QUACKS conference

Shotaro Makisumi (Columbia University), Matthew Hogancamp (Northeastern University), Geordie Williamson (University of Sydney)

Organizer's time: 2020-08-13 08:00 America/Los_Angeles

Duration: 6 hours

Schedule at https://pages.uoregon.edu/belias/QUACKS/schedule.html

Registration at https://pages.uoregon.edu/belias/QUACKS/registration.html

Shotaro Makisumi (Columbia University): Curved Hecke categories I and II

Abstract: The Hecke algebra admits an involution which preserves the standard basis and exchanges the canonical basis with its dual. This involution is categorified by "monoidal Koszul duality" for Hecke categories, studied in previous joint work with Achar, Riche, and Williamson. In this talk I will explain the following rough statement: "The Koszul dual of the Hecke category is equivalent to the derived category of bimodules for a particular Koszul complex in the Hecke category of the Langlands dual." This is motivated by the curved Koszul duality of Positselski and Burke. Based on joint work with Matt Hogancamp.

Matt Hogancamp (Northeastern University): Curved Hecke categories III

Abstract: In this talk I will discuss a certain category of curved complexes (aka the curved Hecke category) whose construction conjecturally possesses a Koszul self-symmetry which is evidently lacking in the usual category of Soergel bimodules (aka the Hecke category). This curved Hecke category is a generalization of a category constructed in type A in joint work with Eugene Gorsky.

The goal of the talk will indicate the relation (known and conjectured) between the curved Hecke category and the following categories: (1) the usual Hecke category, (2) its Koszul dual, and (3) graded category O. Along the way we will learn the "why" of curved complexes. This is based on joint work with Shotaro Makisumi

Geordie Williamson (University of Sydney): Miraculous Treumann-Smith theory and geometric Satake

Abstract: This talk will be about geometric approaches to the representation theory of reductive algebraic groups in positive characteristic p. A cornerstone of the geometric theory is the geometric Satake equivalence, which gives an incarnation of the category of representations as a category of perverse sheaves on the affine Grassmannian. It is surprising that several "easy" algebraic facts have no explanation on the geometric side. This is frustrating, as several deep conjectures appear to point to a clear relation with the geometry of the affine Grassmannian. One dreams of using geometric Satake as a fundamental localisation theorem (akin to Beilinson-Bernstein localisation for complex semi-simple Lie algebras) from which one can deduce structural results in representation theory. There are now several pieces of evidence in the Langlands program for the viability of this philosophy.

I will explain a recent step towards this dream. One of the "easy" algebraic facts alluded to above is the linkage principle, which decomposes the category of representations into "blocks" controlled by the affine Weyl group. In joint work with Simon Riche, we explain this decomposition via a certain mod p version of hyperbolic localization, known as Treumann-Smith theory. The theory has its roots in Smith's study of Z/pZ actions on spheres in the 1930s, and was upgraded to sheaves a few years ago by Treumann. We also deduce a new proof of the Lusztig character formula (for large p) and a conjecture of ours on characters of tilting modules (for all p).

Submitted by: Eugene Gorsky