A tale of two modules

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

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