Irreducible Modular Representations of Finite and Algebraic Groups

In these notes we outline some aspects of the modular representation theories of finite groups of Lie type in defining and cross-characteristics, with particular interest paid to how these theories relate to the modular representation theory of algebraic groups and the (characteristic 0) representation theory of Lie algebras and quantum groups. We begin by summarizing some classical results on the representation theory of complex semisimple Lie algebras and Lie groups, and then compare the classical theory to the representation theory of algebraic groups, discussing some of the issues encountered in moving to fields of positive characteristic and discussing some of the progress that has been in resolving these issues. We then discuss how the study of maximal subgroups leads to the study of linear representations in cross-characteristic, and conclude with a discussion of how the theory of quantum enveloping algebras (quantum groups) helps us to understand this situation.