Tilting modules for a reductive algebraic group G in prime characteristic were formally introduced (in this setting) by Donkin in the early 1990’s. They have since come to play an increasingly central role in the study of G-modules, featuring prominently in modular character formulas and also shedding light on the “Humphreys-Verma Problem.”
In the early days, Donkin made a series of conjectures that predicted properties of the direct summands of tensor products involving the Steinberg module. Some of these conjectures remain open, while others have been resolved only in the last few years. In this talk I will discuss what we presently know and do not know, and why it is important. This is based on joint work with Chris Bendel, Dan Nakano, and Cornelius Pillen.