The spherical and anti-spherical modules are modules for the affine Hecke algebra obtained by inducing the trivial and sign modules from the finite Hecke algebra. They have have an illustrious history (Kazhdan-Lusztig isomorphism, Bezrukavnikov equivalence). They admit canonical and p-canonical bases. The canonical bases in the spherical and anti-spherical modules contain important information in the representation theory of reductive algebraic groups: the canonical basis in the spherical module appears in Lusztig's character formula for simple modules; the canonical basis in the anti-spherical module appears in Soergel's character formula for tilting modules for the quantum group. I will state p-versions of these results: the p-canonical basis in the spherical module controls simple characters; the p-canonical basis in the anti-spherical module controls tilting characters. The first statement is recent work with Riche (p >= 2h-2), the second statement was a conjecture with Riche, and was solved last year in joint work with Achar, Makisumi and Riche (p >= h). It doesn't seem unreasonable to hope that both statements are true for all p
Modular representation theory and the Hecke category
Geordie Williamson (University of Sydney)
MSRI: Simons Auditorium
The Hecke category is a monoidal category which controls several categories in representation theory. For example, appearances of Kazhdan-Lusztig polynomials usually point to an action of the Hecke category. I will discuss progress (partly joint with Ben Elias and Xuhua He) and conjectures (joint with Simon Riche) in the representation of reductive algebraic groups in characteristic p, which arise from studying the Hecke category modulo p.