Reductive Groups over Fields

These are lecture notes that Tony Feng live-TEXed from a course given by Brian Conrad at Stanford in“winter” 2015, which Feng and Conrad edited afterwards. Two substitute lectures were delivered (by Akshay Venkatesh and Zhiwei Yun) when Conrad was out of town. This is a sequel to a previous course on the general structure of linear algebraic groups; some loose ends from that course (e.g., Chevalley’s self-normalizing theorem for parabolic subgroups and Grothendieck’s theorem on geometrically maximal tori in the special case of finite ground fields) are addressed early on. The main novelty of the approach in this course to avoid the two-step process of first developing the structure theory of reductive groups over algebraically closed fields and then using that to establish the refined version over general fields. Instead, systematic use of dynamic techniques introduced in the previous course (and reviewed here) make it possible to directly establish the general structure theory over arbitrary fields in one fell swoop (building on the “geometric” theory of general linear algebraic groups from the previous course, where reductivity was not the main focus). This document sometimes undergoes minor updates to make corrections (typos, etc.) or clarifications. Please send errata/comments [email protected].