Geometric and Modular Representation Theory Seminar

The Hecke category action on the principal block via Smith theory

Wall-crossing functors on the principal block of category $O$ give an action of the (finite) Hecke category. If one knows enough about the Hecke category, one can deduce the Kazhdan-Lusztig conjectures from the existence of this action. This is a simple example of the power of categorification. In 2013, Riche and I conjectured that something similar is true for the principal block of reductive algebraic groups: namely that wall-crossing functors give an action of the (affine) Hecke category. We showed that the conjecture implies several rather deep statements in representation theory (mod $p$ analogues of the Kazhdan-Lusztig conjectures). Recently, this conjecture has been proved in two different ways: the first (by Bezrukavnikov and Riche) via mod $p$ localization and the second (by my student Josh Ciappara) via Smith theory. I will give an outline of Josh's proof. Bezrukavnikov and Riche's proof will be discussed later in the seminar.

Date & Time

January 27, 2021 | 3:00pm – 5:00pm

Location

Simonyi Hall 101 and Remote Access

Affiliation

University of Sydney; Distinguished Visiting Professor, School of Mathematics

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