Maximality phenomena in p-Kazhdan-Lusztig theory

Abstract

Over the past decade the paradigm has emerged that the structure of many important objects in modular representation theory are encoded in p-Kazhdan-Lusztig bases of Hecke algebras. Determining these bases remains a major open problem. In the first hour of this talk, I will explain how to compute these bases using the diagrammatic Hecke category. Then in the second half, I’ll introduce an alternate way of describing these bases; those bases which are maximal with respect to certain properties resulting from the monoidal structure of the Hecke category. Surprisingly this provides a uniform way of describing all currently known p-Kazhdan-Lusztig bases arising from finite flag varieties.