabanes and Späth, towards proving the McKay conjecture, reduce the new criterion to considering a local property and establishing a suitable bijection.The local property concerns the action of group automorphisms on local representations, which should follow from explicitly constructing these representations, something that to date has received little attention. The required bijection relates the local and global height zero representations, while also taking into account the action of various automorphisms. Späth and the applicant have been successful in validating this criterion in one of the easier cases and the methods used present a blueprint to approach the other Lie types.Hopefully this project will also provide a deeper understanding of why such a local-global conjecture holds for simple groups by developing a simultaneous parametrisation for the local and global representations of groups of Lie type.